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Variations on a theorem of Tate

Stefan Patrikis

Princeton University, Department of Mathematics, Fine Hall, Washington Road, Princeton,NJ 08544

Current address: Department of Mathematics, MIT, Cambridge, MA 02139E-mail address: patrikis@math.mit.eduURL: http://math.mit.edu/˜patrikis/

2000Mathematics Subject Classification.Primary 11R39, 11F80, 14C15

Key words and phrases.Galois representations, algebraic automorphic representations, motivesfor motivated cycles, monodromy, Kuga-Satake construction, hyperkahler varieties

Partially supported by an NDSEG fellowship and NSF grant DMS-1303928.

Abstract. Let F be a number field. These notes explore Galois-theoretic, automorphic, and mo-tivic analogues and refinements of Tate’s basic result that continuous projective representationsGal(F/F)→ PGLn(C) lift to GLn(C). We take special interest in the interaction of this resultwith al-gebraicity (for automorphic representations) and geometricity (in the sense of Fontaine-Mazur). Onthe motivic side, we study refinements and generalizations of the classical Kuga-Satake construction.Some auxiliary results touch on: possible infinity-types ofalgebraic automorphic representations;comparison of the automorphic and Galois “Tannakian formalisms”; monodromy (independence-of-ℓ) questions for abstract Galois representations.

Contents

Chapter 1. Introduction 11.1. Introduction 11.2. What is assumed of the reader: background references 101.2.1. Automorphic representations 101.2.2. ℓ-adic Galois representations 101.2.3. Motives 111.2.4. Connecting the dots 111.3. Notation 131.4. Acknowledgments 14

Chapter 2. Foundations & examples 152.1. Review of lifting results 152.2. ℓ-adic Hodge theory preliminaries 212.2.1. Basics 212.2.2. Labeled Hodge-Tate-Sen weights 232.2.3. Induction andℓ-adic Hodge theory 242.3. GL1 252.3.1. The automorphic side 252.3.2. Galois GL1 302.4. Coefficients: generalizing Weil’s CM descent of typeA Hecke characters 332.4.1. Coefficients in Hodge theory 332.4.2. CM descent 342.5. W-algebraic representations 36

2.6. Further examples: the Hilbert modular case and GL2 ×GL2⊠−→ GL4 39

2.6.1. General results on the GL2 ×GL2⊠−→ GL4 functorial transfer 39

2.6.2. The Hilbert modular case 422.6.3. A couple of questions 472.7. Galois lifting: Hilbert modular case 472.7.1. Outline 472.7.2. GL2(Qℓ)→ PGL2(Qℓ) 482.8. Spin examples 51

Chapter 3. Galois and automorphic lifting 573.1. Lifting W-algebraic representations 573.1.1. Notation and central character calculation 573.1.2. Generalities on lifting fromG(AF) to G(AF) 593.1.3. Algebraicity of lifts: the ideal case 63

iii

3.2. Galois lifting: the general case 663.3. Applications: comparing the automorphic and Galois formalisms 733.3.1. Notions of automorphy 733.3.2. Automorphy of projective representations 743.4. Monodromy of abstract Galois representations 813.4.1. A general decomposition 823.4.2. Lie-multiplicity-free representations 84

Chapter 4. Motivic lifting 914.1. Motivated cycles: generalities 914.1.1. Lifting Hodge structures 914.1.2. Motives for homological equivalence and the Tannakian formalism 934.1.2.1. Neutral Tannakian categories 944.1.2.2. Homological motives 974.1.3. Motivated cycles 994.1.4. Motives with coefficients 1034.1.5. Hodge symmetry inMF,E 1044.1.6. Motivic lifting: the potentially CM case 1094.2. Motivic lifting: the hyperkahler case 1114.2.1. Setup 1114.2.2. The Kuga-Satake construction 1134.2.3. A simple case 1164.2.4. Arithmetic descent: preliminary reduction 1194.2.5. Arithmetic descent: the generic case 1214.2.6. Non-generic cases: dim(TQ) odd 1244.2.7. Non-generic cases: dimTQ even 1264.3. Towards a generalized Kuga-Satake theory 1274.3.1. A conjecture 1274.3.2. Motivic lifting: abelian varieties 1284.3.3. Coda 132

Index of symbols 135

Index of terms and concepts 137

Bibliography 139

iv

CHAPTER 1

Introduction

1.1. Introduction

Let F be a number field, withΓF = Gal(F/F) its Galois group relative to a choice of algebraicclosure. Tate’s theorem thatH2(ΓF ,Q/Z) vanishes (see [Ser77, Theorem 4]) encodes one of thebasic features of the representation theory ofΓF: since coker(µ∞(C) → C×) is uniquely divisible,H2(ΓF ,C

×) vanishes as well, and therefore all obstructions to lifting projective representationsΓF → PGLn(C) vanish. More generally, as explained in [Con11], for any surjectionH′ ։ H oflinear algebraic groups overQℓ with central torus kernel, any homomorphismΓF → H(Qℓ) lifts toH′(Qℓ). The simplicity of Tate’s theorem is striking: replacingQ/Z by some finite groupZ/m ofcoefficients, the vanishing result breaks down.

Two other groups, more or less fanciful, extendΓF and conjecturally encode the key struc-tural features of, respectively, automorphic representations and (pure) motives. The more remoteautomorphic Langlands group,LF, at present exists primarily as heuristic, whereas the motivicGalois groupGF would take precise form granted the standard conjectures, and in the meantimecan be approximated by certain unconditional substitutes1. In either case, however, we can ask foran analogue of Tate’s theorem and attempt to prove unconditional results. The automorphic ana-logue is simpler to formulate, amounting to studying the fibers of the functorial transferLG→ LGwhereG andG are connected reductiveF-groups,G formed fromG by extending the (possiblynon-connected) centerZG to a central subgroupZ, i.e. G = (G× Z)/ZG, so thatZ/ZG is a torus. Forsimplicity, we always take theF-groupsG (andZ) to be split, but the results should also hold in thequasi-split case.2 We show (Proposition 3.1.4) that cuspidal representationsof G(AF) do indeedlift to G(AF). The analogue is false forGF, anda priori there is really no reason to believe that‘lifting problems’ for the three groupsΓF ,LF, andGF should qualitatively admit the same answer:a general automorphic representation has seemingly no connection with eitherℓ-adic representa-tions or motives; and a generalℓ-adic representation has no apparent connection with classicalautomorphic representations or motives. Two quite distinct kinds of transcendence– one complex,oneℓ-adic– prevent these overlapping theories from being reduced to one another. A key problem,therefore, is to identify the overlap and ask how the liftingproblem behaves when restricted to this(at least conjectural) common ground; it is only in posing this refined form of the lifting problemthat the relevant structures emerge.

So we take a detour to discuss notions of algebraicity for automorphic representations. Weillaid the foundation for this discussion in his paper [Wei56], by showing that for Hecke characters(automorphic representations of GL1(AF)), an integrality condition on the archimedean componentsuffices to imply algebraicity of the coefficients of the correspondingL-series (i.e. algebraicity of

1Namely, Deligne’s theory of absolute Hodge cycles, or Andr´e’s theory of motivated cycles; see§4.1.3.2To be specific, the precise description of the fibers should bea feature of the quasi-split case; see Conjecture

3.1.8 and the subsequent discussion. The fact that the fibersare merely non-empty should be completely general.

1

its Satake parameters). Waldschmidt ([Wal82]) later proved necessity of this condition. Weil,Serre, and others also showed that a subset, the ‘typeA0’ Hecke characters (Definition 2.3.2),moreover give rise to compatible systems of Galois characters ΓF → Q

×ℓ (and, via the theory of

CM abelian varieties, motives underlying these compatiblesystems). The general feeling sincehas been that the most obvious analogue of Weil’s typeA0 condition should govern the existenceof associatedℓ-adic representations. To be precise, letπ be an automorphic representation of ourF-groupG, and letT be a maximal torus ofG. We will use the terminology ‘L-algebraic’ of[BG11] as the general analogue of typeA0 characters. That is, fixing at eachv|∞ an isomorphismιv : Fv

∼−→ C, we can write (in Langlands’ normalization of [Lan89]) the restriction toWFvof its

L-parameter asrecv(πv) : z 7→ ιv(z)

µιv ιv(z)νιv ∈ T∨(C).

with µιv, νιv ∈ X•(T)C andµιv − νιv ∈ X•(T) (here and throughout, ¯ιv denotes the complex conjugateof ιv). Unless there is risk of confusion, we will omit reference to the embeddingιv, writingµv = µιv, etc.

Definition 1.1.1. π is L-algebraic if for allv|∞, µv andνv lie in X•(T).

The naıve reason for focusing on this condition is that, forG = GLn, it lets us see the Hodgenumbers of the (hoped-for) corresponding motive. Clozel ([Clo90]), noticing that cohomologicalrepresentations need not satisfy this condition, but have parametersµv, νv ∈ ρ + X•(T), whereρis the half-sum of the positive roots,3 studied this alternative integrality condition forG = GLn.Buzzard and Gee ([BG11]) have recently elaborated on this latter condition (‘C-algebraic’) and itsrelation withL-algebraicity. For GL1, the two notionsC andL-algebraic coincide, and in generalthey are both useful and distinct generalizations of Weil’stypeA0 condition.

But Weil’s ‘integrality condition,’ which he calls ‘typeA’ (Definition 2.3.2), is more generalthan the typeA0 condition, and its analogues in higher rank seem to have beenlargely neglected.4

It is of independent interest to resuscitate this condition, but we moreover find it essential forunderstanding the interaction of descent problems and algebraicity, including our original liftingquestion. By this we mean the following problem: given connected reductive (say quasi-split)F-groupsH andG with a morphism ofL-groupsLH → LG, and given a cuspidalL-algebraicrepresentationπ of G(AF) which is known to be in the image of the associated functorial transfer,is π the transfer of anL-algebraic representation? The answer is certainly ‘no,’ even for GL1, butit fails to be ‘yes’ in a tightly constrained way. I believe the most useful general notion is thefollowing:

Definition 1.1.2. We say thatπ is W-algebraicif for all v|∞, µv andνv lie in 12X•(T).

This is more restrictive than Weil’s typeA condition (which allows twists| · |r for r ∈ Q aswell), and it is easy to concoct examples of automorphic representations with algebraic Satakeparameters that are not even twists ofW-algebraic representations. It is one of our guiding princi-ples, suggested by the (archimedean) Ramanujan conjecture, that all such examples are degener-ate, and up to unwinding these degeneracies, all representations with algebraic Satake parameters

3For some choice of Borel containingT. Such a choice was implicit in defining a dual group, with its Borel B∨

containing a maximal torusT∨, and the above archimedeanL-parameters.4One substantial use of typeA but not A0 Hecke characters since the early work of Weil and Shimura occurs

in the paper [BR93], in which Blasius-Rogawski associate motives to certain tensor products of Hilbert modularrepresentations.

2

should be built up fromW-algebraic pieces. Moreover, forL-homomorphismsLH → LG withkernel equal to a central torus, the distinction betweenW- andL-algebraicity underlies the obstruc-tions to preservingL-algebraicity in the lift. Some initial evidence comes fromthe tensor productGLn ×GL′n→ GLnn′ (Proposition 2.5.8). Specializing to the context of Hilbert modular forms, we

then discussW-algebraicity and the GL2 × GL2⊠−→ GL4 transfer (its fibers and algebraicity prop-

erties) in detail, moreover linking it to the Galois side.5 Focusing finally on a transferLG → LGas in Tate’s lifting theorem, we prove the following algebraic refinement of the already-mentionedlifting result:

Proposition 1.1.3. Let F be CM field, and for simplicity let G be a split semi-simple F-group,and letZ be a torus. Letπ be a cuspidal representation of G(AF). Assumeπ∞ is tempered.

(1) If π is L-algebraic, then there exists an L-algebraic lift to a cuspidal automorphic repre-sentationπ of G(AF).

(2) If π is W-algebraic, then there exists a W-algebraic liftπ.

In §2.4 we develop a conjectural framework that allows this result to be extended to all totallyimaginaryF. See page 9 of this introduction. In contrast, over totally real fields, we have thefollowing:

Proposition 1.1.4. Now suppose either F is totally real, withπ as before, or F is arbitrary,but that the central characterωπ admits a finite-order extension to a Hecke character ofZ(AF).Continue to assumeπ∞ is tempered.

(1) If π is L-algebraic, then it admits a W-algebraic liftπ.(2) Assume F is totally real, and for the ‘only if ’ direction of the following statement assume

Conjecture 1.1.5 below. Then the images ofµv and νv under X•(T) → X•(ZG) lie inX•(ZG)[2], andπ admits an L-algebraic lift if and only if these images are independent ofv|∞.

To give a few examples, whenF is totally real there is no obstruction to findingL-algebraic liftsfor G a simple split group of typeA2n,E6,E8, F4 orG2. The natural question to ask, then, is whetherfor other groups there actually are counter-examples. Corollary 3.1.15 applies limit multiplicityformulas (as in [Clo86]) to produce many discrete-series examples generalizing the basic case ofmixed-parity Hilbert modular forms.6 The Conjecture 1.1.5 mentioned here is a problem in localrepresentation theory that plays a key role in the comparison of L-packets (both local and global)onG andG; we require it to deduce a description of the fibers of the transferLG→ LG. In its mostgeneral form, Adler-Prasad conjectured ([AP06]) the following forG andG quasi-split:

Conjecture 1.1.5 ([AP06] Conjecture 2.6). Let v be a place of F, and let G andG be quasi-split groups, related as above. Then for any irreducible admissible (smooth for finite v, Harish-Chandra module for infinite v) representationπv of G(Fv), the restrictionπv|G(Fv) decomposes withmultiplicity one.

5Various interesting examples arise, such as the construction of ℓ-adic Galois representations which are pure buthave no geometric twists.

6For the relationship of these results with Arthur’s conjectural construction, which requires modification, of amorphismLF → GF , see Remark 3.1.16.

3

The motivation for this conjecture is the uniqueness of Whittaker models for quasi-split groups.It is now known (by Adler-Prasad and others) for pairs (G,G) = (GLn,SLn) or (GU(V),U(V))whereV is a symplectic or orthogonal space overFv, as well as for generic ˜πv.

7 We also note thatthe assumptions on temperedness at infinity are not so serious as might appear: for a cuspidal au-tomorphic representation of GLn(AF), W-algebraic implies (unconditionally) tempered at infinity.Certainly some results for non-temperedπ are also possible, especially now that Arthur ([Art ]) hasproven his conjectures for classical groups.

The Galois question parallel to these refined automorphic lifting results has been raised byBrian Conrad in [Con11]. In that paper, Conrad addresses lifting problems of the form

H′(Qℓ)

ΓF

ρ<<③

③③

③③ ρ

// H(Qℓ)

,

whereH′ ։ H is a surjection of linear algebraic groups with central kernel. He discusses existence(a local-global principle), ramification control, andℓ-adic Hodge theory properties, and the resultsare comprehensive, except for one question (see Remark 1.6 and Example 6.8 of [Con11]):

Question 1.1.6. Suppose that the kernel ofH′ ։ H is a torus. Ifρ is geometric, when doesthere exist a geometric lift ˜ρ?

This, in the caseH = G∨, H′ = G∨, is the natural Galois analogue of Propositions 1.1.3 and1.1.4; indeed, it provided much of the motivation to understand those automorphic questions. By‘geometric,’ we mean almost everywhere unramified and de Rham at places aboveℓ.8 Conrad’sExample 6.8 shows the answer is at least ‘not always;’ he produces a characterψ : ΓL → Q

×ℓ over

certain CM fieldsL (with F the totally real subfield) such that IndFL (ψ) reduces to a geometric pro-

jective representation that has no geometric lift. We have an essentially general, but unfortunatelyconditional, solution to this problem. Having assumed thatcertain (GL-valued) Galois representa-tions are either automorphic or motivic, or merely satisfy certain Hodge-Tate weight symmetries(Hypothesis 3.2.4) that are consequences of these assumptions, we prove (Theorem 3.2.10 andremarks following):

Theorem 1.1.7. Let F be totally imaginary, and letρ : ΓF → H(Qℓ) be a geometric representa-tion satisfying Hypothesis 3.2.4. Thenρ admits a geometric liftρ : ΓF → H′(Qℓ).

There is an analogue in the totally real case (Corollary 3.2.8 and Remark 3.2.11), which is anexact parallel of Proposition 1.1.4. It is somewhat surprising that the Galois theory is not morecomplicated than the automorphic theory: roughly speaking, the automorphic input of tempered-ness is missing, and the obstruction to finding a geometric lift, say for the simple yet decisive caseGLn→ PGLn, seems to be a question of1

n-integrality rather than12-integrality; this first appearanceis, however, a red herring, andW-algebraicity remains the condition of basic importance ontheGalois side as well.

7For archimedeanv, it is obvious in many cases, and in this context it should be accessible in general.8As part of a general definition, we should also include reductive algebraic monodromy group– the Zariski closure

of the image– but that is not essential for this question.

4

We also include a local version of this result; it uses the same argument but requires no addi-tional assumption on ‘Hodge-Tate symmetry.’ ForK/Qℓ finite, we can ask the same sorts of liftingquestions withΓK in place ofΓF. A theorem of Wintenberger ([Win95]), in the case ofH′ ։ H acentral isogeny, generalized by Conrad toH′ ։ H with kernel of multiplicative type, asserts thatfor anyℓ-adic Hodge theory propertyP (i.e. Hodge-Tate, crystalline, semi-stable, or de Rham) aρ

of typeP admits a typeP lift if and only if ρ restricted to the inertia groupIK admits a Hodge-Tatelift. This need not hold in the isogeny case, but we can complete this story by showing it holdsunconditionally for central torus quotients: see Corollary 3.2.12. This question too was suggestedby Conrad, and a more difficult variant will appear in§3.6 of [CCO12].

Having developed the automorphic and Galois sides in parallel, and having found the sameanswer to the two algebraic variants of Tate’s lifting problem, we can use known results on theexistence of automorphic Galois representations to construct, unconditionally, infinite families ofGalois representations over totally real fields that fail tolift: §2.8 treats GSpin2n+1→ SO2n+1 in theregular case (where on the automorphic side we try to liftL-algebraic, discrete series at infinity,representationsπ of Sp2n(AF) to GSp2n(AF)). In other cases, we can use the conjectural Fontaine-Mazur-Langlands correspondence for GLn– we specify what we mean by this in Conjecture 1.2.1below– to develop parallel results for other groups. Here isan example from§3.3:

Proposition 1.1.8. Let F be any number field. Assume the Fontaine-Mazur-Langlands conjec-ture (1.2.1) and Conjecture 2.4.7. Then any geometricρ : ΓF → PGLn(Qℓ) is automorphic, in theweak sense that there exists an L-algebraic automorphic representationπ of SLn(AF) such thatρcorresponds almost everywhere locally toπ.9

For F imaginary, we can invoke Theorem 3.2.7. ForF totally real, a rather delicate descentargument is required. Similarly, we can deduce (see Proposition 3.3.7) from Fontaine-Mazur-Langlands some cases of the converse, providing evidence for a conjecture of Buzzard and Gee(Conjecture 3.2.1 of [BG11]).

More generally, we are led to ask whether descent questions on the automorphic and Galoissides obey the same formalism: if an automorphic representation π of G(AF) is L-algebraic withassociated Galois representationρπ : ΓF → LG(Qℓ) (as conjectured in [BG11], for instance), thenwe can ask two parallel questions for a givenL-homomorphismLH → LG:

• Is π the functorial transfer of an automorphic representation of H?• Doesρπ lift through LH(Qℓ)?

Closely related to these are the parallel questions:• If π descends toH, then does it have anL-algebraic descent?• If ρπ lifts throughLH(Qℓ)→ LG(Qℓ), then does it have a geometric factorization?

We have already discussed some cases (arising from Tate’s lifting question) of this general prob-lem. Here is another example:

Proposition 1.1.9. Continue to assume Fontaine-Mazur-Langlands and Conjecture 2.4.7. LetΠ be a cuspidal L-algebraic representation ofGLn(AF), and suppose thatρΠ ρ1 ⊗ ρ2, whereρi : ΓF → GLni (Qℓ). Then there exist cuspidal automorphic representationsπi of GLni (AF) suchthatΠ = π1 ⊠ π2.

9This side-steps the following subtlety, which is at the heart of the failure of multiplicity one for SLn(AF): twoprojective representations of a finite group can be element-by-element conjugate but not globally conjugate. SeeExample 2.1.10 and the discussion in§3.3.1.

5

The interest of this result is that examples generated from mixed-parity Hilbert modular formsshow that theπi need not beL-algebraic; we observe in passing that in that context (F totally real,n1 = n2 = 2), certain cases of this result are, allowing a finite base-change, unconditional (viapotential automorphy theorems). The structural point hereis that the two Tannakian formalismsseem to be compatible in this tensor product example. In contrast:

Proposition 1.1.10.Let F be totally real. Letρ : ΓF → PGL2(Qℓ) be geometric but have nogeometric lift, and supposeAd(ρ) : ΓF → SO3(Qℓ) satisfies the (potential automorphy) hypothesesof Corollary 4.5.2 of [BLGGT10]. Then there exists a liftρ : ΓF → GL2(Qℓ) and, after a totallyreal base change F′/F, a mixed-parity Hilbert modular representationπ onGL2/F′, such that wehave matching of local parameters

Sym2(ρ)|ΓF′ ∼w Sym2(π) ⊗ χfor some finite order, necessarily non-trivial, Hecke character χ of F′. In contrast, restricting toL′ = F′K where K/Q is a quadratic imaginary extension in whichℓ is inert, we can find a liftρsuch thatSym2(ρ) corresponds toBCL′/F′(Sym2(π)).

Conversely, starting with a mixed-parityπ, we can produce such aχ and ρ.

This proposition gives a precise example of the Fontaine-Mazur-Langlands correspondencebreaking down past theL-algebraic/geometric boundary. It also quantifies a difference betweendescent problems (in this case, characterizing the image ofSym2) on the automorphic and Galoissides.10 In all, these examples motivate a comparison of the images ofr : LH → LG on the au-tomorphic and Galois sides, whenr is anL-morphism with central kernel. The most optimisticexpectation (forH andG quasi-split) is that if ker(r) is a central torus, then the two descent prob-lems for (L-algebraic)Π and (geometric)ρΠ are equivalent; whereas if ker(r) is disconnected, thereis an obstruction to the comparison, that nevertheless can be killed after a finite base-change.

Despite these comparisons of the Galois and automorphic formalisms, the theory we develophere is deeply inadequate. Note that Proposition 1.1.8 (as in the Buzzard-Gee conjecture) onlymakes claims about weak (almost everywhere local) equivalence of automorphic and Galois rep-resentations for groups other than GLn. The following two simple questions point toward just howmuch we are missing:

Question 1.1.11. (1) Suppose thatρ : ΓF → G∨(Qℓ) is weakly-equivalent to an automor-phic representationπ of G(AF). Do there exist weakly-equivalent lifts ˜ρ : ΓF → G∨(Qℓ)andπ onG(AF)?11

(2) When does a weakly compatible system of representationsρℓ : ΓF → G∨(Qℓ) lift to aweakly compatible system ˜ρℓ : ΓF → G∨(Qℓ)? (For the precise definition of ‘weaklycompatible system,’ see Definition 1.2.3 below.)

The difficulties in answering the first question are reflected in the second, which in generalhas a negative answer; see Example 2.1.10 for the prototypical counterexample, which is familiarfrom the study of automorphic multiplicities and endoscopy. The only way I know to address thesecond problem involves both assuming and proving much more: granting that theρℓ areℓ-adicavatars of a (pro-algebraic) representation of the motivicGalois groupGF,E of motives overF with

10More simply, such examples exist forn: GL1→ GL1.11We do not return to this question in these notes, but certain cases are manageable, and will be explained in a

subsequent paper.

6

coefficients in a number fieldE (implicit here are specifiedE-forms ofG∨ andG∨), and showingthat this representation lifts toG∨, possibly after enlargingE. We have therefore come full-circleto the motivic lifting-problem raised at the beginning of this introduction. This is far more difficultthan the corresponding automorphic and Galois questions, but we can unconditionally treat certainexamples.

In order to pose the question precisely, we need a category of‘motives’ in which the Galoisformalism ofGF is unconditional. There are two common constructions, one based on Deligne’stheory ([DMOS82]) of absolute Hodge cycles, the other based on Andre’s theory ([And96b]),much inspired by Deligne’s work, of motivated cycles. We work with motivated cycles, since theinclusions

algebraic cycles⊆ motivated cycles⊆ absolute Hodge cycles⊆ Hodge cycles

more or less imply that results about motivated cycles (eg, ‘Hodge cycles are motivated’) followa fortiori for absolute Hodge cycles. Note that the Hodge conjecture asserts that each⊆ is anequality; Deligne’s ‘espoir’ ([Del79, 0.10]) asserts that the last⊆ is an equality. We letMF,E

denote the category of motives for motivated cycles overF with coefficients inE (see§4.1.3 and4.1.4). This is a semi-simpleE-linear Tannakian category, and, choosing an embeddingF → C,we can associate (via theE-valued Betti fiber functor) a (pro-reductive) motivic Galois groupGF,E.We will prove a lifting result of the form

GSpin(VE)

GF,E

ρ99s

ss

ss ρ

// SO(VE)

for certainρ arising from degree 2 primitive cohomologyVQ of a hyperkahler variety (see Defini-tion 4.2.1) overF (or, rather, forρ having analogous formal properties); the extension of scalarsVE = VQ ⊗Q E is essential for the lifting result to hold.

The starting-point for this refined motivic lifting result is work of Andre ([And96a]) on the‘motivated’ theory of hyperkahlers; indeed, his results imply a version of this lifting statement withF replaced by a large but quantifiable (with considerable work: see Theorem 8.4.3 of [And96a])finite extensionF′/F. Our contribution is the arithmetic descent fromF′ to F, and for this we usea variant of a technique introduced by Ribet ([Rib92]) to study so-called ‘Q-curves,’ elliptic curvesoverQ that are isogenous to all of theirΓQ-conjugates. Passing from a softer geometric statement(compare: ‘a variety overF has a point over some finite extension’) to a precise arithmetic version(compare: ‘when does a variety overF have a point overF?’) typically requires some deep input;in this case, Faltings’s isogeny theorem ([Fal83]) does the hard work. The method requires a case-by-case analysis (depending on the motivic group of the transcendental lattice), and I have decidednot to pursue all the cases here, but here is a partial result (see Theorem 4.2.11, Theorem 4.2.29,and Proposition 4.2.31 and following for more cases and moreprecise versions):

Theorem 1.1.12.Let(X, η) be a polarized variety over F satisfying Andre’s conditions Ak, Bk, B+k(see§4.2.1). For instance, with k= 1, X can be a hyperkahler variety with b2(X) > 3. Supposethat the transcendental lattice TQ ⊂ VQ = Prim2k(XC,Q)(k) satisfies either

• EndQ−Hodge(TQ) = Q; or• TQ is odd-dimensional.

7

For simplicity, moreover assume thatdetVℓ = detTℓ = 1 asΓF-representations.12 Then after somefinite scalar extension to VE = VQ ⊗Q E there is a lifting of the motivic Galois representationρV : GF → SO(VQ):

GSpin(VE)

GF,E

ρ99ssssssssss ρV

// SO(VE).

Moreover, this lifting gives rise to a weakly compatible system of liftsρλ : ΓF → GSpin(Vλ) onλ-adic realizations, for all finite placesλ of E.

For more cases in whichTQ is even-dimensional, see§4.2.7; the omitted cases should yieldto similar methods. In particular, we obtain in some cases a positive answer to the second part ofQuestion 1.1.11; this compatibility ofλ-adic realizations isnotautomatic for Andre’s motives (norfor absolute Hodge motives), so we need to exploit an explicit description of the lift. Moreover,the excluded caseb2(X) = 3 of Theorem 1.1.12 is related to a more general result for potentiallyabelian motives: we prove such a lifting result across an arbitrary surjectionH′ → H with centraltorus kernel: see Proposition 4.1.30 and Lemma 4.2.5.

Our next example seems to be a novel result even in complex algebraic geometry, where manyauthors have studied variants of the Kuga-Satake construction for Hodge structures of ‘K3-type’(see, for example, [Mor85], [Voi05], [Gal00]). Generalizing the well-known case ([Mor85]) ofH2 of an abelian surface– which has a Hodge structure ofK3 type, so falls within the ken ofclassical Kuga-Satake theory13– we prove an analogous GSpin→ SO motivic lifting result forH2

of any abelian variety (see Corollary 4.3.3). Moreover, we can describe the corresponding motive(in the spin representation) as a Grothendieck motive, without assuming the Standard Conjectures.

Note that Serre has asked ([Ser94, 8.3]) whether homomorphismsGF → PGL2 lift to GL 2.This is closely related to the question, also raised by Serre, of whether the derived group ofGFis simply-connected, but Serre notes that questions of thissort do not have obvious conjecturalanswers, even if we assume we are inle paradis motivique. Our abstract Galois lifting results, aswell as the handful of motivic examples, suggest and provideevidence for the following sharpeningand generalization of Serre’s question, that this motivic lifting phenomenon is utterly general:

Conjecture 1.1.13. Let F and E be number fields, and let H′ ։ H be a surjection, withcentral torus kernel, of linear algebraic groups over E. Suppose we are given a homomorphismρ : GF,E → H. Then if F is imaginary, there is a finite extension E′/E and a homomorphismρ : GF,E′ → HE′ lifting ρ ⊗E E′. If F is totally real, then such a lift exists if and only if theHodgenumber parity obstruction of Corollary 3.2.8 vanishes.

This is consistent with the Fontaine-Mazur conjecture, Serre’s expectation forGF , and The-orem 1.1.7, and indeed the three of them taken together implythis conjecture (mimic the proofof Proposition 4.1.30). As we have seen, the cases of potentially abelian motives, hyperkahler

12This can be achieved in each case after a quadratic, independent-of-ℓ extension onF. It is only so we canwork with SO(Vℓ) rather than O(Vℓ), but since our abstract Galois-lifting results apply to non-connected groups (seeTheorem 1.1.7, for instance), this hypothesis is not essential.

13The paper [Gal00] treats the case of a particular family of abelian four-folds, where a constraint on theMumford-Tate group allows one to extract a Hodge structure of K3-type; as we show, though, the lifting phenom-enon is completely general.

8

motives, and abelian varieties provide some evidence both for this conjecture and for Serre’s orig-inal question. In current work-in-progress, we study examples of this motivic lifting phenomenonfor (unlike the examples in this paper) motives not lying in the Tannakian category generated byabelian varieties and Artin motives. There is clearly vast terrain waiting to be discovered here,some of which is not entirely hostile to exploration.

A few other general themes recur throughout these notes, in automorphic, Galois-theoretic, andmotivic variants. The first is the systematic exploitation of ‘coefficients,’ and the general principlethat our arithmetic objects will naturally have coefficients in CM fields. In§2.4, we use thisprinciple to formulate (and prove, for regular representations; see Proposition 2.4.7) a conjecturalgeneralization of Weil’s result that a typeA Hecke character of a number fieldF descends, up tofinite order twist, to the maximal CM subfieldFcm; this generalization asserts that theinfinity-typeof anL-algebraic cuspidal automorphic representation of GLn(AF), over a totally imaginary fieldF, necessarily descends toFcm. In §4.1.5, we use this principle, which in the motivic contextis more or less the Hodge index theorem, to establish for Andre’s motives the Hodge-Tate weightsymmetries needed for the Galois lifting Theorem 1.1.7. Finally, Corollary 3.4.14 gives an exampleof how having CM coefficients can be exploited even in the study of abstract compatible systemsof ℓ-adic representations.14

This last result is based on the following abstract independence-of-ℓ result:15

Proposition 1.1.14.Letρλ : ΓF → GLn(Eλ) be a compatible system of irreducible, Lie-multiplicity-free representations ofΓF with coefficients in a number field E, so thatρλ IndF

Lλ(rλ) for someLie-irreducible representation rλ of ΓLλ , where the number field Lλ a priori depends onλ. Thenthe set of places of F having a split factor in Lλ is independent ofλ. If we further assume that theLλ/F are Galois, then Lλ is independent ofλ.

By Lie-multiplicity-free, we mean multiplicity-free after all finite restrictions. One applica-tion is a converse to a theorem of Rajan (Theorem 4 of [Raj98]), also generalizing a result of Serre(Corollaire 2 to Proposition 15 of [Ser81]). See§3.4 for further discussion, as well as the aformen-tioned application (Corollary 3.4.14), which is a weak Galois-theoretic shadow of the automorphicProposition 2.4.7. In§3.4, we also record a variant for number fields of a result of Katz (for affinecurves over a finite field; see [Kat87]), which in particular clarifies the place in the general theoryoccupied by Lie-multiplicity-free (or more specifically, Lie-irreducible) representations:

Proposition 1.1.15. Let ρ : ΓF → GLn(Qℓ) be an irreducible representation. Then eitherρis induced, or there exists d|n, a Lie irreducible representationτ of dimension n/d, and an Artinrepresentationω of dimension d such thatρ τ ⊗ ω. This decomposition is unique up to twistingby a finite-order character. Consequently, any (irreducible)ρ can be written in the form

ρ IndFL (τ ⊗ ω)

for some finite L/F and irreducible representationsτ andω of ΓL, with τ Lie-irreducible andωArtin.

This is a very handy result, which despite its basic nature does not seem to be widely-known.Some of our descent arguments in§4.2 rely on it.

14Since the writing of this paper, the same principles have been applied in [PT] to establish new results on thepotential automorphy of regular motives, and the irreducibility of automorphic Galois representations.

15See Proposition 3.4.9 for a more general statement.

9

Finally, time and again our arguments are buttressed by someexplicit knowledge of the con-straints on infinity-types of automorphic representations, and on Hodge-Tate weights of Galoisrepresentations. More generally, we find that asking finer structural questions about the interac-tion of functoriality and algebraicity naturally leads to existence (and non-existence) problems thatare often dissociated from functoriality.16 We certainly have many more questions than answers(scattered through§’s 2.3-2.7), although fortunately our main results depend largely on detailedstudy of GL1 (§2.3), which provides both the technical ingredients for later arguments and themotivation for higher-rank results (namely, after a close reading of Weil’s [Wei56], the definitionof W-algebraicity and the key descent principle of Proposition2.4.7). In this latter respect, I donot believe that GL1 has yielded all of its fruit– see the discussion surroundingCorollary 2.3.9–but the extreme difficulty of establishing the (non-)existence of automorphic representations withgiven infinity-types, even in qualitative form, necessarily tempers further conjecture.

For a reader interested only in certain aspects of these notes, I hope that the table of contents isa clear reference to the points of interest.

1.2. What is assumed of the reader: background references

Because this monograph studies all three vertices– and, to afar lesser extent, the edges– ofthe mysterious Galois-automorphic-motivic triangle, thereader will need some limited familiaritywith all three subjects. In this section, we indicate what background will be assumed, give someuseful references, and also explain that while the background required may be broad, it is notterribly deep; indeed, the arguments of this paper are elementary, once some basic definitions areassimilated. Let us deal individually with each of these three topics.

1.2.1. Automorphic representations.Preliminary to the study of automorphic representa-tions, one must have some familiarity with the theory of reductive algebraic groups. For sim-plicity, we always restrict to split groups, so it is sufficient to know the theory over algebraicallyclosed fields. Many arguments in this paper rest on the manipulation of root data; Springer’s sur-vey [Spr79] and his book [Spr09] (especially chapters 7-10) are very clear, and provide morethan enough background. We also assume familiarity with thealgebraic representation theory of(connected) reductive groups (i.e., elementary highest-weight theory).

We use a little of the representation theory of reductive groups over local fields. Familiaritywith unramified representations of splitp-adic groups and the Satake correspondence ([Gro98]is a beautiful guide; [Car79, III] is a thorough treatment of unramified representations; [Cas] isthe best general introduction to the theory of admissible smooth representations), and with someaspects of the formalism of the archimedean local Langlandscorrespondence will suffice. As forthe global theory of automorphic representations themselves, and the theory of the L-group, thestandard and more than adequate reference is [Bor79]; [Bor79, §9-11] would be particularly usefulbackground reading, sketching the archimedean theory and,crucially for our purposes, explainingthe desired, in some cases proven, connection between central characters and L-parameters.

1.2.2. ℓ-adic Galois representations.Apart from some very elementary notions, we onlyrequire some familiarity with the formal, non-geometric, aspects ofℓ-adic Hodge theory, whichis to say the study ofℓ-adic representations ofΓK for a finite extensionK/Qℓ (when studied in a

16As we will see in §2.1, this theme is present even in the proof of Tate’s original vanishing theoremH2(ΓF ,Q/Z) = 0.

10

purely local context,ℓ is traditionally replaced byp, as in ’p-adic Hodge theory’). An excellentoverview, with references to detailed proofs, is [Ber04, §I-II]. A ‘text-book-style’ reference (veryuseful but not quite in final form) for everything we do is [BC].17 Finally, we should remark to areader new top-adic Hodge theory that the subject has been much in flux in thelast 10 years, andthat one should take heed of recent conceptual advances (eg [Bei12], [Bha12], [FF12], [Sch13])before readingtoodeeply into the ‘classical’ theory.

1.2.3. Motives. Here we recommend some familiarity with the notion of motive, as conceivedby Grothendieck, and with Grothendieck’s Standard Conjectures on algebraic cycles. Kleiman’sarticles [Kle68] and [Kle94] provide a clear, careful, and concise introduction. Certainly familiar-ity with the various cohomological realizations (Betti, deRham,ℓ-adic), and the relations betweenthem, of a smooth projective variety is necessary; the first section, ‘Review of Cohomology,’ ofDeligne’s article [DMOS82, §I.1] will provide quick and easy orientation for someone newto thesubject. We will also require (some of) the theory of Tannakian categories; we will give a briefintroduction that should be enough for a reader to follow allof our arguments, but a thorough treat-ment is [DM11]. Finally, although not necessary for the present work, theexcellent book [And04]provides a broad survey of the theory of motives, from its inception to more recent developments.

1.2.4. Connecting the dots.These notes rely on the systematic transfer of intuition back andforth between these three areas; it may be helpful, then, to collect in precise form the toweringconjectures that dominate this conceptual landscape. The following three-part conjecture, and theconsequence of combining parts 2 and 3, summarize the principal problems in the field.

Conjecture 1.2.1 (Fontaine-Mazur-Langlands-Tate).Let ρ : ΓF → GLn(Qℓ) be an irreduciblegeometric Galois representation. Recall from the introduction that this meansρ is almost every-where unramified, and is de Rham at all places aboveℓ. Then:

(1) (Fontaine-Mazur) There exists a smooth projective varietyX/F and an integer r such thatρ is isomorphic to some sub-quotient of Hj(XF ,Qℓ)(r).

(2) (Fontaine-Mazur-Tate) Such aρ is motivic in the sense that it is cut out byQℓ-linearcombinations of (homological) algebraic cycles. More precisely, for any embeddingι : Q → Qℓ, there is an idempotent project e in the algebra C0

hom(X,X)Q (see[Kle68,

§1.3.8], where this is denotedA(X)) of self-correspondences of X withQ-coefficients,such that

ρ e(H∗(X)(r)Q) ⊗Q,ι QℓasΓF-representations.

(3) (Fontaine-Mazur-Langlands) There exists a cuspidal automorphic representationΠ ofGLn(AF) such that for almost all v unramified forρ, the eigenvalues ofρ( f rv) agreewith the Satake parameters ofΠv, viewed inQℓ via the compositionιℓ ι−1

∞ (recv(Πv)).18

Moreover forι : F → C, the archimedean Langlands parameter

recv(Πv)|Fv× : WFv

→ GLn(C),

17It is also highly recommended to visit Laurent Berger’s website to check on the status of the course-notes fromhis course at IHP in the Galois Trimestre of 2010.

18A stronger version moreover asks that for all finite placesv,

WD(ρ|ΓFv) f r−ss

ιℓ ι−1∞ (recv(Πv))

11

landing in a maximal torus Tn, is of the form z7→ zµι zνι for µι, νι ∈ X•(Tn), and for anyτ : F ⊂ Fv → Qℓ, the Hodge-Tate co-characterµτ of ρ|ΓFv

is (conjugate to)µι∗∞,ℓ(τ).19

Conversely, given a cuspidal automorphic representationΠ of GLn(AF) with integralarchimedean Langlands parameters, there exists an irreducible geometric Galois repre-sentationρΠ : ΓF → GLn(Qℓ) (depending onι∞, ιℓ) satisfying the above compatibilities.

Remark 1.2.2. (1) Note that the Standard Conjectures are implicit in part 2 of Conjecture1.2.1. We will rarely work directly with motives for homological equivalence in thesenotes, substituting instead Andre’s category of motivated motives. Keeping one’s faith inthe Standard Conjectures, but wanting to assume less at the outset, one could substitutethe algebra of motivated correspondencesC0

mot(X,X) (Definition 4.1.13 and following) forC0

hom(X,X) in Conjecture 1.2.1.(2) This is not a historically faithful presentation of these conjectures; for our purposes in

these notes, however, this formulation is convenient.(3) Part 3 implies a similar correspondence between semi-simple (not necessarily irreducible)

geometric Galois representations and (suitably algebraic) isobaric automorphic represen-tations of GLn(AF). Alternatively, one can begin from this more general conjecture anddeduce that cuspidal automorphic representations must correspond to irreducible Galoisrepresentations using standard properties of Rankin-Selberg L-functions.

The Fontaine-Mazur-Tate conjecture challenges us, given ageometricℓ-adic representationρℓ,to produce a motive withρℓ asℓ-adic realization, and in particular to produce a family ofℓ′-adicrealizations (for allℓ′) that are ‘compatible’ withρℓ. It is often convenient to abstract this notionof compatible system of Galois representations, as an intermediary between the isolatedρℓ and therobust motive.

Definition 1.2.3. LetF and E be number fields, and letN be a positive integer. A rankNweakly compatible system ofλ-adic (or, informally,ℓ-adic)representations ofΓF with coefficientsin E is a collection

R = (ρλλ,S, Qv(X)v<S),

consisting of:(1) for each finite placeλ of E, a continuous semi-simple geometric representation

ρλ : ΓF → GLN(Eλ);

(2) a finite set of placesS of F, containing the infinite places, such that for allv < S and forall λ of different residue characteristic fromv, ρλ|ΓFv

is unramified;(3) for all v < S, a polynomialQv(X) ∈ E[X] such that for allλ of different residue character-

istic fromv, Qv(X) is the characteristic polynomial ofρλ( f rv).We will sometimes use a similar notion where GLN is replaced by a, for simplicity, split connectedreductive groupH over the number field of coefficientsE. In this case, repeat the above definitionverbatim, except replace the characteristic polynomial ofρλ( f rv) with the analogous point of thespace of Weyl-invariant functions on a maximal torus inH, i.e. the image ofρλ( f rv) ∈ H(Eλ)under the Chevalley map induced by the map on coordinate rings

E[H] ⊃ E[H]H ∼−−→res

E[T]W,

19ι∗∞,ℓ(τ) is the pullback ofτ to an archimedean embedding viaιℓ, ι∞; see§1.3.

12

whereT is anE-split maximal torus, andW is the Weyl group of (H,T).

We will sometimes impose apurity hypothesis onR:

Definition 1.2.4. Fix an integerw. For any finite placev of F, let qv denote the order of theresidue field ofF at v. We say that a weakly compatible systemR is pureof weightw ∈ Z if for adensity one set of placesv of F, each rootα of Qv(X) in E satisfies|ι(α)|2 = qw

v for all embeddingsι : E → C.

Another standard variant of the definition of weakly compatible system would also specifythat theℓ-adic Hodge numbers are suitably ‘independent ofℓ,’ but we will not require this; seefor instance [BLGGT10, §5.1]. Finally, an elementary argument (see, eg, [CHT08, Lemma2.1.5]) shows that any continuousEλ-representation ofΓF (a compact group) takes values in someGLN(E′) for some finite extensionE′ of Eλ.

1.3. Notation

For a number fieldF, we always choose an algebraic closureF/F and letΓF denote Gal(F/F).We writeCF = A×F/F

× for the idele class group ofF.If L/F is a finite extension insideF andW a representation ofΓL (= Gal(F/L)), we abbreviate

IndΓFΓL

(W) to IndFL (W). Forg ∈ ΓF , we write (gW) for the conjugate representation ofgΓLg−1.

We fix separable closuresQ andQℓ of Q andQℓ for all ℓ. We fix throughout embeddingsιℓ : Q → Qℓ and ι∞ : Q → C. An archimedean embeddingι : F → C thus induces anℓ-adicembeddingτ∗

ℓ,∞(ι) = ιℓ ι−1∞ ι : F → Qℓ; and similarly anℓ-adic embeddingτ : F → Qℓ induces

ι∗∞,ℓ(τ) : F → C. If there is no risk of confusion, we just writeτ∗(ι) andι∗(τ). These embeddingswill be invoked (often implicitly) whenever we associate automorphic forms and Galois represen-tations.

For a connected reductiveF-groupG, we construct dual andL-groupsG∨ and LG overQ(having chosen a maximal torus, Borel, and splitting, although we will only ever make the maximaltorus explicit), and then useιℓ andι∞ to regard the dual group overQℓ orC as needed.

If v is a place ofF, we denote by recv the local reciprocity map from irreducible admissiblesmooth representations (v finite) or irreducible admissible Harish-Chandra modules (v infinite) to(frobenius semi-simple) representations of the Weil(-Deligne) group ofFv. We use this in theunramified and archimedean cases, and for GLn. For the group GL1, this is local class field theory,normalized so that uniformizers correspond to geometric frobenii, which we denotef rv.

It will be convenient to have a short-hand for the assertion that an automorphic representationand Galois representation ‘correspond’ under the conjectural global Langlands correspondence.We will elaborate on a number of related notions in§3.3.1, but for now we give two that will comeup frequently. Recall that we have fixed embeddingsιℓ and ι∞ of Q into Qℓ andC, respectively.If π is an automorphic representation ofG(AF), andρ : ΓF → LG(Qℓ) is a continuous, almosteverywhere unramified, representation, then we writeρ ∼w π if for almost every unramified placev of F (at which we may assumeρ is unramified), recv(πv) : WFv → LG(C) can be realized (up to

G∨-conjugation) overQι∞−→ C, and that then

ιℓ ι−1∞ (recv(πv)) ∼

(ρ|ΓFv

)fr−ss,

13

where∼ here denotesG∨(Qℓ)-conjugacy, and ‘fr-ss’ means we replaceρ( f rv) with its semi-simplepart. More concretely, we restrict to placesv at which bothπ andρ are unramified, so that recv(πv)andρ|ΓFv

are both just given by their evaluations atf rv; after usingιℓ ι−1∞ to regard them as defined

over the same field, we ask that these two elements beG∨(Qℓ)-conjugate.As in the Fontaine-Mazur-Langlands Conjecture 1.2.1, we also often want to compare the

archimedean component ofπ with the Hodge-Tate weights ofρ; whenπ is L-algebraic (see Def-inition 1.1.1), andρ is de Rham, we may write, for allι : F → C, the archimedean Langlandsparameter recv(πv)|F×v in the form

z 7→ ι(z)µι ι(z)νι

for µι, νι ∈ X•(T∨). Then writeρ ∼w,∞ π if ρ ∼w π, and moreover for all suchι : F → C, and forall τ : F ⊂ Fv → Qℓ, theτ-labeled Hodge-Tate co-character (see§2.2.2) ofρ|ΓFv

is (conjugate to)µι∗∞,ℓ(τ).

By a CM field we mean as usual a quadratic totally imaginary extension of a totally real field;these, and their real subfields, are the number fields on whichcomplex conjugation is well-defined,independent of the choice of complex embedding. For any number field, we writeFcm for themaximal subfield ofF on which complex conjugation is well-defined; thus it is the maximal CMor totally real subfield, depending on whetherF contains a CM subfield. We also writeQcm for theunion of all CM extensions ofQ insideQ.

For any topological groupA, we denote byAD the group Homcts(A,S1) (S1 is the unit circle);whenA is abelian and locally compact, we topologize this as the Pontryagin dual.

For any ground fieldk (always characteristic zero for us) with a fixed separable closureks, anda (separable) algebraic extensionK/k insideks, we write K for the normal (Galois) closure ofKoverk insideks.

We always denote base-changes of schemes by sub-scripts: thus,Xk′ is the base-changeX×Speck

Speck′ for ak-schemeX and a base-extension Speck′ → Speck. Notation such asM ⊗k k′ will bereserved (see§4.1.4) to describe extension of scalars from an objectM of a k-linear abelian cate-gory to one of ak′-linear abelian category; when there is no risk of confusionwe will sometimesdenote this also byMk′ .

Finally, we will signal either a significant change in running hypotheses, or an essential yeteasily-overlooked point, with the ‘dangerous bend’ symbol

1.4. Acknowledgments

Some of this work was carried out with the support of an NDSEG fellowship, and final revisionswere made with the support of NSF grant DMS-1303928. I thank the Institut Henri Poincare,which I visited for the 2010 Trimestre Galois, and the OxfordUniversity mathematics department,which I visited in 2011− 2012, for their hospitality. I of course am grateful to the Princeton mathdepartment, both for its official support, and for its singular camaraderie.

I thank the anonymous referee(s) for many helpful comments improving the readability of thispaper. It also gives me great pleasure to thank the followingpeople. My debt to Brian Conrad, whowrote the paper ([Con11]) that catalyzed much of the present work, will be clear. He also providedvery helpful comments on an earlier draft. I am grateful to Peter Sarnak for his encouragementand several enjoyable conversations about this material, and to Christopher Skinner for reading anearlier draft. Most of all, I thank Andrew Wiles, my advisor,for his ongoing support and patientencouragement over the last few years.

14

CHAPTER 2

Foundations & examples

This chapter discusses the motivating examples and technical ingredients that underly the gen-eral arguments of Chapter 3. After (§2.1) recalling foundational lifting results of Tate, Winten-berger, and Conrad, and (§2.2) setting up the tools we will need fromℓ-adic Hodge theory, weundertake a detailed discussion (§2.3) of Hecke characters andℓ-adic Galois characters, especiallywith reference to their possible infinity-types and Hodge-Tate weights. In§2.4.1 we first explainan abstract principle that is important for ‘doing Hodge theory with coefficients;’ in§2.4 we applythis to generalize Weil’s result on descent of typeA Hecke characters. Drawing further inspirationfrom [Wei56], we then discuss (§2.5) what seems to be the most useful generalization of the typeA condition to higher-rank groups, which we callW-algebraic representations. The subsequentsections (§2.6 and§2.7) discuss the simplest non-abelian example, that ofW-algebraic Hilbertmodular forms, their associated Galois representations, and the first interesting cases of Conrad’sgeometric lifting question (Question 1.1.6); although certain results in these sections are super-seded by the general theorems of Chapter 3, we can also prove much more refined statements in

the Hilbert modular case, as well as some complementary results about the GL2 × GL2⊠−→ GL4

functorial lift.1 Another case where more precise results are accessible withthe current technologyis for certain automorphic representations on symplectic groups, and their associated (orthogonaland spin) Galois representations (§2.8).

2.1. Review of lifting results

In this section, we review the basic Galois lifting results due to Tate, Wintenberger, and Con-rad. We then highlight some of the basic problems that remainunaddressed by these foundationalresults. The two main elements of Tate’s proof will come up repeatedly throughout these notes,so they are worth making explicit: he requires information coming both from the ‘automorphic-Galois correspondence’ and from the bare automorphic theory, which in our work amounts to somequestion of what infinity-types automorphic representation can have.2 In Tate’s proof, the formeris global class field theory– in the form of the local-global structure of the Brauer group– and thelatter takes the form of precise results about the structureof the connected component of the ideleclass groupCF. We will give a slightly different proof that emphasizes the continuity with some ofour other arguments, especially Lemma 2.3.6.

Theorem 2.1.1 (Tate; see [Ser77, Theorem 4]).Let F be a number field3. Then H2(ΓF ,Q/Z) =0.

1Some of the results of§2.7, among other things, have been independently obtained by Tong Liu and Jiu-KangYu in their preprint [LY ].

2A very simple aspect of this extremely difficult problem, that is.3As Tate observes, the same holds for global function fields, but we are not concerned with this case.

15

Proof. It suffices to proveH2(ΓF,Qp/Zp) = 0 for all primesp, and then an easy inflation-restriction argument shows we may assume thatF containsµp. SinceH2(ΓF,Qp/Zp) is p-powertorsion, to show it is zero we may instead show that multiplication byp is injective, or equivalentlythat the boundary map

H1(ΓF ,Qp/Zp)δ−→ H2(ΓF ,Z/p) Br(F)[p]

is surjective (the identification with the Brauer group is possible sinceµp ⊂ F). First we notethat the corresponding claim holds for the completionsFv: since Br(Fv)[p] Z/p (for v finite;for v archimedean the argument is even simpler), we only need the corresponding (atv) boundarymapδv : H1(ΓFv,Qp/Zp)→ H2(ΓFv,Z/p) to be non-zero, and thus only that multiplication byp onH1(ΓFv,Qp/Zp) should not be surjective. But a characterF×v → Qp/Zp is a pth-power if and only ifit is trivial on µp(Fv) ⊂ F×v , and there are obviously characters not satisfying this condition. Notealso that forφv ∈ H1(ΓFv,Qp/Zp), the imageδv(φv) depends only on the restrictionφv|µp(Fv).

For globalF, let α be ap-torsion element of Br(F) ⊂ ⊕vBr(Fv), with local componentsαv.By the local theory, for allv we have charactersφv : F×v → Qp/Zp such thatδv(φv) = αv, and thecollection ofαv is equivalent to the collection of restrictionsφv|µp(Fv), i.e. to the correspondingcharacterφ : µp(AF) → Qp/Zp. The fact that theαv arise from a global Brauer classα impliesthatφ factors throughµp(F)\µp(AF), and we need only produce a finite-order extension to a Heckecharacterφ : CF → Q/Z. The archimedean classesαv are trivial for v complex, so Lemma 2.3.6implies the existence of such an extension (note that it suffices to produce, as in the Lemma, anextension to a characterφ : CF → Q/Z, since we can then project down toQp/Zp and still have anextension ofφ).

Example 2.1.2. As mentioned in the introduction, Tate’s result certainly breaks down with finitecoefficients. For example, consider the projectivization of theℓ-adic Tate module of an ellipticcurve over a totally real fieldF. The obstruction to lifting to SL2(Qℓ) lives in H2(ΓF ,Z/2), and itvanishes if and only if theℓ-adic cyclotomic character admits a square-root, which cannot happenfor F totally real.

The papers [Con11] and [Win95] both study the problem of lifting Galois representationΓF →H(Qℓ) through a surjection of linear algebraic groupsH′ → H with central kernel. The problemnaturally breaks into two cases: that of finite kernel (an isogeny), and that of connected (torus)kernel. It is helpful to contrast these cases using an algebraic toy model:

Example 2.1.3. LetH′ ։ H be a surjection oftori over an algebraically closed field of char-acteristic zero, with kernelD, and suppose we are given an algebraic homomorphismρ : Gm→ H.Via the (anti-) equivalence of categories between diagonalizable algebraic groups and abeliangroups (taking character groups), we see that the obstruction to lifting ρ as an algebraic morphismlives in Ext1(X•(D),Z); in particular, whenD is a torus, all suchρ lift. For instance, we can fill inthe dotted arrow in the diagram

G2m

(w,z)7→wr zs

Gm

>>⑥⑥

⑥⑥

z7→zn// Gm

precisely whengcd(r, s)|n. The kernel ofG2m→ Gm is connected precisely whengcd(r, s) = 1, so

there is no obstruction in that case, and in general there is an explicit congruence obstruction.

16

Since it is the basis of all that follows, I recall the application of Theorem 2.1.1 to liftingthrough central torus quotients (see Proposition 5.3 of [Con11]).

Proposition 2.1.4 (5.3 of [Con11]). Let H′ ։ H be a surjection of linear algebraic groupsoverQℓ with kernel a central torus S∨.4 Then any continuous representationρ : ΓF → H(Qℓ) liftsto H′(Qℓ), i.e. we can fill in the diagram

H′(Qℓ)

ΓF

ρ<<③

③③

③③

ρ// H(Qℓ).

Proof. By induction, we may assumeS∨ = Gm. There is an isogeny complementH′1 in H′

to S∨: that is, we still have a surjectionH′1 ։ H, but nowH′1 ∩ S∨ is finite,5 and thus equal toµn0 ⊂ Gm for somen0. For any integern divisible byn0, we can enlargeH′1 to H′n := H′1 · S∨[n],which now surjects ontoH with kernel S∨[n] µn. These isogenies yield obstruction classescn ∈ H2(ΓF ,Z/n) (asΓF-module, theµn ⊂ S∨[n] is of course trivial) that are compatible under thenatural mapsZ/n→ Z/n′ for n|n′. Tate’s theorem ([Ser77]) tells us that

lim−−→n

H2(ΓF ,Z/n) = H2(ΓF,Q/Z) = 0,

so for sufficiently largen, there exists a lift ˜ρ : ΓF → H′n(Qℓ).

Remark 2.1.5. • Note that crucial to lifting here is the ability to enlarge the coefficients:if H′ ։ H is a morphism of groups overQℓ andρ lands inH(Qℓ), then we only obtain alift to H′(Qℓ(µn)) for sufficiently largen.• For example (and similarly in general; compare Lemma 3.2.2), in the case of GLn →

PGLn, this proof produces lifts with finite-order determinant. If ρ is Hodge-Tate, it ispossible that no such lift is also Hodge-Tate. These non-Hodge-Tate Galois lifts haveno parallel in the ‘toy model’ of Example 2.1.3, but we will use them as a stepping-stone toward finding geometric lifts, just as in the automorphic context we can choosean initial, possibly non-algebraic, lift, and then modify it to something algebraic (seeProposition 3.1.12). By contrast, in the ‘motivic’ versionof this problem (see§4.2), themotivic Galois group does not have the extra flexibility to admit such ‘non-algebraic’ lifts.• If ρ is almost everywhere unramified, then it is easy to see that ˜ρ is almost everywhere

unramified; see Lemma 5.2 of [Con11].

The problem of lifting through the isogenyH′1 → H (or any isogenyH′ → H) is taken upin [Win95]. Before describing this work, we state results of Wintenberger and Conrad related tolifting p-adic Hodge theory properties through central quotients.6

Theorem 2.1.6 (Corollary 6.7 of [Con11]). Let H′ ։ H be a surjection of linear algebraicgroups overQℓ with central kernel of multiplicative type. For K/Qℓ finite, letρ : ΓK → H(Qℓ) be

4I use this awful notation to remain consistent with the ‘dualpicture’ that we will later use to think about thesequestions.

5The simplest example is SLn ⊂ GLn.6Wintenberger treated isogenies; the general case is due to Conrad.

17

a representation satisfying a basicℓ-adic Hodge theory property7 P. Provided thatρ admits a liftρ : ΓK → H′(Qℓ), it has a lift satisfyingP if and only ifρ|IK has a Hodge-Tate lift IK → H′(Qℓ).

Remark 2.1.7. When the kernel ofH′ → H is a central torus, Proposition 2.1.4 shows thatρ

always has some lift. We will see later (Corollary 3.2.12) that it even always has a Hodge-Tate lift,so there is no obstruction to finding a lift ˜ρ : ΓK → H′(Qℓ) satisfyingP. Even simple cases of thistheorem yield very interesting results: for instance, applied toH′ = GLn1×GLn2 → H ⊂ GLn1n2, forH the image of the exterior tensor product, it shows that if a tensor product ofΓK-representationssatisfiesP, then up to twist they themselves satisfyP. We will use this example in Proposition3.3.8.

We now sketch the proof of Wintenberger’s main global result. This provides an occasion tosimplify and generalize the arguments using subsequent progress inp-adic Hodge theory; it alsoserves to make clear what these methods do and do not prove, and to set up a contrast with thequite different techniques that we will use.

Theorem 2.1.8 (2.1.4 of [Win95]). Let H′π−→ H be a central isogeny of linear algebraic groups

overQ. Let S be a finite set of non-archimedean places of F. Then there exist two extensions ofnumber fields F′′ ⊃ F′ ⊃ F, and a finite set of finite places S′ of F′ such that for any prime numberℓ and any representationρℓ : ΓF → H(Qℓ) satisfying

• ρℓ has good reduction8 outside S ;• For all v|ℓ, the one-parameter subgroupsµv : Gm→ HCFv

giving the Hodge-Tate structureof ρℓ|ΓFv

lift to H ′CFv

;9

then the restrictionρℓ|ΓF′ admits a geometric liftρ′ℓ, unramified outside S′, to H′(Qℓ); if ρℓ is

crystalline (resp. semistable) at places aboveℓ, then the lift may be chosen crystalline (resp.semistable). Moreover, the restrictionρ′

ℓ|ΓF′′ is unique, i.e. independent of the initial choice of lift

ρ′ℓ.

Proof. We follow Wintenberger’s arguments in detail, simplifying where the technology al-lows. First we explain some generalities: since the map onQℓ-pointsπℓ : H′(Qℓ) → H(Qℓ) neednot be surjective, we have an initial obstructionO1(ρℓ) in Hom(ΓF,H1(ΓQℓ , kerπ)) given by thecomposition

ΓFρℓ−→ H(Qℓ)/ im(πℓ) → H1(ΓQℓ , kerπ(Qℓ)).

We have the choice of killing this obstruction by enlargingQℓ to some finite extension, or byrestrictingF; following Wintenberger, we will do the former. Having dealt with this obstruction,we will face the more serious lifting obstructionO2(ρℓ) in H2(ΓF , kerπ(Qℓ)).

With these preliminaries aside, we construct the fieldF′, first dealing with all but the (finite)set ofℓ lying below a place inS. Leta(π) be the annihilator of kerπ. The fieldF′ (and set of placesS′) will be defined, independent ofℓ, in the following three steps:

• Let F1 be the maximal abelian extension of exponenta(π), unramified outsideS. For allℓ andρℓ, the classO1(ρℓ) dies after restriction toΓF1: outside ofSℓ = S ∪ v|ℓ, it is of

7Hodge-Tate, de Rham, semistable, crystalline8Meaning unramified for places not dividingℓ, and crystalline for places aboveℓ.9When some local liftIFv → H′(Qℓ) exists, this condition is equivalent to the existence of a local Hodge-Tate lift.

Whenρℓ|IFvis moreover crystalline, or semistable, it is equivalent tothe existence of a crystalline, or semistable, lift.

18

course unramified, and ifv|ℓ but v < S (i.e. ρℓ|ΓFvis crystalline), then Theoreme 1.1.3 of

[Win95]) shows the existence of a crystalline lift, valued inH′(Qℓ), of ρℓ|IFv.10

• Let F2/F1 be a totally imaginary extension containingQ(ζa(π)) and such that for all placesv′ above a placev in S, a(π) divides the local degree [F2,v′ : F1,v(ζa(π))]. If we write S2 forthe set of places ofF2 aboveS, ℓ, or a(πℓ) (the annihilator of kerπ(Qℓ)), then this implieslocal triviality of the obstruction classes, as long as we assumeS contains no places aboveℓ. That is,

O2(ρℓ) ∈ ker

H2(ΓF2,S2, kerπ(Qℓ))→

⊕

v∈S2∪S∞

H2(ΓFv, kerπ(Qℓ))

.

At v′ aboveS, the fact thatF2 containsζa(π) makes the local obstruction class a Brauerobstruction, which is killed overF2 by the assumptiona(π)|[F2,v′ : F1,v(ζa(π))]. At placesnot in S but abovea(πℓ) there is no obstruction, since unramified reprsentations alwayslift (granted thatO1(ρℓ) is trivial). Finally, at places aboveℓ, which we have assumed fornow do not lie inS, Wintenberger’s local result implies the existence of a (crystalline,even) lift.11

• Let F3 be the Hilbert class field ofF2. By global duality, the above kernel is Pontryagindual to

ker

H1(ΓF2,S2, kerπ(Qℓ)

∗)→⊕

v∈S2∪S∞

H1(ΓF2,S2, kerπ(Qℓ)∗)

The Galois module kerπ(Qℓ)∗ is trivial sinceF2 containsζa(π), so this is just the space ofhomomorphismsΓF2 → kerπ(Qℓ) that are unramified everywhere and trivial at the placesS2. Restriction of the classO2(ρℓ) to F3 corresponds to (via global duality forF3 now)the *transfer* on the Pontryagin dual of theH1’s, so by the Hauptidealsatz (trivializationof ideal classes upon restriction to the Hilbert class field), O2(ρℓ) dies upon restriction toF3.

We conclude that, independent ofℓ not below places ofS, and ofρℓ satisfying the hypothesesof the theorem, there exist lifts over the number fieldF3. To deal withℓ for which the localrepresentations are not crystalline, we apply Proposition0.3 of [Win95] to find a finite extensionF(ℓ)/F, unramified outsideSℓ, and such that allρℓ|ΓF(ℓ) will lift. 12 EnlargingF3 by the compositeof theF(ℓ) (for ℓ below places ofS), we obtain a number fieldF4 and a set of placesS4 equal to allplaces aboveS or a(π), such that forall ℓ, anyρℓ as in the theorem has a lift with good reductionoutsideS4 after restriction toΓF4.

The final step of the argument makes one final extension to showthat these lifts can be takencrystalline (or semistable) wheneverρℓ is, and even unramified outside all primes aboveSℓ (in-cluding those abovea(π), although this is not necessary for the theorem’s conclusion).13 This will

10Since weakly-admissible is now known to be equivalent to admissible, Wintenberger’s argument applies withoutrestriction on the Hodge-Tate weights, or the ramification of the local fieldFv/Qℓ.

11On the full decomposition groupΓFv , this is the Proposition of 1.2 of [Win95]; to extend fromIFv to ΓFv , onemust use the fact that a crystalline lift on inertia is unique.

12This is a ‘soft’ result. It is easy to see that for fixedℓ, all ρℓ unramified outsideS lift after restriction tosomeF(ℓ). The point of Wintenberger’s theorem is that, with theℓ-adic Hodge theory hypotheses,F(ℓ) can be takenindependent ofℓ.

13We omit the details. See Lemme 2.3.5 and following of [Win95].

19

be the number fieldF′, and the set of placesS′ can then be taken to be those aboveS. Finally, allsuch liftsρ′

ℓ(with good reduction outsideS′) are the same over the maximal abelian extension of

exponenta(π), unramified outsideS′, of F′; this is ourF′′.

Remark 2.1.9. We can apply the theorem to a system of representationsρℓ all having good re-duction outsideS, and all having liftable Hodge-Tate cocharacters (which inthe ‘motivic’ case wemay assume all arise from a common underlying geometric Hodge structure), to obtain a commonnumber fieldF′, with finite set of placesS′, and a further extensionF′′ such that the whole systemlifts to F′ (with good reduction outsideS′), and such that all the lifts are unique overF′′. The cru-cial limitation of this result is that, even if theρℓ are assumed to be the system ofℓ-adic realizationsof some motive (for absolute Hodge cycles, as in [Win95], or for motivated cycles), the proof ofthe theorem produce liftsρ′

ℓthat are neither (proven to be) motivic nor even weakly compatible

(almost everywhere locally conjugate). So, even with the strong uniqueness properties ensured byWintenberger’s theorem, it is still not at all clear how to lift a compatible (or even motivic) systemof ℓ-adic representations to another compatible (or motivic) system.

Here is the simplest cautionary example of how weakly compatible systems need not lift toweakly compatible systems; it arises from an example due to Serre (see [Lar94]) involving pro-jective representations that are everywhere locally, but not globally, conjugate:

Example 2.1.10. For any positive integern, let Hn denote the Heisenberg group of upper-triangular unipotent elements in GL3(Z/nZ). This is a non-split extension

1→ Z/n→ Hn→ Z/n× Z/n→ 1,

and writingA andB for lifts of the generators of the quotient copies ofZ/n, andZ for a generatorof the center, we have the commutation relation [A, B] = Z. Now let ζ be a primitiventh root ofunity; for anyα ∈ Z/n, we can then define a representationρα : Hn → GLn(C) by, in the standardbasisei, i = 1, . . . , n, and abusively lettinge0 denoteen as well,

A(ei) = ei−1

B(ei) = ζ(i−1)αei

Z(ei) = ζαei

for i = 1, . . . , n. One can check that for allα ∈ (Z/nZ)×, the associated projective representationsρα : Hn → PGLn(C) are everywhere locally conjugate. Nevertheless, for distinct α, they are notglobally conjugate: if they were, then the corresponding GLn(C)-representations would be twist-equivalent. The implications for our problem are the following: takeρα andρβ for α , β, andform a weakly-compatible systemρℓ : ΓF ։ Hn → PGLn(Qℓ) (Hn certainly arises as Galois groupof a finite extension of number fields.) in which someρℓ are formed fromρα and others fromρβ.There is then no system of weakly-compatible lifts ˜ρℓ : ΓF → GLn(Qℓ), since any character ofΓF

is trivial onZ (viewed as an element of Gal(FabF′/F), whereF′/F is the givenHn-extension; notethat Gal(FabF′/Fab) is generated byZ). Also note that we can even produce such examples where

20

the lifts to GLn(C) have the same determinant:

det(ρα) :

A 7→ (−1)n−1;

B 7→ 1 if n is odd, or ifα andn are even;−1 otherwise;

Z 7→ 1.

Thus, even weakly compatible systemsΓF → PGLn(Qℓ) × Q×ℓ need not have compatible lifts

through the isogeny GLn→ PGLn × Gm.To connect this example more explicitly with endoscopy, note that CentPGLn(C)(ρα) is the (cyclic

ordern) subgroup generated byρα(A), whereasρα itself is irreducible. These centralizers partiallygovern the multiplicities of discrete automorphic representations in Arthur’s conjectures. Explic-itly, ρα(ABA−1) = ζαB.

2.2. ℓ-adic Hodge theory preliminaries

2.2.1. Basics.In this section, we recall some background and prove some simple lemmas inℓ-adic Hodge theory. Throughout, letK/Qℓ be a finite extension. Choose an algebraic closureK/K,and letCK denote the completion ofK. SinceΓK acts by isometries onK, CK inherits a continuousΓK-action. We will mainly use only the simplest of Fontaine’s period rings, BHT and BdR, and eventhese only formally. Again, we refer the reader to the references in§1.2.2 for background, but herewe quickly recall the little needed to follow future arguments. BHT is the gradedCK-algebra

BHT =⊕

i∈ZCK(i),

with the obvious continuousCK-semilinear action ofΓK, CK(i) denoting a twist ofCK by the i th

power of the cyclotomic character. BdR is the fraction field of a complete DVR andK-algebra B+dRwhose residue field isCK. We do not recall the construction, but note that BdR is a filtered K-algebra (the filtration associated to the maximal ideal of B+

dR) with a continuousK-linear action ofΓK. Two of the fundamental results of the theory assert that gr•(BdR) BHT (as gradedCK-algebraswith semi-linearΓK-action) and BΓK

dR = BΓKHT = K. From this second point it follows that ifV is a

finite-dimensional representation ofΓK overQℓ, then

DHT(V) = (BHT ⊗Qℓ V)ΓK

is aK-vector space of dimension at most dimQℓ(V), and similarly

DdR(V) = (BdR ⊗Qℓ V)ΓK

is aK-vector space of dimension at most dimQℓ(V).

Definition 2.2.1. LetV be a finite-dimensional representation ofΓK overQℓ. V is said to beHodge-Tateif dimK DHT(V) = dimQℓ(V), andde Rhamif dimK DdR(V) = dimQℓ(V).

If V is de Rham, then it is Hodge-Tate, but not conversely.14

14The standard example is a non-split extension ofΓK- representations

0→ Qℓ → V → Qℓ(1)→ 0.

Standard results on Galois cohomology of local fields show such aV exists. It’s not so hard to showV is Hodge-Tate,but to show that it cannot be de Rham seems to require quite deep input; see for instance [BC, Example 6.3.5].

21

We will usually considerΓK-representationsV with coefficients in some extensionE/Qℓ, typ-ically either some unspecified finite extension, orE = Qℓ. Then DdR(V) (mutatis mutandis forDHT) is a filteredK ⊗Qℓ E-module; ifV, viewed as aQℓ representation by forgetting theE-linearstructure, is de Rham, then DdR(V) is a freeK ⊗Qℓ E-module. To see this, we may assumeE islarge enough to contain all embeddings ofK intoQℓ. Then for allQℓ-embeddingsτ : K → E, wehave orthogonal idempotentseτ ∈ K ⊗Qℓ E, giving rise to an isomorphism

K ⊗Qℓ E(eτ−−→∼

⊕

τ : K→E

E

x⊗ α 7→ (τ(x)α)τ .

As K ⊗Qℓ E-module,

DdR(V) ⊕

τ

eτDdR(V) ⊕

τ

(BdR ⊗K,τ V)ΓK

(compare Lemma 2.2.6), and each space on the right-hand sidehasE-dimension at most dimE(V).The sum of theirE-dimensions is then at most dimQℓ(K) dimE(V), so theQℓ-dimension of the right-hand side is at most dimQℓ(K) dimQℓ(V), which is, sinceV is de Rham, theQℓ-dimension of theleft-hand side. Therefore equality holds everywhere, and it easily follows that DdR(V) is free overK ⊗Qℓ E ⊕τE.

Example 2.2.2. Note that even whenV is de Rham, other steps of the filtration need not be freeover K ⊗Qℓ E. We sketch a basic example, omitting the details because they require introducingmore about BdR than we will subsequently need. Take any extension to a characterψ of ΓK of thecharacter

IKrecK−−−→ O×K

τ−→ Q×ℓ ,

whereτ : K → Qℓ is any fixedQℓ-embedding. IfK = Qℓ, so that there is only one such embedding,thenψ|IK is just the restriction to inertia of the cyclotomic character, and DdR(ψ) has filtrationconcentrated in degree -1. IfK , Qℓ, then for all embeddingsτ′ , τ, the filtered one-dimensionalQℓ-vector spaceeτ′DdR(ψ) has its filtration concentrated in degree zero, whereas theτ piece isconcentrated in degree -1.

It will be crucial for us to have a refined notion of Hodge-Tateweight that takes into accountthe different embeddingsτ : K → Qℓ:

Definition 2.2.3. LetV be a de Rham representation ofΓK on aQℓ-vector space of dimensiond, so that DdR(V) is a freeK ⊗Qℓ Qℓ-module of rankd. Then for eachτ : K → Qℓ, we define thed-element multi-set ofτ-labeled Hodge-Tate weights, HTτ(D), to be the collection of integersh(with multiplicity) such that

grh(eτ(DdR(V)) , 0.

We could also make this definition forV having coefficients in a subfieldE ⊂ Qℓ, possiblythen having to enlargeE in order to contain the embeddings ofK (and thus define the labeledHodge-Tate weights). In§2.4.1, we will discuss the elementary but significant implications ofEnot containing all embeddings ofK.

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2.2.2. Labeled Hodge-Tate-Sen weights.Later we will work with ΓK-representations thatare not even Hodge-Tate and will need a generalized notion of(labeled) Hodge-Tate weight. First,again letV be a de Rham representation ofΓK on aQℓ-vector space. The identification of gr•(BdR)with BHT implies that the multiplicity ofq in HTτ(V) also equals dimQℓ(V ⊗τ,K CK(−q))ΓK . We usethis formulation to extend the definition ofτ-labeled weights to the Hodge-Tate case. Note thatif K′/K is any finite extension, andτ′ : K′ → Qℓ is any embedding extendingτ : K → Qℓ, thenHTτ′(V|ΓK′ ) = HTτ(V).

Now we drop the assumption thatV is Hodge-Tate. For Sen’s theory, which among other thingsintroduces a generalized notion of Hodge-Tate-Sen weight,[Ber04, II.1.2] is a very brief, andvery illuminating, overview; see the references there or [BC, Part IV] for a detailed introduction.Sen’s theory gives much more precise results than what we usehere; the most important thingfor our purposes is that it associates to anyCK-semi-linear representationV of ΓK a CK-linearendomorphismΘV (the ‘Sen operator’) ofV (see [BC, Corollary 15.1.6, Theorem 15.1.7]). Theeigenvalues ofΘV are the ‘Hodge-Tate-Sen’ weights; this terminology is justified by the fact thatVis Hodge-Tate15 if and only ifΘV is semi-simple with integer eigenvalues ([BC, Exercise 15.5.4]).

We will now be more precise about how to work withΓK-representations with coefficients inQℓ. We will also describe Sen operators for representations valued in a general linear algebraicgroup. For some finite extensionE/Qℓ insideQℓ, our (arbitrary)ΓK-representationV descendsto an E-linear representationVE. For any embeddingι : Qℓ → CK, we have the Sen operatorΘρ,ι ∈ EndCK (V ⊗Qℓ ,ι CK). More precisely, letK′ = ι(E)K, so thatVE ⊗E,ι CK is aCK-semilinearrepresentation ofΓK′, to which we can in the usual way associate the Sen operatorΘρ,ι. Notethat the Sen operator of aCK semi-linear representation is insensitive to finite restriction, so thisconstruction is independent of the choice of (sufficiently large)K′. Moreover, choosing a modelVE′ of V with coefficients in some finite extensionE′/E insideQℓ also yields the sameΘρ,ι, sincewe don’t change theCK-semilinear representation ofΓK′ (this may be a biggerK′, having enlargedE). This allows the following Tannakian observation about Sen operators (§6 of [Con11]).

Lemma 2.2.4.Let G be a linear algebraic group overQℓ with Lie algebrag andρ : ΓK → G(Qℓ)a (continuous) representation. Then for each embeddingι : Qℓ → CK, there exists a unique SenoperatorΘρ,ι ∈ g ⊗Qℓ ,ι CK such that for all linear representations r: G −−→

/Qℓ

GLV, Lie(r)(Θρ,ι) is

equal to the previously-definedΘrρ,ι.

Proof. This follows from Tannaka duality for Lie algebras (a precise reference is [HC50]).Namely, for eachQℓ-representationG→ GLV we get an elementΘV,ι ∈ glV⊗ιCK , and these satisfythe functorial properties

• If V1T−→ V2 is aG-morphism, then Lie(T) ΘV1,ι = ΘV2,ι Lie(T);

• ΘV1⊕V2,ι = ΘV1,ι ⊕ ΘV2,ι;• ΘV1⊗V2,ι = ΘV1,ι ⊗ idV2 + idV1 ⊗ ΘV2,ι.

(If different fieldsE and K′ as above are needed to define theΘVi ,ι, we can pass to a commonextension and apply the remarks preceding the lemma.) Tannaka duality then implies that allΘV,ι

arise from a unique elementΘρ,ι ∈ g ⊗ι CK.

15Strictly speaking, in§2.2.1 we only defined Hodge-Tate representations on finite-dimensionalQℓ-vector spaces.This is a special case of a more general notion forCK-semi-linearΓK-representations: see [BC, Definition 2.3.4].

23

Remark 2.2.5. The necessary functorial properties of the Sen operators are not automatic. Allthe relevant statements are in§15 of [BC].

As mentioned previously,V⊗Qℓ ,ιCK is Hodge-Tate if and only ifΘρ,ι is semi-simple with integereigenvalues. We will need the following comparison:

Lemma 2.2.6.Supposeρ : ΓK → AutQℓ(V) is Hodge-Tate. Any embeddingι : Qℓ → CK induces

τι : K → Qℓ, and we then have

HTτι(ρ) = eigenvalues ofΘρ,ι.

Proof. Let E and K′ be as above. TheΓK-representationVE is Hodge-Tate, and there is anatural isomorphism of gradedE ⊗Qℓ K′-modules

(VE ⊗Qℓ BHT)ΓK ⊗K K′∼−→ (VE ⊗Qℓ BHT)ΓK′ .

Restricting to theqth graded pieces, we get anE ⊗Qℓ K′-isomorphism

(VE ⊗Qℓ CK(−q))ΓK ⊗K K′∼−→ (VE ⊗Qℓ CK(−q))ΓK′ ,

which in turn is anE ⊗Qℓ K′-isomorphism⊕

τ : K→E

(VE ⊗τ,K CK(−q))ΓK ⊗K K′∼−→

⊕

ι : E→CK

(VE ⊗E,ι CK(−q))ΓK′ .

Projecting to theτ-component, we obtain anE ⊗τ,K K′-isomorphism

(VE ⊗τ,K CK(−q))ΓK ⊗K K′∼−→

⊕

ι : E→CK :τι=τ

(VE ⊗E,ι CK(−q))ΓK′ .

Writing mq,τ for the multiplicity of q in HTτ(V), the left-hand side is a freeE ⊗τ,K K′-module ofrankmq,τ. Meanwhile, theK′-dimension of theι-factor of the right-hand side is by definition themultiplicity of q as a Hodge-Tate weight ofVE ⊗E,ι CK, hence isq’s multiplicity as an eigenvalueof Θρ,ι. We deduce (applying theι-projection inE⊗τ,K K′

∏ι:τι=τ K′) that for allι the eigenvalues

of Θρ,ι match the multi-set HTτι(V).

Remark 2.2.7. We will only apply this whenV is in fact de Rham, with the exception ofCorollary 3.2.12.

2.2.3. Induction and ℓ-adic Hodge theory. We will later need a result on compatibility ofFontaine’s functors with induction. The following lemma iscertainly well-known, but I don’tknow of a proof in the literature, so I record some details. For a finite extensionK/Qℓ, we let (asis standard notation in the subject)K0 ⊂ K denote the maximal unramified sub-extension ofK.

Lemma 2.2.8. Let L/K/Qℓ be finite, and let W be a de Rham representation ofΓL. Then V=IndK

L W is also de Rham, andDdR(V) is the image under the forgetful functorFilL → FilK ofDdR(W).Moreover,IndK

L W is crystalline if and only if W is crystalline and L/K is unramified.

Proof. This follows almost immediately from Frobenius reciprocity if one uses contravariantFontaine functors:

D∗dR(V) := HomΓK (V,BdR) HomΓL(W,BdR|ΓL) = D∗dR(W).

24

Since D∗dR(V) DdR(V∗), and IndKL (W∗) IndKL (W)∗, this shows that DdR(V) is simply the filtered

K-vector space underlying DdR(W). Comparing dimensions, it is clear thatV is de Rham if andonly if W is. In the crystalline case, the same observation yields D∗

cris(V) D∗cris(W), so

dimK0 Dcris(V) = dimK0 Dcris(W) = dimL0 Dcris(W)[L0 : K0],

and comparing dimensions we see thatV is crystalline if and only ifW is crystalline and [L0 :K0] = [L : K], i.e. L/K is unramified.

2.3. GL1

A detailed analysis of the GL1 theory provides both motivation and technical tools for address-ing the more general Galois and automorphic lifting problems.

2.3.1. The automorphic side.We begin by reviewing some basic facts about Hecke charac-tersψ : A×F/F

× → C×. Any suchψ is the twist by| · |rAF, for somer ∈ R, of a unitary Hecke

character. At each placev of F, ψv = ψ|F×v can be decomposed via projection to the maximalcompact subgroup inF×v ; at v|∞, this lets us write any unitaryψv in terms of a given embeddingιv : Fv → C as

ψv(xv) = (ιv(xv)/|ιv(xv)|)mv|ιv(xv)|itvC ,with mv = mιv an integer, andtv a real number. To avoid cumbersome notation, we will sometimesomit reference to the embeddingιv. Hecke characters encode finite Galois-theoretic informationas well as rather subtle archimedean information; the latter will be particularly relevant in whatfollows, and the basic observation ([Wei56]) is:

Lemma 2.3.1. There is a (unitary, say) Hecke characterψ of F with archimedean parametersmv, tvv|∞, as above, if and only if for some positive integer M, all global unitsα ∈ O×F satisfy

∏

v|∞

(ιv(α)|ιv(α)|

)mv

|ιv(α)|itv

M

= 1.

Equivalently, the inside product is trivial for allα in some finite-index subgroup ofO×F .

Using the notation we have just established, we recall Weil’s notions of typeA0 and typeAHecke characters:

Definition 2.3.2. A Hecke characterψ is said to be of typeA if r ∈ Q andtv = 0 for all v|∞. ψis said to be typeA0 if tv = 0 andmv

2 + r is an integer for allv|∞. Equivalently, for allv|∞, thereexist integerspv andqv such thatψv(z) = ιv(z)pv ιv(z)qv.

Then Weil ([Wei56]) shows:

Lemma 2.3.3 (Weil). Let χ be a type A Hecke character locally trivialized by the modulusm.χ then induces a characterχm of Im, the group of non-zero, prime tom, fractional ideals, whosevalues are all algebraic numbers.

Later we will be interested in a higher-rank version of the typeA condition; in current discus-sions of algebraicity of automorphic representations, people tend to focus on analogues of typeA0.We recall a basic result of Weil-Artin ([Wei56]), which will later inspire much of our discussionin higher-rank as well.

25

Lemma 2.3.4 (Weil-Artin). Letψ be a type-A Hecke character of a number field F. Write Fcm

for the maximal CM subfield of F. Then there exists a finite-order Hecke characterχ0 of F and atype-A Hecke characterψ′ of Fcm such thatψ = χ0 · BCF/Fcm(ψ′). If Fcm is totally real, thenψ is afinite-order twist of some power| · |r , r ∈ Q, of the absolute value.

Remark 2.3.5. We emphasize that the true content of this result, which should fully generalizeto higher-rank (see Proposition 2.4.7), is that the infinity-type of a type-A Hecke character descendsto the maximal CM subfield.

Proof. Weil leaves it as an exercise, so we give a proof. We then indicate a second proof,which we will apply more generally in Proposition 2.4.7. We may assume thatψ is typeA0. FirstassumeF/Q is Galois, and setG = Gal(F/Q). We may regardF as a subfield ofC (i.e., choose anembedding), so the above relation becomes

∏

g∈Gg(α)mg = 1

for all α in some finite-index subgroup ofO×F and some integersmg. Now take any embeddingι : F → C, using it to define a complex conjugationcι on F, and thus onG. In particular, whenFis totally imaginary we can rewrite the above as

∏

G/cι

g(α)mg(cιg)(α)mcιg = 1.

Applying log|ι(·)|, we get (for allα)∑

G/cι

(mg +mcιg) log |ι(g(α))|C = 0,

which by the unit theorem is only possible whenmg + mcιg equals a constantwι independent ofg; the analogous argument in the totally real case shows that the mg themselves are independentof g, concluding that case of the proof. Butwι is independent ofι as well,16 hencemg + mcιg isindependent both ofg and the choice of conjugation. This implies that for any twoι, ι′, mg = mcιcι′g,and sinceFcm is precisely the fixed field (inF) of the subgroup generated by all productscιcι′ , weare done.

Now let F be arbitrary, and letψ be a typeA Hecke character ofF. As usual lettingF denotethe Galois closure, we know from the Galois case thatψ|F has infinity-type descending to (F)cm.SettingH1 = Gal(F/F) andH2 = Gal(F/(F)cm), the valuesmιv (for embeddingsιv : F → C) areconstant onH1-orbits (by construction) andH2-orbits (by the Galois case), hence onH1H2-orbits.But F ∩ (F)cm = Fcm, soH1H2 = Gal(F/Fcm), and the infinity-type descends all the way down toFcm, proving the lemma in general.

Later, we will generalize the following argument (see Proposition 2.4.7 for more details) inthe totally imaginary case: for allσ ∈ Aut(C), there is a Hecke characterσψ whose finite part isσψ f (x) = σ(ψ f (x)) and whose infinity-type is given by the integersmσ−1ιv (the key but easy check isF×-invariance). We may assumeψ is unitary, socψ = ψ−1, and therefore the fixed fieldQ(ψ f ) ⊂ Qof all σ ∈ Aut(C) such thatσψ f ψ f is contained inQcm. The result follows from some Galoistheory.

16For instance, whateverι, 12 |G|wι =

∑g∈G mg.

26

As pointed out to me by Brian Conrad, a third, more algebraic,approach is possible, whereinfinity-types of algebraic Hecke characters are related to(algebraic) characters of the connectedSerre group; compare the discussion of potentially abelianmotives in§4.1.30. Of course, in allarguments, the unit theorem is the essential ingredient.

We will have to analyze Hecke characters in the following setting: An automorphic represen-tationπ of an F-groupG has a central characterωπ : ZG(F)\ZG(A) → C×, and we will want tounderstand the possible infinity-types of extensions

ZG(F)\ZG(A)ωπ

//

C×

Z(F)\Z(A)ω

99ss

ss

s

whereZ is anF-torus containingZG. The basic case to which this is reduced, for split groups atleast, is whereZG = µn andZ = Gm, where we have to extend a character fromµn(F)\µn(A) to thefull idele class groupF×\A×.

Lemma 2.3.6.Let F be any number field. Given a continuous characterω : µn(F)\µn(A)→ S1,fix embeddingsιv : Fv → C and write

ω∞ : (xv)v|∞ 7→ ιv(xv)mv

for some set of residue classes mv ∈ Z/nZ. Then:

• There exist type A (unitary) Hecke character extensionsω : F×\A× → S1 if and only ifthe images mv ∈ Z/nZ depend only on the restriction v|Fcm. There exists a finite-orderextensionω if and only if for all complex v|∞, the classes mv ∈ Z/nZ are all zero.• The extensionω is unique up to the nth power of another type A Hecke character, except

in the Grunwald-Wang special case, where one first has two choices of extension to CF [n](then the nth power ambiguity).• In particular, if F is totally real or CM, then (unitary) typeA extensions ofω always exist.

If F is totally real, they are finite-order, and if F is CM, a finite-order extension can bechosen if and only ifω|F×∞ is trivial.

Proof. For the time being, letF be arbitrary. The cokernel ofµn(F)\µn(A) → CF[n] is eithertrivial or order 2 (the Grunwald-Wang special case), so we first choose an extensionω′ to CF [n](we will use this flexibility in Proposition 3.1.4. Then Pontryagin duality gives an isomorphism

CDF /nCD

F∼−→ (CF [n])D,

so there exists some ˜ω ∈ CDF extendingω′, and it is unique up tonth powers. Restricted toF×∞, we

can write (implicitly invokingιv)

ω∞ : (xv)v|∞ →∏

v|∞(xv/|xv|)mv|xv|itvC

for integersmv and real numberstv. Then a typeA lift of ω′ exists if and only if there is a Heckecharacterψ with infinity-type

ψ∞ : (xv) 7→∏

v|∞(xv/|xv|) fv |xv|itv/nC

27

for some integersfv (ωψ−n is then the desired typeA character). Following Weil, the existence of aHecke character ˜ω corresponding to the infinity-datamv, tvv is equivalent to the existence of someintegerM such that for allα ∈ O×F ,

∏

v|∞(α/|α|)mvM |α|itvM

v = 1,

and, similarly, such aψ can exist if and only if there existsM′ ∈ Z such that∏

v|∞(α/|α|) fvMM′n|α|itvMM′

v = 1

for all α ∈ O×F. Substituting, the left-hand side is∏

v|∞(α/|α|)( fvn−mv)MM′ , which can equal 1 for allα ∈ O×F if and only if there exists a typeA Hecke character with infinity-type

(xv) 7→∏

v|∞(xv/|xv|) fvn−mv.

By Weil’s classification of typeA Hecke characters, this is possible if and only iffvn−mv dependsonly on v|Fcm. This in turn is possible (for some choice offv) if and only if mv ∈ Z/nZ dependsonly onv|Fcm. In particular, over CM and totally real fieldsF, there is no obstruction, so a typeAlift of ω always exists.

The claim about existence of finite-order extensions is a simple variant: if allmv (v complex)are zero inZ/nZ, then there is a typeA lift ω with infinity-type

ω∞ : (xv)v|∞ →∏

v|∞(xv/|xv|)mv

for some integersmv, all divisible byn and only depending onιv|Fcm. But then there is also a typeA Hecke characterψ of F with infinity-type

ψ∞ : (xv)v|∞ →∏

v|∞(xv/|xv|)mv/n,

soωψ−n is a finite-order extension ofω.As for uniqueness, once a typeA lift of ω′ is chosen, any other is (by the discussion at the

start of the proof) a twist by thenth power of another Hecke character, which must itself clearlybeof typeA. This is then the ambiguity in extendingω itself, except in the Grunwald-Wang specialcase, in which there is the additionalZ/2Z ambiguity noted above.

Remark 2.3.7. • Of course any quasi-character ofµn(A) is unitary, and to understandall extensions it suffices (twisting by powers of| · |A) to understand unitary extensions–hence the restriction to S1 instead ofC×.• This raises a tantalizing question: certainly over a non-CMfield F we can produce char-

actersω that have no typeA extensions (although they certainly all have some extension toa Hecke character, by Pontryagin duality). What ifω actually arises as the central charac-ter of a suitably algebraic cuspidal automorphic representation (on SLn(A), for instance)?We return to this in§2.4.

A few consequences, either of the result or the method of proof, will follow; first, though, letus make a definition:

28

Definition 2.3.8. LetF be any number field, and letψ ∈ CDF be a unitary Hecke character. We

say thatψ is of Maass typeif for all v|∞, the restrictionψv : F×v → S1 has the formx 7→ |x|itv forreal numberstv.

Corollary 2.3.9. Letψ : CF → C× be a Hecke character of a number field F.

• If F is totally real or CM, thenψ can be decomposed as

ψ = ψalgψMaassψ f in| · |w,where w is the unique real number twistingψ to a unitary character,ψalg is unitary typeA,ψMaassis of Maass type, andψ f in is finite order. The last three characters are all uniqueup to finite-order characters.• If ψ is of Maass type (after twisting to a unitary character), with F arbitrary, thenψ is

‘nearly divisible’: for any n∈ Z there exist Hecke charactersχ of Maass type andχ0 offinite order such thatχnχ0 = ψ. In particular, after a finite base-changeψ is n-divisible.• WriteAF(1) for the space (topological group) of all Hecke characters ofF. Suppose F is

CM, of degree2s overQ. Then there is an exact sequence

1→ ΓDF →AF(1)→ R × Zs× Qs−1→ 1.

(As always,ΓDF denotesHomcts(ΓF ,S1); it is the space of Dirichlet characters.) For the totally real

subfield F+ of F, we have a similar sequence

1→ ΓDF+ →AF+(1)→ R × Qs−1→ 1.

It is also possible (via the unit theorem) to compute all possible infinity-types of Hecke char-acters of any number field, but we will make no use of this calculation. Roughly speaking, thenew transcendentals that arise in the ‘mixed’ case where algebraic and Maass parts cannot be sep-arated are the arguments (angles) of fundamental units. It is very tempting to ask whether for CM(or totally real fields) algebraic and spherical (‘Maass’) infinity-types on higher-rank groups cantwist together in a non-trivial way. If this question is too naıve, is there any other higher-rankgeneralization of part 1 of Corollary 2.3.9?

We include a couple other related useful results. For the first, note that in contrast with thesituation forℓ-adic Galois characters (see Lemma 2.3.15 below), a generalHecke character cannotbe written as annth-power up to a finite-order twist.17

Lemma 2.3.10. Let ψ be a Hecke character of F, and suppose that L/F is a finite Galoisextension over whichBCL/F(ψ) = ψ NL/F is an nth-power. Then up to a finite-order twist,ψ itselfis an nth-power.

Proof. Letω be a Hecke character ofL such thatψNL/F = ωn. Thenω is Gal(L/F)-invariant,

and the next lemma shows thatω descends toF up to a finite-order twist

The next lemma completes the previous one, and also enables aslight refinement of a result ofRajan (see below):

Lemma 2.3.11. Let L/F be a Galois extension of number fields, and letψ be a Gal(L/F)-invariant Hecke character of L. Then there exists a Hecke character ψF of F and a finite-orderHecke characterψ0 of L such thatψ = (ψF NL/F) · ψ0.

17Consider, for example, a typeA Hecke character whose integral parameters at infinity are not divisible byn.

29

Proof. We may assumeψ is unitary, and we may choose, for all infinite placesw of L, embed-dingsιw : Lw → C such that allιw for w above a fixed placev of F restrict to the same embeddingιv : Fv→ C. Gal(L/F) acts transitively on the placesw|v, so when we write

ψw(xw) =

(ιw(xw)|ιw(xw)|

)mw

· |ιw(xw)|itw,

Gal(L/F)-invariance implies that themw andtw depend only on the placev beloww (from now onin the proof, the embeddingsιw will be implicit). We therefore denote these bymv andtv. Lemma2.3.1 implies there is a Hecke character ofF with infinity-type given by the datamv, tvv. Namely,there is an integerM such that for allα ∈ O×L,

1 =

∏

w|∞

(α

|α|

)mw

|α|itwC

M

.

Restricting toα ∈ O×F , this becomes

1 =

∏

v|∞

∏

w|v

(α

|α|

)mv

|α|itvC

M

=

∏

v|∞

(α

|α|

)mv

|α|itvC

M#w|v

,

(#w|v is independent ofv) which is simply the criterion for there to be a Hecke character of Fwith infinity-type given by the collectionmv, tvv. Any such character has base-change differingfrom ψ by a finite-order character, so we are done.

Remark 2.3.12. The aforementioned theorem of Rajan describes the image of solvable base-changesolvable base-change: precisely, for a solvable extensionL/F of number fields and a Gal(L/F)-invariant cuspidal automorphic representationπ of GLn(AL), Theorem 1 of [Raj02] asserts thatthere is a cuspidal representationπF of GLn(AF) and a Gal(L/F)-invariant Hecke characterψ of Lsuch that

BCL/F(π0) ⊗ ψ = π.Lemma 2.3.11 shows that thisψmay be chosen finite-order; in particular, for some cyclic extensionL′/L (soL′/F is still solvable), BCL′/L(π) descends toF.

2.3.2. GaloisGL1. We now discuss (continuous) Galois charactersψ : ΓF → Q×ℓ . For the time

being,F is any number field. These are necessarily almost everywhereunramified, and we focuson ℓ-adic Hodge theory aspects. The following is well-known, and is proven in [Ser98, ChapterIII]:

Theorem 2.3.13.Suppose thatψ is de Rham. Then for all places v not dividingℓ, ψ|ΓFvassumes

algebraic values inQιℓ−→ Qℓ, and there exists a type A0 (i.e. L = C-algebraic) Hecke character

ψ : AF/F× → C× corresponding toψ (via ι∞ : Q → C). Moreover,ψ is motivic: the Fontaine-Mazur conjecture holds forGL1/F.

Proof. We indicate how this follows from the arguments of [Ser98, III]; see Remark 2.3.14 forthe explanation of a simple case. Sinceψ is Hodge-Tate at allv|ℓ, it is (by a theorem of Tate: see[Ser98, III-A6]) locally algebraic in the sense of [Ser98, III-1.1]. Then the argument of [Ser98,III.2.3 Theorem 2] implies that, for a suitable modulusm, ψ is theℓ-adic Galois representationassociated to an algebraic homomorphismS

m,Qℓ→ Gm,Qℓ

, whereSm is theQ-torus of [Ser98, II-2.2]. Up to isomorphism, this algebraic representation canbe realized over some finite extension

30

E of Q insideQℓ. Taking the ‘archimedean realization’ as in [Ser98, II-2.7], we obtain the desiredtype A0 Hecke character. It then follows by Lemma 2.3.3 thatψ|ΓFv

assumes algebraic valuesfor v ∤ ℓ. The Fontaine-Mazur conjecture forψ follows from a theorem of Deligne ([DMOS82,Proposition IV.D.1]), which shows the somewhat stronger statement thatψ is theℓ-adic realizationof a motive for absolute Hodge cycles.

Remark 2.3.14. Consider the simple case in whichF = Q. Thenψ|ΓQℓ has a single labeledHodge-Tate weight; it therefore has the same (labeled) Hodge-Tate weights as an integer powerωr

ℓ

of theℓ-adic cyclotomic character. It follows then from the above-cited theorem of Tate ([Ser98,III-A6]) and global class field theory thatψω−r

ℓis finite-order, hence is theℓ-adic realization (via a

fixed isomorphismC∼−→ Qℓ) of some finite-order Hecke characterχ. The Hecke character corre-

sponding toψ is thenχ| · |rAQ . It is also easy to show thatψ is motivic:ωrℓ

is theℓ-adic realization of

the Tate motiveQ(r), and ifψω−rℓ

factors through a finite quotient Gal(F/Q), then it is a suitable di-rect factor ofH0(XQ,Qℓ), for the zero-dimensional varietyX = ResF/Q(SpecF) whoseQ-points are

naturally indexed by embeddingsτ : F → Q, with theΓQ-action onH0 arising from permutationof the set of embeddings (for more details, see the discussion of Artin motives in§4.1.2).

More generally, the idea behind establishing the Fontaine-Mazur conjecture in the abelian caseis to find a sub-quotient of the cohomology of a CM abelian variety having ‘labeled Hodge num-bers’ matching those ofψ; that this is possible essentially follows from the constraints on the∞-type of the typeA0 Hecke character underlyingψ (as in Lemma 2.3.4. If the CM abelian varietywere actually defined overF, this would realizeψ inside some finite-order twist of its cohomology.The subtle part of the theorem is the need to have some controlover the field of definition of theCM abelian variety.

We make repeated use of the following well-known observation (see for instance Lemma 3.1of [Con11]).

Lemma 2.3.15.Let χ : ΓF → Q×ℓ be anℓ-adic Galois character. For any non-zero integer m,

there are charactersχ1, χ0 : ΓF → Q×ℓ , with χ0 finite-order, such thatχ = (χ1)mχ0.

As on the automorphic side, we want to consider Galois characters somewhat outside the al-gebraic range. As is customary, we write HTτ(ρ) to indicate theτ : F → Qℓ-labeled Hodge-Tateweights of anℓ-adic Galois representationρ; see§2.2 for details, as well as for what we meanwhen we write non-integralτ-labeled Hodge-Tate-Sen weights.

Corollary 2.3.16. Let F be a totally imaginary field, and n a non-zero integer. For allτ : F → Qℓ, fix an integer kτ. Then there exists a Galois characterψ : ΓF → Q

×ℓ with HTτ(ψ) = kτ

n

(respectively, withHTτ(ψ) ≡ kτn modZ) if and only if

(1) kτ depends only onτ0 := τ|Fcm (respectively, only depends modulo n onτ0);(2) and there exists an integer w such that kτ0+kτ0c = w (respectively, kτ0+kτ0c ≡ w mod n)

for all τ, and c the complex conjugation on Fcm.18

Proof. Suppose such aψ exists, with weightskτn . ψn is geometric, and so (viaιℓ andι∞) there

exists a typeA0 Hecke characterψ of F corresponding toψn. Forv|∞ andιv : Fv∼−→ C, ψ|F×v is given

18If F is Galois, we can rephrase this askτ + kτc = w for all choices of complex conjugationc on F.

31

byψv(xv) = ιv(xv)

kτ∗(ιv) ιv(xv)kτ∗(ιv) ,

whereτ∗(ιv) = τ∗ℓ,∞(ιv) denotes the embeddingF → Qℓ induced byιv, ιℓ, andι∞. By purity for

Hecke characters, there is an integerw such thatkτ∗(ιv) + kτ∗(ιv) = w; moreover, by Weil’s descentresult,kτ∗(ιv) depends only onιv|Fcm. The constraint on the weights follows.

Conversely, given a set of weights satisfying the purity constraint, we form a putative infinity-type pιv = kτ∗(ιv) for a Hecke character ofF; that this is in fact an achievable infinity-type followsfrom Weil. We form the associated geometric Galois character and then use the fact that, up to afinite-order twist, we can always extractnth roots ofℓ-adic characters.

The modn statement is a simple modification. For instance, to construct a character withgiven weights satisfying the purity constraint modulon, proceed as follows. Choose a maximal setmoduloc of embeddingsτ0 : Fcm → C. Choose lifts toQ of the congruence classes

kτ0n modZ,

and declarekτ = kτ0 for all τ lying above theseτ0. Then choosew ∈ Z lifting kτ0 + kτ0c (mod n),which is possible by hypothesis, and setkτ0c = w− kτ0.

The same technique yields an easy example of the constraintson Galois characters over totallyreal fields:

Lemma 2.3.17.Suppose F is totally real, andψ : ΓF → Q×ℓ is a character with all Hodge-Tate-

Sen weights inQ. Then all of these weights are equal. Conversely, for x, d ∈ Z, there are globalcharacters with all HTS weights equal toxd.

Proof. For some integerd, all the HTS weights lie in1dZ, so ψd is geometric, and therefore

corresponds to a typeA0 Hecke character.F is totally real, soψd must be, up to a finite-order twist,an integer power of the cyclotomic character.

It is worth remarking that the most general answer to the question ‘what Galois charactersψ : ΓF → Q

×ℓ exist’ is essentially Leopoldt’s conjecture.

Finally, we will need a lemma refining the construction of certain Galois characters over CMfields:

Lemma 2.3.18.Let F be a totally real field in which a primeℓ , 2 is unramified. Let L= KFbe its composite with an imaginary quadratic field K in whichℓ is inert, and letψ be any unitarytype A Hecke character of L. Then:

(1) There exists a Galois characterψ : ΓL → Q×ℓ such thatψ2 corresponds toψ2 (which is

type A0).(2) Moreover, the frobenius eigenvaluesψ( f rv) at all unramified v lie inQcm.

Proof. First, letmιv as before denote the integers giving the infinity-type ofψ. Using the knownalgebraicity ofψ ([Wei56]) and the fixed embeddingsι∞, ιℓ, we can define theℓ-adic representationassociated toψ2:

(ψ2)ℓ rec−1L : A×L/(L×L×∞)→ Qℓ

(xw) 7→∏

w∤ℓ∞ψw(xw)2

∏

w|ℓ

ψw(xw)2∏

τ : Lw→Qℓ

τ(xw)mι∗∞,ℓ (τ)

.

32

By the dual Grunwald-Wang theorem, to show that (ψ2)ℓ is the square of some characterΓL → Q×ℓ ,

it suffices to check that for all placesw of L, the above character is locally onL×w a square. This in

turn immediately reduces to seeing whether, for eachw|ℓ, the characterχw : L×w → Q×ℓ given by

χw(xw) =∏

τ : Lw→Qℓ

τ(xw)mι∗∞,ℓ (τ)

is a square. Writing

L×w = 〈ℓ〉 × µ∞(L×w) × (1+ ℓOw),

it suffices to check on each component of this factorization. On the〈ℓ〉 factor, this is clear (choosea square root ofχw(ℓ)). On the 1+ ℓOw factor, theℓ-adic logarithm, using our hypotheses thatℓ , 2 is unramified, lets us define a single-valued square-root function, and thus extract a squareroot ofχw|1+ℓOw. Now note that the product overτ : Lw → Qℓ is a product over pairs of complex-conjugate embeddings extending a givenFv → Qℓ, andmι∗(τ) = mι∗(τ) = −mι∗(τ) (ψ is unitary).µ∞(L×w) is isomorphic toZ/(q− 1)Z, whereq is the order of the residue field atw. Let ζ 7→ 1 givethe isomorphism, forζ a primitive (q − 1)st root of unity. Complex conjugation must identify tomultiplication by an elementr ∈ Z/(q− 1) satisfyingr2 ≡ 1 modq− 1; in particular,r is an oddresidue class. Thenx 7→ τ(x)mι∗ (τ) · (τ c)(x)−mι∗ (τ) takesζ to an even power ofτ(ζ), henceχw|µ∞(L×w)

is a square.19

The second part of the lemma follows from the construction ofψ and the corresponding state-ment ([Wei56]) for the typeA Hecke characterψ.

2.4. Coefficients: generalizing Weil’s CM descent of typeA Hecke characters

In the coming sections we pursue higher-rank analogues of two aspects of Weil’s paper [Wei56].This section extends the important observation that typeA Hecke characters of a number fieldFdescend, up to a finite-order twist, to the maximal CM subfieldFcm (see Lemma 2.3.4). Of interestin its own right, this generalization also provides some of the intuition necessary for a generalsolution to Conrad’s lifting question (Question 1.1.6).

2.4.1. Coefficients in Hodge theory. The guiding principle that allows us to reinterpret, andcorrespondingly generalize, Weil’s result is that carefulattention to the ‘field of coefficients’ of anarithmetic object can yield non-trivial information aboutits ‘field of definition,’ or that of certainof its invariants. We begin by recording in an abstract setting a lemma whose motivation is ‘doingHodge theory with coefficients.’ Letk be a field of characteristic zero, and letF andE be extensionsof k with F/k finite. Let D be a filteredF ⊗k E-module that is free of rankd. Let E′ be a Galoisextension ofE large enough to splitF over k. For all (k-embeddings)τ : F → E′, we defineas in Definition 2.2.3 theτ-labeled Hodge-Tate weights HTτ(D) as follows: for suchτ we haveorthogonal idempotentseτ ∈ F ⊗k E′ giving rise to projections

F ⊗k E′(eτ)−−→∼

∏

τ

E′

x⊗ α 7→ (τ(x)α)τ .

19Note also that ifℓ is split in L/Q, χw cannot be a square whenmι∗∞,ℓ(τ)is odd.

33

The projectioneτ(D⊗E E′) is then a filteredE′-vector space of dimensiond, and we define HTτ(D)to be the collection of integersh (with multiplicity) such that

grh(eτ(D ⊗E E′)) , 0.

In §2.2.1 we applied this formalism to the filteredK ⊗Qℓ Qℓ-module DdR(V), whenV was a rep-resentation ofΓK (K/Qℓ finite) on aQℓ-vector space. In this section, however, we emphasize thegeneral formalism: one should really keep in mind notℓ-adic Hodge theory, but rather the de Rhamrealization of a motive overF with coefficients inE.

Lemma 2.4.1. HTτ(D) depends only on theGal(E′/E)-orbit of τ : F → E′. If E/k is Galois,thenHTτ(D) depends only on the restrictionτ|τ−1(E).

Proof. We decomposeF ⊗k E into a product of fields∏

Ei, writing qi : F ⊗k E ։ Ei for thequotient map. This yields filteredEi-vector spacesDi for all i. Any E-algebra homomorphismτ : F ⊗k E→ E′ factors throughqi(τ) for a unique indexi(τ), and then HTτ(D) is simply the multi-set of weights ofDi(τ). This implies the first claim, since the Gal(E′/E)-orbit of τ is simply allembeddingsτ′ for which i(τ′) = i(τ).

Now we assumeE/k Galois and address the second claim. Having fixed a ‘reference’ embed-ding τ0 : F → E′, it makes sense to speak ofF ∩ E := F ∩ τ−1

0 (E) insideE. ObviouslyE splitsF ∩ E overk, so we can write

F ⊗k E F ⊗F∩E (F ∩ E ⊗k E) ∏

σ

F ⊗F∩E,σ E,

where theσ range over all embeddingsF∩E → E. These factorsF⊗F∩E,σE are themselves fields,sinceE/k is Galois, and so this decomposition realizes explicitly the decomposition ofF ⊗k E intoa product of fields. In particular, by the first part of the Lemma, HTτ(D) depends only oni(τ), i.e.only on theσ : F ∩ E → E to whichτ restricts.

Definition 2.4.2. We call a filteredF ⊗k E-moduleD regular if the multi-sets HTτ(D) aremultiplicity-free.

A very simple application that we will use later is:

Corollary 2.4.3. Let D be a filtered L⊗k E-module (globally free) for some finite extensionL/F, and suppose L does not embed inE. Then the restriction of scalars (image under the forgetfulfunctor)ResL/F(D) is not regular.

2.4.2. CM descent.We will see that Weil’s result (Lemma 2.3.4) is the conjunction of Lemma2.4.1 with the fact that algebraic Hecke characters have CM ‘fields of coefficients.’ This observa-tion will lead us naturally to the desired higher-rank generalization.

The following notation will be in effect for the rest of the paper. LetG be a connected reductiveF-group. For eachv|∞ fix an isomorphismιv : Fv

∼−→ C. For π an automorphic representation ofG(AF), we can write (in Langlands’ normalization) the restriction toWFv

of its L-parameter as

recv(πv) : z 7→ ιv(z)µιv ιv(z)

νιv ∈ T∨(C).

with µιv, νιv ∈ X•(T)C andµιv − νιv ∈ X•(T). For v imaginary,µιv = νιv. Unless there is risk ofconfusion, we will omit reference to the embeddingιv, writing µv = µιv, etc. Recall that, in theterminology of [BG11], π is L-algebraic if for allv|∞, µv andνv lie in X•(T); it is C-algebraic if

34

µv andνv lie in ρ + X•(T), whereρ denotes the half-sum of the positive roots (with respect to ourfixed Borel containingT used to define a based root datum ofG).

Our starting point is a result (and, more generally, conjecture) of Clozel. TakeG = GLn/Fand, for simplicity,F to be totally imaginary. We collect the data ofπ’s archimedeanL-parametersasM = µιι, which we will loosely refer to as the ‘infinity-type’ ofπ. Forσ ∈ Aut(C), define theactionσM = µσ−1ιι. Recall thatπ is said to beregular if all roots of GLn are non-vanishing on allof the co-charactersµι.

Theorem 2.4.4 (Theoreme 3.13 of [Clo90]). Let F be any number field, and supposeπ isa cuspidal, C- or L-algebraic automorphic representation of GLn(AF) that is moreover regular.Thenπ f has a model over the fixed fieldQ(π f ) ⊂ Q ⊂ C of all automorphismsσ ∈ Aut(C) suchthatσπ f π f , withQ(π f ) in fact a number field forπ C-algebraic. For eachσ ∈ Aut(C) there is acuspidal representationσπ with finite partσπ f = π f ⊗C,σ C and with infinity-typeσM.

More generally, Clozel conjectures this for anyC-algebraic isobaric automorphic representa-tion of GLn(AF). Clozel’s theorem and conjecture are stated forC-algebraic representations, buttheL-algebraic analogue follows easily by twisting.

Hypothesis 2.4.5.Throughout the rest of this section, we will assume thatπ is an isobaric C- orL-algebraic automorphic representation ofGLn(AF) satisfying the conclusion of Clozel’s theorem.

We first note an elementary but crucial refinement of Hypothesis 2.4.5:

Corollary 2.4.6. The fieldQ(π f ) is CM.

Proof. It suffices to check for each factor in an isobaric decomposition, sowe may assumeπis cuspidal. Consider a unitary twistπ| · |w, with w necessarily some half-integerw. Its field ofdefinition is CM if and only if that ofπ is. Therefore we may assumeπ is unitary. TheL2 innerproduct then implies thatπ∨ cπ, which in turn implies that for allσ ∈ Aut(C), cσπ σcπ, andtherefore that the fixed fieldQ(π f ) of σ ∈ Aut(C) : σπ π is CM.

We can now formulate (and prove under Hypothesis 2.4.5) the appropriate higher-rank gener-alization of Weil’s result that typeA Hecke characters ofF, up to a finite-order twist, descend toFcm.

Proposition-Conjecture 2.4.7.Letπ be as in Hypothesis 2.4.5. Conjecturally, this should holdfor all isobaric C- or L-algebraic automorphic representations ofGLn(AF). Then the infinity-typeof π descends to Fcm.

Proof. Let E denote the Galois closure (inC) of Q(π f ); E is also a CM field. Extend eachι : F → C to an embedding ˜ι : F → C of the Galois closureF. The image ˜ι(F) ⊂ C doesnot depend on the extension. By the corollary, ˜ι(F) is linearly disjoint fromE over ι((F)cm), andtherefore we can findσ ∈ Aut(C/E) restricting (via ˜ι) to any element we like of Gal(F/(F)cm);the collection of suchσ acts transitively on the set of embeddingsF → C lying above a fixedF ∩ (F)cm = Fcm → C.20 For any two suchι, ι′ : F → C (related byσ ∈ Aut(C/E)), we deduce,sinceσπ π and henceσM = M, thatµι = µσ−1ι = µι′ .

Any study of algebraic automorphic forms over non-CM fields will need to take this result intoaccount.

20G = Gal(F/Fcm) is generated byH = Gal(F/F) andH′ = Gal(F/Fcm), with H′ normal. The set HomFcm(F,C) ispermuted transitively byG, with H acting trivially, so for any such embeddingx, H′x = H′Hx = Gx= HomFcm(F,C).

35

Remark 2.4.8. • If π is regular, this yields an unconditional descent result forits infinity-type.• Let F be a CM field andπ be a regularL or C-algebraic cuspidal representation of

GLn(AF). Suppose thatπ = IndFL (π0) for some extensionL/F. ThenL is CM. This modest

consequence of the proposition suggests that in the study ofregular automorphic repre-sentations/motives/Galois representations over CM fields, we will never have to grapplewith the (less well-understood) situation over non-CM fields. In §3.4.2 we discuss anabstract Galois-theoretic analogue.• If we know more aboutQ(π f ) than that it is CM (an extreme case:Q(π f ) = Q), the proof

of the proposition yields a correspondingly stronger result. For the related formalism, see§2.4.1.

Remark 2.4.9. Let us also note that Proposition 2.4.7 is a ‘seed’ result (under Hypothesis2.4.5); if we moreover assume functoriality (in a form that requires functorial transfers to be strongtransfers– i.e. of archimedeanL-packets– at infinity), then it implies:

• Let G be a connected reductive group over a totally imaginary fieldF, and letπ be acuspidal temperedL-algebraic automorphic representation ofG(AF). Then the infinity-type ofπ descends toFcm.• LetG andπ be as above. LetG ⊃ G be a connected reductiveF-group obtained by enlarg-

ing the center ofG to a torusZ (see§3.1). Then the central characterωπ : ZG(F)\ZG(AF)→C× extends to anL-algebraic Hecke character ofZ(AF). (Compare Proposition 13.3.1.)

2.5. W-algebraic representations

The current section generalizes a second aspect of [Wei56], discussing a higher-rank analogueof the typeA, but not necessarilyA0, condition, and begins to motivate its arithmetic significance.As always, letG be a connected reductive group over a number fieldF. We continue with theinfinity-type notation of§2.4.

For many interesting questions about algebraicity of automorphic representations, and espe-cially the interaction of algebraicity and functoriality,the framework ofC andL-algebraic repre-sentations does not suffice. Motivated initially by Weil’s study of type-A Hecke characters, wemake the following definition:

Definition 2.5.1. Letπ be an automorphic representation ofG(AF). We say thatπ is W-algebraic if for allv|∞, µιv andνιv in fact lie in 1

2X•(T).

Example 2.5.2. For GL1, a unitaryW-algebraic representation is precisely a unitary Heckecharacter of typeA in the sense of Weil ([Wei56]). Weil’s typeA characters also include arbitrarytwists | · |r for r ∈ Q, since these also yieldL-series with algebraic coefficients. TheL- andC-algebraic representations are the Hecke characters of typeA0.

Example 2.5.3. Consider a Hilbert modular formf on GL2/F with (classical) weightskττ : F→R,where the positive integerskτ are not all congruent modulo 2; these are called ‘mixed-parity’. fthen gives rise to an automorphic representation (in the unitary normalization, say) that isW-algebraic but neitherL- nor C-algebraic. The previous example yields a special case: choose aquadratic CM (totally imaginary) extensionL/F, and letψ be a unitary Hecke character ofL thatis typeA but not typeA0. Then the automorphic induction IndF

L ψ yields an example on GL2/F.We will elaborate on the case of mixed-parity Hilbert modular forms in§2.6.

36

We are led to ask (compare§2.4 and Conjectures 3.1.5 and 3.1.6 of [BG11]):

Question 2.5.4. • Let π be aW-algebraic automorphic representation onG. Does theG(AF, f )-moduleπ f have a model overQ (or Qcm)?. Note that we do not seek a modelover a fixed number field. Alternatively, isπv defined overQ for almost all finite placesv? By analogy with the terminology of [BG11], we might call the latter conditionW-arithmetic, and ask whether some condition (which will not quite beW-algebraicity!) oninfinity-types characterizesW-arithmeticity.• Similarly, we can ask the CM descent question of§2.4 for W-algebraic representations.

Example 2.5.6 below will be a cautionary tale: if non-induced cuspidalΠ with infinity-types as in that example exist, then there is no evidence thatthey would satisfy the ana-logue of Clozel’s algebraicity conjecture. If they do not exist, or if they miraculouslystill satisfy the conclusion of Clozel’s theorem, then we could confidently extend the CMdescent conjectures to theW-algebraic case.

For tori, the most optimistic conjecture holds:

Lemma 2.5.5. For arbitrary F-tori, W-algebraic implies W-arithmetic.

Proof. After squaring, this follows from the corresponding result for L-algebraic/L-arithmetic(§4 of [BG11]).

Continuing with Example 2.5.2, let us emphasize that theW-algebraic condition does notcapture all automorphic representations with algebraic Satake parameters; we nevertheless wantto make a case for isolating this condition, rather than allowing arbitrary rational parametersµv, νv ∈ X•(T)Q, which would be the naıve analogue of typeA. First, note that if a unitaryπis tempered at infinity with real parametersµv, νv ∈ X•(T)R, then these parameters in fact lie in12X•(T). In particular, the Ramanujan conjecture implies that allcuspidal unitaryπ on GLn/F withreal infinity-type are in factW-algebraic. It is easy to construct non-cuspidal automorphic repre-sentations with rational Satake parameters that do not twist to W-algebraic representations, but thepoint is that all such examples will be degenerate, soW-algebraicity is the condition of basic impor-tance. A more interesting question, concerning the difference betweenW, L, andC-algebraicity, isthe following:

Example 2.5.6. LetG = GL4/Q (for example; there are obvious analogues for any totally realfield). LetF/Q be real quadratic, and letπ as in Example 2.5.3 be a mixed parity (unitary) Hilbertmodular representation. It cannot be isomorphic to its Gal(F/Q)-conjugate, soΠ = IndQF(π) iscuspidal automorphic on GL4/Q, and at infinity its Langlands parameter (restricted toC×) lookslike

z 7→

(z/z)k1−1

2 0 0 0

0 (z/z)1−k1

2 0 0

0 0 (z/z)k2−1

2 0

0 0 0 (z/z)1−k2

2 ,

wherek1 andk2 are the classical weights ofπ at the two infinite places ofF. This exhibits the‘parity-mixing’ within a single infinite place, which implies the following:

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Lemma 2.5.7. For any functorial transferLGL4r−→ LGLN arising from an irreducible represen-

tation r, not equal to any power of the determinant, ofGL4(C), the transferLift r(Π) cannot beL-algebraic.

Proof. We have stated the lemma globally, and therefore conjecturally, but of course the anal-ogous archimedean statement (which is well-defined since the local transfer ofΠ∞ via r is knownto exist) is all we are really interested in. The proof is elementary highest-weight theory.

We will see (in§2.6) that mixed-parity Hilbert modular representations areW-arithmetic, usingthe fact that they haveL-algebraic functorial transfers; by contrast, if a representationΠ with theinfinity-type of the above example isW-arithmetic, it may be harder to establish. Of course, in theabove example,Π is automorphically induced, so itsW-arithmeticity is an immediate consequenceof that ofπ. We are led to ask whether there exist non-automorphically induced examples of suchΠ, or more generally of cuspidal, non-inducedΠ on GLn/F that exhibit the ‘parity-mixing’ withina single infinite place. The trace formula does not easily yield them, since such aΠ is not thetransfer from a classical group of a form that is discrete series at infinity (in these cases there is aparity constraint on regular elliptic parameters).

One further motivation for consideringW-algebraic representations comes from studying thefibers of functorial lifts, and their algebraicity properties. For instance, given two mixed-parityHilbert modular formsπ1 andπ2, with the (classical) weight ofπ1,v andπ2,v having the same parityfor eachv|∞, the tensor productΠ = π1 ⊠ π2 is L-algebraic. Theπi are not themselves twists ofL-algebraic automorphic representations, soΠ cannot be expressed as a tensor product ofL-algebraicrepresentations (this is proven in Corollary 2.6.3). This is, however, the ‘farthest’ fromL-algebraicthat theπi ’s can be:

Proposition 2.5.8. LetΠ be an L-algebraic cuspidal automorphic representation ofGLnn′(AF)

for some number field F. Assume thatΠ = π⊠π′ is in the image ofGLn×GLn′⊠−→ GLnn′ for cuspidal

automorphic representationsπ andπ′ of GLn(AF) andGLn′(AF).21 If F is totally real or CM, thenthere are quasi-tempered (i.e., tempered up to a twist) W-algebraic automorphic representationsπof GLn(AF) andπ′ of GLn′(AF) such thatΠ = π ⊠ π′.

Proof. The first part of the argument is similar to, and will make useof, Lemme 4.9 of [Clo90](Clozel’s ‘archimedean purity lemma’), which shows thatΠ∞ itself is quasi-tempered. First assumeF is totally imaginary. LetwΠ (the ‘motivic weight’) be the integer such thatΠ| · |−wΠ/2 is unitary,and writezµvzνv andzµ

′vzν

′v for theL-parameters atv|∞ of π andπ′. Temperedness ofΠ implies that

for all i, j = 1, . . . , n,Re(µv,i + µ

′v, j + νv,i + ν

′v, j) = wΠ.

Fixing j and varyingi, we find that Re(µv,i + νv,i) is independent ofi– call it wπv. Similarly,for wπ′v = wΠ − wπv, we have Re(µ′v,i + ν

′v,i) = wπ′v for all i. For each ofπ and π′ there is a

unique real numberr, r ′ such that each ofπ| · |−r/2, π′| · |−r ′/2 is unitary. Thus,zµv−r/2zνv−r/2 (likewisefor π′v) is a generic (by cuspidality) unitary parameter.22 This implies (see the proof of Lemme4.9 of [Clo90]) that these parameters are sums of the parameters ofunitary characters and 2× 2complementary series blocks (what Clozel denotesJ(χ, 1) andJ(χ, α, 1), whereα ∈ (0, 1/2) is thecomplementary series parameter). Since Re(µv,i + νv,i) is independent ofi, we can immediately

21To be precise, we wantΠ to be a weak lift that is also a strong lift at archimedean places.22On GLn, L-packets are singletons, and we can identify which parameters correspond to generic representations.

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rule out any complementary series factors, and we deduce that eachzµv−r/2zνv−r/2 is a direct sumof (parameters of) unitary characters, and thatwπv = r is independent ofv (andwπ′v = r ′), and

r+r ′ = wΠ ∈ Z. We may replaceπ byπ| · | r′−r2 andπ′ byπ′| · | r−r′

2 , so that stillπ⊠π′ = Π but now eachhas half-integral ‘motivic weight’ (namely,12wΠ ∈ 1

2Z). In particular, Re(µv),Re(νv),Re(µ′v), andRe(ν′v) all now consist of half-integers (it is easily seen that allof these half-integers are moreovercongruent moduloZ). Now, Im(µv,i + µ

′v, j) = 0 for all i, j, so there exists atv ∈ R such that

Im(µv,i) = Im(νv,i) = −Im(µ′v,i) = −Im(ν′v,i) = tv

for all i. To conclude the proof, note that if there exists a Hecke character ofF with infinity-componenets|z|itv

Cfor all v|∞, then we can twistπ andπ′ to arrange that both beW-algebraic (in

fact, either bothL-algebraic or bothC-algebraic). Looking at central characters, we are handed aHecke character with infinity-type

z 7→ zdet(Re(µv))+intvzdet(Re(µv))+intv,

and by Corollary 2.3.9, the desired Hecke character exists if and only if the set of half-integersdet(Re(µv)) :=

∑i Re(µv,i) is the infinity-type of a typeA Hecke character ofF, i.e. descends to

Fcm. Of course, whenF itself is CM, this is no obstruction, and the proof is complete.Now assumeF is totally real. By a global base-change argument (as in Clozel’s Lemme 4.9),

we may deduce temperedness ofπ and π′ from the totally imaginary (or even CM) case, andto determine whether there areW-algebraic descentsπ andπ′, as usual it suffices to look at thearchimedeanL-parameters restricted toWFv

. The purity constraint (see Corollary 2.3.9) on thehalf-integers det(Re(µv)) forces these to be independent ofv, and we can argue as in the CM caseto finish the proof.

In the non-CM case, we can describe the infinity-types of possible Hecke characters; this givesrise to an explicit relation between the det(Re(µv)) and tv that needs to be satisfied ifπ andπ′

are to haveW-algebraic twists. Conversely, note that the tensor product transfer (only known

to exist locally, of course) GLn × GLm⊠−→ GLnm clearly preservesL- or W-algebraicity (but not

C-algebraicity).In §3.1, we take up in detail the (algebraicity of the) fibers of the functorial transferLG→ LG,

whereG ⊂ G with torus quotient.

2.6. Further examples: the Hilbert modular case andGL2 ×GL2⊠−→ GL4

In this section we will elaborate on the example of mixed-parity Hilbert modular forms: wediscussW-arithmeticity in this context and make some initial forayson the Galois side. First,

however, we recall known results on the automorphic tensor product GL2 × GL2⊠−→ GL4 and

provide a description of its fibers.

2.6.1. General results on theGL2×GL2⊠−→ GL4 functorial transfer. Ramakrishnan ([Ram00])

has proven the existence of the automorphic tensor product transfer in the case GL2×GL2⊠−→ GL4.

Moreover, he establishes a cuspidality criterion:

Theorem 2.6.1 (Ramakrishnan, [Ram00, Theorem M]).Letπ1 andπ2 be cuspidal automorphicrepresentations onGL2/F. Thenπ1⊠π2 is automorphic. The cuspidality criterion divides into twocases:

39

• If neither πi is an automorphic induction,π1 ⊠ π2 is cuspidal if and only ifπ1 is notequivalent toπ2 ⊗ χ for some Hecke characterχ of F.• If π1 = IndF

L (ψ) for a Hecke characterψ of a quadratic extension L/F, thenπ1 ⊠ π2 iscuspidal if and only ifBCL(π2) is not isomorphic to its own twist byψθψ−1, whereθ is thenon-trivial automorphism of L/F.

Here is the Galois-theoretic heuristic for the first part of the cuspidality criterion. AssumeV1

andV2 are two-dimensional, irreducible, non-dihedral representations. Non-dihedral implies thateach Sym2 Vi is irreducible. Suppose thatV1 ⊗ V2 W1 ⊕W2. Taking exterior squares, we get

(Sym2 V1 ⊗ detV2) ⊕ (detV1 ⊗ Sym2 V2) ∧2W1 ⊕ (W1 ⊗W2) ⊕ ∧2W2.

We may therefore assume dimW1 = 3, dimW2 = 1 (so renameW2 as a characterψ), and thus

V1 ⊗ V2(ψ−1) W1(ψ

−1) ⊕ 1,

which implies thatV1 andV2 are twist-equivalent.We now describe the fibers of this tensor product lift, beginning with a Galois heuristic. Sup-

pose we have four irreducible 2-dimensional representations (overQℓ, say) satisfying

V1 ⊗ V2 W1 ⊗W2.

Taking exterior squares as above, we get

(Sym2 V1 ⊗ detV2) ⊕ (detV1 ⊗ Sym2 V2) (Sym2 W1 ⊗ detW2) ⊕ (detW1 ⊗ Sym2 W2)

AssumeV1 is non-dihedral, so Sym2 V1 is irreducible. Then we may assume

Sym2 V1 ⊗ detV2 Sym2 W1 ⊗ detW2

anddetV1 ⊗ Sym2 V2 detW1 ⊗ Sym2 W2.

Comparing determinants in this and the initial isomorphism, we find detV1 detV2 = detW1 detW2,and so (fori = 1, 2)

Ad0(Vi) Ad0(Wi).

Vi andW1 therefore give rise to isomorphic two-dimensional projective representations, and so, bythe (not very) long exact sequence in continuous cohomologyassociated to

1→ Q×ℓ → GL2(Qℓ)→ PGL2(Qℓ)→ 1, 23

Vi andWi are twist-equivalent: we find a pair of charactersψi such thatV1 W1(ψ1) andV2

W2(ψ2). If V2 is also non-induced, thenψ2 = ψ−11 , elseV1 ⊗ V2, and hence one ofV1 or V2, is an

induction.

Because the functorial transfers GL2Sym2

−−−→ GL3 ([GJ78]) and GL4∧2

−−→ GL6 ([Kim03]) areknown, this argument works automorphically until the last step. To conclude a purely automorphicargument, we have to know the fibers of Ad0. These were determined by Ramakrishnan as aconsequence of his construction of the tensor product lift (and his cuspidality criterion):

Theorem 2.6.2 (Ramakrishnan, [Ram00] Theorem 4.1.2).Let π and π′ be unitary cuspidalautomorphic representations onGL2/F such thatAd0 π Ad0 π′. Then there exists a Heckecharacterχ of F such thatπ π′ ⊗ χ.

23Note that the surjection admits a topological section.

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It would be remiss not to mention the special nature of this fiber description: by [LL79 ], itis essentially multiplicity one for SL2. In any case, from this and the preceding calculation, weconclude:

Corollary 2.6.3. Letπ1, π2, π′1, π

′2 be unitary cuspidal automorphic representations onGL2/F

satisfyingπ1 ⊠ π2 π′1 ⊠ π′2 (it suffices to assume this almost everywhere locally). Assume that

π1 ⊠ π2 is cuspidal (equivalently, satisfies Ramakrishnan’s criterion). Up to reordering, we haveAd0 πi Ad0 π′i for i = 1, 2, and there are Hecke charactersχi of F such thatπi π

′i ⊗χi (i = 1, 2).

Moreover,χ1 = χ−12 , so the fibers of the lift are just twists by Hecke characters.

Proof. If at least one, sayπ1, of theπi is non-dihedral, then it only remains to checkχ1 = χ−12 .

Sinceπ1 ⊠ π2 (π1 ⊠ π2) ⊗ (χ1χ2), we deduce from a theorem of Lapid-Rogawski– which weelaborate on in Lemma 2.6.4 below– that eitherχ1χ2 = 1, orπ1 ⊠ π2 is an automorphic inductionfrom the (quadratic or quartic) cyclic extension ofF cut out byχ1χ2. In the latter case, there is acyclic base changeL/F such that BCL(π1 ⊠ π2) = BCL(π1) ⊠ BCL(π2) is non-cuspidal. If bothπ1

andπ2 are non-dihedral, or ifπ2 = IndFK(ψ2) with K a quadratic extension ofF not contained inL,

then the cuspidality criterion implies there exists a Heckecharacterψ of L such that BCL(π1)⊗ψ BCL(π2). Sinceπ1 is non-dihedral,ψ is invariant under Gal(L/F); for cyclic extensions, there isno obstruction to descending invariant Hecke characters, soψ descends to a character ofF that wealso write asψ, i.e. BCL(π1 ⊗ ψ) BCL(π2). By cyclic descent (also TheoremB of [LR98]), π1

andπ2 are twist-equivalent, contradicting cuspidality ofπ1 ⊠ π2. We conclude thatχ1χ2 = 1.We resume the above argument whenπ2 is dihedral withK ⊂ L. If L = K, thenχ1χ2 is the

quadratic characterχ of Gal(K/F), soπ2 = π′2 ⊗ (χ−1

1 χ), and thusπ2 = π′2 ⊗ χ−1

1 sinceπ2 ⊗ χ = π2.If L/F is quartic, then the result of Lapid-Rogawski still impliesπ1 ⊠ π2 = IndF

L (ψ), whereψ isnow a Hecke character ofL. Base-changing toL and comparing isobaric constituents, we againcontradict the assumption thatπ1 is non-dihedral.

Finally, we treat the case where bothπ1 andπ2 are dihedral. Base-change implies that bothπ′1andπ′2 are dihedral as well, and moreover that we may assume thatπi andπ′i are both induced fromthe same fieldLi , say by charactersψi andψ′i . Write σi for the non-trivial element of Gal(Li/F).Then

IndFL1

(ψ1 ⊗ BCL1(IndF

L2(ψ2))

)= IndF

L1

(ψ′1 ⊗ BCL1(IndF

L2(ψ′2))

),

and up to replacingψ1 by ψσ11 , we may assume that on GL2/L1 we have

ψ1

ψ′1⊗ BCL1(IndF

L2(ψ2)) = BCL1(IndF

L2(ψ′2)).

If L1 , L2, then we find

IndL1L1L2

(ψ1

ψ′1· ψ2) = IndL1

L1L2(ψ′2),

and possibly replacingψ2 by its conjugateψσ22 , we may assume that as Hecke characters ofL1L2,

ψ1/ψ′1 = ψ′2/ψ2. Let α be the Hecke character ofL1 given byψ1/ψ

′1. This equality showsα is

σ1-invariant, and therefore descends to a Hecke character ofF. Consequently,π1 = π′1 ⊗ α, and

π2 = π′2 ⊗ α−1 as automorphic representations on GL2/F. The caseL1 = L2 is similar, but easier,

and we omit the details.

The above corollary made use of a characterization of the image of cyclic automorphic induc-tion (in the quadratic and quartic case). This characterization in the case of non-prime-degree does

41

not follow from the results of [AC89], but it is now known thanks to work of Lapid-Rogawski.The following is stated without proof in [LR98], as an easy application of their Statement B24. Wefill in the details:

Proposition 2.6.4.[Lapid-Rogawski] LetΠ be a cuspidal automorphic representation onGLn/F,and let L/F be a cyclic extension. ThenΠ is automorphically induced from L if (and only if)Π Π ⊗ ω for a Hecke characterω of F that cuts out the extension L/F.

Proof. Lapid-Rogawski establish the following, which implies the Proposition:25

Theorem 2.6.5 (Statement B, part (a) of [LR98]). Let E/F be a cyclic extension of numberfields, withσ a generator ofGal(E/F), and letω be a Hecke character of E. Denote byωF itsrestriction toA×F ⊂ A×E. SupposeΠ is a cuspidal automorphic representation onGLn/E satisfyingΠσ Π⊗ω. Let K/F be the extension (necessarily of order dividing n) of F cut out byωF, and letL = KE. Then E∩ K = F, and[K : F] divides n.

Supposeω cuts outL/F, cyclic of orderm, and supposem is divisible by a primeℓ (andm, ℓ).We see thatΠ Π ⊗ ωm/l as well, and lettingE/F be the (cyclic degreeℓ) extension cut out byωm/l, the case of prime degree implies thatΠ = IndF

E(π) for some cuspidal representationπ onGLn/E. Moreover,

IndFE(π) IndF

E(π ⊗ (ω NE/F)),so the description of the fibers in the prime case impliesπ andπ⊗ (ω NE/F) are Galois-conjugate:writing τ for a generator of GalL/F (or for its restriction toE), there exists an integeri such that

πτi π ⊗ (ω NE/F).

Part (a) of Theorem 2.6.5 implies that the restriction ofω NE/F to A×F/F×, which is justωℓ, cuts

out an extensionK/F that is linearly disjoint fromE/F,26 so in particular (ωNE/F)ℓ is a non-trivialHecke character ofE. But now, sinceτℓ is trivial on E, we can iterate the previous conjugationrelation to deduce

π π ⊗ (ω NE/F)ℓ.

It now follows from induction on the degree [L : F] that π IndEL (π0) for some cuspidalπ0 on

GL n[L:F]

/L, so we’re done.

2.6.2. The Hilbert modular case.Now we take up the case of Hilbert modular forms, start-ing with an observation of Blasius-Rogawski ([BR93]): while a ‘mixed parity’ Hilbert modularrepresentationπ (Definition 2.6.6) does not itself twist to anL-algebraic representation, its basechanges to every CM field will.27 Since we work up to twist, we may assume the parameters ofπ

at placesv|∞ have the form (restricted toF×v ⊂WFv and implicitly invokingι as above)

z 7→(z/z)

kv−12 0

0 (z/z)1−kv

2

.

24The proof of which was conditional on versions of the fundamental lemma that are now known.25Which is in turn used to prove the more refined parts (b) and (c) of Statement B of [LR98].26The point is thatΠ is cuspidal. For instance, in the simple caseℓ2 ∤ m, there’s no need to appeal to the

Lapid-Rogawski result:ℓ divides mℓi, soℓ|i, and thenπ π ⊗ (ω NE/F ).

27This observation of Blasius-Rogawski is the only time I haveseen substantive use made of typeA but notA0

Hecke characters.

42

Definition 2.6.6. In the above notation,π is mixed parityif as v varies over allv|∞, kv takesboth even and odd values.

For a quadratic CM extensionL/F, let ψ : A×L/L× → S1 be a Hecke character whose infinity

type atv is given byz 7→ (z/|z|)kv−1

for all v. Then BCL(π) ⊗ ψ is L-algebraic, with infinity type

z 7→((z/z)kv−1 0

0 1

)

Given two suchπ1 andπ2, bothW- but notL-algebraic, and withΠ = π1⊠ π2 L-algebraic, Blasius-Rogawski (see Theorem 2.6.2 and Corollary 2.6.3 of [BR93]) can then associate anℓ-adic GaloisrepresentationρΠ,ℓ toΠ via the identity

BCL(Π) (BCL(π1)(ψ1)) ⊠ (BCL(π2)(ψ2)) ⊗ (ψ1ψ2)−1,

since all three tensor factors on the right-hand side areL-algebraic with associated Galois repre-sentations. (To get a Galois representation forΠ overF itself, they varyL and use the now-famouspatching argument.) We pursue this a little farther, emphasizing the ‘W-algebraic’ and non-deRham/geometric aspects of this construction.

Proposition 2.6.7. LetΠ be the tensor productπ1 ⊠ π2 for πi as above corresponding to mixedparity Hilbert modular forms (butπ1 andπ2 having common weight-parity at each infinite place).Assume thatΠ is cuspidal, and for simplicity assume that theπi are non-dihedral.

(i) Theℓ-adic Galois representationρΠ,ℓ is Lie irreducible.(ii) ρΠ,ℓ is isomorphic to a tensor productρ1,ℓ ⊗ ρ2,ℓ of two-dimensional continuous, almost

everywhere unramified, representations ofΓF . No twist of eitherρi,ℓ is de Rham (or evenHodge-Tate), andρΠ,ℓ is not a tensor product of geometric Galois representations(al-though it is after every CM base change, by construction).28

(iii) Ad 0(ρi,ℓ) is geometric, corresponding to the L-algebraicAd0(πi) (suitably ordering therepresentations). In particular, a mixed-parity Hilbert modular representationπ givesrise to a geometric projective Galois representationsρπ,ℓ : ΓF → PGL2(Qℓ); these arethe representations predicted by Conjecture3.2.2 of [BG11] for irreducible SL2(AF)-constituents ofπ|SL2(AF ).

(iv) With theπi normalized to be W-algebraic, they are also W-arithmetic (see Question 2.5.4).

Proof. To showρΠ,ℓ is a tensor product, observe that it is essentially self-dual, since

Π∨ π∨1 ⊠ π∨2 (π1 ⊗ ω−1

π1) ⊠ (π2 ⊗ ω−1

π2) Π ⊗ (ωπ1ωπ2)

−1.

On the Galois side, we deduce thatρ∨Π,ℓ ρΠ,ℓ ⊗ µ−1

Π, whereµΠ is the (geometric) Galois character

corresponding to the (L-algebraic) Hecke characterωπ1ωπ2. Since after one of these quadraticbase changesL/F ρΠ,ℓ|ΓL is irreducible and orthogonal (rather than symplectic), wesee that it (asΓF-representation) is orthogonal, i.e.

ρΠ,ℓ : ΓF → GO4(Qℓ).

28Compare Proposition 3.3.8.

43

Writing µ for the multiplier character on GO4, and observing that det2= µ4 on this group, we find

an exact sequence of algebraic groups

1→ Gm→ GL2 ×GL2⊠−→ GO4

det·µ−2

−−−−−→ ±1 → 1.

The image ofρΠ,ℓ is contained in the kernel GSO4 of det·µ−2: for this it suffices to know thatρΠ,ℓ isisomorphic to a tensor product after two disjoint quadraticbase changes, for then (det·µ−2)ρΠ,ℓ isa character ofΓF that is trivial on a set of primes of density strictly greaterthan 1/2, and therefore itis trivial. We can then apply Proposition 2.1.4 to deduce that ρΠ,ℓ lifts across GL2(Qℓ)×GL2(Qℓ)→GSO4(Qℓ), i.e. it is isomorphic to a tensor productρ1,ℓ ⊗ ρ2,ℓ. The same exact sequence impliesthat any expression ofρΠ,ℓ as a tensor product has the form (ρ1,ℓ ⊗ χ) ⊗ (ρ2,ℓ ⊗ χ−1) for somecontinuous characterχ : ΓF → Q

×ℓ . Lie-irreducibility also follows since the Zariski closure of

the image of eachρi,ℓ contains SL2 (otherwiseρΠ,ℓ, and henceΠ, is automorphically induced), andthus the Zariski closure of the image ofρΠ,ℓ contains SO4, which acts Lie-irreducibly in its standard4-dimensional representation.

Now we show all twistsρi,ℓ ⊗ χ are almost everywhere unramified, but none are geometric(they fail to be de Rham). This follows from the Galois-theoretic arguments of the next section,but here we give a more automorphic argument. Note that both the de Rham and almost everywhereunramified conditions can be checked after a finite extension. By construction, for anyL = FKfor K/Q imaginary quadratic, there exist (L-algebraic) cuspidal automorphic representationsτi

(i = 1, 2) on GL2/L with associated geometric Galois representationsσi : ΓL → GL2(Qℓ) such that

ρ1,ℓ|ΓL ⊗ ρ2,ℓ|ΓL ρΠ,ℓ|ΓL σ1 ⊗ σ2.

This implies there is a Galois characterχ : ΓL → Q×ℓ such that

σ1(χ) ρ1,ℓ|ΓL ,

σ2(χ−1) ρ2,ℓ|ΓL ,

and every abelianℓ-adic representation is unramified almost everywhere (easy), so eachρi,ℓ isalmost everywhere unramified. We then have the equivalences

ρi,ℓ is geometric⇐⇒ ρi,ℓ|ΓL is geometric⇐⇒ χ is geometric.29

But if χ is geometric, then (Theorem 2.3.13) there exists a typeA0 Hecke characterχA of L suchthat

τ1 ⊗ χA ∼w,∞ ρ1,ℓ

τ2 ⊗ χ−1A ∼w,∞ ρ2,ℓ,

where∼w,∞ is the notation of§1.3. Gal(L/F)-invariance and cyclic base change then imply thereare cuspidal automorphic representations ˜τi on GL2/F lifting τ1 ⊗ χA andτ2 ⊗ χ−1

A , and thereforeBCL(τ1 ⊠ τ2) BCL(Π). This impliesτ1 ⊠ τ2 andΠ are twist-equivalent, and Corollary 2.6.3ensures that (up to relabeling)πi andτi are twist-equivalent. But ˜τi is L-algebraic sinceτi andχA

are, yet it is easy to see (becauseπ is mixed-parity andF is totally real) that no twist ofπi can beL-algebraic.

29Moreover,ρi,ℓ is Hodge-Tate⇐⇒ ρi,ℓ|ΓL is Hodge-Tate⇐⇒ χ is Hodge-Tate⇐⇒ χ is geometric!

44

That, suitably ordered, Ad0(ρi,ℓ) is a geometric Galois representation whose local factors matchthose of theL-algebraic cuspidal representation Ad0(πi) follows easily from the earlier∧2 argument(see§2.6.1)and the fact that our forms (representations) are allnon-dihedral.

Part (iv): Normalizeπi to beW-algebraic. Thenπi ⊠ πi = Sym2(πi)⊞ωπi is L-algebraic, and by[BR93] it has Satake parameters inQ at unramified primes. Writingαi,v, βi,v for the parametersof πi,v (when unramified), we conclude thatα2

i,v, β2i,v ∈ Q, henceαi,v, βi,v ∈ Q. This impliesπi,v has

a model overQ for all suchv,30 answering the weaker version of Question 2.5.4. In this casethestronger version (thatπi, f is defined overQ) follows as well, by a check (omitted) at the ramifiedprimes.

Now we prove a more refined result in the caseπ1 = π2:

Proposition 2.6.8. Let π be a unitary, cuspidal, non-induced, mixed-parity Hilbertmodularrepresentation, so thatΠ = π ⊠ π has an associated Galois representation, the sum of Galoisrepresentations associated to the L-algebraic representations Sym2(π) andωπ. As before, thereare ℓ-adic representationsρi,ℓ such thatρΠ,ℓ = ρ1,ℓ ⊗ ρ2,ℓ.

• We can normalize theρi,ℓ so thatρi,ℓ( f rv) has characteristic polynomial inQ[X] for allunramified v. Moreover, its eigenvalues in fact lie inQcm, and are pure of some weight,which we may take to be zero.• For quadratic CM extensions L= KF with K/Q a quadratic imaginary field in whichℓ is inert, there is aρℓ : ΓF → GL2(Qℓ) for which Sym2(ρℓ) ∼w,∞ Sym2(π). Later on(Corollary 2.7.8), we will see this is not possible over a totally real field.• Nevertheless, we can chooseρℓ and a finite-order characterχ such thatSym2(ρℓ) : ΓF →

GO3(Qℓ) is, viewingGO3(Qℓ) as the dual group(Gm × SL2)∨(Qℓ), associated to an auto-morphic representation(ωπχ, π0) of (Gm × SL2)(AF), as in3.2.2 of [BG11], with π0 anyirreducible constituent ofπ|SL2(AF ).

Proof. If we takeπ1 = π2 = π in the unitary normalization (this will ensureπ ⊠ π has ‘motivicweight zero,’ with an implicit– and provable– application of Ramanujan at infinity) and decomposethe Galois representationρΠ,ℓ associated toΠ = π⊠π asρ1⊗ρ2, then the now-familiar∧2 argumentshows that Ad0(ρ1) Ad0(ρ2). We can therefore writeρΠ,ℓ ρ⊗ (ρ⊗ χ) for some Galois characterχ. This χ need not be a square, but we can normalizeρ so thatχ is finite order (by Lemma2.3.15). Now, the Ramanujan conjecture is known forΠ31, and of course forχ, soρ ⊗ ρ is pureof weight zero (at unramified primes, say). Letαv andβv be frobenius eigenvalues ofρ( f rv) atan unramified primev, so thatα2

v andβ2v (being eigenvalues of Sym2(ρ)) have absolute value one

under any embeddingQℓ → C. The same then holds forαv andβv, soρ itself is pure of weightzero. Therefore the frobenius eigenvalues ofρ lie in Qcm.

We conclude by showing that, after certain CM base-changes,we can take the finite-ordercharacterχ to be trivial. Restricting toL = KF as above, we can find a type A Hecke characterψ of L such thatπ ⊗ ψ is L-algebraic, and then letting (ψ2)ℓ be the geometricℓ-adic representationassociated toψ2, we claim that, possibly replacingψ by ψ−1, there exists a Galois characterλ of L

30Note that the fields of definition ofπv and its Satake parameters may be different; but if one is algebraic, thenboth are.

31Of course,Π is not cuspidal, so literally this is for Sym2(π) andωπ.

45

such that

ρ|ΓL ⊗ λ ρ|ΓL ⊗ (χ(ψ2)ℓλ−1).

OverL, π · ψ−1 has associated Galois representationr, and thenr · (ψ2)ℓ corresponds toπ · ψ. Thenthere exists aλ such that eitherr = ρ ·λ andr · (ψ2)ℓ = ρ(χλ−1), or r = ρ · (χλ−1) andr · (ψ2)ℓ = ρ ·λ.Either way, plugging one equation into the other we get the claim.

ρ is not an induction, soχ|ΓL = λ2(ψ2)−1

ℓ. In particular,χ|ΓL is a square if and only if (ψ2)ℓ admits

a Galois-theoretic square-root. This was discussed in Lemma 2.3.18 above: it need not always bethe case (for instance, ifℓ is split in L/Q), but it is if ℓ is inert inK. Choosing such anL = KF, wecan now findχL : ΓL → Qℓ such that

ρΠ,ℓ|ΓL (ρ|ΓL ⊗ χL)⊗2.

HereχL = λψ for ψ a square-root of (ψ2)ℓ, and we’re done.For the final point, note that the inclusionGm × SO3 → GO3 is an isomorphism, with inverse

mapg 7→ (detµ

(g), detµ

(g)−1g). In particular, Sym2 : GL2 → GO3 becomesg 7→ (det(g),Ad0(g) inthese coordinates. The SL2(AF)-constituents ofπ have Langlands parameters corresponding tothe projectivization of those ofπ, and since Ad0 : GL2 → SO3 PGL2 is just the quotient mapGL2։ PGL2, the claim follows.

Remark 2.6.9. In particular, the Proposition tells us that there are pureℓ-adic Galois represen-tations no twist of which are geometric. They do not, however, live in compatible systems.

These propositions would have no content if the only examples of mixed parity Hilbert modularforms were the inductions described in Example 2.5.3; we endthis section by showing

Lemma 2.6.10.Over any totally real field, non-CM mixed parity Hilbert modular forms exist.

Proof. Considering the (semi-simple, connected)Q-groupG = ResF/Q(SL2/F), we can applyTheorem 1B from§4 of Clozel’s paper [Clo86]. We fix a discrete series representationπ∞ ofG(R)

∏τ : F→R SL2(R) corresponding to the desired mixed-parity infinity-type.Then fix an

auxiliary supercuspidal levelKp0 ⊂ G(Qp0) for some primep0. At some other finite primep, letπp

be a fixed (twist of) Steinberg representation. Finally, letS be a set of finite primes (disjoint fromp0, p) at which we will let the level go to infinity in the manner of Clozel’s paper (writtenKS → 1).Letting K be any fixed compact open subgroup away fromS, p, p0,∞, Clozel proves that

lim infKS→1

[vol(KS) ·mult

(π∞ ⊗ πp, L

2cusp(G(Q)/G(AF))Kp0×KS×K

)]> 0.

(He also determines an explicit but non-optimal constant.)This suffices for the Lemma: it impliesthe existence of (infinitely many) cuspidal automorphic representations onG that are Steinberg atp, and therefore not automorphically induced, and have the desired mixed parity at infinity.

Remark 2.6.11. In [Shi12], Shin derives, as a corollary of very general existence results forautomorphic representations, an exact limit multiplicityformula forcohomologicalHilbert mod-ular forms. Unfortunately, this excludes precisely the mixed-parity forms we are interested in. Itseems, however, that his methods should extend to cover all discrete series infinity types.

46

2.6.3. A couple of questions.In Proposition 2.6.7, the key point was an understanding ofwhich typeA Hecke characters– or their Galois analogues– could exist. Iwould like to raise herea couple questions in higher rank about the existence of automorphic forms with certain infinity-types.

Question 2.6.12. Are there automorphic representationsπ on GL2/F (F totally real) such thatat two infinite placesv1 andv2,

• πv1 is discrete series andπv2 is limit of discrete series (excluding the easily-constructeddihedral cases, as in Example 2.5.3)? Clozel’s result does not apply to this case;• πv1 is discrete series (or limit of discrete series), andπv2 is principal series (looks like a

Maass form)?• as in the previous item, but with algebraic infinity-type (soa ‘Plancherel density’-type

result will not suffice)?

2.7. Galois lifting: Hilbert modular case

2.7.1. Outline. We continue to elaborate on the examples of the previous section, now turningto some preliminary cases of a problem raised by Conrad in [Con11]. Recall from the introductionthat he addresses lifting problems

H′(Qℓ)

ΓF

ρ<<③

③③

③③ ρ

// H(Qℓ)

,

whereH′ ։ H is a surjection of linear algebraic groups with central kernel, and the key remainingquestion is:

Question 2.7.1. Suppose that the kernel ofH ։ H is a torus. Ifρ is geometric, when doesthere exist a geometric lift ˜ρ?

Here is an outline of the coming sections:

• First we address, in the regular case, withF totally real,32 Conrad’s question for liftingacross GL2(Qℓ)→ PGL2(Qℓ), finding that all examples ofρ not having geometric lifts areaccounted for by mixed-parity Hilbert modular forms (a converse to Propositions 2.6.7and 2.6.8).• To give a higher-rank example in which potential automorphytheorems still allow us to

link the automorphic and Galois sides, we then carry out in§2.8 an analogous discussionfor GSpin2n+1(Qℓ) → SO2n+1(Qℓ) (§2.2 introduces the necessary background about Senoperators and ‘labeled Hodge-Tate-Sen weights’ inℓ-adic Hodge theory; in the currentsection, we use more elementary terminology, although somemanipulations are justifiedby the general theory).• Before proceeding to develop the general framework (§3.2), we take a detour (§3.1) to

discuss the automorphic analogue. This is in some sense moreintuitive, and it motivates

32Modulo the difference between potential automorphy and automorphy!

47

the Galois-theoretic solution. Note that in general we haveneither constructions of au-tomorphic Galois representations nor (potential) automorphy theorems, so we cannot un-conditionally bridge the automorphic-Galois chasm. Proving a ‘modular lifting theorem’in this context is an important remaining question (see Question 1.1.11).

2.7.2. GL2(Qℓ)→ PGL2(Qℓ). In this subsection we prove:

Theorem 2.7.2. Let F be totally real. Supposeρ : ΓF → PGL2(Qℓ) is a geometric representa-tion with no geometric lift toGL2(Qℓ). Then

• There exists a liftρ : ΓF → GL2(Qℓ) such that for all CM L/F, ρ|ΓL is the twist of ageometric Galois representation (in particular,ρ|ΓL has a geometric lift, which ought tobe automorphic).

Now assume moreover thatAd(ρ) : ΓF → SO3(Qℓ) ⊂ GL3(Qℓ) satisfies the hypotheses of (thepotential automorphy theorem) Corollary4.5.2 of [BLGGT10]33 Then:

• After a totally real base-change F′/F, Ad(ρ) is automorphic, and more precisely• we may normalizeρ such thatSym2(ρ)|ΓF′ is a geometric Galois representation corre-

sponding toSym2(π) ⊗ χ for some mixed parity Hilbert modular representationπ onGL2/F and *non-trivial* finite order characterχ. Furthermore,ρ is totally odd.

Remark 2.7.3. If Ad0(ρ) is not regular, there are two possibilities: either it fails to be regulareverywhere, in which case Fontaine-Mazur conjecture thatρ (up to twist) has finite image. Suchρ always have finite-image (hence geometric) lifts by Tate’s theorem (Proposition 2.1.4). Ifρ istotally odd, and Ad0(ρ) exhibits a mixture of regular and irregular behavior, it should be related tothe existence of a mixed-parity Hilbert modular form that islimit of discrete series at some infiniteplaces, and genuine discrete series at others. I know of no such examples not arising from Heckecharacters. The question of whether there exist non-dihedral ρ that are even at some and odd atother infinite places is related to Question 2.6.12.

We now make our first use of non-Hodge-Tate Galois representations; recall that in§2.2 wehave described the theory of (non-integral) Hodge-Tate-Sen weights. In fact, a reader unfamiliarwith this general theory will have no difficulty following the arguments of the current section.

Lemma 2.7.4. For any number field F and any geometricρ : ΓF → PGL2(Qℓ), there is a liftρ : ΓF → GL2(Qℓ) of ρ whose Hodge-Tate-Sen (HTS) weights at all v|ℓ belong to1

2Z. In this case,Sym2(ρ) anddet(ρ) are geometric.

Proof. Proposition 2.1.4 ensures that we can find some lift ˜ρ. By assumption, Ad0(ρ) is geo-metric, so

ρ ⊗ ρ ⊗ det(ρ)−1 Ad0(ρ) ⊕ 1

is geometric. We ask whether det( ˜ρ) is a square. There at least exists a Galois characterψ : ΓF →Q×ℓ such that det( ˜ρ) = ψ2χ with χ finite order (Lemma 2.3.15). Thus

[ρ ⊗ ψ−1]⊗2 (Ad0(ρ) ⊗ χ) ⊕ χ

is also geometric, and in particular Hodge-Tate. Replacingρ with the new lift ρ ⊗ ψ−1 we con-clude that Sym2(ρ) and det ˜ρ are Hodge-Tate. The theorem of Wintenberger and Conrad (Theorem

33Essentially: there is a lift ˜ρ of ρ such that ( ˜ρ mod ℓ)|ΓF(ζℓ ) is irreducible and ˜ρ is regular (if true for one lift, thisis true for all), andℓ is sufficiently large.

48

2.1.6) then proves they must be de Rham, since the original projectiveρ was. Any lift ρ is almosteverywhere unramified by Lemma 5.3 of [Con11], so the proof is complete.

We now proceed in the same spirit as in the construction ofW-algebraic forms via typeAHecke characters. The following lemma is anad hocversion of Theorem 3.2.7; it may help innavigating that rather formal proof:

Lemma 2.7.5. Considerρ : ΓF → GL2(Qℓ) as produced by Lemma 2.7.4. Let L/Q be a qua-dratic CM extension. Then for all such L, the restriction ofρ to ΓL is the twist of a geometricGalois representation.

Proof. Recall that our fixed embeddingsQι∞−→ C andQ

ιℓ−→ Qℓ yield a map

ι∗∞,ℓ :⋃

v|ℓHomQℓ(Fv,Qℓ)→ HomQ(F,C).

Sym2(ρ) is de Rham, so for each placev|ℓ of F and eachτ : Fv → Qℓ, it hasτ-labeled Hodge-Tateweights34

HTτ

(Sym2(ρ|ΓFv

))= Aτ,

Aτ + Bτ

2, Bτ,

where theAτ andBτ are integers, necessarily having the same parity. We can distinguish the subset

Aτ, Bτ ⊂ HTτ

(Sym2(ρ|ΓFv

))

(trivially in the non-regular case, easily otherwise). It follows that ρ|ΓFvis Hodge-Tate (hence

de Rham) if and only if allAτ and Bτ are even. Consider the mapι∗∞,ℓ for L as well asF; thetwo versions are compatible, so conjugate archimedean embeddings ι, ι : L → C pull back toembeddingsL → Qℓ that lie above a commonτ : F ⊂ Fv → Qℓ. To eachτwe then unambiguouslyassociate the parityǫτ ∈ 0, 1 of Aτ (andBτ), and at the unique archimedean placev∞,τ of L aboveι∗∞,ℓ(τ) we define a character

ψv∞,τ : L×v∞,τ → C×

z 7→ ι(z)kτ2 ι(z)−

kτ2 ,

wherekτ is any integer with parityǫτ.L is a CM field, so Weil ([Wei56]) tells us that there exists a typeA Hecke characterψ of

L with components at archimedean places given by theseψv∞,τ . We wish to twist ˜ρ|ΓL by a Galoisrealization ofψ, shifting all of its HTS weights to be integers (recall this can be checked by lookingat Sym2(ρ|ΓL)), but care is needed: there is no Galois representation canonically associated to a typeA (not A0) Hecke character. Instead, we associate (canonically, having specifiedι∞ and ιℓ) an ℓ-adic representation (ψ2)ℓ to the typeA0 Hecke characterψ2, and then we non-canonically extract,up to a finite-order character, a (Galois-theoretic, non-geometric) square rootψ. Thenρ|ΓL ⊗ ψ isHodge-Tate, since the analogues of theAτ andBτ, but now for Sym2(ρ|ΓL ⊗ ψ) = Sym2(ρ|ΓL)⊗ (ψ2)ℓ,are all even. Thus ˜ρ|ΓL ⊗ ψ is geometric (again by Theorem 2.1.6).

This completes the proof of the first part of Theorem 2.7.2. Weneed deeper tools to makefurther progress. Let ˜ρ be the lift as above, with both Sym2(ρ) and det( ˜ρ) geometric. Sym2(ρ)

34By duality, or by the Hodge-Tate-Sen theory.

49

factors through GO3(Qℓ) with totally even multiplier character det( ˜ρ)2. We now make furtherassumptions on ˜ρ (or ρ) in order to apply a potential automorphy theorem:

Hypothesis 2.7.6. (1)Assumeℓ > 7.(2) AssumeSym2(ρ) is ‘potentially diagonalizable’ and regular at all v|ℓ. For instance, we

require that for each v|ℓ it be crystalline withτ-labeled Hodge-Tate weights distinct andfalling in the Fontaine-Laffaille range for allτ : Fv → Qℓ.

(3) Assume that the reduction modℓ of Sym2(ρ)|ΓF(ζℓ) is irreducible. (We have takenℓ , 2, sothis is equivalent to the analogous assumption forρ.)

Then we can can apply TheoremC of [BLGGT10] to deduce that after base-change to sometotally real fieldF′, Sym2(ρ)|ΓF′ is automorphic, corresponding to a RAESDC automorphic rep-resentationΠ on GL3/F′. Let us abusively denote by det( ˜ρ) the Hecke character (of typeA0)corresponding to det( ˜ρ). ThenΠ ⊗ det(ρ)−1 is self-dual with trivial central character, and (by, forinstance, [Art ]– details, in greater generality, appear in Lemma 2.8.6– although in this case theresult is older) there exists a cuspidalπ on GL2/F′ such that Ad0(π) Π ⊗ det(ρ)−1. We mayassume thatωπ has finite order (applying Lemma 2.3.6 and Proposition 3.1.4, sinceF′ is totallyreal), because the descent is originally to SL2/F′, and we then have control over the choice ofextension to GL2/F′.

35

Lemma 2.7.7. Such aπ on GL2/F′ satisfyingAd0(π) Π ⊗ det(ρ)−1 is necessarily a mixedparity Hilbert modular representation.

Proof. It is clear from the archimedean L-parameters thatπ is W-algebraic. If it wereL-algebraic, there would be a corresponding Galois representationρπ36 with Ad0(ρπ) Ad0(ρ), andthusρπ and ρ|ΓF′ would be twist-equivalent. Lemma 2.3.17 then implies ˜ρ has all integral or allhalf-integral Hodge-Tate weights, which contradicts the assumption thatρ has no geometric lift(note that bothF′ andF are totally real). Similarly, ifπ wereC-algebraic, then there would be ageometric representation corresponding toπ| · |1/2, and we can use the same argument. We concludethatπ must be mixed parity.

Corollary 2.7.8.For anyρ : ΓF → PGL2(Qℓ) as in Theorem 2.7.2, and satisfying the auxiliarypotential automorphy Hypothesis 2.7.6, there exists a liftρ : ΓF → GL2(Qℓ) and, after a totally realbase change F′/F, a mixed-parity Hilbert modular representationπ onGL2/F′, such that

Sym2(ρ)|ΓF′ ∼w,∞ Sym2(π) ⊗ χfor some finite order Hecke characterχ of F′. Moreover, any liftρ is totally odd, and the char-acterχ is necessarily non-trivial. In contrast, restricting to L′ = F′K where K/Q is a quadraticimaginary extension in whichℓ is inert, we can find a liftρ such thatSym2(ρ) corresponds toBCL′/F′(Sym2(π)).

Conversely, starting with a mixed-parityπ, we can produce such aχ and ρ.

Proof. In the notation of the previous Lemma, we have

Sym2(π) ⊗ det(ρ)ωπ

Π ∼w,∞ Sym2(ρ)|ΓF′ .

35We only sketch this here, because these matters will be discussed in greater detail and generality in§3.1.36By [BR93].

50

The character det( ˜ρ)/ωπ is L-algebraic, hence equalsχ| · |m for a finite-order characterχ and anintegerm. Replacingπ by π ⊗ | · |m/2, which is still mixed-parity, we are done except for the lastpart.

Assume instead that we can takeχ = 1. In particular,ωπ ∼w,∞ det(ρ), and thus det( ˜ρ)(cv) =ωπv(−1). By the description of discrete series representations of GL2(R) and purity ofωπ, we knowthat the Langlands parameterφv : WR → GL2(C) takes the form

z 7→(z/z)

kv−12 0

0 (z/z)1−kv

2

|z|−w/2C

j 7→(0 (−1)kv−1

1 0

).

Henceωπv(−1) = det(φv)( j) = (−1)kv. Since Ad0(ρ)|ΓF′ ∼w,∞ Ad0(π) is regular algebraic self-dualcuspidal, PropositionA of [Tay12] shows that Ad0(ρ) is odd, and thus for allv|∞, the eigenvaluesof Ad0(ρ)(cv) are−1, 1,−1. We conclude that det( ˜ρ)(cv) = −1 for all v|∞, proving the oddnessclaim for ρ, and contradicting, when we assumeχ = 1, the fact that thekv are both odd and even.

The remaining claims, constructing fromπ such a ˜ρ andχ, were established in Proposition2.6.8.

This concludes the proof of Theorem 2.7.2.

2.8. Spin examples

To address Conrad’s question in general, we will have to re-cast the arguments of§2.7 interms of root data. In the meantime we give another pleasantly concrete example, but one thatrelies on root-theoretic manipulation, in the hopes of easing the transition to the general case;moreover, specializing to discrete series/regular examples, we can also still exploit known resultsabout automorphic Galois representations. The previous example (lifting across the surjectionGL2(Qℓ)→ PGL2(Qℓ)) is really just the first case of a family of spin examples,

GSpin2n+1(Qℓ)

ΓF

ρ

99ss

ss

ss

ρ// SO2n+1(Qℓ),

whereρ is geometric but has no geometric lift ˜ρ. F will be a totally real field, and we will againsee that over CM fields, geometric lifts do in fact exist. The casen = 1 will amount to the contentsof §2.7.

We first recall the basic setup for (odd) Spin groups. We will also make heavy use of thisnotation, and its obvious analogues for even Spin groups, later on in some of our examples of

51

‘motivic lifting’ (see §4.2 and§4.3).

1

1

1 // ±1 //

Spin2n+1//

SO2n+1// 1

1 // Gm//

z7→z2

GSpin2n+1//

ν

SO2n+1// 1

Gm

Gm

1 1

whereν is the Clifford norm. We can then identify

GSpin2n+1

ν

Gm×Spin2n+11,(−1,c)∼

oo

(x,g)7→x2

Gm Gm

wherec is the non-trivial central element of Spin2n+1. This yields an identification of Lie algebras

gspin2n+1∼←− gl1 × so2n+1,

wheregl1 is identified with the center ofgspin2n+1. Alternatively, GSpin2n+1 is the dual group toGSp2n, and it will be convenient to keep both normalizations of (based) root data for GSpin2n+1–one from the spin description, one from the dual description– in mind. Let (X,∆,X∨,∆∨) be thebased root datum for GSp2n (say defined with respect toJ2n =

(0 1n−1n 0

)), with its diagonal maximal

torusT and the BorelB ⊃ T for which ei − ei+1 and 2en − e0 form a set of positive simple roots,with

ei : diag(t1, . . . , tn, νt−11 , . . . , νt

−1n ) 7→ ti

e0 : diag(t1, . . . , tn, νt−11 , . . . , νt

−1n ) 7→ ν.

Then for GSpin2n+1, we have the based root datum (withX∨ the character group)

X∨ =n⊕

i=0

Ze∗i

∆∨ = α∨i = e∗i − e∗i+1n−1i=1 ∪ α∨n = e∗n

X =n⊕

i=0

Zei

∆ = αi = ei − ei+1n−1i=1 ∪ αn = 2en − e0,

with X andX∨ in the duality〈ei , e∗j 〉 = δi j .Alternatively, since Spin2n+1 is the connected, simple, simply-connected group with Lie algebra

so2n+1, we can write its character group as the weight lattice ofso2n+1, i.e., as the submoduleof ⊕n

i=1Qχi spanned byχ1, . . . , χn,χ1+...+χn

2 , and its co-character lattice as those∑n

i=1 ciλi such that

52

ci ∈ Z and∑

ci ∈ 2Z. The duality is〈χi , λ j〉 = δi j . Letting χ0 andλ0 generateX•(GL1) andX•(GL1), respectively, the description of GSpin2n+1 as GL1×Spin2n+1

1,(−1,c) then leads to another descriptionof the root datum as

Y• =n⊕

i=1

Zχi ⊕ Z(χ0 +χ1 + . . . + χn

2) ⊂

n⊕

i=0

Qχi

∆• = χi − χi+1n−1i=1 ∪ χn

Y• =n⊕

i=1

Z(λi +λ0

2) ⊕ Zλ0

∆• = λi − λi+1n−1i=1 ∪ 2λn.

We summarize by comparing the two descriptions:

Lemma 2.8.1. There is an isomorphism of based root data

(X∨,∆∨,X,∆) (Y•,∆•,Y•,∆•)

given by

e∗i 7→ χi for i = 1, . . . , n;

e∗0 7→ χ0 −χ1 + . . . + χn

2;

ei 7→ λi +λ0

2for i = 1, . . . , n− 1;

e0 7→ λ0.

For example, the center of GSpin2n+1 is generated by the co-charactere0 ↔ λ0. The Cliffordnorm is given by the character 2χ0↔ 2e∗0 +

∑n1 e∗i .

Returning to our representationρ, recall that to eachv|ℓ andι : Qℓ → CFv we can associate theSen operator

Θρ,ι := Θρ|ΓFv,ι ∈ so2n+1 ⊗Qℓ ,ι CFv.

(The placev is implicit in ι.) Sinceρ is Hodge-Tate, we can identifyΘρ,ι up to conjugation with adiagonal element

m1

.. .mn

0−mn

.. .−m1

,

wheremj = mj(ι) are all integers.37

Proposition 2.8.2. Let ρ : ΓF → SO2n+1(Qℓ) be a geometric representation as above, with Ftotally real. Thenρ lifts to a geometric,GSpin2n+1-valued representation if and only if the parityof

∑j mj(ι) is independent of v|ℓ andι : Qℓ → CFv.

37We choose the ‘anti-diagonal’ orthogonal pairing, so that SO2n+1 has a maximal torus consisting of diagonalmatrices.

53

Remark 2.8.3. By itself, this result is formal. Later we will see howsuchρ arise, at least whenρ is regular; but note that this Proposition makes no such assumption.

Proof. By Proposition 2.1.4, some lift ˜ρ, necessarily unramified almost everywhere, exists.All possible lifts differ from thisρ by e0 χ for someχ : ΓF → Q

×ℓ . As before, we can write the

multiplier characterν(ρ) : ΓF → Q×ℓ asχ2χ0, whereχ0 has finite order, and then replace ˜ρ by a

twist havingν(ρ) of finite-order (Lemma 3.2.2 will explain this procedure ingeneral). Then, sincefinite-order characters have Hodge-Tate weights zero, functoriality of the Sen operator implies thatfor all ι : Qℓ → CFv, Θρ,ι maps to (0,Θρ,ι) in (gl1)ι ⊕ (so2n+1)ι.38

Now consider the spin representation GSpin2n+1

rspin−−−→ GL2n. In the above root datum notationthis corresponds to the highest weight−e∗0↔ −χ0+

χ1+...+χn

2 . The image ofΘρ,ι is then a semi-simpleelement with one eigenvaluem1+...+mn

2 , and all eigenvalues congruent to this (half-integer) modulo

Z. In particular,rspin ρ|ΓFvis Hodge-Tate, and thus de Rham, if and only if for allι : Qℓ → CFv,

m1(ι) + . . . + mn(ι) is even. If all (for all v|ℓ and all ι) of these sums are odd, then we twistby a characterχ with all Hodge-Tate-Sen weights 1/2 (see Lemma 2.3.17) to get a new ˜ρ, nowgeometric, liftingρ. On the other hand, Lemma 2.3.17 shows that forF totally real we cannottwist ρ in a similar way if some

∑j mj(ι) are even and others are odd.

Corollary 2.8.4. Let ρ : ΓF → SO2n+1(Qℓ) be as in Proposition 2.8.2. Let L/F be a CMextension. Thenρ|ΓL has a geometric lift.

Proof. With the framework of the previous proof, this follows by the same argument as Lemma2.7.5.

If we make additional assumptions so that potential automorphy theorems apply, we can ofcourse say more. The next two lemmas merely cash in on some very deep recent results.

Lemma 2.8.5. Assume thatρ : ΓF → SO2n+1(Qℓ) ⊂ GL2n+1(Qℓ) as in the previous propositionmoreover satisfies:

• For all v|ℓ, ρ|ΓFvis regular and potentially diagonalizable;

• ρ|ΓF(ζℓ)is irreducible.

• ℓ ≥ 2(2n+ 2).

Then after some totally real base change F′/F, there exists a regular L-algebraic self-dual cuspidalautomorphic representationπ of GL2n+1(AF′) such thatπ ∼w,∞ ρ.

Proof. This is immediate from TheoremC (=Corollary 4.5.2) of [BLGGT10], sinceρ is au-tomatically totally odd self-dual.39

Lemma 2.8.6. Let ρ ∼w,∞ π be as in Lemma 2.8.5. Thenπ descends to a cuspidal automor-phic representation onSp2n(AF′), in a way compatible with unramified and archimedean localL-parameters.

Proof. This follows from Arthur’s classification of automorphic representations of classicalgroups ([Art ]). Namely,ρ∨ ρ implies thatπ∨ π, and det(ρ) = 1 implies thatωπ = 1. π is

38Writing gι as a short-hand forg ⊗Qℓ ,ιCFv .

39Note that those authors always work withC-algebraic automorphic representations, so the statements of theirtheorems always have an extra, but easily unraveled, twist.

54

cuspidal, so the associated formalA-parameterφ (see§1.4 of [Art ]) is simple generic and so comesfrom a unique simple twisted endoscopic datumGφ as in Theorem 1.4.1 of [Art ]; since 2n+ 1 isodd, the parameterφ therefore factors through either SO2n+1(C) or O2n+1(C), but the latter case isruled out sinceωπ = 1. It follows thatGφ = Sp2n/F.40 The local statement follows from [Art ,Theorem 1.4.2].

Proposition 2.8.7.Continuing with the assumptions (and conclusions) of the previous two lem-mas,ρ|ΓF′ has a geometric lift toGSpin2n+1(Qℓ) if and only ifπ admits an L-algebraic extension toGSp2n(AF). In the other direction, we have, as in Lemma 2.6.10, an automorphic construction ofinfinitely many suchρ, whose local behavior41 we can additionally specify at any finite number ofplaces, that do not admit a geometric lift.

Proof. (Some details of this argument are omitted, deferring to more general arguments in thenext section.) At eachv|∞, we can write theL-parameter ofπv as

φv : WFv → SO2n+1(C) ×WFv

φv|WFv: z 7→

zm1 zl1

. ..zmn zln

1z−mnz−ln

.. .z−m1 z−l1.

,

with all mi, l i ∈ Z. Since the transfer ofπ to GL2n+1(AF) is cuspidal, Clozel’s archimedean puritytheorem (see Lemme 4.9 of [Clo90]) implies thatmi + l i = 0 for all i (and allv). To extendπ inan L or W-algebraic fashion, the key point is to construct an appropriate extension of the centralcharacter. Any liftφv to GSpin2n+1(C) of the parameterφv must, onWFv

, take the formz 7→ zµvzνv,where

µv =

n∑

i=1

mv,i(λi +λ0

2) + µv,0λ0

νv = −n∑

i=1

mv,i(λi +λ0

2) + νv,0λ0.

The central character of this extension is (by pairing with 2χ0, the Clifford norm; this procedurefor computing central characters is explained in general in[Lan89])

ωv : z 7→ z∑n

i=1 mv,i+2µv,0z−∑n

i=1 mv,i+2νv,0.

If we can choose an extension ofπ to an automorphic representation ˜π of GSp2n(AF′) with finite-order central character, then this calculation shows thatφv (the localL-parameter for ˜πv) is always‘W-algebraic,’42 and it is ‘L-algebraic’ if and only if

∑ni=1 mv,i is even. The result would follow

40One would like to say that sinceρ lands in SO2n+1(Qℓ), Sym2(ρ) contains the trivial representation and thereforeL(s, π,Sym2) has a pole ats = 1; this would allow us to descendπ to Sp2n(AF) using [CKPSS04]. Unfortunately,nothing is known a priori aboutL(s,V) where Sym2(ρ) = 1 ⊕ V; in particular, we can’t say immediately it is non-vanishing ats = 1, so this argument doesn’t seem to work. The argument we havegiven allows us to deduce thatL(s, π,Sym2) has a simple pole ats= 1, and thereforeL(s,V) is non-vanishing ats= 1.

41i.e. inertial type42Using this term abusively for the obvious local analogue.

55

easily, twisting our given ˜π by | · |1/2 if all∑

i mv,i are odd (just as in the Galois case). That we canfind such an extension ˜π with finite-order central character will be proven in much more generalityin Proposition 3.1.14.

The construction of geometricρ : ΓF → SO2n+1(Qℓ) having no geometric lift to GSpin2n+1(Qℓ)(and with specified local behavior) follows as in Lemma 2.6.10, applying Clozel’s limit multiplicityformula toG = Sp2n overF, transferring these forms to GL2n+1 (via [Art ]), and then invoking theParis Book Project, or, more precisely, Remark 7.6 of [Shi11].

We have therefore generalized some of the results of§2.7 (the casen = 1).

56

CHAPTER 3

Galois and automorphic lifting

This chapter begins (§3.1) by addressing the natural automorphic analogue of Conrad’s liftingquestion. It is much easier to see what should be true in this setting, and the proofs are simpleras well. Equipped with the intuition coming from the automorphic case, we address the originalGalois-theoretic question in§3.2. In §3.3, we combine the results of§3.1 and 3.2 to compare,modulo the Fontaine-Mazur-Langlands conjecture, descent problems for certain automorphic rep-resentations and Galois representations. The closing section, §3.4, is of a rather different nature,assembling a few results about the images of compatible systems ofℓ-adic Galois representations.The results of this section continue the attempt to compare aspects of the automorphic and Galois-theoretic formalisms.

3.1. Lifting W-algebraic representations

In this section, we make some simplifying assumptions:G will be a connected semi-simplesplit group overF, andZ will be an F-split torus containingZG. Then letG = (Z × G)/ZG, asbefore, with maximal torusT = (Z × T)/ZG and centerZ. In each caseZG is embedded anti-diagonally. In fact, the cases of greatest interest are whenG is simply-connected (soG∨ is adjoint),such as SLn or Sp2n, but we do not assume this. The assumptions thatG is semi-simple and thatZis a torus are not essential, and in fact in§3.2 we will work more generally.

Now letπ be a (unitary) cuspidal automorphic representation ofG(AF) that isW-algebraic. Weare interested in the problem of liftingπ to aW-algebraic automorphic representation ˜π of G(AF);whenπ is moreoverL-algebraic, we similarly ask whether anL-algebraic lift exists. By ‘lift,’ wesimply mean the most naive thing: the restriction ˜π|G(AF) containsπ. Corollary 3.1.6 justifies thisconvention, showing that under theL-morphismLG→ LG, theL-packet ofπ is a functorial transferof theL-packet ofπ.

3.1.1. Notation and central character calculation.In order to do computations in terms ofroot data, we choose coordinates (using invariant factors)such that

X•(Z) =r⊕

i=1

Zwi ⊕s⊕

j=1

Zw′j ,

and the kernel ofX•(Z)→ X•(ZG) is

r⊕

i=1

diZwi ⊕s⊕

j=1

Zw′j .

57

This isX•(S) for the torusS = Z/ZG. We then writeX•(ZG) = ⊕r1(Z/diZ)wi. To relate parameters

for G to those forG, we use the Cartesian diagram

X•(T) //

X•(Z)

X•(T) // X•(ZG),

representing elements ofX•(T) as pairs (χT , χZ) that are congruent inX•(ZG). Extending scalarsto Q (or any characteristic zero field),X•(T)Q

∼−→ X•(T)Q ⊕ X•(Z)Q. We will usually (out of sadnecessity) takeF to be either CM or totally real. In the former case (or whenever an archimedeanplace is complex) we compute local central characters (see page 21ff. of [Lan89]) as follows:

Supposev is complex, so there is an isomorphismιv : Fv∼−→ C. Let

recv(πv) : z 7→ ιv(z)µv ι(z)νv ∈ T∨(C)1

be the associated Langlands parameter, withµv, νv ∈ X•(T)C with µv− νv ∈ X•(T). Write∑r

i=1[µv−νv] iwi for the projection ofµv − νv to X•(ZG). Choose anyµv,i , νv,i ∈ C with µv,i − νv,i an integerprojecting to [µv − νv] i ∈ Z/diZ. We can then regard ˜µv = (µv,

∑r1 µv,iwi) andνv = (νv,

∑r1 νv,iwi) as

elements ofX•(T)C parametrizing an extension of the localL-parameter ofπv to a parameter forG(Fv). Identifying, via our chosen basis,Z(Fv) with (C×)r+s, the central characterωπv of this lift issimply

ωπv : (z1, . . . , zr , z′1, . . . , z

′s) 7→

r∏

i=1

zµv,i−νv,i

i |zi |νv,i

C.

Restricting toZG(Fv) µd1 × · · · × µdr ⊂ (C×)r , we find that the central characterωπv is given by

ωπv : (ζ1, . . . , ζr) 7→r∏

i=1

ζµv,i−νv,i

i ,

where eachζi is adthi root of unity. Clearly this is independent of the choice of lifts µv, νv.

Definition 3.1.1. If F is a CM field, andπ is a (unitary) automorphic representation ofG(AF)with archimedean parametersµv, νv as above at eachv|∞, then we define ˜ω = ω(π) to be any choiceof Hecke character ofZ lifting ωπ and such that

ωv(z1, . . . , z′s) =

r∏

i=1

(zi/|zi |)µv,i−νv,i .

This is a unitary, typeA Hecke character whose existence is assured by Lemma 2.3.6.

Whenv is real, the lifting process is more complicated (see [Lan89]), and the central charactercomputation depends very much on where theL-parameter ofπv sends an element ofWR −WC.We can avoid this, however:

Definition 3.1.2. Letπ be as in the previous definition, but now supposeF is totally real. Thenwe defineω to be any choice of finite-order Hecke character ofZ extendingωπ (existence again byLemma 2.3.6).

1From now on, we writez, z in place ofιv(z), ιv(z).

58

3.1.2. Generalities on lifting from G(AF) to G(AF). We now try to find an automorphicrepresentation ˜π ⊂ L2

cusp(G(F)\G(AF), ω)2 lifting π, whereω is any (unitary) lift of the centralcharacterωπ. We say that we are in the Grunwald-Wang special case if one ofthe pairs (F, di) isin the usual Grunwald-Wang special case. LetHAF = G(AF)Z(AF) (a normal subgroup ofG(AF)),and letHF = HAF ∩ G(F).

Lemma 3.1.3. We are in the Grunwald-Wang special case if and only if HF strictly containsG(F)Z(F).

Proof. Recall the charactersdiwi ∈ X•(S) = X•(G) (sinceG is semi-simple). These induce anisomorphism

HF/G(F)Z(F)∏

diwi−−−−→r∏

i=1

(F× ∩ (A×F)di

)/(F×)di .

There are various ways to show forms onG(AF) extend toG(AF); the argument we use here ismodeled on one of Flicker for the case (G,G) = (GSp2n,Sp2n) (see Proposition 2.4.3 of [Fli06]).

Proposition 3.1.4.Letπ be a cuspidal automorphic representation of G(AF), with central char-acterωπ. If we are not in the Grunwald-Wang special case, then for anyextensionω of ωπ to aHecke character ofZ, there exists a cuspidal automorphic representationπ of G(AF) lifting π, andhaving central characterω. If we are in the Grunwald-Wang special case, then for at least oneextension ofωπ to

r∏

i=1

CF[di] ⊃r∏

i=1

µdi (F)\µdi (AF),

and for any extension of this character to a Hecke characterω of Z, there exists a cuspidalπ liftingπ with central characterω.3

Remark 3.1.5. In particular, in all cases, for all Hecke charactersω extendingωπ, there exists aHecke character ˜ω′ having the same infinity-type, and a cuspidal representation π of G(AF) liftingπ, with central character ˜ω′.

Proof. Choose an extension ˜ω : Z(F)\Z(AF)→ S1 of ωπ. By extending functions alongZ(AF)via ω, we embed the spaceVπ of π into the space of cusp-forms onG(F)Z(F)\HAF , and therebyobtain an extension ofπ to a representationπω of HAF . We construct an intertwining map

IndG(AF )HAF

(πω)U−→ L2

cusp(G(F)\G(AF), ω),

where the induction consists of functionsF : G(AF) → Vπ with the usual left-HAF -equivariance,and the requirement of compact support moduloHAF . Write δ1 : Vπ → C for evaluation at 1 of thecusp forms inVπ, and set

L(F) =∑

G(F)Z(F)\G(F)∋u

δ1(Fu),

2That is, the space of measurable functions onG(F)\G(AF) with (unitary) central character ˜ω, and square-integrable moduloZ(F∞)0– or, equivalently, moduloAG(R)0, where AG is the maximalQ-split central torus inResF/Q(G).

3This odd modification in the special case is presumably an artifact of my clumsy proof.

59

where we writeFu for the value atu of F ∈ Ind(πω). This sum is in fact finite:G(F) is discretein G(AF), the fibers ofG(F)Z(F)\G(F) → HAF\G(AF) are finite, and we have assumedF hascompact support moduloHAF . It is also well-defined because forγ ∈ G(F)Z(F),

δ1(Fγu) = δ1(γ · Fu) = Fu(γ) = Fu(1) = δ1(Fu).

The mapU is then given byU(F) : g 7→ L(I (g)F),

where I (·) denotes theG(AF)-action on the induction.U is clearly G(AF)-equivariant and hasoutput U(F) which is left-G(F)-equivariant (by its construction as an average) and has centralcharacter ˜ω; U(F) is a cusp form by a simple calculation using the fact that theunipotent radicalof any parabolic ofG is in fact contained inG.

To see thatU , 0, take a non-zero formf ∈ Vπ; we may assume (by translating)f (1) , 0.Then defineF ∈ Ind(πω) by

Fh : h′ 7→ f (h′h)

if h ∈ HAF , andFg = 0 for g < HAF . Then

U(F)(1) =∑

u

δ1(Fu) =∑

u

f (u) if u ∈ HAF ;

0 otherwise.

So, if we are not in the special case, we just getU(F)(1) = f (1) , 0. If we are in the specialcase, then for each pair (F, di) in the special case there is an elementαi ∈ F× which is everywherelocally adth

i -power– sayαi = βdii , but not globally adth

i -power. By abuse of notation, we also writeαi for an element ofG(F) such that (diwi)(αi) is this element ofF× and the other charactersd jwj

( j , i) andw′j of G are trivial onαi.4 Regardingβi as an element of thei th componenet ofZ(AF),

we can therefore writeαi = βi · γi, where eachγi lies in G(AF). If only one pair (F, di) is in thespecial case, then the above expression forU(F)(1) becomesf (1) + f (αi) = f (1) + ω(βi) f (γi).This time we normalizef so that f (γi) , 0 (rather thanf (1) , 0), and so necessarily eitherf (1) + ω(βi) f (γi) , 0, or this expression is non-zero for any ˜ω extending the *other* extensionof ωπ to a character ofCF [di]. Thus we can choose at least one initial extension toCF [di], andthereafter the argument proceeds as in the non-exceptionalcase. If multiple (F, di) are in thespecial case, the same argument applies: arrange somef (γi) , 0, and ifU(F)(1) = 0, change theextension ofωπ to CF [di] just in this i th component.

Finally, given thatU , 0, we take ˜π to be the image of any irreducible constituent of IndG(AF)HAF

(πω)that survives underU, and we claim this is the desired extension ofπω. The isomorphism classesof HAF -representations appearing in the restriction toHAF of the full induction are precisely theconjugatesπg

ω, for g ∈ G(AF), andG(AF)-stability of π implies that all of these in fact appear inπ|HAF

. In particular, this latter restriction containsπω.

We need to supplement this by understanding to what extentπ is a ‘functorial transfer’ of ˜π,

with respect to theL-homomorphismLG ։ LG. Somewhat more generally, letGη−→ G be a

morphism of connected reductive groups overFv, with abelian kernel and cokernel; here we eithertakev to be archimedean, or finite such thatG andG are unramified overFv. By assumption,there is a dualL-homomorphismLη : LG → LG. In both the archimedean and unramified cases,

4We use the assumption thatG is split overF.

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theL-parametersφv andφv, as well as the correspondingL-packetsΠφv andΠφv, of πv andπv have

been defined (as will be explained in Corollary 3.1.6).

Corollary 3.1.6. Continue in the setting of the proposition. Thenπ is a weak transfer ofπ,which is also a strong transfer at archimedean places. More precisely, for all places v of F that areeither archimedean or such that G is unramified at v, the restriction πv|G(Fv) is a finite direct sum ofelements ofΠφv, andφv reduces toφv (i.e. Lη φv = φv up to G∨(C)-conjugacy).

Proof. For infinite placesv, the assertion follows from desideratum (iv) on page 30 of [Lan89].Since the construction of the correspondence is inductive,the verification necessarily stretches outthrough§3 of that paper; for the first (and most important) case of discrete series, see page 43.

Now we treat the unramified case. Suppose ˜πv is unramified, i.e. there exists a hyperspecialmaximal compact subgroupKv of G(Fv) such that ˜πKv

v , 0. Then for some unramified character

χv of T(Fv), πv is a sub-quotient ofI G(Fv)B(Fv)

χv. The natural mapG(Fv)/B(Fv) → G(Fv)/B(Fv) is anisomorphism, so restriction of functions gives an isomorphism ofG(Fv) representations

(I G(Fv)B(Fv)

χv

)|G(Fv)

∼−→ IG(Fv)B(Fv) (χv|B(Fv)).

Write χv for χv|B(Fv). It is an unramified character ofT(Fv), and under the identification of unrami-fied characters ofT(Fv) with Hom(X•(T)ΓFv ,C×) (and the analogue forT), and thus with the spaceof unramifiedL-parameters,χv corresponds to the parameterφv =

Lη φv. By definition of unram-ified L-packets (see [Bor79, §10.4]), to see that any constituentπv of πv|G(Fv) lies in the packetΠφv,we need only check thatπv has invariants under some hyperspecial maximal compact subgroup ofG(Fv). To see this, letu be a non-zero vector in ˜π

Kvv . Decomposing ˜πv|G(Fv) = ⊕M

1 π⊕mii (with theπi

distinct isomorphism classes; in fact, the multiplicitiesmi are all equal), we see thatu lies in one ofthe isotypic componenetsπ⊕mi

i , since the inductionI (χv) can only have oneKv := G(Fv) ∩ Kv- in-variant line. Fixing a decompositionu =

∑mij=1 u j with u j in the j th copy ofπi, we find that theu j are

themselves alsoKv-invariant, henceπi contains aKv-invariant vector (this in turn implies the mul-tiplicities mi must equal 1). SinceG(Fv) acts transitively on the isomorphism classesπi, eachπi,and in particular our givenπv, contains some vector of the form ˜πv(g)u, which isgKvg−1-invariant;gKvg−1 is also a hyperspecial maximal compact subgroup, so we’re done.

Remark 3.1.7. According to expected properties of localL-packets,π should be a strong trans-fer of π (see§10.3 of [Bor79]), but of course this statement is meaningless until the local Langlandscorrespondence is known forG andG.

Eventually, we will also want to understand the ambiguity inthe choice of ˜π. Strictly speak-ing, we only need this in one direction of Proposition 3.1.14below, but as a general problem, itsimportance is basic. We will need to assume something from local representation theory, whichhas also previously been conjectured, more generally for quasi-splitG andG, by Adler and Prasad([AP06].

Conjecture 3.1.8.Let v be any place F, and let G andG be as above. Then for any irreduciblesmooth representationπv of G(Fv), the restrictionπv|G(Fv) decomposes with multiplicity one.

The motivation for this conjecture is the uniqueness of Whittaker models for quasi-split groups,and indeed the conjecture holds for generic ˜πv. The analogous statement forv archimedean isstraightforward in many cases: for instance, for a simpleG, it is easy becauseR×/(R×)n is cyclic for

61

all n. Moreover, the structure of archimedean L-packets may be well-enough understood to deducethe conjecture in general for archimedeanv, but we do not pursue this here. Forvnon-archimedean,we have verified it in the unramified case (see the proof of Lemma 3.1.6), and in general the conjec-ture is known for the following pairs (G,G): (GLn,SLn) (from the theory of, possibly degenerate,Whittaker models; see [LL79 ] for the n = 2 case), and any pair (GU(V),U(V)) where (V, 〈, 〉) is avector space overFv equipped with a non-degenerate symmetric or skew-symmetric form 〈, 〉, andGU(V), respectivelyU(V), denote the similitude and isometry groups of the pairing (see Theorem1.4 of [AP06]). So, whatever the status of the general conjecture, the discussion below applies tosome interesting cases.

Lemma 3.1.9. Assume Conjecture 3.1.8. Supposeπ and π′ are two lifts ofπ (i.e. their restric-tions containπ) with the same central characterω.

• There exist continuous idele charactersαi : A×F → C×, for i = 1, . . . , r, such that, in thenotation of§3.1.1,

π π′ ·

r∏

i=1

αi diwi

,

and eachαdii factors as a genuine Hecke characterα

dii : A×F/F

× → C×.• Since Hecke characters satisfy purity, so do the charactersαi.

Remark 3.1.10. The point of the Lemma is that the idele charactersαi need not be Heckecharacters, i.e. need not factor throughCF . We are particularly interested in the constraint on theinfinity type ofαi, which by the lemma must be just as rigid as the constraints for Hecke characters.This will be applied in Proposition 3.1.14.

Proof. As before, writeHAF = G(AF)Z(AF), and writeHv = G(Fv)Z(Fv) for its local analogue.By hypothesis, both restrictions ˜π|HAF

andπ′|HAFcontainπ⊠ ω, so this holds everywhere locally as

well. Lemma 2.4 of [GK82] (applying Conjecture 3.1.8) implies that for all finitev, there exists asmooth characterαv : G(Fv)/Hv → C× such that ˜πv π

′v ·αv. By duality (for finite abelian groups),

we can extend this to a character ¯αv:

1 // G(Fv)/Hv

αv

//∏r

i=1 F×v /(F×v )di

αv

vv♥♥♥♥♥♥♥♥♥♥♥♥♥♥

C×

We expressαv in coordinates asαv =

(αi,v (diwi)

)ri=1

for smooth charactersαi,v of F×v trivial on (F×v )di . Globally, we then have

π π′ ·

r∏

i=1

(⊗′vαi,v) diwi

.

Here theαi := ⊗′vαi,v are continuous5 charactersA×F → C×, but they need not be Hecke characters.Taking central characters, however, we see that eachα

dii is in fact a Hecke character.

5Almost allαi,v are unramified, since the same holds for ˜πv andπ′v.

62

Remark 3.1.11. Under Conjecture 3.1.8, one should be able to refine the arguments of thissection to produce a multiplicity formula for cuspidal automorphic representations ofG(AF) interms of those ofG(AF), as in Lemma 6.2 of [LL79 ]). One can also axiomatize the passagebetween local Langlands conjectures forG and forG; this would include verifying compatibilityof the conjectural formulae for the sizes ofL-packets (a template, in the case of (GSp,Sp), is givenin the paper of [GT10]; their arguments will clearly apply much more generally).

3.1.3. Algebraicity of lifts: the ideal case.Now we return to questions of algebraicity, han-dling the CM and totally real cases in turn. In each case, we carefully choose an extension ofthe central character ofπ (as described in Definitions 3.1.1 and 3.1.2), and then find a ˜π as inProposition 3.1.4 with that central character, and whose restriction toG(AF) containsπ. This willyield enough information about the archimedeanL-parameters of ˜π to deduce the desired alge-braicity statements. To give a clean argument with broad conceptual scope, we will assume thatπ

is tempered at infinity6. As remarked before, this is not a serious assumption for algebraic repre-sentations: it is satisfied for forms having cuspidal transfer to some GLN. For non-tempered forms,analogous results for the discrete spectrum can be deduced from Arthur’s conjectures.

Proposition 3.1.12.Let F be CM, withπ, π as above, and withω the (type A) extension ofωπ

described in Definition 3.1.1. Assumeπ∞ is tempered.

(1) If π is L-algebraic, thenπ is L-algebraic. In particular, there exists an L-algebraiclift ofπ.

(2) If π is W-algebraic, thenπ is W-algebraic.

Proof. We use the notation of§3.1.1. We may of course take allµv,i andνv,i to be integers. Bythe choice of central character ˜ω, the archimedeanL-parameter for ˜πv corresponds to

µv = (µv,∑ µv,i − νv,i

2wi) ∈ X•(T)C ⊕ X•(Z)C,

and

νv = (νv,∑ νv,i − µv,i

2wi) ∈ X•(T)C ⊕ X•(Z)C.

We write this parameter as an obviously integral term plus a defect:

µv = (µv,∑

µv,iwi) + (0,−∑ µv,i + νv,i

2wi),

and likewise for ˜νv. Note that this lies inX(T) if and only if for all i = 1, . . . , r, we haveµv,i+νv,i

2 ∈ diZ.The discussion is so far general; if we now assumeπv is tempered, then for allλ ∈ X•(T), thecharacter

z 7→ z〈µv,λ〉z〈νv,λ〉

is unitary, i.e.Re(〈µv + νv,X•(T)〉) = 0.

W-algebraic representations of course have real infinitesimal character, soµv = −νv, and thereforein the initial choice of lift we may assumeµv,i = −νv,i for all i. Then obviously ˜µv is L-algebraic.

If π is only W-algebraic, then we have to check that 2˜µv = (2µv,∑

(µv,i − νv,i)wi) lies in X•(T).This element ofX•(T) ⊕ X•(Z) represents an element ofX•(T) if and only if 2µv maps to

∑i[µv −

6No assumption at finite places.

63

νv] iwi in X•(ZG). The latter is also the image ofµv−νv, so it is equivalent to ask thatµv+νv ∈ X•(T)map to zero inX•(ZG). As above, temperedness ofπv guarantees this, so we are done.

Remark 3.1.13. This result would immediately extend to totally imaginary fields if we knewthatωπ extended to a typeA0 Hecke character ofZ(AF). As noted in Remark 2.4.9, this wouldfollow from the (‘CM descent’) conjectures of§2.4.

So we turn to the case of totally realF. I am grateful to Brian Conrad for urging a coordinate-free formulation of the obstruction in part 2 of the proposition.

Proposition 3.1.14. Now suppose F is totally real, withπ, ω, π as before (ω finite-order asin Definition 3.1.2). Alternatively, let F be arbitrary, butassume thatωπ admits a finite-orderextensionω. Continue to assumeπ∞ is tempered.

(1) If π is L-algebraic, then it admits a W-algebraic liftπ.(2) Assume Conjecture 3.1.8 for the ‘only if ’ direction of this statement. Assume F is totally

real, and consider this W-algebraic liftπ. Then the images ofµv andνv under X•(T) →X•(ZG) lie in X•(ZG)[2], andπ admits an L-algebraic lift if and only if these images areindependent of v|∞.

Proof. We continue with the parameter notation of the previous proof. Let π be L-algebraic.First we check that since ˜ω can be chosen finite-order (automatic in the totally real case, but anadditional assumption at complex places)µv − νv maps to zero inX•(ZG). If v is imaginary, then ˜ωcannot be finite order unlessωπv is trivial, henceµv andνv themselves are trivial inX•(ZG). If v isreal, then inT∨ ⊂ G∨ we have the relation

φ( j)zµvzνvφ( j)−1 = zνvzµv,

writing φ for theL-parameter andj ∈WFv −WFv. Now,φ( j) ∈ NG∨(T∨) represents an elementw of

the Weyl group ofG∨, and this yields the relationswµv = νv andwνv = µv. Thusµv−νv = µv−wµv,which lies in the root latticeQ of G. [This holds forwχ−χ for anyχ ∈ X•(T): reduce to the case ofsimple reflections, where it is clear from the defining formula.] Restricting characters toZG factorsthrough a perfect dualityZG × X•(T)/Q→ Q/Z, soµv − wµv ∈ Q has trivial image inX•(ZG).

Any lift with finite-order central character has parametersµv = (µv, 0) andνv = (νv, 0); this pairyields a well-defined representation ofWFv

, i.e. µv − νv ∈ X•(T), and its projections toT andZare what they have to be, so they are the only possible parameters; that this extends (possibly non-uniquely) to anL-parameter on the whole ofWFv follows from general theory (Langlands Lemma),but we do not need the details of this extension. Now we use theassumption thatπv is tempered:as in the previous proposition, we see thatµv = −νv. Hence 2µv maps to zero inX•(ZG), and so2(µv, 0) ∈ X•(T), i.e. (µv, 0) isW-algebraic.

For F totally real, we now show the second part of the proposition.By Lemma 3.1.9, if asecond liftπ′ is L-algebraic, then

π′ π ·

r∏

i=1

αi diwi

,

where eachαdii is a Hecke character of our totally real field, and therefore (sinceπ andπ′ both have

rational infinitesimal character) takes the formχi · | · |diyi for a finite-order characterχi and a rationalnumberyi. The infinitesimal character of ˜π′v (for v|∞) then corresponds to (µv,

∑yidiwi) ∈ X•(T)Q,

64

which is integral if and only ifyi ∈ 1diZ andyidi is the image ofµv in Z/diZ for eachi. The claim

follows easily.

Corollary 3.1.15.Let F be totally real andπ be L-algebraic on the F-group G.

(1) Supposeπ∞ is tempered. If all di are odd, thenπ is L-algebraic. In particular, if the simplefactors of G are all type A2n (i.e. SL2n+1), or of type E6, E8, F4, G2,

7 then any L-algebraicπ has an L-algebraic lift.

(2) Assume F, Q (still totally real). For F-groups G that are simple simply-connected (andsplit) of type Bn, Cn, D2n, and E7, there exist, assuming Conjecture 3.1.8 for the pair(G, G), L-algebraicπ on G, tempered at infinity, that admit no L-algebraic lift toG.

Proof. The first part is immediate from the proof of the proposition: for di odd, if the image of2µv in Z/diZ is trivial, then so of course is the image ofµv. For the second part, we use existenceresults for automorphic forms that are discrete series at infinity as in Lemma 2.6.10 and Proposition2.8.7. For a semi-simple, simply-connected group,ρ lies in the weight lattice, so discrete seriesrepresentations will beL-algebraic. Choose a quotientG։ G′ ։ Gad, with T′ the induced maxi-mal torus ofG′, such thatX•(T′)/Q X•(ZG)[2] under the isomorphismX•(T)/Q

∼−→ X•(ZG).Thenwe can find a discrete series representation ofG(F∞) such that at two infinite placesv1 andv2, theparameters have the form (onC×)

z 7→ zρ+µvi z−ρ−µvi ,

where theµvi are distinct inX•(T′)/Q.8 This ensures that theρ + µvi have distinct images inX•(ZG)[2]. It remains therefore to check that the split real formsof these groups, except forA2n−1,actually admit discrete series. TypeCn was treated previously. For typeBn, the (real) split group isSpin(n, n+ 1), which always admits discrete series (real orthogonal groups SO(p, q) have discreteseries wheneverpq is even). Likewise, for typeDn, the split form Spin(n, n) has discrete series ifand only ifn is even. The split real form ofE7 also admits discrete series: its maximal compactsubgroup is SU(8), which contains a compact Cartan isomorphic to (S1)7 (see the table in Appendix4 of [Kna02]).

Remark 3.1.16. • Arthur constructs in [Art02 ] a conjectural candidate for the automor-phic Langlands groupLF, also giving an ‘automorphic’ construction of a candidate forthe motivic Galois groupGF. (In §4.1 we will discuss motivic Galois groups in detail,explaining both conjectural and unconditional constructions.) The ability to extendL-algebraic representations ofG to L-algebraic representations ofG is essential to Arthur’sconjectural construction (see [Art02 , §6]) of a morphismLF → GF. Roughly, the factorof the motivic Galois groupGF associated toπ (or rather its almost-everywhere system ofHecke eigenvalues; this construction is restricted to suitably ‘primitive’ π) is defined as asub-groupGπ of G∨×TF , withTF the Taniyama group (see§4.1.6 for background onTF);thenLF andGF are constructed as suitable fiber products over varying pairs (G, π).9 Thenatural motivic Galois representation corresponding to ‘the motive ofπ’ is the projection

7A2n andE6 are the interesting examples here, since in the other cases the adjoint group is simply-connected. Theindex [X•(T) : Q] for the simply-connected simple group of typeE6 is Z/3Z.

8We have shifted what was previously denotedµv by ρ in this case because of the description of discrete seriesL-parameters as arising from elements ofρ + X•(T). Note thatρ mapst toX•(ZG)[2] since 2ρ is in the root latticeQ.

9For non-splitG, one works with the semi-direct productG∨ ⋊ TF , with TF acting via its projection to the globalWeil groupWF .

65

to G∨, and implicit in the construction is the fact that this liftsto G∨. Corollary 3.1.15then gives concrete examples of automorphic representations (including ones that oughtto be ‘primitive,’ arranging the local behavior suitably) for which the Tannakian formal-ism cannot behave in this way; that is, while Arthur’s construction is consistent with ourdiscussion in the CM case (Proposition 3.1.12), it must be modified for totally real fieldsF.• If our lifting results were generalized to quasi-split groups, we could presumably include

Dn for n odd in the second part of the Proposition, since the non-split quasi-split form hassignature (n+ 1, n− 1), for which the associated orthogonal group admits discrete series.Similarly, we could include typeA2n−1 by using the quasi-split unitary group SU(n, n)instead of SL2n.

3.2. Galois lifting: the general case

Now we discuss a general framework for Conrad’s lifting problem. We consider lifting prob-lems of the form

H′(Qℓ)

ΓF

ρ<<③

③③

③③

ρ// H(Qℓ),

for any surjectionH′ ։ H, with central torus kernel, of linear algebraic groups overQℓ. In thispicture, we want to understand when a geometricρ does or does not admit a geometric lift ˜ρ.We will see that the general case reduces to the case whereH′ andH are connected reductive,and so to reap the psychological benefits of the work of§3.1, we will begin by considering thecaseH′ = G∨, H = G∨, whereG ⊂ G is an inclusion of connected reductive10 split F-groups,constructed by extending the centerZG of G to a central torusZ; that is

G = (G× Z)/ZG.

The quotientZ/ZG is a torusS, and we get an exact sequence

1→ S∨ → G∨ → G∨ → 1,

and a canonical isomorphism of Lie algebrasg∨ g∨ ⊕ s∨. We caution the reader that, even ifH′ and H are assumed connected reductive, this isnot the most general situation: namely, thecenterZ need not be connected in order to haveG∨ ։ G∨ a surjection with central torus kernel.11

Nevertheless, at least forF totally imaginary, we will reduce the general analysis to this case. Tostart, we elaborate on some of the associated group theory, slightly recasting the notation of theprevious section to take into account the fact thatG may have positive-dimensional center. Theexact sequence

1→ ZG → Z→ S→ 1

gives rise, by applyingX•(·) and then Hom(·,Z), to an exact sequence

1→ Hom(X•(ZG),Z)→ X•(Z)→ X•(S)→ Ext1(X•(ZG),Z)→ 1.

10Unlike in §3.1, whose results were largely intended to motivate the results of this section, we no longer requireG to be semi-simple.

11Consider for instanceG = Spin2n ⊂ G = GSpin2n, which dualizes to a surjection GSO2n։ PSO2n.

66

Using invariant factors, it is convenient to fix a basis ofX•(Z) such that the inclusionX•(S) ⊂ X•(Z)is in coordinates

X•(S) // X•(Z) // X•(ZG)

⊕ri=1Zdiwi

//⊕r+s

i=1 Zwi//⊕r

i=1Z/diZwi ⊕⊕s

j=1Zwr+ j

Let w∗i denote the dual basis forX•(Z∨), and letν∗i denote the basis ofX•(S∨) dual todiwi; inparticular,w∗i maps todiν

∗i underX•(Z∨)→ X•(S∨).

We work with the maximal torusT = (T × Z)/ZG of G, and deduce from its definition an exactsequence

1→ X•(T)→ X•(T) ⊕ X•(Z)→ X•(ZG)→ 1,

and thus a (crucial) exact sequence

(1) 1→ Hom(X•(ZG),Z)→ X•(T∨) ⊕ X•(Z∨)→ X•(T∨)→ Ext1(X•(ZG),Z)→ 1.

Note also that there is a canonical isomorphismX•(Z∨)∼−→ X•(G∨), which follows from exactness

of the row in the diagram:

1

S∨

1 // G∨sc//

G∨ //

Z∨ // 1

G∨

1

whereG∨sc denotes the simply-connected cover of the derived group ofG∨. Recall that we indexSen operatorsΘρ,ι by embeddingsι : Qℓ → CFv, wherev is a place aboveℓ. Given a geometricρ : ΓF → G∨(Qℓ), Conrad’s result (see Theorem 2.1.6, and Remark 2.1.5) implies that a lift ρ isgeometric if and only if it is Hodge-Tate at all places aboveℓ; we will use this from now on withoutcomment. That is, we need to arrange ˜ρ such that each Sen operatorΘρ,ι ∈ Lie(G∨)ι is conjugateto an element of Lie(T∨)ι that pairs integrally with all ofX•(T∨)12 under the natural map

X•(T∨)Lie−−→ Hom(Lie(T∨),Qℓ).

Here is our starting-point:

12This condition is independent of the wayΘρ,ι is conjugated into Lie(T∨)ι, since: (1) we already know thatΘρ,ιpairs integrally with the roots, which all lie inX•(T∨); (2) the ambiguity in conjugating into Lie(T∨)ι is an element ofthe Weyl group; (3) for any weightλ ∈ X•(T∨) andw in the Weyl group,wλ − λ lies in the root lattice. Compare theproof of Proposition 3.1.14.

67

Lemma 3.2.1. There exists some liftρ of ρ. Any other lift is of the form

ρ(r∑

i=1

νi χi) : g 7→ ρ(g) ·r∏

i=1

(νi χi)(g),

where theνi = diwi range over the above basis of X•(S∨), and eachχi : ΓF → Q×ℓ is a continuous

character.

Proof. A lift exists by Proposition 2.1.4. Although continuous cohomology does not in generalhave goodδ-functorial properties, short exact sequences do give (notvery) long exact sequenceson H0 andH1, so any lift has the form ˜ρ(χ) for someχ : ΓF → S∨(Qℓ). We compose with the dualcharactersν∗i in X•(S∨) to putχ in the promised form.

The following lemma is the general substitute for choosing lifts with finite-order Clifford normin the spin examples; this result is also implicit in the proof of 2.1.4, but a little warm-up with ournotation is perhaps helpful.

Lemma 3.2.2. Let ρ : ΓF → G∨(Qℓ) be a geometric representation. Then there exists a liftρ : ΓF → G∨(Qℓ) such that, for all v|ℓ and all ι : Qℓ → CFv, the Sen operatorΘρ,ι pairs integrallywith all of X•(Z∨) X•(G∨).

Proof. It suffices to find a lift whose composition with all elements ofX•(Z∨) is Hodge-Tate.We use the bases of the various character groups specified above. In particular, composing aninitial lift ρ with the variousw∗i ∈ X•(G∨), i = 1, . . . , r, we can write

w∗i ρ = χdii χi,0 : ΓF → Q

×ℓ ,

where theχi andχi,0 are Galois characters withχi,0 finite-order. Then we consider the new lift

ρ′ = ρ(r∑

i=1

(diwi) χ−1i ),

which has the advantage thatw∗i ρ′ = (w∗i ρ) · χ−dii is finite-order for alli = 1, . . . , r. Moreover,

for the charactersw∗r+ j, j = 1, . . . , s, namely, the sub-module Hom(X•(ZG),Z) ⊂ X•(Z∨), the

compositionsw∗r+ j ρ′ are all geometric, sinceρ is. Thereforeα ρ′ is geometric for allα ∈ X•(Z∨),as desired.

Returning to equation (1) on page 67, we see that the obstruction to geometric lifts then comesfrom Ext1(X•(ZG)tor,Z). For any weightλ ∈ X•(T∨), there is a positive integerd ∈ Z such thatdλ ∈ X•(T∨) ⊕ X•(Z∨), so with aρ as produced by Lemma 3.2.2,Θρ,ι pairs integrally withdλ forall ι. We obtain a well-defined class, independent of the choice ofρ as constructed in the proof ofLemma 3.2.2 (namely, with the compositionsw∗i ρ finite-order fori = 1, . . . , r),

〈λ,Θρ,ι〉 ∈ Q/Z,which can also be interpreted as the common value moduloZ of the eigenvalues ofΘrλρ,ι =

Lie(rλ) Θρ,ι; here, and in what follows, we denote byrλ the irreducible representation ofG∨

associated to the (dominant) weightλ ∈ X•(T∨). This pairing factors through a map

Ext1(X•(ZG),Z)→ Q/Z,68

and since for any finite abelian groupA, the long exact sequence associated to 0→ Z → Q →Q/Z→ 0 yields an isomorphism

Hom(A,Q/Z)∼−→ Ext1(A,Z),

we can make the following definition:

Definition 3.2.3. Letθρ,ι be the element ofX•(ZG)tor canonically corresponding to the abovemap Ext1(X•(ZG),Z)→ Q/Z.

To make further progress, we need to assume thatρ satisfies certain Hodge-Tate weight sym-metries.

Hypothesis 3.2.4. Let H be a linear algebraic group andρ : ΓF → H(Qℓ) a geometric Ga-lois representation with connected reductive algebraic monodromy group Hρ = (ρ(ΓF))Zar. Weformulate the following Hodge-Tate symmetry hypothesis for such aρ:

• Let r be any irreducible algebraic representation r: Hρ → GL(Vr). Then:(1) For τ : F → Qℓ, the setHTτ(r ρ) depends only onτ0 = τ|Fcm.(2) Writing HTτ0 for this set common to allτ aboveτ0, there exists an integer w such

that13

HTτ0c(r ρ) = w− HTτ0(r ρ),for c the unique complex conjugation on Fcm.

For ρ whose algebraic monodromy group is reductive but not necessarily connected, the corre-sponding hypothesis is simply that some finite restriction (with connected monodromy group) ofρsatisfies the above.

In particular, we note for later use that the lowest weight inHTτ0c(r ρ) is w minus the highestweight in HTτ0(r ρ). We will see that to establish lifting results for geometric representationsρ : ΓF → G∨(Qℓ), we will only need to know Hypothesis 3.2.4 for some easily identifiable finitecollection of compositionsrλ ρ, λ ∈ X•(T∨), but we do not make that explicit here. Most impor-tant, Hypothesis 3.2.4 should in fact be no additional restriction onρ, because of the following con-jecture, which would follow from various versions of the conjectural Fontaine-Mazur-Langlandscorrespondence and Tate conjecture (Conjecture 1.2.1):

Conjecture 3.2.5. Let F be a number field, and letρ : ΓF → GLN(Qℓ) be an irreducible geo-metric Galois representation. Then:

(1) For τ : F → Qℓ, the setHTτ(ρ) depends only onτ0 = τ|Fcm. (This will still hold if ρ isgeometric but reducible.)

(2) Writing HTτ0 for this set common to allτ aboveτ0, there exists an integer w such that

HTτ0c(ρ) = w− HTτ0(ρ),

for c the unique complex conjugation on Fcm.

Unfortunately, for an abstract Galois representation, this conjecture will be extremely diffi-cult to establish. The next lemma explains it in the automorphic case; for a motivic variant, seeCorollary 4.1.26.14

13Interpreted in the obvious way.14But note for now that in the most basic motivic cases, where the Galois representation is given byH j(XF ,Qℓ)

for some smooth projective varietyX/F, the Hodge-Tate symmetries are immediate (even when this representation is

69

Lemma 3.2.6. Supposeρ : ΓF → GLN(Qℓ) is an irreducible geometric representation. Ifρ isautomorphic in the sense of Conjecture 1.2.1, corresponding to a cuspidal automorphic represen-tation π of GLN(AF), and if we assume that Proposition 2.4.7 is unconditional for π (i.e., admitHypothesis 2.4.5), then Conjecture 3.2.5 holds forρ.

Proof. This is immediate from the passage between infinity-types and Hodge-Tate weights,Conjecture 2.4.7, and Clozel’s archimedean purity lemma (which was proven as part of Proposition2.5.8).

We can now understand when geometric lifts ought to exist; the proof proceeds by reductionto the following key case:

Proposition 3.2.7. Let F be a totally imaginary field, and letρ : ΓF → G∨(Qℓ) be a geometricrepresentation with algebraic monodromy group equal to thewhole of G∨. Assumeρ satisfiesHypothesis 3.2.4. Thenρ admits a geometric liftρ : ΓF → G∨(Qℓ).

Proof. Choose a lift ˜ρ as supplied by Lemma 3.2.2. Recall that our weight-bookkeeping isdone within the diagram

0

X•(Z∨) // X•(S∨)

OO

// Ext1(X•(ZG),Z) // 0

X•(T∨)

OO

X•(T∨)

OO

0

OO

We have the elementsν∗i ∈ X•(S∨) dual todiwi ∈ X•(S). Their images in Ext1(X•(ZG),Z) form abasis. Letλi be a (dominant weight) lift toX•(T∨) that also satisfies〈λi , d jwj〉 = δi j . Note that thevalue〈Θρ,ι, λi〉 ∈ Q/Z does not depend on the choice of lift ofν∗i , and it clearly lies in1

diZ/Z, so we

write it in the form kι,idi+ Z for an integerkι,i. By considering the geometric representationrdiλi ρ,

we find thatΘrdiλi ρ,ι has eigenvalues that depend only onτι (by Lemma 2.2.6; see that lemma forthe notationτι as well), and thus we can writekτ,i in place ofkι,i. (Equivalently, we can work withthe elementsθρ,τ ∈ X•(ZG)tor.) These classeskτ,idi

modZ serve as both highest and lowestτ-labeledHodge-Tate weights (moduloZ) for rλi ρ; we deduce that, modulodiZ, the highest and lowestτ-labeled weights ofrdiλi ρ are both congruent tokτ,i

15.

reducible) from theℓ-adic comparison isomorphism of [Fal89]; if j is even, the symmetries similarly hold for primitivecohomology. What is not obvious is that if this Galois representation decomposes, that the irreducible factors all satisfythe conjecture.

15Note that all eigenvalues ofΘrλi ρ are congruent moduloZ; this does not imply that all elements of HTτ(rdiλi ρ)are congruent modulodiZ, but this congruence does hold for the highest and lowest weights.

70

Now, the geometric Galois representations (rdiλi ) ρ for i = 1, . . . , r are irreducible, because wehave assumed thatG∨ is the monodromy group ofρ. Applying Hypothesis 3.2.4, we deduce thatkτ depends only onτ0 = τ|Fcm, along with the symmetry relation

kτ0,i + kτ0c,i ≡ wi mod di

for some integerwi (for all τ0 : Fcm → Qℓ).This relation allows us, by Lemma 2.3.16, to find Galois charactersψi : ΓF → Q

×ℓ with HTτ(ψi) ∈

kτ,idi+ Z for all τ. We then form the twist ˜ρ′ := ρ(

∑i(diwi) ψ−1

i ); recall that〈diwi , λ j〉 = δi j . Thisnew lift is then geometric:

〈Θρ′,ι, λi〉 = 〈Θρ,ι, λi〉 + 〈Θ∑(djwj )ψ−1

j ,ι, λi〉 ≡

kτι,idi− kτι,i

di≡ 0 modZ.

(Recall it suffices to check ˜ρ′ is Hodge-Tate, by Conrad’s result, quoted here as Theorem 2.1.6.)

Corollary 3.2.8. Let ρ : ΓF → G∨(Qℓ) be geometric with algebraic monodromy group G∨;maintain the notation of the previous proof, but now supposeF is totally real. Thenρ has ageometric lift if and only if for varyingι, the elementsθρ,ι (see Definition 3.2.3) in X•(ZG)tor areindependent ofι.

Remark 3.2.9. More concretely, to determine whetherρ has a geometric lift, apply the follow-ing criterion:

(1) Ignore anyi for whichdi is odd; these do not obstruct geometric lifting;(2) Then for fixedi the integerskτ,i mod di are all wi

2 translated by a two-torsion class inZ/diZ;

(3) ρ has a geometric lift if and only if each of these classeskτ,i (or, equivalently, the associatedtwo-torsion class) is independent ofτ.

Proof. By the previous proof and Lemma 2.3.17,ρ has a geometric lift if and only if the classeskτ,i mod di (fixed i, varying τ) are independent ofτ. The weight-symmetry relation becomes2kτ,i ≡ wi mod di, and the claim follows easily.

Over totally imaginary fields, we can now reduce the general lifting problem to the special caseof full monodromy:

Theorem 3.2.10.Let F be totally imaginary, and letπ : H′ ։ H be any surjection of linearalgebraic groups with central torus kernel. Supposeρ : ΓF → H(Qℓ) is a geometric representation,with arbitrary image, satisfying Hypothesis 3.2.4. Thenρ admits a geometric liftρ : ΓF → H′(Qℓ).

Proof. We may assumeH′ andH are reductive sinceπ induces an isomorphism on unipotentparts (we take this observation from Conrad, who has exploited this reduction in the arguments of[Con11]).

We next show that the theorem holds ifHρ := ρ(ΓF)Zar ⊂ H is connected. In that case, letH′ρ be

the preimage inH′ of Hρ. ThenH′ρ → Hρ is a surjection of connected reductive groups with central

torus kernel, and we may writeHρ = G∨, H′ρ = G∨ whereG ⊂ G is an inclusion of connected

reductive groups of the formG = (G × Z)/ZG for some inclusion ofZG into a multiplicative groupZ. If Z is not connected, Proposition 3.2.7 does not immediately apply, so we embedZ into atorus Z, with corresponding inclusionsG ⊂ G ⊂ G. Dually, G∨ ։ G∨ is a quotient to which

71

we can apply Proposition 3.2.7, and then projecting fromG∨ to G∨, we obtain a geometric liftρ : ΓF → H′ρ(Qℓ) ⊂ H′(Qℓ) of ρ.

For arbitraryρ (i.e. Hρ not necessarily connected), letF′/F be a finite extension such that

ρ(ΓF′)Zar= H0

ρ is connected. OverF′, we can therefore find a geometric lift ˜ρF′ : ΓF′ → H′ρ(Qℓ) ⊂H′(Qℓ), lettingH′ρ as before denote the preimage ofHρ in H′. Thus, forany lift ρ0 : ΓF′ → H′(Qℓ)

of ρ|ΓF′ , there exists a characterψ : ΓF′ → S∨(Qℓ) such that ˜ρ0 · ψ is geometric. In particular,letting ρ : ΓF → H′(Qℓ) be any lift overF itself (with rational Hodge-Tate-Sen weights), there isa ψF′ : ΓF′ → S∨(Qℓ) such that ˜ρ|ΓF′ · ψF′ is geometric. But by Corollary 2.3.16, there is a Galoischaracterψ : ΓF → S∨(Qℓ) whose labeled Hodge-Tate-Sen weights descend those ofψF′ . It followsimmediately that ˜ρ · ψ : ΓF → H′(Qℓ) is a geometric lift ofρ.

Remark 3.2.11. (1) The theorem also lets us make explicit preciselywhich sets of labeledHodge-Tate weights can be achieved in a geometric lift ˜ρ. We will exploit this in§4.2.

(2) We will not treat the general totally real case here, since a somewhat different approachseems more convenient in that case. For a succinct, coordinate-free treatment of the totallyreal case, and another perspective on the arguments of this section, see [Pat13], where thefollowing general result is obtained: for anyH′ ։ H as in our lifting setup, we writeH0 = G∨ and (H′)0 = G∨, whereG = (G × Z)/ZG; hereZ is not necessarily connected.We can define in this generality elements

θρ,τ ∈ coker(X•(Z)tor → X•(ZG)tor

),

and ρ admits a geometric lift toH′ if and only if the θρ,τ are independent ofτ. Theargument described in [Pat13] has the disadvantage of not making explicit the parityobstruction (assuming ‘Hodge symmetry’) found in Proposition 3.2.8; for this reason wehave retained the two different expositions of the totally real case.

The method of proof of Proposition 3.2.7 and Theorem 3.2.10 also implies the following localresult, which emerged from a conversation with Brian Conrad. A proof of a more difficult result,whereFv is replaced by ap-adic field with algebraically closed residue field, will appear in§3.6of [CCO12].

Corollary 3.2.12. Let H′ ։ H be a central torus quotient, and letρ : ΓFv → H(Qℓ) be aHodge-Tate representation ofΓFv, for Fv/Qℓ finite. Then there exists a Hodge-Tate liftρ : ΓFv →H′(Qℓ).

Proof. We sketch a proof, replacing appeal to Hypothesis 3.2.4 andthe existence of certainHecke characters by the simpler observation that local class field theory lets us find the necessarytwisting charactersψ : ΓFv → Q

×ℓ by hand. That is,OFv sits in Iab

Fvin finite index, and on the former

we can define the characterx 7→

∏

τ : K→Qℓ

τ(x)kτ

for any integerskτ; then up to a finite-order character we take adth root for an integerd to build(having extended to all ofΓFv) characters with any prescribed set of rationalτ-labeled Hodge-Tateweights (τ running over allFv → Qℓ). This observation suffices (invoking Lemma 2.2.6 in the fullgenerality of Hodge-Tate, rather than merely de Rham, representations) for the previous argumentsto carry through.

72

This Corollary combined with [Con11, Proposition 6.5] implies the following stronger result:

Corollary 3.2.13. Let H′ ։ H be a central torus quotient, and letρ : ΓFv → H(Qℓ) be arepresentation ofΓFv, for Fv/Qℓ finite, satisfying a basic p-adic Hodge theory property16 P. Thenthere exists a liftρ : ΓFv → H′(Qℓ) also satisfyingP.

3.3. Applications: comparing the automorphic and Galois formalisms

In §3.2, we took the Fontaine-Mazur-Langlands conjecture (or rather its weakened form Hy-pothesis 3.2.4) relating geometric Galois representations ΓF → GLn(Qℓ) to L-algebraic automor-phic representations of GLn(AF) as input to establish some of our lifting results. Now we want toapply these lifting results to give some evidence for the relationship between automorphic formsand Galois representations on groups other than GLn. We will touch on the Buzzard-Gee conjec-ture, certain cases of the converse problem, and some general thoughts about comparing descentproblems on the (ℓ-adic) Galois and automorphic sides. We first digress to discuss what ‘automor-phy’ of anLG(Qℓ)-valued representation even means.

3.3.1. Notions of automorphy.As usual we have fixedι∞ : Q → C and ιℓ : Q → Qℓ. It issometimes more convenient simply to fix an isomorphismιℓ,∞ : C → Qℓ, and to regardQ as thesubfield of algebraic numbers inC. G is a connected reductiveF-group, and we take aQ-formof the L-groupLG. For an automorphic representationπ of G(AF) andρ : ΓF → LG(Qℓ), alwaysassumed continuous and composing withΓF-projection to the identity, there are (at least) fourrelations between suchρ andπ that might be helpful, of which only the first two can be statedunconditionally. First we describe analogues that restrict to either the automorphic or Galois side,borrowing some ideas from [Lap99] (and, inevitably, [LL79 ]). For automorphic representationsπandπ′ of G(AF), three notions of similarity are:

• π ∼w π′ if almost everywhere locally, the (unramified, say)L-parameters areG∨(C)-conjugate.• π ∼w,∞ π′ if almost everywhere locally, and at infinity, theL-parameters are conjugate.

This also makes sense unconditionally, since archimedean local Langlands is known.• π ∼ew π′ if everywhere locally, theL-parameters areG∨(C)-conjugate; this only makes

sense if one knows the local Langlands conjecture forG.• π ∼s π

′ is the most fanciful condition: if the conjectural automorphic Langlands groupLF

exists, soπ andπ′ give rise to representationsLF → LG(C), then this condition requiresthese representations to be globallyG∨(C)-conjugate.

A fifth notion would compareL-parameters in a particular finite-dimensional representation of G∨.We can unconditionally make the same sort of comparisons betweenℓ-adic Galois representa-

tions, writingρ ∼w ρ′, ρ ∼w,∞ ρ

′, ρ ∼ew ρ′, andρ ∼s ρ

′. By equivalence ‘at infinity’ here, we meanthat at real places the actions of complex conjugation are conjugate, and at places aboveℓ, theassociated Sen operators (i.e. labeled Hodge-Tate data) are conjugate. Since it is conjectured, buttotally out of reach, that frobenius elements act semi-simply in a geometric Galois representation,we should only compare ‘frobenius semi-simplifications’ inthese definitions of local equivalence.For some nice examples, Lapid’s paper ([Lap99]) studies the difference between∼w,∼ew, and∼s

16Namely: crystalline, semi-stable, de Rham, or Hodge-Tate.

73

for certain Artin representations (and, when possible, thecorresponding comparison on the auto-morphic side).

We then have corresponding ways to relate anℓ-adicρ and an automorphicπ:• ρ ∼w π

• ρ ∼w,∞ π• ρ ∼ew π

• ρ ∼s π

First, write ρ ∼w π if for almost all unramifiedv (for ρ and π), ρ|ssWFv

is G∨(Qℓ)-conjugate to

recv(πv) : WFv → G∨(C)⋊ΓF; implicit is the assumption that the local parameter lands inG∨(Q)⋊ΓF,so that we can applyιℓ ι−1

∞ . Note that this definition does not distinguish betweenπ and otherelements of its (conjectural) globalL-packetL(π). The other relations are straightforward modifi-cations (for compatibility with complex conjugation, we take the condition in Conjecture 3.2.1 of[BG11]), except forρ ∼s π. WritingGF,E(σ) for the motivic Galois group for motives17overF withE-coefficients, using a Betti realization viaσ : F → C, one might hope that after fixingE → Cas well, one would obtain a map of pro-reductive groups overC, LF → (GF,E)(σ) ⊗E C (compareRemark 3.1.16). Ifπ corresponds to a representation rec(π) of LF, andρ arises from (completingat some finite place ofE) a representationρE of GF,E(σ), it makes sense to ask whether rec(π) fac-tors throughGF,E(σ)(C), and whether the resulting representation is globallyG∨(C)-conjugate to(the complexification viaE → C of) ρE. Of course, any discussion of the automorphic Langlandsgroup and its relation with the motivic Galois group is pure speculation; but these heuristics doprovide context for the basic problems raised in Question 1.1.11 and Conjecture 1.1.13, as well asthe work of§4.2.

Although we don’t actually require it, it is helpful to keep in mind a basic lemma of Steinberg,which implies that∼w can be checked by checking in all finite-dimensional representations:

Lemma 3.3.1. Let x and y be two semi-simple elements of a connected reductive group G∨ overan algebraically closed field of characteristic zero. If x and y are conjugate in every (irreducible)representation of G∨, or even merely have the same trace, then they are in fact conjugate in G∨.

Proof. For semi-simple groups, this is Corollary 3 (to Theorem 2) in Chapter 3 of [Ste74];the proof extends to the reductive case (and even more generally, see Proposition 6.7 of [Bor79]).The key point is that the characters of finite-dimensional representations ofG∨ restrict to a basis ofthe ring of Weyl-invariant regular functions on a maximal torus; these in turn separate conjugacyclasses in the torus.

3.3.2. Automorphy of projective representations.

Throughout this section,we assume the Fontaine-Mazur-Langlands conjecture on automorphyof geometric(GLN-valued)Galois representations. It suffices to take a version that matches unram-ified (almost everywhere) and Hodge-theoretic parameters;to be precise, assume Part 3 of Conjec-ture 1.2.1, and note that this includes the requirement thatcuspidality is equivalent to irreducibilityunder the automorphic-Galois correspondence. We will showhow our lifting results– both au-tomorphic and Galois-theoretic– give rise to a ‘Fontaine-Mazur-Langlands’-type correspondence

17Either for absolutely Hodge cycles, for motivated cycles, or, assuming the standard conjectures, for homologicalcycles. Again, we will deal more precisely with motivic Galois groups in§4.1.

74

between algebraic automorphic representations of SLN(AF) and PGLN(Qℓ)-valued geometricΓF-representations.18 The starting point is the following consequence of the results of§3.2:

Corollary 3.3.2.Let F be totally imaginary. Then any geometricρ : ΓF → PGLn(Qℓ) is weaklyautomorphic, i.e. there exists an L-algebraic automorphicrepresentationπ of SLn(AF) such thatρ ∼w,∞ π (or ρ ∼ew π, if we assume a form of Fontaine-Langlands-Mazur that matches local factorseverywhere).

Proof. We have seen thatρ lifts to a geometric ˜ρ : ΓF → GLn(Qℓ), which by assumption isautomorphic, corresponding to some ˜π on GLn/F. The irreducible constituents of ˜π|SLn(AF ) form aglobalL-packet whose local unramified parameters correspond to those ofρ.

We now give descent arguments that extend this automorphy result toF totally real. First weneed a couple of elementary lemmas.

Lemma 3.3.3. Let L/F be a cyclic, degree d, extension of number fields, withσ a generator ofGal(L/F). Let χ be a Hecke character of L, and letδ be any Hecke character whose restrictionto CF ⊂ CL is δ = δL/F, a fixed order d character that cuts out the extension L/F. Assume thatχ1+σ+...+σd−1

= 1. Then for a unique integer i= 0, . . . , d − 1, χδi is of the formψσ−1 for a Heckecharacterψ of L.

Proof. We may assumeχ is unitary. WriteCD as usual for the Pontryagin dual of a locallycompact abelian groupC. We have the following exact sequences:

1

Gal(L/F)

OO

1 // CF

OO

// CLσ−1

// CL

CL

NL/F

OO

,

dualizing to

1

Gal(L/F)D

1 // CDF

NL/F

oo CDL

resoo CD

Lσ−1oo

CDL .

18For some unconditional results in this direction, see [Pat13, §4].

75

By assumption,NL/F res(χ) = 1, sores(χ) = δ−i ∈ Gal(L/F)D for some integeri, unique modulod. Thenres(χδi) = 1, and we are done by exactness of the horizontal diagram.

We need a special case of anℓ-adic analogue of the remark after StatementA of [LR98];19 thatremark is in turn the (much easier) analogue, for complex representations of the Weil group, of themain result of their paper. We first record the simple case that we need, and then out of independentinterest we prove a generalℓ-adic analogue of the Lapid-Rogawski result.

Lemma 3.3.4. Let L/F be a quadratic CM extension of a totally real field F, withσ ∈ ΓF

generatingGal(L/F) Supposeψ : ΓL → Q×ℓ is a Galois character such thatψ1−σ is geometric.

Then there exists a unitary, type A Hecke characterψ of L such thatψ1−σ is the type A0 Heckecharacter associated toψ1−σ.

Proof. Write (ψ1−σ)A for the Hecke character associated toψ1−σ. By the previous lemma, itsuffices to check that (ψ1−σ)A is trivial on CF ⊂ CL. If a finite placev of L is split over a placevF of F, and unramified forψ, then for a uniformizer v of FvF (embedded into theLv andLσv

components ofA×L),

(ψ1−σ)A(v, v) = ψ1−σ( f rv)ψ

1−σ( f rσv) =ψ( f rv)

ψ( f rσv)· ψ( f rσv)

ψ( f rv)= 1.

Similarly for v inert, (ψ1−σ)A(v) =ψ( f rv)

ψ(σ f rvσ−1)= 1. The Hecke character (ψ1−σ)A |CF is therefore

trivial. To see that we may chooseψ to be unitary and typeA, we invoke Corollary 2.3.9: decom-posingψ as in that result, both the| · |w and ‘Maass’ components descend to the totally real subfieldF, so dividing out by them yields a newψ, now unitary typeA, and withψ1−σ unchanged.

Lemma 3.3.5.Let L/F be cyclic of degree d, withσ ∈ ΓF restricting to a generator ofGal(L/F).Supposeρ : ΓL → GLn(Qℓ) is an irreducible continuous representation satisfyingρσ ρ · χ forsome characterχ : ΓL → Q

×ℓ . Suppose further thatχ is geometric (in particular, this holds ifρ is

geometric), necessarily of weight zero, so we may regard it as a characterχA =∏

w∈|L| χw : CL →C×. Then the (finite-order) restriction ofχA to CF ⊂ CL cannot factor through a non-trivialcharacter ofGal(L/F).

Proof. Iterating the relationρσ ρ · χ, we obtainρ ρ · χ1+σ+...+σd−1, so thatχ1+σ+...+σd−1

is finite-order. Each of the charactersχσi

has some common weightw, so d · w = 0, and thusw = 0. Moreover, writing as usualpιw for the algebraic parameter ofχw (with respect to a choiceιw : L → C representing the placew), we have

∑pιw = 0 asιw ranges over a Gal(L/F)-orbit of such

embeddings. IfL has a real embedding, thenχ has finite-order, and the passage fromχ : ΓabL → Q

×ℓ

to χA : CL → C× is simply via the reciprocity mapCL ։ ΓabL .20 In particular,χ(x) = χA(x′) for any

representativex′ in CL of the image ofx in ΓabL . If on the other handL is totally imaginary, then

continuing to writex′ = (x′w)w∈|L|, we have

χ(x) =∏

w∤ℓ∞χw(x′w)

∏

w|ℓ

χw(x′w)∏

τ : Lw→Qℓ

τ(x′w)pι∗∞,ℓ (τ)

.

19In the proof of Corollary 3.3.6 below, we could replace appeal to this lemma by simply citing StatementB of[LR98].

20Throughout this argument we implicitly use our fixed embeddingsιℓ, ι∞, but omit any reference to them fornotational simplicity.

76

If we further assume that the representativex′ can be chosen inCF ⊂ CL, with elementsx′v ∈ F×vgiving rise to allx′w for w|v, then we can rewrite

∏

w|ℓ

∏

τ : Lw→Qℓ

τ(x′w)pι∗∞,ℓ (τ)

as ∏

v|ℓ

∏

τ : Fv→Qℓ

∏

w|v,τ|ττ(x′v) =

∏

v|ℓ

∏

τ : Fv→Qℓ

τ(x′v)∑

w|v,τ|τ pι∗∞,ℓ (τ)= 1.

The last equality follows sinceχ1+σ+...+σd−1is finite-order, and we are summingpι∗∞,ℓ(τ) over a full

Gal(L/F)-orbit. A similar argument shows that∏

w|∞χw(x′w) = 1.

We conclude that in this case (x′ ∈ CF), χ(x) can be computed simply asχA(x′).Let V be the space on whichρ acts. Then the isomorphismρσ ρ · χ yields an operator

A ∈ Aut(V) satisfyingAρσ = ρ · χA. Fix g ∈ ΓL, and for anyx ∈ ΓL we compute

tr(ρ(g)A) = tr(ρσ(x)ρ(g)Aρσ(x)−1)

= tr(ρσ(x)ρ(g)ρ(x−1)χ(x−1)A)

= χ(x−1) tr(ρ(σxgx−1)A).

(Hereσx = σxσ−1.) So, if we can find anx ∈ ΓL such thatσxgx−1 = g andχ(x−1) , 1, then wewill have tr(ρ(g)A) = 0. Doing this for allg ∈ ΓL, we see that by Schur’s Lemmaρ cannot beirreducible, elseA = 0. Now,y = g−1σ ∈ ΓF satisfiesσygy−1 = g, sox = yd ∈ ΓL does as well. Itsuffices to show that ifχ|CF cuts out the extensionL/F, thenχ(x) , 1. In fact, the image ofx inΓab

L lies in the image of the transfer Ver :ΓabF → Γab

L . Explicitly,

Ver(y) =d−1∏

i=0

σi(g−1σ)φ(σig−1σ)−1,

whereφ : ΓF → σii=0,...,d−1 records the representative of theΓL-coset of an element ofΓF . It isthen easily seen21 that

Ver(y) = (g−1σ)d = x,

so by class field theoryx ∈ ΓabL is represented by an elementx′ of CF ⊂ CL under the reciprocity

map recL. This element is a generator ofCF/NL/FCL, sincey lifts a generator of Gal(L/F), andthusχ(x) = χA(x′) , 1 if χA |CF factors through a non-trivial character of Gal(L/F).

Finally we can (conditionally) prove automorphy of geometric projective representations overtotally real fields.

Corollary 3.3.6.Let F be totally real. Continue to assume Part 3 of Conjecture1.2.1 (Fontaine-Mazur-Langlands). Then for any geometricρ : ΓF → PGLn(Qℓ), there exists an L-algebraicπ onSLn/F such thatρ ∼w π.

21Note thatφ(σd−1g−1σ) = 1, while otherwiseφ(σig−1σ) = σi+1.

77

Proof. We will first treat the case ofρ having irreducible lifts to GLn(Qℓ). Choose a lift ˜ρ withfinite-order determinant, a CM quadratic extensionL/F, and, by Theorem 3.2.7, a Galois characterψ : ΓL → Q

×ℓ such that ˜ρ|ΓL · ψ−1 is geometric. Let ˜π be the cuspidal automorphic representation of

GLn(AL) corresponding to this geometric twist. Writeσ for the nontrivial element of Gal(L/F),so thatπσ π · χ whereχ is the Hecke character corresponding to the geometric Galois characterψ1−σ. Appealing either to Lemma 3.3.4 or Lemma 3.3.5, we can writeχ = ψ1−σ for a unitarytype A Hecke characterψ. Thenπ · ψ is σ-invariant, and by cyclic (prime degree) descent, thereis a cuspidal representationπ of GLn(AF) whose base-change is ˜π · ψ. We want to compare theprojectivization of the unramified parameters ofπ with the unramified restrictionsρ|ΓFv

.To do so, we repeat the argument but instead with infinitely many (disjoint) quadratic CM

extensionsLi/F, showing that in all cases the descentπ to GLn(AF) gives anL-packet for SLn(AF)that is independent of the fieldLi. ρ is still a fixed lift with finite-order determinant, and we canwrite, for eachτ : F → Qℓ, HTτ(ρ) ∈ kτ

n + Z for some integerkτ, which we fix (rather than justits congruence class modn). For some integerw, we have the purity relation 2kτ ≡ w mod n,as follows, for instance, from (geometric) liftability after a CM base-change;22 as with kτ, wefix an actual integerw, not just the congruence class. Now, for each suchτ, let ι : F → C bethe archimedean embedding associated viaι∞, ιℓ (elsewhere denotedι∗∞,ℓ(τ)). For eachLi, fix an

embeddingτ(i) : Li → Qℓ extendingτ, so that the other extension isτ(i)c. Likewise writeι(i) andι(i) c = ι(i) for the corresponding complex embeddings. We can then construct Galois charactersψi : ΓLi → Q

×ℓ such that

HTτ(i)(ψi) =kτn

HTτ(i)c(ψi) =w− kτ

n,

such that ˜ρ|ΓLi· ψ−1

i is geometric, corresponding to anL-algebraic cuspidal ˜πi on GLn/Li. Asbefore, we find a Hecke characterψi of Li such thatψ1−σi

i is the typeA0 Hecke character associatedto ψ1−σi

i ; here we writeσi for the non-trivial element of Gal(Li/F), but of course all theσi are justinduced by complex conjugation. Again, for alli we find cuspidal automorphic representationsπi

of GLn(AF) such that BCLi/F(πi) = πi · ψi. Restricting to compositesLiL j , we have the comparison

ρ|LiL j ψ−1i |Li L j ·

(ψi

ψ j

|Li L j

)= ρ|LiL j ψ

−1j |Li L j ,

and thus

BCLi L j/Li (πiψi) · BCLi L j

(ψ j

ψi

)· BCLi L j

(ψi

ψ j

)= BCLi L j/L j (π jψ j),

23

so finally

BCLiL j (πi) · BCLi L j

(ψ j

ψi· ψi

ψ j

)= BCLi L j (π j).

22After such a base-changeL/F, the characterψ twisting ρ to a geometric representation will have Hodge-Tate-Sen weights congruent tokτn ∈ Q/Z at both embeddingsL → Qℓ aboveτ; the integerw is then the weight of the Heckecharacter associated toψn.

23Here ψi

ψ jrestricted toΓLi L j is geometric, so we abusively write this for the associated Hecke character as well.

78

If the characterψ j

ψi· ψi

ψ jis finite-order– in the next paragraph, we check that we may assume

this– it cuts out a cyclic extensionL′/LiL j, and we have BCL′(πi) = BCL′(π j). L′/F is solvable,however, so the characterization of the fibers of solvable base-change in [Raj02] implies thatπi

andπ j are twist-equivalent, hence thatπi |SLn(AF ) andπ j |SLn(AF ) define the sameL-packet of SLn(AF).Let us denote byπ0 any representative of this globalL-packet. Now consider placesv of F thatare split in a givenLi/F. The semi-simple partρ( f rv)ss is equal (in PGLn(Qℓ)) to (ρψ−1( f rw))ss

for any w|v, and this is conjugate in GLn(Qℓ) to ιℓ,∞ (recw(πw)( f rw)), whose projectivization liesin the same PGLn(Qℓ)-conjugacy class asιℓ,∞

(recv(π0,v)( f rv)

). This verifies that for all suchv,

ρ( f rv)ss is PGLn(Qℓ)-conjugate toιℓ,∞(recv(π0,v)( f rv)

). Varying Li/F, and remembering thatπ0

is independent of this variation, we get the same result for all v split in any single quadratic CMextensionLi/F (we have to throw out a finite number of suchLi to ensure our representationsremain cuspidal/irreducible), we conclude thatρ ∼w π.

To finish the proof, we must check thatψ j

ψi· ψi

ψ jmay indeed be assumed finite-order. First,

recall that eachψi may be taken unitary and typeA; in this case, the infinity-type is determinedby the relationψ1−σi

i = ψ1−σii . Explicitly (using the above notation for the various embeddings),

ψin

corresponds to a Hecke character ofLi with infinity-type (where we abusively denoteι(i)(z) bysimplyz)

recι(i)(ψin) : z 7→ zkτ zw−kτ ,

so

recι(i)(ψ1−σii ) : z 7→ z

2kτ−wn z

w−2kτn .

(Recall that 2kτ ≡ w mod n.) We then have, under our assumptions,

recι(i)(ψi) : z 7→(

z|z|

) 2kτ−wn

.

Of course, recι(i)c is the same but withw−2kτn in the exponent. To make the parameter comparison

after restriction to a compositeLiL j, we use the following notation for embeddings ofLiL j intoQℓandC, lying above the givenτ andι:

τ1 extendsτ(i) andτ( j),

τ2 extendsτ(i) andτ( j) c,

ι1 extendsι(i) andι( j),

ι2 extendsι(i) andι( j) c.

We then have the conjugate embeddingsτ1 c, etc. Computing theτk-labeled weights ofψi

ψ j,

and translating them to the infinity-type at the place corresponding toιk, with ιk as the chosenembeddingLiL j → C, we then find

recι1

(ψi

ψ j

): z 7→ 1

recι2

(ψi

ψ j

): z 7→ (z/z)

2kτ−wn ,

79

whereas

recι1

(ψ j

ψi

): z 7→ 1

recι2

(ψ j

ψi

): z 7→ (z/z)

w−2kτn .

We conclude thatψ j

ψi· ψi

ψ jis, with our normalization of theψi , in fact finite-order, and the proposition

follows.Finally, we quickly treat the case of generalρ, having reducible lifts. If ˜ρ as above decomposes

ρ = ⊕mi=1ρi, say withρi of dimensionni, geometricity ofρ implies that over any CML/F the same

Galois characterψ twists ρi, for all i, to a geometric representation. We can therefore use, for alli, the same Hecke characterψ such thatψ1−σ = ψ1−σ. As above, we invoke automorphy of ˜ρiψ

−1,and, twisting byψ, descend to a cuspidal automorphic representationΠi of GLni/F. The samelocal check (forv split in L/F) as above, but now crucially relying on the fact thatψ andψ wereindependent ofi, showsρ( f rv) is PGLn(Qℓ)-conjugate toι

(recv(⊞m

i=1Πi,v)( f rv)).

By a similar argument, we can ‘construct’ the Galois representations (assuming of course theGLN correspondence) associated to (tempered)L-algebraicπ on SLn/F for F CM (or imaginary,assuming Conjecture 2.4.7) or totally real. By Proposition3.1.12, we are reduced to the case ofFtotally real.

Proposition 3.3.7. Continue to assume Fontaine-Mazur-Langlands. Let F be a totally realfield, and letπ be an L-algebraic cuspidal automorphic representation ofSLn(AF). Assume thatπ∞ is tempered. Then there exists a (not necessarily unique) projective representationρ : ΓF →PGLn(Qℓ) satisfyingρ ∼w π.

Proof. By Proposition 3.1.14, there exists aW-algebraic cuspidal (and tempered at∞) π onGLn/F lifting π. For all but finitely many quadratic CML/F, we can find a typeA Hecke characterψ such that BCL/F(π) ·ψ is L-algebraic and cuspidal on GLn/L, hence corresponds to an irreduciblegeometric representation ˜ρL : ΓL → GLn(Qℓ). Conjugating, we find ˜ρσL ≡ ρL · (ψσ−1), the twistbeing by the geometric character associated to the typeA0 Hecke characterψσ−1. We wish towrite χ = (ψσ−1) in the formψσ−1 for some Galois characterψ : ΓL → Q

×ℓ (reversing the process

in Corollary 3.3.6). Once this is managed, we have ( ˜ρL · ψ−1) is σ-invariant, hence descendsto ρ : ΓF → PGLn(Qℓ). Arguing as in Corollary 3.3.6, we find that this projectivedescent isindependent ofL/F, and, again as in that proof, by varyingL/F we obtain the compatibilityρ ∼w π.

To constructψ, we take as first approximation a Galois characterψ such thatψ2 equals(ψ2) upto a finite-order characterχ0; recall thatψ2 is typeA0, so we can attach the Galois character(ψ2).Then

(ψσ−1)2 = (ψ2)σ−1 = ((ψ2)χ0)σ−1 = χ2χσ−1

0 ,

and consequently ˆχ agrees withψσ−1 up to a finite-order character. Twisting ˜ρL, we may thereforeassume that ˆχ has finite-order. It still satisfies ˆχ1+σ = 1, so invoking Lemma 3.3.3 and (a simple

80

case of) Lemma 3.3.5 we find a finite-order Hecke character,24 which may therefore be directlyregarded as a Galois character, that casts ˆχ in the desired form.

Here is another example comparing the Tannakian formalisms:

Proposition 3.3.8. Continue to assume Fontaine-Langlands-Mazur and, in the totally imag-inary but non-CM case, Conjecture 2.4.7. LetΠ be a cuspidal L-algebraic representation ofGLn(AF), and suppose thatρΠ ρ1 ⊗ ρ2, whereρi : ΓF → GLni (Qℓ). Then there exist cuspidalautomorphic representationsπi of GLni (AF) such thatΠ = π1 ⊠ π2.

Remark 3.3.9. As the examples in§2.6 show, sometimes theπi cannot be takenL-algebraic.Nevertheless, by Proposition 2.5.8, they can always be taken W-algebraic.

Proof. First supposeF is totally imaginary. For allv|ℓ, the fact thatρ1|ΓFv⊗ ρ2|ΓFv

is deRham implies that locally theseΓFv-representations are twists of de Rham representations. Tosee this, we apply Corollary 3.2.12 (to find a Hodge-Tate lift) and Theorem 2.1.6 (to find a deRham lift, given that a Hodge-Tate lift exists) to the lifting problem (with central torus kernel)

GLn1 × GLn2

⊠−→ Gn1,n2 ⊂ GLn1n2, whereGn1,n2 denotes the image of the tensor product map. Inparticular, the projectivizations of theρi |ΓFv

are de Rham, so the global projective representationsρi : ΓF → PGLni (Qℓ) are geometric. We know that these geometric projective representations havegeometric lifts, and we may therefore assume our originalρi were in fact geometric. They thencorrespond toL-algebraicπi, and we haveΠ = π1 ⊠ π2.

ForF totally real, we perform a descent similar to previous arguments. Restricting to CML/F,we find a Galois characterψ such thatρ1·ψ−1 andρ2 ·ψ are geometric, corresponding toL-algebraiccuspidalπi on GLni/L. Writing ψ1−σ = ψ1−σ for a Hecke characterψ, we findπ1·ψ andπ2·ψ−1 areσ-invariant, so descend to cuspidal representations ¯πi of GLni/F. Since BCL/F(Π) = BCL/F(π1 ⊠ π2),we deduce thatΠ and π1 ⊠ π2 are twist-equivalent, from which the result follows. (The sameargument applies toF that are neither totally real nor totally imaginary: just replace the restrictionto CM extensionsL/F with restrictions to totally imaginary quadratic extensionsL/F.)

These examples (and, for instance, Corollary 2.7.8) motivate a comparison of the images ofr : LH → LG on the automorphic and Galois sides, whenr is anL-morphism with central kernel.The most optimistic expectation (forH andG quasi-split) is that if ker(r) is a central torus, then thetwo descent problems for (L-algebraic)Π and (geometric)ρΠ are equivalent; whereas if ker(r) isdisconnected, there is an obstruction to the comparison, that nevertheless can be killed after a finitebase-change. IfG is not GLn/F, then one will have to decide whether weak equivalence (π ∼w ρ)suffices to connect the descent problems, or whether some stronger link (the mysteriousπ ∼s ρ)must be postulated.

3.4. Monodromy of abstract Galois representations

In this section we discuss some general results about monodromy of ℓ-adic Galois represen-tations. Much of the richness of this subject comes from its blending of two kinds of representa-tion theories, that of finite groups, and that of connected reductive algebraic groups. We will see

24Writing χ = ψσ−1 where the infinity-components ofψ have the formzpzq, we seezp−qzq−p is the correspondingcomponent of ˆχ, hence thatp = q at each infinite place. Twistingψ by the base-change of a character of the totallyreal fieldF, we can then assume it is finite-order.

81

(Proposition 3.4.1) that the basic lifting result (Proposition 2.1.4) allows us to some extent to un-derstand how these two representation theories interact. In §3.4.2 we develop more refined resultsin the ‘Lie-multiplicity-free’ case (see Definition 3.4.5); this situation encapsulates the essentialdifficulties of independence-of-ℓ questions, such questions being trivial for Artin representations.

3.4.1. A general decomposition.The following result is a simple variant of a result of Katz([Kat87, Proposition 1]), which he proves for lisse sheaves on affine curves over finite fields. Wecan replace Katz’s appeal to the Lefschetz affine theorem by Proposition 2.1.4. Recall that a Galoisrepresentation isLie irreducibleif it is irreducible after restriction to every finite-indexsubgroup(i.e., the connected component or Lie algebra of its algebraic monodromy group acts irreducibly).

Proposition 3.4.1. Let F be any number field, and letρ : ΓF → GLQℓ(V) be an irreduciblerepresentation of dimension n. Then eitherρ is induced, or there exists d|n, a Lie irreduciblerepresentationτ of dimension n/d, and an Artin representationω of dimension d such thatρ τ ⊗ ω. Consequently, any (irreducible)ρ can be written in the form

ρ IndFL (τ ⊗ ω)

for some finite L/F and irreducible representationsτ andω of ΓL, with τ Lie-irreducible andωArtin.

Proof. LetG denote the algebraic monodromy group ofρ, with G0 the connected componentof the identity. Abusively writingρ for the representationG → GL(V), we may assumeρ|G0 isisotypic (elseρ is induced, and we are done). Ifρ|G0 is irreducible, thenρ itself is Lie-irreducible,so again we are done. Therefore, we may assume thatρ|G0 τ⊕d

0 for somed ≥ 2, with τ0 an irre-ducible representation ofG0, and consequently a Lie-irreducible representation ofΓL for anyL/Fsufficiently large thatρ(ΓL) ⊂ G0(Qℓ). Since the irreducibleΓL-representationτ0 is ΓF-invariant,it extends to a projective representation ofΓF . By the basic lifting result (Proposition 2.1.4), thisprojective representation lifts to an honestΓF-representationτ1, so for some characterχ : ΓL → Q

×ℓ ,

τ⊕d1 |ΓL ρ|ΓL ⊗ χ.

The characterα := det(ρ)/ det(τ⊕d1 ) of ΓF hasΓL-restriction equal toχ−n, and overF itself we

can find charactersα1, α0 : ΓF → Q×ℓ , with α0 finite-order, such thatα = αn

1α0. Then (χα1|ΓL)n =

(α−1αn1)|ΓL = α

−10 |ΓL, and replacingτ1 by τ1 ⊗ α1, andL by a finite extension trivializingα0, we find

a Lie-irreducible representationτ of ΓF and a finite extensionL of F such thatτ⊕d|ΓL ρ|ΓL. TheΓF-representation

ω := HomΓL(τ, ρ).

is therefore ad-dimensional Artin representation,25 and the natural mapτ ⊗ ω → ρ (i.e. v⊗ φ 7→φ(v)) is an isomorphism ofΓF-representations.

Corollary 3.4.2. Letρ : ΓF → GLQℓ(V) be a semi-simple representation (not necessarily irre-ducible), and suppose thatρ is Lie-isotypic, i.e. for all F′/F sufficiently large,ρ|ΓF′ is isotypic. Thenthere exists a Lie-irreducible representationτ and an Artin representation (possibly reducible)ω,both ofΓF , such thatρ τ ⊗ ω.

25As ΓL-representation,ω HomΓL (τ|ΓL , τ|⊕dΓL

) Q⊕dℓ .

82

Proof. Decomposeρ into irreducibleΓF-representations as⊕r1ρi. Eachρi is Lie-isotypic: there

existsL/F and integersmi such thatρi |ΓL τ⊕mi0 for all i, whereτ0 is a Lie-irreducible representation

independent ofi. By the argument of the previous proposition, after possibly enlargingL wefind a ΓF-representationτ whose restriction toL is isomorphic toτ0, and then there are Artinrepresentationsωi of ΓF such thatτ ⊗ ωi ρi. Consequently,

ρ τ ⊗ (r⊕

1

ωi).

Remark 3.4.3. • In general, the fieldL in Proposition 3.4.1 is not unique, even up toΓF-conjugacy. Examples of such non-uniqueness should not arise in the Lie-multiplicityfree case (see§3.4.2), but in the Artin case there are easy examples arisingfrom therepresentation theory of finite groups. Consider, for instance, the quaternion groupQ8 =

±1,±i,± j,±k. The (unique) irreducible two-dimensional representation of Q8 can bewritten in the form IndQ8

〈x〉(εx), where〈x〉 denotes one of the subgroups generated byi, j, ork, andεx is a generator of the character group of〈x〉. None of these subgroups is conjugateto any of the others.26 It would be interesting to achieve a more systematic understandingof these ambiguities.• These structure theorems for Galois representations should have an analogue on the auto-

morphic side. In fact, Tate has shown (2.2.3 of [Tat79]) the analogue of Proposition 3.4.1for representations of the Weil groupWF , where it takes the particularly simple form thatany irreducible, non-inducedρ : WF → GLn(C) is isomorphic toω⊗χ, whereω is an Artinrepresentation andχ : WF → C× is a character. A basic question is whether we shouldexpect Proposition 3.4.1 to hold for ‘representations ofLF.’ If we had the formalism ofLF, then to carry out the argument of the proposition with complex representations ofLF

in place ofℓ-adic representations ofΓF requires two ingredients:– That a homomorphismLF → PGLn(C) lifts to a homomorphismLF → GLn(C); but

this is ‘implied’ by Proposition 3.1.4. I should mention in this respect the theoremof Labesse ([Lab85]), which establishes the analogue for lifting homomorphismsWF → LG across surjectionsLG→ LG with central torus kernel.

– That the analogue of the characterα in the proof of Proposition 3.4.1 can be writtenas a finite-order twistα0 of the nth power of a characterα1. This isnot automatic,as it is forℓ-adic characters, but in this case we can exploit Lemma 2.3.10, whichapplies to theα of the Proposition, since there the restriction toL is (continuing withthe notation of the Proposition)χ−n.

This discussion motivates the following conjecture, whoseformulation of course requiresassuming deep cases of functoriality:

Conjecture 3.4.4. Let π be a cuspidal automorphic representation ofGLn(AF); as-sumeπ is not automorphically induced from any non-trivial extension L/F. Then there ex-ist cuspidal automorphic representationsτ andω of, respectively,GLd(AF) andGLn/d(AF)such thatπ = τ ⊠ ω, with the following properties:

26Although these groups are not conjugate, they are related by(outer) automorphisms ofQ8, but applying outerautomorphisms in this fashion will not in general preserve an irreducible induced character: consider the principalseries of GL2(Fp) and the outer automorphism ofA1.

83

– for all finite extensions L/F, the base-changeBCL/F(τ) remains cuspidal;– for some finite extension L/F, BCL/F(ω) is isomorphic to the isobaric sum of n/d

copies of the trivial representation.

3.4.2. Lie-multiplicity-free representations. In this section, we focus on the cases antithet-ical to that of Artin representations, putting ourselves inthe following situation. LetF be anynumber field, and recall from definition 1.2.3 the definition of a (weakly) compatible system ofλ-adic representations ofΓF , with coefficients in a number fieldE. Let

ρλ : ΓF → GLn(Eλ)

be such a (semi-simple, continuous) compatible system. WriteVλ for the space on whichρℓ acts.

Definition 3.4.5. We say thatVλ is Lie-multiplicity free if after any finite restrictionL/F, Vλ|ΓL

is multiplicity-free. Equivalently,lim−−→L/F

EndEλ [ΓL](Vλ)

is commutative. We will often abbreviate ‘Lie-multiplicity-free’ to ‘LMF.’

Cases to keep in mind are Hodge-Tate regular (see Definition 2.4.2 and the preceding discus-sion of§2.4.1)Vλ, or Vλ of the formH1(AF ,Qℓ) whereA/F is an abelian variety with End0(AF)a commutativeQ-algebra (by Faltings’s proof of the Tate conjecture). Elementary representationtheory yields:

Lemma 3.4.6. (1)Suppose Vλ is irreducible. Then Vλ is LMF if and only if it can bewritten

Vλ IndFL(λ)(Wλ),

where we write L(λ) to show the a priori dependence onλ if Vλ belongs to a compatiblesystem, and where Wλ is a Lie-irreducibleEλ-representation ofΓL(λ), all of whoseΓF-conjugates remain distinct after any finite restriction.

(2) Let Wλ be an irreducible representation ofΓL, and assume that Vλ = IndFL (Wλ) is LMF.

Then Vλ is irreducible.

Proof. For (1), restrict to a finite-index subgroup ofΓF over whichVλ decomposes into adirect sum of Lie-irreducible representations; take one such factor, and consider its stabilizer inΓF– Vλ is then induced from this subgroup. For (2), Mackey theory implies we need to check thatWλ|gΓLg−1∩ΓL and (gWλ)|gΓLg−1∩ΓL are disjoint for allg ∈ ΓF − ΓL. These two representations occur asdistinct factors in theVλ|gΓLg−1∩ΓL , so they are disjoint sinceVλ is LMF.

For general (possibly reducible) LMF representations, there is a decomposition into a sum ofterms as in the lemma. IfVλ belongs to a compatible system, we expect that the number of suchfactors should be independent ofλ; this is an extremely difficult problem (unlike the correspondingquestion for Artin representations). Let us indicate the difficulties through an example.

Example 3.4.7. Supposef is a holomorphic cuspidal Hecke eigenform on the upper half-planeof some weightk ≥ 2 and levelN and nebentypusǫ. We normalizef so that itsq-expansion at thecusp∞, f =

∑an( f )qn, has leading coefficienta1( f ) = 1. Then work of Eichler-Shimura-Deligne

(see [Del71]) yields a compatible system

ρ f ,λ : ΓQ → GL2(Eλ)

84

of λ-adic representations (hereE is the number field generated by thean( f )) characterized by theproperty that for allp ∤ N, the characteristic polynomial ofρ f ,λ( f rp) is equal to

X2 − ap( f )X + pk−1ǫ(p).

Moreover, the eigenvalues ofρ f ,λ( f rp) arep-Weil numbers of weightk−1, i.e. have absolute valuep

k−12 in all complex embeddings, andρ f ,λ is Hodge-Tate of weights 0, 1 − k. Since f is cuspidal,

we expect these Galois representations to be irreducible. This can be provenbecause the Fontaine-Mazur-Langlands conjecture is known for the groupGL1. Supposeρ f ,λ χ1 ⊕ χ2. Then (up tore-orderingχ1, χ2) Theorem 2.3.13 implies27 thatχ1 must be a finite-order character andχ2 mustbe the product of a finite-order character and the (1− k)th power of the cyclotomic character. Thiscontradicts the fact that the eigenvalues ofρ f ,λ( f rp) are p-Weil numbers of weightk − 1, so wehave proven the asserted irreducibility.

Therefore in this section we pursue a much more modest goal: restricting to the case of irre-ducible compatible systems, we will be able to say somethingabout independence ofλ of the fieldsL(λ) of Lemma 3.4.6.

Our basic strategy is that the placesv for which tr(ρλ( f rv) equals zero should detect the fieldL(λ). This is in marked contrast to the case of Artin representations: any irreducible (non-trivial)representation of a finite group has elements acting with trace zero. Our main tool will be thefollowing (slight weakening of a) theorem of Rajan:

Theorem 3.4.8 (Theorem 3 of [Raj98]). Let E be a mixed characteristic non-archimedeanlocal field, and let H/E be an algebraic group. Let X be a subscheme of H (over E), stable underthe adjoint action of H. Supposeρ : ΓF → H(E) is a Galois representation, unramified almosteverywhere, and let C= X(E) ∩ ρ(ΓF). Denote by Hρ ⊂ H the algebraic monodromy group

ρ(ΓF)Zar

, and letΦ = Hρ/H0ρ denote its group of connected components. Forφ ∈ Φ, we write Hφ

for the corresponding component.• LetΨ = φ ∈ Φ|Hφ ⊂ X. Then the density of the set of places v of F withρ( f rv) ∈ C is

precisely|Ψ|/|Φ|.Rajan applies this to prove28 (Theorem 4 of [Raj98]) that an irreducible, butLie reducible,

representation necessarily has a positive density of frobenii acting with trace zero; note that thisalso follows immediately fromCebotarev and Proposition 3.4.1, which is a more robust versionof Rajan’s result (basically combining his argument with Proposition 2.1.4). Our next two resultsestablish a converse, also extending Corollaire 2 to Proposition 15 of [Ser81] to its natural level ofgenerality: that result handles the case of connected monodromy groups.

Proposition 3.4.9. Let ρλ : ΓF → GLn(Eλ) be a continuous, semi-simple, LMF representation.Decompose Vλ as above, so

Vλ

rλ⊕

i=1

IndFL(λ)i

(Wλ,i)

27This special case of Theorem 2.3.13 is notably easier than the general case. It suffices to show that a characterχ : ΓQ → Q

×ℓ with all labeled Hodge-Tate weights equal to zero is finite-order; this follows from global class field

theory and the corresponding statement for charactersΓQℓ → Q×ℓ . This latter statement is a slight improvement of a

fundamental theorem of Tate: see [Ser98, §III.A-3].28In addition to the main result of his paper, a beautiful ‘strong multiplicity one’ theorem forℓ-adic

representations.

85

for Lie-irreducible representations Wλ,i of ΓL(λ)i . Then:

(1) Up to a density zero set of places,

v ∈ |F | : tr(ρλ( f rv)) = 0 = v : f rv <

⋃

i

⋃

σ∈Sλ,i

σΓL(λ)iσ−1,

where Sλ,i is a set of representatives ofΓF/ΓL(λ)i .(2) Further assume thatρλ belongs to a compatible systemρλ of λ-adic representations

of ΓF (although in contrast to definition 1.2.3, we need not assumehere that theρλ aregeometric). Then up to a set of density zero, the set of placesof F which have a split factorin L(λ)i for some i is independent ofλ. If we further assume that all Vλ are absolutelyirreducible(rλ = 1) and all L(λ)/F are Galois, then L(λ) is independent ofλ.29

Proof. For the first part of the Proposition, we ignore the underlying ‘coefficient’ number fieldE and just viewρλ as valued in GLn(E) for some sufficiently large finite extensionE ofQℓ. The “⊇”direction follows from the usual formula for the trace of an induced representation. To establishthe reverse inclusion, let us consider, for each non-empty subsetI ⊂ ∐

i Sλ,i, the setXI of placesv such that tr(ρλ( f rv)) = 0, and f rv ∈ σΓL(λ)i(σ)σ

−1 if and only if σ ∈ I [Notation: if σ ∈ I , thenσ ∈ Sλ,i for a uniquei =: i(σ)]. Also set

ΓI =⋂

σ∈IσΓL(λ)i(σ)σ

−1,

and letGI , resp.G, denote the algebraic monodromy group ofρλ|ΓI , resp.ρλ. To establish the “⊆”(up to density zero) direction, we must show thatXI has density zero for every non-emptyI . For allI , GI containsG0, the identity component ofG. We apply Rajan’s Theorem (3.4.8 above) toρλ|ΓI :if XI has positive density, then there is a full connected component TG0 ⊂ GI on which the tracevanishes (hereT is some coset representative for the component). Representing endomorphismsof Vλ in block-matrix form corresponding to the decomposition

Vλ|ρ−1λ

(G0) =⊕

i,σ

σWλ,i,

we have

tr

T ·

∗ 0 · · · 00 ∗ · · · 0...

.... . . 0

0 0 · · · ∗

= 0,

where the∗’s representarbitrary elements of the End(σWλ,i). For this, we use the fact that theseconstituents are (absolutely) irreducible and distinct: either apply Wedderburn theory to the semi-simpleEλ-algebraEλ[[ρλ(ρ−1

λ (G0))]], or apply Schur’s lemma (using absolute irreducibility) and analgebra version of Goursat’s lemma (using multiplicity-freeness). Then the above matrix equationimplies the automorphismT, written in block-matrix form, has all zeros along the (block)-diagonal.But I is non-empty, soT ∈ GI preserves at least one subspaceσWλ,i, and so its block-diagonalentry corresponding to that sub-space must be non-zero (invertible). This contradiction forcesall components ofGI to have non-zero trace (generically), for all non-emptyI , and thus the “⊆”direction is established.

29In the non-Galois case, see Exercise 6 in [Cp86]!

86

Part 2 now follows from independence ofλ of tr(ρλ( f rv)) andCebotarev, noting that

v : f rv ∈⋃

i

⋃

σ∈Sλ,i

Γσ(L(λ)i )

is the set of placesv of F that have at least one split factor in someL(λ)i .

We make a few remarks about the limitations of this method:

Remark 3.4.10. (1) Without any information about how the variousVλ decompose intoirreducible sub-representations, this result yields frustratingly little, since it is easy to finddisjoint collections of number fieldsLii∈I andL′i i∈I ′ such that the union of primes witha split factor (or even split) in the variousLi equals the union of those with a split factorin the variousL′i . For, the simplest example, takeL1 = Q, L′1 = Q(i), L′2 = Q(

√2),

L′3 = Q(√−2).

(2) Even when theVλ are irreducible, and even assuming that one fieldL(λ0) is Galois, caremust be taken when the other inducing fieldsL(λ) are not (known to be) Galois. We seethat L(λ0) is contained inL(λ) for all λ, but equality does not follow, as the followingexample from finite group theory shows. We want an inclusionH ≤ K ⊳ G of groupswith K normal inG, H a proper subgroup ofK, and∪g∈GgHg−1 = K. TakingK ⊳ S4 tobe the copy of the Klein four-group given by the (2, 2)-cycles, andH to be the subgroupgenerated by one of these permutations, meets the requirements. Note that such examplesrequireK to have non-trivial outer automorphism group: ifG-conjugation acts byK-inner automorphisms onK, then∪GgHg−1 = ∪KkHk−1 = K implies H = K, since nofinite group is the union of conjugates of a proper subgroup.

(3) The group-theoretic counterexample of the previous item should not arise in practice: ifwe assume thatWλ0 can be put in a compatible-system, say withλ-adic realizationWλ,0,then conjecturallyWλ,0 will be Lie irreducible as well, and then the isomorphism

IndFL(λ)(Wλ) IndF

L(λ0)(Wλ,0)

implies, by Mackey theory, that there is a non-zeroΓL(λ0)-morphism

IndL(λ0)s(L(λ))(sWλ)։Wλ,0

for somes ∈ ΓF . By part (2) of Lemma 3.4.6, this induction is irreducible, so this map isan isomorphism. ButWλ,0 is (conjecturally) Lie-irreducible, soL(λ0) = L(λ).

In any case, the following corollary is the promised converse to Rajan’s result:

Corollary 3.4.11.An irreducible, LMF representationsρℓ : ΓF → GLn(Qℓ) is Lie irreducibleprecisely when the set of v withtr(ρℓ( f rv)) = 0 has density-zero. In particular, in a compatiblesystem of irreducible, LMF representations, Lie-irreducibility is independent ofλ.

Remark 3.4.12. If we know general automorphic base-change for GLn, we can formulate theconditions ‘Lie irreducible’ and ‘Lie multiplicity free’ on the automorphic side. This result thensuggests how to tell whether a ‘LMF’ automorphic representation is automorphically induced.Finding an intrinsic characterization of the image of automorphic induction, even conjecturally,is a mystery (in contrast to its close cousin base-change), so it may come as a surprise that thereshould be such a simple condition at the level of Satake parameters, for this broad class of LMFrepresentations.

87

I originally developed Proposition 3.4.9 to prove that a regular compatible system of represen-tations ofΓF for F a CM field, if induced, is necessarily induced from a CM field. See Remark 2.4.8for an automorphic analogue. Here is a partial result; it is another application of the ’Hodge-theorywith coefficients’ in§2.4.1. For the most natural formulation of the result, recall the definition ofpurity of a weakly compatible system (Definition 1.2.4).

Lemma 3.4.13.Let F be a CM field, and letR = ρλ : ΓF → GLn(Eλ) be a weakly compatiblesystem ofλ-adic representations with coefficients in a number field E. Assume that theρλ are(almost all) Hodge-Tate regular, and pure. Finally, assumethat there is a single number field Lsuch that forλ above a set of rational primesℓ of density one,

ρλ IndFL (rλ),

for someEλ-representation rλ of ΓL. Then L is CM.

Proof. We may assumeE/Q is Galois. In yet another variant of the theme of§2.4, purityimplies we may take the number fieldE to be CM: simply observe that for any choicec of complexconjugation in Gal(E/Q), the characteristic polynomialsQv(X) of ρλ( f rv) satisfy

cQv(X) = XnQv(qwv /X)/Qv(0),

and thus for any two complex conjugationsc, c′, cc′R R. It follows that for a density one set ofv, Qv(X) has coefficients inEcm. TheCebotarev density theorem then yields a well-defined weaklycompatible system (in the sense of Definition 1.2.3) ofλ-adic representations with coefficientsin Ecm. (This argument is taken from [PT, Lemma 1.1, 1.2].30). Thus we takeE to be CM.By regularity, we may also assume that there is a CM extensionE′/E such that for all finite-indexsubgroupsH of ΓF , and all primesλ of E′, all sub-representationsr ⊂ ρλ|H are actually defined overE′λ: this is an elementary argument (see [BLGGT10, Lemma 5.3.1(3)], with the CM refinementof [PT, Lemma 1.4]), the key idea being that regularity gives us an abundant supply of elementsin the image ofρλ with distinct eigenvalues, and thatρλ can then be defined over the extension ofE generated by these eigenvalues. Thus, after enlargingE, we may assume allρλ andrλ act onEλ-vector spaces.

For simplicity enlargeE to containLcm, the maximal CM subfield ofL, and take its Galoisclosure– the result remains CM. IfL , Lcm, then we can find a (positive-density) set ofℓ (unram-ified in L and for the systemρλ) which are split inE (with, say,λ|ℓ) but not in (the Galois closureof L, hence)L. Consider a non-split primew|ℓ of L, above a placev of F, and the restriction

rλ|ΓLw: ΓLw → GLn(Eλ) = GLn(Qℓ);

By Lemma 2.2.8, DdR(ρλ|ΓFv) is the image under the forgetful functor (from filteredLw-vector

spaces to filteredFv-vector spaces) of DdR(rλ|ΓLw). SinceLw does not embed inEλ = Qℓ, we can

invoke Corollary 2.4.3 to showρλ is not regular, a contradiction. ThereforeL = Lcm.

Combining this with Proposition 3.4.9, since regular clearly implies LMF,31 we deduce:

30Our definition of compatible system is somewhat weaker than that given in [PT]; for our purposes, we can justignore the part of the proof of [PT, Lemma 1.1] that computes Hodge numbers.

31This is the one case in which the LMF condition is provably independent ofℓ.

88

Corollary 3.4.14.Let F be a CM field, and letR = ρλλ be an absolutely irreducible, pure,Hodge-Tate regular weakly compatible system of representations ofΓF. Suppose that when wewrite

ρλ IndFL(λ)(Wλ),

where Wλ is Lie irreducible, the extensions L(λ)/F are Galois. Then the field L= L(λ) is indepen-dent ofλ, and L is itself CM.

Remark 3.4.15. One way of interpreting this result is that to study regular motives, compatiblesystems, or algebraic automorphic representations over CMfields, we will never have to leave thecomfort of CM fields; essentially all progress in the study ofautomorphic Galois representationsis currently restricted to this context. This is in particular the case for Galois representationsoccurring as irreducible sub-quotients of the cohomology of a Shimura variety.

89

CHAPTER 4

Motivic lifting

As noted in the introduction (see Question 1.1.11), the results of Part 3 raise more questionsthan they resolve. In this chapter, we discuss some cases of the motivic analogue of Conrad’slifting question; this is also the natural framework for theproblem of findingcompatiblelifts of acompatible system of Galois representations. Ideally, we would be able to work in Grothendieck’scategory of pure motives, defined using the relation of homological equivalence on algebraic cycles(see§4.1.2 for a brief review). This category can only be proven tohave the desirable categoricalproperties– namely, equivalence to the category of representations of some pro-reductive (‘motivicGalois’) group– if we assume Grothendieck’s Standard Conjectures on algebraic cycles, which arefar out of reach. (For the basic formalism (as relevant for the theory of motives) of algebraic cyclesand precise statement of the Standard Conjectures, see [Kle68].) We therefore need an uncondi-tional variant of the motivic Galois formalism, and we adoptAndre’s approach, using his theory ofmotivated cycles ([And96b]). In §4.1 we review Andre’s theory, prove some supplementary resultsneeded for the application to motivic lifting, and then treat the motivic lifting problem in the po-tentially abelian case. In§4.2 we prove an arithmetic refinement of Andre’s work ([And96a]) on,roughly speaking, the motivated theory of hyperkahler varieties. This provides a motivic analogueof Theorem 3.2.10 in many non-abelian examples. Finally, in§4.3, we speculate on a generalizedKuga-Satake construction, of which the results of§4.2 are the ‘classical’ case; we then prove thisfor H2 of an abelian variety, generalizing known results for abelian surfaces.

4.1. Motivated cycles: generalities

4.1.1. Lifting Hodge structures. In trying to produce a motivic analogue either of Winten-berger’s or of my lifting theorem, one is naturally led to tryto lift all cohomological, rather thanmerely theℓ-adic, realizations of a ‘motive.’ The easiest such liftingproblem is for real Hodge-structures, which are parametrized by representations of the Deligne torusS = ResC/R(Gm). RecallthatX•(S) = Zα1 ⊕ Zα2, whereα1 andα2 are the first and second projections in the isomorphism

S(C) = (C ⊗R C)×∼−→ C× × C×

z1 ⊗ z2 7→ (z1z2, z1z2).

HereS(R) ⊂ S(C) is C× = (C ⊗R R)×, and the Gal(C/R)-action, induced byz1 ⊗ z2 7→ z1 ⊗ z2, isgiven byc: (w, z) 7→ (z, w). In particular,

(c · α1)(w, z) = c(α1(z, w) = z,

i.e. (c · α1) = α2, and similarly (c · α2) = α1. The group of characters overR, denotedX•R(S), is

thenZ(α1 + α2), whereα1 + α2 = N is the norm, onR-points satisfyingN(z) = zz.

91

Let H′0 → H0 be a surjection of linear algebraic groups overR with kernel equal to a centraltorusZ0. We are interested in the lifting problem for algebraic representations overR:

H′0π

S

h??⑧

⑧⑧

⑧

h// H0.

Any suchh lands in some (typically non-split) maximal torusT0, and any lift will land in thepreimageπ−1(T0) =: T′0, which is a maximal torus ofH′0. We are therefore reduced to studying thedual diagram of freeZ-modules with Gal(C/R) = ΓR-action:

0

X•(Z0)

OO

X•(T′0)

zz

OO

X•(S) X•(T0)oo

OO

0

OO

As short-hand, we denote the vertically-aligned charactergroups, from bottom to top, byY, Y′,andL, so we in fact are studying the sequence

0→ HomΓR(L,X•(S))→ HomΓR(Y

′,X•(S))→ HomΓR(Y,X•(S))→ Ext1ΓR(L,X

•(S))→ . . .

Any real torus is isomorphic to a product of copies ofGm, S, andS1 = ker(N : S → Gm). Thecharacter groupX•(S1) is X•(S)/Z(α1+α2). A generator is the image ofα1−α2, which onR-pointsis simply the characterz 7→ z/zof the (analytic) unit circle S1. Complex conjugation acts as−1 onX•(S1). We can therefore completely address the lifting problem by understanding morphisms andextensions between these three basicZ[ΓR]-modules. The case of immediate interest to us will bewhenZ0 is split, soL is just some number of copies ofZ with trivial ΓR-action. Now,

Ext1ΓR(Z,X•(S)) = H1(ΓR,X

•(S)) = 0,

since ker(1+c) = im(c−1) = Z(α1−α2). Therefore anyh: S→ H0 lifts, and the ambiguity in liftingis a collection of elements (of order equal to the rank ofZ0) of HomΓR(Z,X

•(S)) Z(α1 + α2).1

1Note that if we dealt with representations ofS1 we would find an obstruction inH1(ΓR,X•(S1)) = Z/2Z.

92

Example 4.1.1. In§4.2.2 we will consider the following setup:VR will be an orthogonal spacewith signature (m− 2, 2),2. Write m= 2n or m= 2n+ 1. The lifting problem will be

GSpin(VR)

π

S

h::

h// SO(VR),

whereh lands in a maximal anisotropic torusT0 (S1)n. We can write

X•(T′0) =

n⊕

i=1

Zχi

⊕ Z(χ0 +

∑χi

2), 3

where conjugation acts by−1 on eachχi, i = 1, . . . , n, and trivially onχ0. Here⊕ni=1Zχi is the

submoduleX•(T0). A morphismS → T0 ⊂ SO(VR) is given in coordinates byχi 7→ mi(α1 − α2),for some integersmi. A lift to a morphismS→ GSpin(VR) then amounts to an extension

χ0 +

∑ni=1 χi

27→ ǫ0

2(α1 + α2) +

∑ni=1 mi

2(α1 − α2),

whereǫ0 is any integer having the same parity as∑

mi. The Clifford normN is given by thecharacter 2χ0, so the composition of such a lift with the Clifford norm isǫ0(α1 + α2). In theK3 (orhyperkahler) examples to be considered in the next section, m1 = 1 and all othermi = 0, so we findthere is a unique lift

GSpin(VR)

π

S

h::

h// SO(VR),

whereN h: S→ Gm is anyoddpower of the usual normS→ Gm. The Kuga-Satake theory takesthe norm itself, which then gives rise to weight 1 Hodge structures.

4.1.2. Motives for homological equivalence and the Tannakian formalism. Our aim in thissub-section is to review Grothendieck’s construction of the categoryMhom

F of pure motives for ho-mological equivalence over a fieldF. As a preliminary, we provide some background on the theoryof neutral Tannakian categories. We can then describe the output of the Standard Conjectures, tak-ing for simplicityF to be an abstract (i.e., not embedded) field of characteristic zero, small enoughto be embedded inC: Mhom

F is (conjecturally) a graded, semi-simple,Q-linear neutral Tannakiancategory. With the exception of§4.3.2, in the rest of this paper we will not work directly withMhom

F , so much of this discussion serves only to orient the reader.For more background, the readershould consult [And04], especially Chapter 4, or [Sch94].

2This part of the discussion applies to any signature (p, q) with at least one ofp or q even, so that SO(VR) has acompact maximal torus.

3The charactersχi are conjugate in SO(VC) to the characters denotedχi in §2.8 (see page 52). The torusT0 is builtout of copies of SO(2) embedded in SO(m− 2, 2), and these are just the usual characters

(cos(θ) sin(θ)− sin(θ) cos(θ)

)7→ eiθ.

93

4.1.2.1. Neutral Tannakian categories.We begin with an overview of the theory of neutralTannakian categories; [DM11] is a very readable and thorough introduction to which we referthe reader for details (the original source is [SR72]). Note that we will always work with neutralTannakian categories, which simplifies the theory considerably; for deeper aspects in the non-neutral case, see [Del90]. Let E be a field. The prototypical neutral Tannakian category is thecategory VecE of finite-dimensional vector spaces overE. VecE is an abelian category with anotion of tensor product (namely, the usual tensor product of E-vector spaces) satisfying certainnatural requirements: associativity, commutativity, andexistence of a unit (E regarded as anE-vector space) for the tensor product. Moreover, every object of VecE has a dual (more generally,internal Hom’s exist), and the endomorphisms of the unit object are just the fieldE itself. Themain theorem of the theory in this context is the trivial observation that VecE is equivalent to thecategory of finite-dimensional representations (overE) of the trivial group.

The general definition merely formalizes the notions in the previous paragraph; we will quicklymake the necessary definitions, and then give a number of examples (Example 4.1.6 below).

Definition 4.1.2. LetC be a category, and let⊗ : C × C → C be a functor satisfying the usualaxioms of a symmetric monoidal category, namely:

(1) (Associativity constraint) There is a functorial isomorphism

AX,Y,Z : X ⊗ (Y⊗ Z) → (X ⊗ Y) ⊗ Z

satisfying thepentagon axiom([DM11, 1.0.1]).(2) (Commutativity constraint) There is a functorial isomorphism

CX,Y : X ⊗ Y→ Y⊗ X

such thatCY,X CX,Y = idX⊗Y, and that is compatible with the associativity constraint inthe sense of thehexagon axiom([DM11, 1.0.2]).

(3) (Unit object) There is a pair (1, e) consisting of an object1 and an isomorphisme: 1→1⊗ 1 such that the functorC → C given byX 7→ 1⊗ X is an equivalence of categories.

We call such a (C,⊗), equipped with its associativity and commutativity constraints (but omittedfrom the notation), atensor category, for short.

If (C,⊗) and (C′,⊗′) are tensor categories (whose associativity and commutativity constraintswe will write asA,C andA′,C′, respectively; unit elements will be (1, e) and (1′, e′)), then atensorfunctor from (C,⊗) to (C′,⊗′) is a pair (F, k) consisting of a functorF : C → C and a functorialisomorphismkX,Y : F(X) ⊗ F(Y)→ F(X ⊗ Y) satisfying the following three compatibilities:

(1) For all objectsX,Y,Z of C, the diagram

FX ⊗ (FY ⊗ FZ)

A′

id⊗k// FX ⊗ F(Y ⊗ Z) k

// F(X ⊗ (Y⊗ Z))

F(A)

(FX ⊗ FY) ⊗ FZk⊗id

// F(X ⊗ Y) ⊗ FZk

// F((X ⊗ Y) ⊗ Z)

is commutative.

94

(2) For all objectsX,Y of C, the diagram

FX ⊗ FYk

//

C′

F(X ⊗ Y)

F(C)

FY⊗ FXk

// F(Y ⊗ X)

is commutative.(3) If (1, e) is a unit object ofC, then (F(1), F(e)) is a unit object ofC′.

We have spelled out the precise conditions for (F, k) to be a tensor functor because of its rel-evance for the structure ofMhom

F : the Kunneth Standard Conjecture essentially ‘corrects’the factthat the Betti realization

HB : MhomF → VecQ

with the natural Kunneth isomorphismkX,YHB(X) ⊗ HB(Y)∼−→ HB(X × Y) doesnot intertwine the

(obvious) commutativity constraints onMhomF and VecQ. See the discussion surrounding Conjec-

ture 4.1.8 for details.For any unit object (1, e), we obtain isomorphismslX : 1⊗ X

∼−→ X andrX : X ⊗ 1∼−→ X.

For any additive tensor category (C,⊗) (for which we require⊗ to be bi-additive), and anyunit object (1, e), EndC(1) = R is a commutative ring, unique up to unique isomorphism, andC isnaturally anR-linear category.

Definition 4.1.3. Let (C,⊗) be a tensor category. It isrigid if for every objectX there is a‘dual’ objectX∨ along with evaluation and co-evaluation morphisms

X∨ ⊗ XevX−−→ 1

1coevX−−−−→ X ⊗ X∨

such that the composites (we suppress the unit isomorphismsand the associators)

XcoevX⊗idX−−−−−−−→ X ⊗ X∨ ⊗ X

idX ⊗evX−−−−−−→ X

X∨idX∨ ⊗coevX−−−−−−−−→ X∨ ⊗ X ⊗ X∨

evX⊗idX∨−−−−−−→ X∨

are idX and idX∨, respectively.

Note that in the absence of a commutativity constraint,X∨ is what would be called a ‘rightdual.’

Finally, we come to the main definition:

Definition 4.1.4. LetE be a field. Aneutral Tannakian categoryoverE is a rigid abelian tensorcategory (C,⊗) such that End(1) = E, and for which there exists a faithful, exact,E-linear tensorfunctorω : C → VecE. Such anω is called afiber functor.

The main theorem of the theory of neutral Tannakian categories just says that every Tannakiancategory is equivalent to the category of representations of some affine group scheme:

95

Theorem 4.1.5 ([DM11, Theorem 2.11]).Let (C,⊗) be a neutral Tannakian category over E,equipped with a fiber functorω : C → VecE. Then the functor of E-algebras given by tensor-automorphisms ofω (see[Del90, 1.9, 1.11]) is represented by an affine group scheme G, and thefunctorC → RepE(G) defined byω is an equivalence of tensor categories.4

Informally, we might callG the ‘Galois group’ of (C,⊗), whence the later terminology ‘motivicGalois group’ in the (conjecturally Tannakian) case ofMhom

F .At last, some examples:

Example 4.1.6. (1) The following example is crucial for the theory ofmotives. The cate-gory VecZ/2E of Z/2Z-graded vector spaces overE, with the usual graded tensor product,

and with commutativity constraint given by the Koszul sign rule, i.e. CV,W : V ⊗ W∼−→

W⊗ V given on homogeneous tensors byv⊗w 7→ (−1)deg(v)·deg(w)w⊗ v, is a rigidE-linearabelian tensor category. Note, however, that the forgetfulfunctor VecZ/2ZE → VecE (withthe obvious isomorphismskX,Y) is not a tensor functor, because it fails to intertwine thecommutativity constraints. Indeed, VecZ/2ZE is not a Tannakian category. Any rigid tensorcategory has an intrinsic notion of rank for every objectX, given by the element of End(1)arising as the composition

1coevX−−−−→ X ⊗ X∨

CX,X∨−−−−→ X∨ ⊗ XevX−−→ 1.

For aZ/2Z-graded vector spaceV = V0⊕V1, it is easy to check that the rank in VecZ/2ZE isdimV0−dimV1. But this notion of rank is preserved by any tensor functor, so an obviouslynecessary condition to admit a fiber functor is that all objects have non-negative rank. (Infact, a deep result of Deligne gives a converse: [Del90, 7.1 Theoreme].)

(2) The categoryQ − HSpol of pure polarizableQ-Hodge structures is a neutral Tannakiancategory overQ, with fiber functor just given by taking the underlyingQ-vector spaceof a Hodge structure (here we take the commutativity constraint on Hodge structures tobe the same as the usual constraint on vector spaces). This isa useful ‘toy model’ forthe theory of pure motives overC. In particular, Theorem 4.1.5 identifiesQ − HSpol

with the category of representations of some affine group scheme MT overQ; MT isthe ‘universal Mumford-Tate group.’ We use this example to explain how properties ofa Tannakian category reflect those of its Galois group (see [DM11, §2 ‘Properties ofGand Rep(G)’]). From its Hodge-theoretic description, MT is obviously connected. This isequivalent ([DM11, Corollary 2.22]) to the condition that for every non-trivial represen-tationX of MT, the strictly full subcategory ofQ−HSpol whose objects are isomorphic tosub-quotients of someX⊕n is not stable under⊗. But if X is a non-trivial Hodge structure,which we may assume pure of weight zero (otherwise the resultis clear by comparingweights), then someHp,−p(X) is non-zero, for somep , 0. ThenHmp,−mp(X⊗m) , 0,which form large enough cannot be the case for any sub-quotient of anyX⊕n. In a similarspirit, MT is pro-reductive, since ([DM11, Proposition 2.23]), since the categoryQ−HSpol

is semi-simple (by polarizability).

4That is, it is a tensor functor that is also an equivalence of categories; this ensures ([DM11, Proposition 1.11])that there exists an inverse ofF that is also a tensor functor.

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4.1.2.2. Homological motives.We begin by sketching the construction of Grothendieck’s cat-egory of (pure) homological motives over a fieldF. References for more details are [Kle68, §1]and [Sch94, §1]. Recall that for simplicity we takeF to be a field that can be embedded inC (oreven a specified subfield ofC)– this does not affect the construction, but does affect whether thecategory of motives is (conjecturally)neutralTannakian. Recall that for a fixed Weil cohomologytheoryH∗ (see [Kle68, §1.2]), homological equivalence defines anadequate equivalence relationon algebraic cycles on smooth projective varieties; for example, sinceF ⊂ C, we will always haveat our disposal Betti cohomology,H∗B. For a smooth projectiveX/F, we let A∗hom(X) denote thegraded, by codimension,Q-algebra of (Q-linear combinations of) algebraic cycles for homologi-cal equivalence. ‘Adequate’ ensures that the intersectionproduct is well-defined as a linear mapAr(X) ⊗ As(X) → Ar+s(X). For any two smooth projectiveF-varietiesX andY, with X connectedof (equi-)dimensiond, we can then define the space of degreer correspondences

Crhom(X,Y) = Ad+r

hom(X × Y).

We obtain a composition of correspondences

Crhom(X,Y) ⊗Cs

hom(Y,Z)→ Cr+shom(X,Z)

given by

f ⊗ g = g f 7→ p13,∗(p∗12 f · p∗23g

),

where thepi j are the projections fromX × Y × Z to the products of two factors. In particular,C0

hom(X,X) is aQ-algebra.

Definition 4.1.7. The categoryMhomF of motives overF for homological equivalence has as

objects triples (X, p,m), whereX is a smooth projective variety overF, m is an integer, andp is anidempotent correspondence inC0

hom(X,X). Morphisms inMhomF are defined by

HomMhomF

((X, p,m), (Y, q, n)) = qCn−mhom(X,Y)p.

MhomF is an additive,Q-linear, pseudo-abelian category ([Sch94, Theorem 1.6]). There is a

bi-additive functor⊗ : MhomF ×Mhom

F →MhomF given on objects by

(X, p,m) ⊗ (Y, q, n) = (X × Y, p⊗ q,m+ n)

(the fiber product is overF; we do not here specify⊗ on morphisms). This satisfies associativityand commutativity constraints induced by the tautologicalisomorphisms (X×Y)×Z X× (Y×Z)andC : X × Y Y × X. A unit object for⊗ is given by (SpecF, id, 0), and we can define dualobjects by (again for simplicity takeX of equi-dimensiond)

(X, p,m)∨ = (X, t p, d −m);

In sum, these structures make (MhomF ,⊗) into a rigidQ-linear tensor category with EndMhom

F(1) =

Q. Grothendieck’s Standard Conjectures anticipate that moreoverMhomF should be a semi-simple

neutral Tannakian category. One must prove that

• MhomF is abelian;

• the abelian categoryMhomF is moreover semi-simple;

• MhomF possesses a fiber functor to VecQ.

97

We begin by discussing the question of a fiber functor. The answer is simple: (MhomF ,⊗), with the

commutativity constraint defined above, doesnot have a fiber functor! For simplicity, let us takeF ⊂ C, so that there is a Betti realization

HB : MhomF → VecQ.

HB is not a tensor functor, however: the diagram

HB(X) ⊗ HB(Y)

C′

k∼

// HB(X × Y)

HB(C)

HB(Y) ⊗ HB(X) k∼

// HB(Y× X),

given by applying the naıve commutativity constraints andthe Kunneth isomorphism, only com-mutes up to sign, since cup-product is anti-commutative. More precisely,HB gives a tensor functorvalued in VecZ/2Z

Q, and since objects ofMhom

F can then have negative rank, it cannot be neutral

Tannakian with the commutativity constraint induced byX × Y∼−→ Y × X. The Kunneth Stan-

dard Conjecturewould giveMhomF a grading (corresponding to cohomological degree), allowing

for a modified commutativity constraint, for whichHB : MhomF → VecQ is a tensor functor. More

precisely, for any Weil cohomology theoryH∗, the isomorphism (again takingd = dimX)

H2d(X × X)(d) ⊕

i

H2d−i(X) ⊗ Hi(X)(d) ⊕

i

Hi(X)∨ ⊗ Hi(X) ⊕

i

End(Hi(X))

gives a cohomology classπiX corresponding to the compositionH∗(X) ։ Hi(X) → H∗(X).

Conjecture 4.1.8. For all i = 0, 1, . . . , 2d, the cohomology classπiX is algebraic, i.e. lies in the

image of the cycle class map Adhom(X × X) → H2d(X × X)(d).

The modified commutativity constraint onMhomF would then be given as follows: if the original

constraint isC : M ⊗ N∼−→ N ⊗ M, we can decomposeC = ⊕r,sCr,s with

Cr,s: πr M ⊗ πsN∼−→ πsN ⊗ πr M,

and then the correct commutativity constraint isC′ = ⊕r,s(−1)rsCr,s.The Kunneth Standard Conjecture is in fact implied by a stronger conjecture, the Lefschetz

Standard Conjecture, which would also show that the primitive decomposition of cohomology’makes sense in the categoryMhom

F . To state this, letX be a smooth projective variety overF, andfix an ample line bundleη on X, giving rise to the Lefschetz operator

L = Lη,H∗ : Hi(X)(r) → Hi+2(X)(r + 1),

and the hard Lefschetz isomorphisms

Ld−i : Hi(X)(r)∼−→ H2d−i(X)(d − i + r)

for all i ≤ d. As always with taking cup-product with the class of an algebraic cycle, these isomor-phisms are given by algebraic correspondences.

Conjecture 4.1.9 (Lefschetz Standard Conjecture).For all i ≤ d, the inverse of Ld−i is givenby algebraic correspondences.

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Note thata priori both the Kunneth and Lefschetz Standard Conjectures depend on the choiceof Weil cohomology. For the classical cohomologies (Betti,etale, de Rham) that are related bycomparison isomorphisms (respecting cycle class maps), these conjectures do not depend on thechoice ofH∗.

Now we move to the question of whetherMhomF is abelian. Here there is a marvelous theorem

of Jannsen:

Theorem 4.1.10 ( [Jan92, Theorem 1] ).For an adequate equivalence relation∼, the categoryof motives for∼-equivalence is abelian semi-simple if and only if∼ is numerical equivalence. Inparticular,Mhom

F is abelian if and only if numerical and homological equivalence coincide.

Indeed, long before Jannsen proved his theorem, Grothendieck conjectured:

Conjecture 4.1.11.Homological and numerical equivalence coincide.

We should also remark that semi-simplicity is closely related to the so-called Hodge StandardConjecture (not to be confused with the Hodge conjecture), which in characteristic zero is a simpleconsequence of the Hodge index theorem (see [Kle68, §3]).

In summary, under the Standard Conjectures, we can equipMhomF with a commutativity con-

straint for which it is a semi-simple neutral Tannakian category, hence:

Conjecture 4.1.12.Let F be a subfield ofC, for simplicity. There is a pro-reductiveQ-groupGhom

F , the motivic Galois group for pure motives over F, and an equivalence (given by any choiceof fiber functorMhom

F → VecQ; an example is H∗B)

MhomF RepQ(Ghom

F ).

4.1.3. Motivated cycles.In the present Part 4 of these notes, most of our results can bestatedsolely in terms of abelian varieties, but both the strongestassertions and the proofs will require in-voking some version of the motivic Galois formalism; in the absence of the Standard Conjectures,we use Andre’s theory of motives for motivated cycles ([And96b]). In this subsection, we providea brief review, elaborating on some points for later application. Although Andre’s theory is devel-oped over any fieldF, to simplify we will always takeF to be an abstract (i.e., not embedded) fieldof characteristic zero, small enough to be embedded inC, and which we will eventually specify tobe a number field.

In [And96b], Andre defines aQ-linear category of motives for ‘motivated cycles’ whose con-struction mirrors the classical construction of (Grothendieck) motives for homological equivalence,but circumvents the standard conjectures by formally enlarging the group of ‘cycles’ to include theLefschetz involutions. Here is the precise definition of thespace of motivated cycles:

Definition 4.1.13. Fix a reference Weil cohomologyH•. A motivated cycle onX with coeffi-cients inE is an element ofH•(X) of the form

prXYX,∗(α ∪ ∗Lβ),

where

• Y is a smooth projectiveF-scheme, with polarizationηY giving rise to a ‘product’ po-larizationηX×Y = [X] ⊗ ηY + ηX ⊗ [Y], with corresponding Lefschetz involution∗L, onX × Y;• α andβ are algebraic cycles withE-coefficients onX × Y.

99

We denote byA•mot(X)E theE-vector space of motivated cycles onX with E-coefficients.A•mot(X)will always mean the caseE = Q.

As in §4.1.2.2, we can then define spacesC•mot(X,Y)E of motivated correspondences (see also[And96b, Definition 2]); for X = Y, this construction yields a gradedE-algebra containing theLefschetz involutions and the Kunneth projectorsπi

X.The first main result of this theory asserts that the spaces ofmotivated cycles in a precise sense

do not depend on the Weil cohomologyH• used to define them:

Theorem 4.1.14 (Theoreme 0.3 of [And96b]). For any smooth projective F-scheme X, letA•mot(X) be the gradedQ-algebra of ‘motivated cycles,’ constructed with respect to a fixed Weilcohomology H•. A•mot(X) contains the classes of algebraic cycles modulo homological equivalence,and there is aQ-linear injection

clH : A•mot(X) → H2•(X)

extending the cycle class map for H. A•mot(X) has the following properties:

• A•mot(X) depends bifunctorially (push-forward and pull-back) on X,satisfying the usualprojection formula.• (See[And96b, §2.3]) Two Weil cohomologies related by comparison isomorphismsyield

canonically and functorially isomorphic algebras A•mot(X) of motivated cycles. In partic-ular, all the classical cohomology theories yield the same A•

mot(X).

From here, Andre defines a categoryMF of motives for motivated cycles exactly as in Defini-tion 4.1.7, replacing algebraic correspondences with motivated correspondences.5 Andre shows theendomorphism algebras of objects ofMF are semi-simple (finite-dimensional)Q-algebras, fromwhich it follows (see [Jan92]) thatMF is an abelian semi-simple category. Since the Kunnethprojectorsπi

X are motivated cycles, we can define the modified commutativity constraint describedin §4.1.2.2, thereby makingMF into a neutral Tannakian category overQ:

Theorem 4.1.15 (Theoreme 0.4 of [And96b]). MF is a neutral Tannakian category overQ. Itis graded, semi-simple, and polarized. Every classical cohomology factors throughMF.

Remark 4.1.16. We will often use the short-handH(X) for the object (X, id, 0), andHi(X) for(X, πi

X, 0) (recall the notation from Definition 4.1.7). It will be clear from context whenH(X) refersto an object ofMF, and when it refers to the output of any particular Weil cohomology theoryH∗.

In particular, this sets in motion the formalism of motivic Galois groups, and the theory be-comes a very useful circumvention of the standard conjectures. Perhaps its most unsatisfactoryfeature– present also in the theory of absolute Hodge cycles– is that its ‘motives’ are not known togive rise to compatible systems ofℓ-adic representations.

Now we specify fiber functors and no longer regardF as an abstract field. For anyσ : F →C, MF is Tannakian and neutralized by theσ-Betti fiber functor (denotedHσ), so we obtain itsTannakian group, ‘the’ motivic Galois group,GF(σ). For theℓ-adic fiber functorX 7→ Het(XF ,Qℓ),

5Andre defines more generally a category of motives ‘modeledon’ a full sub-categoryV of the category of smoothprojectiveF-schemes, withV assumed stable under products, disjoint union, and passageto connected components.This amounts to restricting the auxiliary varietiesY permitted in Definition 4.1.13. We will always takeV to be allsmooth projectiveF-schemes.

100

we denote byGF,ℓ the corresponding motivic group (overQℓ). It is more convenient to choose anembeddingσ : F → C, since this allows us, via the comparison isomorphisms

Het(XF ,Qℓ)∼−−→σ∗

Het(XF ⊗F,σ C,Qℓ) Hσ(X,Q) ⊗Q Qℓ,

to deduce an isomorphismGF(σ)⊗QQℓ GF,ℓ. Eventually,F will simply be regarded as a subfieldof C, with F its algebraic closure inC; in that case, we will omit theσ from the notation. For now,however, we retain it.

As a variant on this formalism, to any object (or collection of objects) M of MF, we canassociate the smallest Tannakian subcategory〈M〉⊗ generated byM (fully faithfully embeddedin MF), and we can then look at its Tannakian group, denotedGM

F (σ). A particularly importantexample comes from takingM to range over (the objects ofMF associated to) all finite etaleF-schemes; this defines the sub-category ofArtin motives. The fully faithful inclusion of thesubcategoryMart

F of Artin motives overF induces a surjection

GF(σ)։ GartF (σ) ΓF ,

and when combined with the base-change functorMF → MF, we obtain an exact sequence6 ofpro-algebraic groups

1→ GF(σ)→ GF(σ)→ ΓF → 1.By Proposition 6.23d of [DM11], for all ℓ there are continuous sections (homomorphisms)ΓF →GF(Qℓ) (after unwinding everything, this is simply the statementthatΓF acts onℓ-adic cohomol-ogy).

We can similarly compare the ‘arithmetic’ and ‘geometric’ versions of the motivic Galois groupof any objectM ofMF: there is a commutative diagram, where the vertical morphisms are surjec-tive:

1 // GMF

F(σ) // GM

F (σ)

1 // GF(σ)

OO

// GF(σ)

OO

// ΓF// 1.

Lemma 4.1.17. Assume thatGMF

F(σ) is connected. Then there exists a finite extension F′/F

such thatGMF′F′ (σ) is connected.

Proof. Fix a primeℓ, a sectionsℓ : ΓF → GF(Qℓ), and a finite extensionF′/F such that theZariski closure of theℓ-adic representationρMℓ

: ΓF′ → GL(Mℓ) is connected. The image of

GMF′F′ ⊗Q Qℓ in GL(Mℓ) is then equal to the product of the two connected groupsρMℓ

(ΓF′ )Zar

and(GMF

F⊗Q Qℓ

), hence is itself connected.

Remark 4.1.18. If in〈MF〉⊗ all Hodge cycles are motivated,GMF

F(σ) is connected.

The assertion that Hodge cycles are motivated is a (still fiercely difficult) weakened versionof the Hodge conjecture. We now describe what is essentiallythe only understood case of thisproblem. Many of the results of [And96a] rest on earlier work of Andre (Theoreme 0.6.2 of

6See Proposition 6.23a, cof [DM11]. This is a corrected, TeXed version of the original articlein [DMOS82]; it isavailable at http://www.jmilne.org/math/xnotes/index.html. Note that partb is a modification of the (only conjectural)statement in the original article.

101

[And96b]) showing that on a complex abelian variety, all Hodge cycles are motivated (a variant ofDeligne’s result that Hodge cycles on a complex abelian variety are absolutely Hodge). One usefulconsequence is:

Corollary 4.1.19 (Andre).Let A/C be an abelian variety, and let M be the motive H1(A) (anobject ofMC). Then the motivic groupGM

C(for the Betti realization) is equal to the Mumford-Tate

group MT(A), and in particular is connected.

Proof. MT(A), recall, is the smallestQ-sub-group of GL(H1B(AC,Q)) whoseR-points contain

the image of theS-representation corresponding to theR-Hodge structureH1B(AC,R); by the gen-

eral theory of Mumford-Tate groups, this is equal to

• the Tannakian group for the Tannakian category ofQ-Hodge structures generated byMB =

H1B(AC,Q); and

• the subgroup of GL(MB) fixing exactly the Hodge tensors in every tensor constructionTm,n(MB) := (MB)⊗m⊗ (M∨

B)⊗n.

Similarly, the motivic groupGMC

is the subgroup of GL(MB) fixing exactly the motivated cycles inevery tensor constructionTm,n(MB). Of course (on any variety) all motivated cycles are Hodge cy-cles, so there is a quite general inclusionMT(A) ⊂ GM

C. Applying Andre’s result to all powers ofA,

we can deduce the reverse inclusion: ift ∈ Tm,n(MB) is a Hodge cycle, then (by weight considera-tions)m= n, and viewing this tensor space (via Kunneth and polarization) insideH2m(A2m,Q)(m),we see thatt is motivated.

The following point is implicit in the arguments of [And96a], but we make it explicit in partto explain an important foundational point in the theory of motivated cycles.

Corollary 4.1.20.Let F be a subfield ofC, with algebraic closureF in C, and let A/F be anabelian variety. Let M be the object ofMF corresponding to H1(AF). ThenGM

F(for the F ⊂ C

Betti realization) is connected, equal to MT(AC).

Proof. We deduce this from the previous result and the following general lemma, which is ofcourse implicit in [And96b]:

Lemma 4.1.21.Let L/K be an extension of algebraically closed fields, and let M be an object ofMK, with base-change M|L to L. Then via the canonical isomorphism Het(M,Qℓ) Het(M|L,Qℓ),theℓ-adic motivic groupsGM

K,ℓ andGM|LL,ℓ agree.

Proof. This follows from the fact (2.5 Scolieof [And96b]) that the comparison isomorphism(for M = H∗(X)) identifies the spaces of motivated cyclesA∗(X)

∼−→ A∗(XL). This follows from stan-dard spreading out techniques, but we provide some details,since they are omitted from [And96b].Writing L = lim−−→

λ

Kλ as the directed limit of its finite-typeK-sub-algebras allows all the data required

to define a motivated cycle onXL (a smooth projectiveY/L, algebraic cycles onXL × Y, the Lef-schetz involution on cohomology ofXL ×Y) to be descended to someSλ = Spec(Kλ). The generalmachinery allows us to assume (enlargingλ) that we haveYλ/Sλ smooth projective (of courseXλ = X ⊗K Kλ is smooth projective), a relatively ample invertible sheafηλ of ‘product-type’ onXλ ×Sλ Yλ, and various closedSλ-subschemes

Z → X := Xλ ×Sλ Yλ102

that are smooth overSλ and whose linear combinations define spread-out versions ofthe algebraiccycles onXL × Y. The purity theorem in this context7 yields an isomorphism

H2 jZ (X,Qℓ)( j)

∼−→ H0(Z,Qℓ),

hence a cycle class [Z] in H2 j(X,Qℓ)( j). Regarding ¯x: Spec(L) → Sλ as a geometric pointover some scheme-theoretic pointx, and letting ¯s: Spec(K) → Sλ be a geometric point over ascheme-theoretic closed points lying in the closure ofx, the cospecialization mapH2 j(Xs,Qℓ)

∼−→H2 j(Xx,Qℓ) is an isomorphism (X/Sλ being smooth proper), and it carries the cycle class [Zs] tothe cycle class [Zx], since these are both restrictions of [Z], and the diagram

H2 j(Xs,Qℓ)

H2 j(X,Qℓ)

77♦♦♦♦♦♦♦♦♦♦♦

''

H2 j(Xx,Qℓ)

commutes.

4.1.4. Motives with coefficients. We will need the flexibility of working with related cat-egories of motives with coefficients. LetE be a field of characteristic zero. Given anyE-linearabelian (or in fact just additive, pseudo-abelian) categoryM, and any finite extensionE′/E, we candefine the categoryME′ of objects with coefficients inE′ as either of the following (this discussionis taken from§2.1 of [Del79] and 3.11, 3.12 of [DM11]):

(1) The category of ‘E′-modules inM,’ whose objects are pairs (M, α) of an objectM ofMand an embeddingα : E′ → EndM(M), and whose morphisms are those commuting withtheseE′-structures.

(2) The pseudo-abelian envelope of the category whose objects are formally obtained fromthose ofM (writing ME′ for the object inME′ arising fromM in M), and whose mor-phisms are

HomME′ (ME′ ,NE′) = HomM(M,N) ⊗E E′.

This construction is valid for infinite-dimensionalE′/E.

To pass from the first to the second description, let (M, α) be as in (1), so that EndME′ (ME′) =EndM(M) ⊗E E′ contains, viaα, E′ ⊗E E′. This E′-algebra (via the left factor) is isomorphic to aproduct of fields, and there is a unique projectioneid : E′ ⊗E E′ ։ E′ in which x⊗ 1 and 1⊗ x bothmap tox. Theneid(ME′) is the object of (2) corresponding to (M, α).

There is a functorM → ME′ , which in the first language isM 7→ (M ⊗E E′, idM ⊗ idE′).8 IfM is semi-simple, then so isME′. If M is a neutral Tannakian category overE with fiber functorω, then we can makeME′ into a neutral Tannakian category overE′. Define anE′-valued fiber

7Theoreme 3.7 of Artin’s Exp. XVI of [sga73]8See 2.11 of [DM11] for a precise description ofM ⊗E E′.

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functorωE′ : M→ VecE′ byωE′(M) = ω(M)⊗E E′. There is a diagram commuting up to canonicalisomorphism,

M //

ωE′""

ME′

ω′E′

VecE′ ,

where

ω′E′(M, α) := ωE′(M) ⊗α⊗idE′ ,E′⊗EE′

E′.

Thisω′E′ neutralizesME′, and so we can define the associated Tannakian groupGE′ = Aut⊗(ω′E′).We wish to compareGE′ withG = Aut⊗(ω). Note thatME′ is equivalent to RepE′(G) andG⊗E E′ =Aut⊗(ωE′). Then the composition of functors

ME′ω−→∼

RepE′(G)F−→∼

RepE′(G ⊗E E′)

given onME′ by9

(X, α) 7→ (ω(X), ω(α)) 7→ E′ ⊗E′⊗E′,ω(α)⊗1

(ω(X) ⊗ E′)

is justω′E′ , whence a tensor-equivalence

RepE′(GE′)∼−→ RepE′(G ⊗E E′).

We apply these constructions to Andre’s category of motives overF for motivated cycles, toform the variantMF,E, motives overF with coefficients inE, for E any finite extension ofQ.We denote the corresponding motivic Galois group (for theσ-Betti realization) byGF,E(σ); its E-linear (pro-algebraic) representations correspond to objects ofMF,E, and it is naturally isomorphictoGF,Q(σ)⊗QE. For any objectM ofMF,E, we also have the corresponding motivic groupGM

F,E(σ).As a more concrete variant, we can start with the (Q-linear, semi-simple) isogeny categoryAV0

Fof abelian varieties overF, and form theE-linear, semi-simple categoryAV0

F,E of isogeny abelianvarieties overF with complex multiplication byE. There is a (contravariant) functorAV0

F,E →MF,E. Faltings’ theorem implies the followingE-linear variant: fix an embeddingE → Qℓ, andconsider an objectA of AV0

F,E; then the natural map

EndAV0F,E

(A) ⊗E Qℓ → EndQℓ[ΓF ](H1(AF ,Qℓ) ⊗E Qℓ)

is an isomorphism.10

4.1.5. Hodge symmetry inMF,E. Our next goal is to show that objects ofMF,E satisfy (un-conditionally) the Hodge-Tate weight symmetries, and the more fundamental symmetries in therational de Rham realization, needed for our abstract Galois lifting results. I expect these resultsare well-known to experts, but they do not seem to have been explained in their natural degree of

9F just sends an objectV of RepE′ (G), corresponding to anE-homomorphismG → ResE′/E(GLE′ (V)), toE′ ⊗E′⊗EE′ V.

10Simply take the usual isomorphism withQ in place ofE, restrict to those endomorphism commuting withE→ EndAV0

F(A), and then project to theE → Qℓ component of the resultingE ⊗Q Qℓ-module.

104

generality, and in a context in which they can beprovenunconditionally11. So, letM be an objectofMF,E, of rankr. The de Rham realizationMdR is a filteredF ⊗Q E-module, which is moreoverfree of rankr. As in §2.4.1, we define theτ : F → E-labeled weights ofM as follows: HTτ(M) isanr-tuple of integersh, with h appearing with multiplicity

dimE grh(MdR⊗F⊗QE,τ⊗1 E

).

In talking of τ-labeled weights, there is alway an ambient over-field, in this caseE, but we will attimes want to change this to eitherC or Qℓ; that will requirefixing an embeddingι : E → Qℓ orι : E → C. In either case, we can then speak of HTιτ(M), with E embedded intoC orQℓ via ι, andthere is an obvious equality HTτ(M) = HTιτ(M).

The essential point, which is the motivic analogue of Corollary 2.4.6, is closely related to thefact thatGF splits12 overQcm (see 4.6(iii ) of [And96b] for this assertion). For lack of reference forthe proof, we give some details:

Lemma 4.1.22.Let N be an object ofMF,E. Then there exists an object N0 ofMF,Ecm such thatN N0 ⊗Ecm E. Consequently,HTτ(N) depends only on the restriction ofτ to Fcm.

Proof. By the formalism of Lemma 2.4.1, the second claim follows from the first. By Propo-sition 3.3 (the analogue of the Hodge index theorem) of [And96b], C0

mot(X,X) is endowed with apositive-definite,Q-valued symmetric form, which we call〈·, ·〉. For any sub-objectM ⊂ H(X) inMF, there follows a decompositionH(X) = M ⊕ M⊥, by positivity and the fact thatM is a semi-simple abelian category.〈·, ·〉 therefore restricts to a positive form onM itself, and in particularevery simple object ofMF carries a positive definite form. Again by semi-simplicity,any suchsimple objectM has End(M) isomorphic to a division algebraD, on which we now have an involu-tion ′ (transpose with respect to〈·, ·〉) and a trace form trM : D→ Q (given byφ 7→ tr(φ|HB(M)) suchthat trM(φφ′) > 0 for all non-zeroφ ∈ D. As in the proof of the Albert classification (see [Mum70,§21, Theorem 2]), this implies that the center ofD is either a totally real or a CM field. Thus,End(M) splits over a CM field; in particular, for any number fieldE and any factorN of M ⊗Q E(inMF,E), N can in fact be realized as (the scalar extension of) an objectofMF,Ecm.

Remark 4.1.23. WhenE = Q, Lemma 2.4.1 shows that HTτ(N) is independent ofτ; this isessentially the assertion that for a smooth projectiveX/F, the Hodge numbers ofX do not dependon the choice of embeddingF → C. The next few results (culminating in Corollary 4.1.26) allhave corresponding strengthenings whenE = Q.

We first record the de Rham-Betti version of the desired symmetry:

Lemma 4.1.24.Let N be an object ofMF,E lying in the k-component of the grading. Then forany choice of complex conjugation c inGal(E/Q),

HTcτ(N) = k− h : h ∈ HTτ(N).

11Our discussion is an analogue of the standard discussion in the theory of motives of CM type– see§7 of[Ser94]. The Galois symmetries themselves have also been postulated in§5 of [BLGGT10], but since that paper isonly concerned with Galois representations arising from essentially conjugate self-dual automorphic representations,it has no need to deal with this symmetry as a general principle.

12i.e., for any algebraic quotientGF ։ H overQ, the connected componentH0 has maximal torus that splits

105

Proof. Any such object is built by taking one of the formM = Hk(X) ⊗ E, for X/F a smoothprojective variety and applying an idempotentα ∈ C0

mot(X,X)E.13 Fix an embeddingι : E → C, sothatι τ : F → C. There is a functorial (Betti-de Rham) comparison isomorphism

MdR⊗F,ιτ C MB,ιτ ⊗Q C,

which commutes with the action ofC0mot(X,X)E on MdR and MB,ιτ, in particular making this an

isomorphism of freeE ⊗Q C-modules. It induces an isomorphism on the corresponding gradeds(with q = k− p):

grp(MdR⊗F,ιτ C) Hp,qιτ (M) Hp,q(Xιτ) ⊗Q E,

againE ⊗Q C-linear. Just for orientation amidst the formalism, this says that

HTτ(MdR) = HTιτ(MdR) =p : Hp,q(Xιτ) , 0, counted with multiplicity dimC Hp,q(Xιτ)

.

Now, the projectionXιτ ×C,c C→ Xιτ induces a transfer of structure isomorphism on rational Betticohomology, and then anE ⊗Q C-linear isomorphism

F∞ : Hp,qιτ (M)→ Hq,p

cιτ (M).

Let α ∈ C0mot(X,X)E be anE-linear motivated idempotent correspondence defining an object N =

αM ofMF,E. There is a commutative diagram ofE ⊗Q C-linear morphisms

Hp,qιτ (M)

F∞//

αιτ

Hq,pcιτ (M)

αcιτ

Hp,qιτ (M)

F∞// Hq,p

cιτ (M).

Consider the images of the two vertical maps. Since the horizontal maps are isomorphisms, andall the maps areE ⊗Q C-linear, we deduce an isomorphism

αιτHp,qιτ (M) ⊗E⊗QC,ι C

∼−→ αcιτHq,pcιτ (M) ⊗E⊗QC,ι C.

The left-hand side is isomorphic to

grp((αM)dR⊗F⊗QE,ιτ⊗ι C

),

and lettingc′ be a complex conjugation in Gal(E/Q) such thatιc′τ = cιτ, the right-hand side isisomorphic to

grk−p((αM)dR⊗F⊗QE,ιc′τ⊗ι C

).

It follows that

HTc′τ(αM) = k− h : h ∈ HTτ(αM) .But by the previous lemma, HTc′τ(αM) = HTcτ(M), so we are done.

Next we observe (as in§2.4 of [And96b], but with a few more details) that ‘p-adic’ comparisonisomorphisms hold unconditionally inMF,E; this is one of the main reasons for working withmotivated cycles rather than absolute Hodge cycles.

13And a Tate twist, but the statement of the lemma is obviously invariant under Tate twists.

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Lemma 4.1.25.Let N be an object ofMF,E. Then for all v|ℓ, the free E⊗Q Qℓ-module Mℓ isa de Rham representation ofΓFv, and there is a functorial (with respect to morphisms inMF,E)isomorphism of filtered Fv ⊗Q E Fv ⊗Qℓ (Qℓ ⊗Q E)-modules

Fv ⊗F MdR DdR(Mℓ|ΓFv).

Proof. It suffices to check for objects of the formM = H(X) ⊗Q E, for X/F smooth projective.Faltings’ theorem14 provides the comparison

H∗dR(XFv) ⊗Q E DdR

(H∗(XFv

,Qℓ))⊗Q E,

which is an isomorphism of filteredFv ⊗Q E-modules. Letα ∈ C0mot(X1,X2)E be a motivated

correspondence. Letd1 = dim(X1). α has realizationsαdR andαℓ in cohomology groups that arealso compared by Faltings:15

H2d1dR ((X1 × X2)Fv)(d1)) ⊗Q E DdR

(H2d1((X1 × X2)Fv

,Qℓ)(d1))⊗Q E,

and the claim is thatαdR maps toαℓ. The comparison isomorphism is compatible with cycleclass maps,16 and our cycleα is spanned by elements of the form prX1×X2×Y

X1×X2,∗ (β ∪ ∗γ), with Y/Fanother smooth projective variety andβ andγ algebraic cycles onX1 × X2 × Y. Applying thecomparison isomorphism also toX1 × X2 × Y, and applying compatibility with cycle classes, cup-product, (hence) Lefschetz involution, and the projectionprX1×X2×Y

X2,∗17, we can deduce thatαdR maps

to αℓ.

Corollary 4.1.26.Let M be an object ofMF,E. Fix an embeddingι : E → Qℓ, and use this toidentify E as a subfield ofQℓ. Then for allτ : F → Qℓ, HTτ(Mℓ ⊗EQℓ) depends only onτ0 = τ|Fcm.If M is moreover pure of some weight k, then

HTτ0c(Mℓ ⊗E Qℓ) =k− h : h ∈ HTτ0(Mℓ ⊗E Qℓ)

.

Proof. Write ι−1τ for the embeddingF → E induced byτ and ι. By the previous lemma,HTτ(Mℓ ⊗E Qℓ) = HTι−1τ(M). The latter depends only onτ|Fcm, and whenM is pure satisfies therequired symmetry, by Lemmas 4.1.22 and 4.1.24.

Finally, we can show that the Galois lifting results of§3.2 apply to representations arisingfrom objects ofMF,E. In an idealized situation in which all Hodge cycles are motivated (or motivicgroups overF are connected), this will always be the case; in general there is a complication, whichI have not yet been able to avoid, arising from the fact that the proof of Theorem 3.2.7 requiresan initial reduction to the case of connected monodromy group. Fix an embeddingE → Qℓ,inducing a placeλ of E. For short-hand, we denote byGF,Qℓ

the base-change toQℓ of theEλ-groupGF,λ defined by theλ-adic etale fiber functor on the categoryMF,E, and we make the analogousdefinition ofGM

F,Qℓfor objectsM ofMF,E.

14Of which there are now several proofs, including the necessary corrections to the original [Fal89]. A ‘simple’recent proof is [Bei12].

15Note thatαℓ is fixed byΓFv16See part (b) of the proof of Theorem 3.6 of [Bei12].17This last point because compatible with pull-back induced by morphisms of varieties and, again, Poincare

duality.

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Corollary 4.1.27.Let F be a totally imaginary field. Let M be an object ofMF,E and E → Qℓa fixed embedding, to which we associate theℓ-adic representation Mℓ ⊗E Qℓ, which is a represen-tation both ofΓF and of the motivic groupGM

F,Qℓ. Make the following assumption

• GMF

F,Qℓis connected.

Suppose that H′ → H is any central torus quotient of linear algebraic groups over Qℓ, and thatthere is a factorization

GMF,Qℓ

→ H → GLQℓ(Mℓ ⊗E Qℓ),

where of course the composition of these inclusions is the natural representation. Then there existsa geometric liftρ

H′(Qℓ)

ΓF

ρ

55

ρ// GM

F,Qℓ(Qℓ) // H(Qℓ).

Similarly, if F is any number field, but E isQ, then without any assumption on the motivic groupGMF

F, such aρ exists.

Remark 4.1.28. The assumption thatGMF

F,Qℓis connected is expected always to hold. In partic-

ular, this should not be necessary for the conclusion of the corollary to hold.

Proof. Corollary 4.1.26 tells us that composingρ with any irreducible algebraic representationof GM

F,Qℓyields anℓ-adic representation satisfying the Hodge symmetries of Conjecture 3.2.5. But

the arguments of Proposition 3.2.7 and Theorem 3.2.10 do notapply directly, because they assumethese symmetries after composition with irreducible representations of the connected component

(ρ(ΓF)Zar

)0.18 The simplest (but imperfect) way around this is to assumeGMF

F,Qℓis connected, and

then run through the arguments of Proposition 3.2.7 and Theorem 3.2.10 withGMF′

F′,Qℓ, for F′ suffi-

ciently large (see Lemma 4.1.17), in place ofρ(ΓF′)Zar

: reduce to the connected case (replacingFby F′) as in Theorem 3.2.10, and then note that the proof of Proposition 3.2.7 only requires identi-fying a connected reductive group containing the image ofρ such that composition with irreduciblerepresentations of this group yields Galois representations with the desired symmetry (whether ornot these Galois representations are themselves irreducible).

The second assertion (whenE = Q) follows by the same argument: by Remark 4.1.23, theτ-labeled Hodge-Tate co-characterµτ of ρ is independent ofτ. It can be interpreted as a co-character

of (ρ(ΓF)Zar

)0, and composing with finite-dimensional representations ofthis group, we obtain thenecessary Hodge-Tate symmetries to apply the argument of Proposition 3.2.7.

Remark 4.1.29. Similarly, there is a motivic version of Corollary 3.2.8.

18If this group were, contrary to conjecture, not equal toGMF

F ,Qℓ, then it is not obvious in general how to relate the

two putative Hodge symmetries.

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4.1.6. Motivic lifting: the potentially CM case. One case of the desired motivic lifting resultis immediately accessible, when the corresponding Galois representations are potentially CM. Wedenote byCMF the Tannakian category of motives (for motivated cycles) over F generated byArtin motives and potentially CM abelian varieties. As before an embeddingσ : F → C yields,through the corresponding Betti fiber functor, a Tannakian groupTF(σ) = Aut⊗(Hσ|CMF ). For anumber fieldE, we can also consider, as in§4.1.4, the categoryCMF,E of potentially CM motivesoverF with coefficients inE, with its Tannakian groupTF,E(σ).19 As withGF, there is a sequence

1→ TF,E → TF,E → ΓF → 1,

where again the projectionTF,E → ΓF has a continuous sectionsλ onEλ-points. Deligne showed in[DMOS82] (Chapter IV: ‘Motifs et groupes de Taniyama’) that TQ is isomorphic to the Taniyamagroup constructed by Langlands as an explicit extension ofΓQ by the connected Serre group.20

Proposition 4.1.30.Let F be totally imaginary. Let H′ → H be a surjection of linear algebraicgroups over a number field E with central torus kernel. For anyhomomorphismρ : TF,E → H,there is a finite extension E′/E such thatρ lifts to a homomorphism

H′E′

TF,E′

ρ<<②

②②

② ρ// HE′ .

Proof. Fix a finite placeλ of E and consider theλ-adic representationρλ = ρsλ : ΓF → H(Eλ).SinceTF is isomorphic to the connected Serre group, we can unconditionally apply Corollary4.1.27 to find, for some finite extensionE′λ/Eλ a geometric lift ˜ρλ : ΓF → H′(E′λ) of ρλ. Note thatρλ is potentially abelian, sinceρλ is, and the kernel ofH′ → H is central. Proposition IV.D.1of [DMOS82] implies that ρλ arises from a homomorphism (of groups overE′

λ) TF ⊗Q E′

λ

TF,E ⊗E E′λ→ H′ ⊗E E′

λ. This in turn must be definable over some finite extensionE′/E (E′ is thus

embedded inE′λ), i.e. there is a homomorphism ˜ρ : TF,E′ → H′ ⊗E E′ whose extension toE′

λgives

rise toρλ. We explain this technical point in Lemma 4.1.31 below. To check that ˜ρ is actually a liftof ρ ⊗E E′, it suffices to observe:

• The restriction toTF,E′ is a lift: the restrictions of ˜ρ andρ are simply the algebraic homo-morphisms of the connected Serre group (tensored withE′) corresponding to the labeledHodge-Tate weights of ˜ρλ andρλ.• Theλ′-adic lift, for the placeλ′ of E′ induced byE′ → E′λ, is a lift (by construction).• It suffices to check that ˜ρ is a lift onE′λ′-points. First, it suffices to check that ˜ρ⊗E′ E′λ′ liftsρ ⊗E E′

λ′ . This in turn can be checked onE′λ′-points: we are free to replaceTF,E′ ⊗E′ E′

λ′

by some finite-type quotientT in which theE′λ′-points are Zariski-dense. Then the closed

subschemeT ×H×H

H → T (where the two maps toH × H are the diagonalH → H × H

and the product of the two mapsT → H × H given byρ andρ followed by the quotientH′ → H) containsT (E′

λ′), hence equalsT .Then we are done, sinceTF,E′(E′λ′) = TF,E′(E

′λ′) · sλ′(ΓF).

19From now on,σ will be implicit.20More precisely, equipped with the data of the projection toΓQ, a section onAF, f -points, and the co-character

Gm,C → TC giving the Hodge filtration on objects ofCMC, TQ is uniquely isomorphic to the Taniyama group,equipped with its corresponding structures.

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Here is the promised lemma showing that ˜ρ may be defined over a finite extensionE′ of E:

Lemma 4.1.31. Let K/k be an extension of algebraically closed fields of characteristic zero.Let T be a (not necessarily connected) reductive group over k, and letπ : H′ → H be a surjection(defined over k) of reductive k-groups. Suppose thatρ : T → H is a k-morphism, and that thescalar extensionρK lifts to a K-morphismρ : TK → H′K. Thenρ lifts to a k-morphism T→ H′.

Proof. This is a simple spreading-out argument. Namely, writingK as the direct limit of itsfinite-typek-sub-algebras, we can descend the relationπK ρ = ρK to some finitely-generatedk-sub-algebraRof K, in particular obtaining ˜ρR: TR→ H′R such that ˜ρR⊗R K = ρ. We then choosea non-zerok-pointα : R→ k (a morphism ofk-algebras), and specializing the relationπR ρR = ρR

via α, we obtain a liftρR⊗R,α k of ρ overk.

Remark 4.1.32. • We have stated Proposition 4.1.30 only for imaginary fields for sim-plicity, but of course there is a variant for general number fields taking into account Corol-lary 3.2.8. WhenF is totally real, there will be suchρ that do not lift: simply take atype A Hecke characterψ of a quadratic CM extensionL/F as in Example 2.5.3. ThenAd0(IndF

L (ψ)) will have a potentially abelian Galoisℓ-adic realizationsΓF → SO3(Qℓ),which arises from a representation ofTF by Proposition IV.D.1 of [DMOS82]. As wehave seen, these representations do not lift geometricallyto GSpin3.• The method of checking that ˜ρ– once we know it exists– actually liftsρ will recur in §4.2;

this division into a geometric argument (liftingρ|TF,Eand a Galois-theoretic argument

(lifting ρsλ) seems to be the natural way to make arguments about motivic Galois groupsover number fields (compare Lemma 4.1.17).• On the automorphic side, representations of the global Weilgroup ought to parametrize

‘potentially abelian’ automorphic representations. In that case, the analogous lifting resultis a theorem of Labesse ([Lab85]).

Before proceeding to more elaborate examples, let us clarify here that, even though we studylifting throughcentralquotients, the nature of these problems is in fact highly non-abelian. That is,suppose we have a lift ˜ρ : GF,E → H′E of ρ : GF,E → HE (we may assumeH′ andH are reductive;let us further suppose they are connected), and letr ′ andr be irreducible faithful representations ofH′E andHE. If the derived group ofH is not simply-connected, but its simply-connected coverHsc

injects intoH′, then typicallyr ′ ρ will not lie in the Tannakian subcategory ofMF,E generated byr ρ and all potentially CM motives, i.e. objects ofCMF,E (of course, it does in the example ofProposition 4.1.30). We make this precise at the Galois-theoretic level:

Lemma 4.1.33.Let H′ → H, r′, and r be as above. Letρ : ΓF → H(Qℓ) be a geometric Galoisrepresentation having a geometric liftρ : ΓF → H′(Qℓ). Assume that

• the algebraic monodromy group ofρ is H itself;

• the kernel of Hsc∩ (ρ(ΓF))Zar → H is non-zero.

Then r′ρ is not contained in the Tannakian sub-category of semi-simple geometricQℓ-representationsof ΓF generated by r ρ and all potentially abelian geometric representations.

Proof. If r ′ ρwere contained in this category, then there would exist an irreducible potentiallyabelian representationτ of ΓF and an injectionr ′ ρ → τ ⊗ (r1 ρ) for some irreducible algebraic

110

representationr1 of H. By Proposition 3.4.1,τ has the form IndFL (ψ · ω) for some characterψand irreducible Artin representationω of ΓL. By Frobenius reciprocity, there is a non-zero map(r ′ ρ)|ΓL → ψ · (r1 ρ)|ΓL ⊗ω; both sides are irreducible,21 soω is one-dimensional, and absorbingω into ψ we may assumer ′ ρ|ΓL

∼−→ ψ · (r1 ρ)|ΓL . Comparing algebraic monodromy groups, we

obtain a contradiction, since by assumptionHsc∩ (ρ(ΓF))Zar

cannot inject intoGm× H.

In §4.2, we will see many examples ofρ with full SO monodromy group; their lifts to GSpinwill satisfy the conclusions of Lemma 4.1.33.

4.2. Motivic lifting: the hyperk ahler case

4.2.1. Setup.The aim of this section is to produce a lifting not merely at the level of a singleℓ-adic representation, but of actual motives, in a very special family of cases whose prototype isthe primitive second cohomology of aK3 surface overF. So that the reader has some examples tokeep in mind, we recall the following definitions:

Definition 4.2.1. LetF be a subfield ofC. A hyperkahler variety XoverF is a geometricallyconnected and simply-connected smooth projective varietyoverF of even dimension 2r such thatΓ(X,Ω2

X) is one-dimensional, generated by a differential formω for whichωr is non-vanishing atevery point ofX (i.e., as a linear functional on the top wedge power of the tangent space). AK3surfaceoverF is a hyperkahler varietyX/F of dimension 2.

Example 4.2.2. The simplest example of a K3 surface is a smooth quartic hypersurface inP3.Higher-dimensional examples of hyperkahlers are notoriously difficult to produce. One standardfamily is gotten by starting with any K3 surfaceX, and then for any integerr ≥ 1 consideringthe Hilbert schemeX[r ] of r points onX (more precisely, the moduli of closed sub-schemes oflengthr); eachX[r ] is a hyperkahler variety, with of courseX[1] = X. (This construction is due toBeauville: see [Bea83, §6].)

More generally, we work in the axiomatized setup of [And96a]. By a polarized varietyovera subfieldF of C we mean a pair (X, η) consisting of varietyX/F and anF-rational ample linebundleη on X. For a fixedk ≤ dimX, we will consider the ‘motive’Prim2k(X)(k) (omitting theη-dependence from the notation), as an object ofMF. Cup-product withη lets us endow the motiveH2k(X)(k) with the quadratic form

〈x, y〉η = (−1)kx∪ y∪ ηdim X−2k ∈ H2 dimX(X)(dim X) Q,

and then we can definePrim2k(X)(k) as the orthogonal complement ofH2k−2(X)(k − 1)∪ η.The two Weil cohomologies we use areℓ-adic (with coefficients inQℓ or some extension inside

Qℓ) and Betti cohomology; we will occasionally take integral Betti cohomology, where we willalways work modulo torsion, so thatPrim2k(XC,Z)(k) is a sub-module ofH2k(XC,Z)(k)/torsion.In that case, the primitive lattice defines a polarized (by〈·, ·〉η) integral Hodge structure of weight0; we writehp,q for the Hodge numbers. Andre proves his theorems, which include versions of theShafarevich and Tate conjectures, under the following axioms:

Ak: h1,−1 = 1, h0,0 > 0, andhp,q = 0 if |p− q| > 2.

21For the right-hand side, see Proposition 3.4.1: the tensor product of a Lie-irreducible and an Artin representationis irreducible; of course, for the purposes of this lemma, wecould just further restrictL.

111

Bk: There exists a smooth connectedF-schemeS, a points ∈ S(F), and a smooth projectivemorphismf : X → S such that:

– X Xs;– the Betti classηB ∈ H2(Xan

C,Z)(1)/torsionextends to a section ofR2 f an

C∗Z(1)/torsion;– letting S denote the universal cover ofS(C), and D denote the period domain of

Hodge structures onVZ := Prim2k(XC,Z)(k) polarized by〈·, ·〉η, we require that theimage ofS→ D contain an open subset.

B+k : For eacht ∈ S(C), every Hodge class inH2k(Xt,Q)(k) is an algebraic class.

Ak is essential to the method, which relies on studying the associated Kuga-Satake abelian variety,which exists for Hodge structures of this particular form.Bk is a statement about deformingX intoa ‘big’ family. B+k is of course a case of the Hodge conjecture, which is always known whenk = 1(the theorem of Lefschetz). But we provide these axioms merely for orientation. Of interest is thefollowing collection of varieties for which they are known to hold:

Proposition 4.2.3. The axioms A1, B1, and B+1 are satisfied by: abelian surfaces, polarizedsurfaces of general type with h2,0(X) = 1 andKX · KX = 1, and polarized hyperkahler varietieswith b2 > 3 (in particular, K3 surfaces). Cubic fourfolds (polarized viaOP5(1)) satisfy A2, B2, andB+2 .

This relies on the work of many people; see§2 and§3 of [And96a]. Finally, Andre observesthat these axioms are independent of the choice of embeddingF → C; note that forAk this is aspecial case of Remark 4.1.23.

Remark 4.2.4. The proposition does not apply to hyperkahler varieties with Betti number 3,since their (projective) deformation theory is not sufficiently robust. In fact, it is believed thatsuch hyperkahlers do not exist. Regardless, in that case one can still say something about themotivic lifting problem, since theℓ-adic representationH2(XF ,Qℓ) is potentially abelian. In prin-ciple, this should force the underlying motivic Galois representation to factor throughTF, at whichpoint we would apply Proposition 4.1.30; but this argument would require checking that twoGF-representations with isomorphicℓ-adic realizations are themselves isomorphic, i.e. an unknowncase of the Tate conjecture. At least we have (as in Proposition 4.1.30):

Lemma 4.2.5. Let X/F be a smooth projective variety with b2(X) = 3. If necessary, replaceF with a quadratic extension trivializingdetH2(XF ,Qℓ). Then theℓ-adic realizationρℓ : GF →SO(H2(XF ,Qℓ)) has a lift ρℓ to GSpin(H2(XF ,Qℓ)) that arises as theℓ-adic realization of a repre-sentation ofTF,E for suitable E.

Notation 4.2.6. • From now on we viewF as a subfield ofC, with F its algebraic clo-sure inC. These embeddings will be used implicitly to define Betti realizations, motivicGalois groups (for Betti realizations), and etale-Betti comparisons, for varieties (or mo-tives) over extensions ofF insideF.• We writeVQ for Prim2k(XC,Q)(k), and for a primeℓ, we will write Vℓ for theℓ-adic real-

izationPrim2k(XF ,Qℓ)(k). As before,Prim2k(X)(k) will be used to indicate the underlyingmotive, i.e. object ofMF.• For fieldsE containingQ, we will sometimes denote the extension of scalarsVQ ⊗Q E by

VE (or similarly forQℓ andVℓ). This is the Betti realization of an objectPrim2k(X)(k)E ofMF,E.

112

• The same subscript conventions will hold for other motives we consider, especially thedirect factors ofPrim2k(X)(k) given by the algebraic cycles (to be denotedAlg) and itsorthogonal complement, the transcendental lattice (T). These motives will be discussedin §4.2.4. In the meantime, we recall that the rational Hodge structurePrim2k(XC,Q)(k)has aQ-subspace spanned by its Hodge classes; the orthogonal complementTQ of thissubspace is thetranscendental subspaceis again a polarizedQ-Hodge structure.• For an objectM of MF, we will sometimes writeρM for the associated motivic Galois

representation. For theℓ-adic realization of this motivic Galois representation (i.e., therepresentation ofΓF on Hℓ(M)), we writeρM

ℓ.

• First assume dimVQ = m is odd. The group GSpin(VQ) may not have a rationally-definedspin representation, but it does after some extension of scalars, and over any suitablylarge fieldE, we denote byrspin: GSpin(VE) → GL(WE) this algebraic representation. Ifm is even, we similarly denote byWE = W+,E ⊕W−,E the direct sum of the two half-spinrepresentations. In contrast toVE = VQ ⊗Q E, this WE is not necessarily an extension ofscalars from an underlyingQ-space.

Possibly after replacingF by a quadratic extension, we may assume that theΓF-representationVℓ is special orthogonal, i.e.ρℓ : ΓF → SO(Vℓ) ⊂ O(Vℓ). Since for almost all finite placesv, thefrobenius f rv acts onH2k−2(XF ,Qℓ) andH2k(XF ,Qℓ) with eigenvalues that are independent ofℓ

([Del74]), and trivially on theℓ-adic Chern classηℓ, the eigenvalues off rv on Prim2k(XF ,Qℓ)(k)are independent ofℓ. Consequently, if detρℓ is trivial for oneℓ, then it is for allℓ. We will fromnow on assume, for technical simplicity, that this determinant condition is satisfied.

4.2.2. The Kuga-Satake construction.We review the classical Kuga-Satake constructionand outline Andre’s refinement, which implies a potential version (i.e., after replacingF by afinite extension) of our motivic lifting result. See§4 and§5 of [And96a]. His approach is inspiredby that of [Del72], in which Deligne used the Kuga-Satake construction (in families) to reducethe Weil conjectures forK3 surfaces to the (previously known) case of abelian varieties. LetVZbe the polarized quadratic latticePrim2k(XC,Z)(k) of the previous subsection. Writem = 2n orm= 2n+ 1 for its rank. Basic Hodge theory implies that the pairing onVR is negative-definite onthe sub-space (H1,−1 ⊕ H−1,1)R, and positive-definite onH0,0

R, hence has signature (2,m− 2). Re-

calling the discussion of§4.1.1, this real Hodge structure yields a homomorphismh: S→ SO(VR)that lifts uniquely to a homomorphismh: S → GSpin(VR) whose compositionNspin h with theClifford normNspin is the usual normS→ Gm,R.

Let C(VZ) andC+(VZ) denote the Clifford algebra and the even Clifford algebra associated tothe quadratic spaceVZ. Let LZ be a free leftC+(VZ)-module of rank one. Denote byC+ the ringEndC+(VZ)(LZ)

op. Because of the ‘op,’C+ naturally acts onLZ on the right, and we then correspond-ingly have EndC+(LZ) C+(VZ). The choice of a generatorx0 of LZ induces a ring isomorphism

C+(VZ)φx0−−→ C+

c 7→(bx0 7→ bcx0).

Via h and the tautological representation GSpin(VR) → C+(VR)×, LR := LZ ⊗Z R acquires a Hodgestructure of type (1, 0), (0, 1); the Hodge type is easily read off from the fact that EndC+(LZ) C+(VZ) is also an isomorphism of Hodge-structures,22 or by recalling the remarks of§4.1.1 and

22Giving C+(VZ) the Hodge structure induced from that on the tensor powers of VZ.

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listing the weights of the spin representation. Concretely(as in the original construction of Satake[Sat66] and Kuga-Satake [KS67]), we can choose an orthogonal basise1, e2 of (H1,−1 ⊕ H−1,1)R,normalized so that〈ei , ei〉 = −1 for i = 1, 2. Then the automorphism ofC+(VR) given by multipli-cation bye1e2 is a complex structure, by the defining relations for the Clifford algebra. In fact, thisintegral Hodge structure is polarizable: this can be shown explicitly, or by a very soft argument(apply Proposition 2.11.b.iii ′ of [Del72]). We therefore obtain a complex abelian variety, the Kuga-Satake abelian varietyKS(X) = KS(XC, η, k), associated to our originalVZ. The right-action ofC+

on LZ commutes with the Hodge structure, soC+ acts as endomorphisms ofKS(X). Generically,this will be the full endomorphism ring; we will have to be attentive later to how much bigger theendomorphism ring can be.

One of the main technical ingredients in [And96a] is the following descent result, which usesrigidity properties of the Kuga-Satake construction:

Lemma 4.2.7 (Main Lemma 1.7.1 of [And96a]). Let (X, η) be a polarized variety over a subfieldF ofC, satisfying properties Ak and Bk. Then there exists an abelian variety AF′ over some *finite*extension F′/F such that

• The base-change AC is the Kuga-Satake variety KS(X);• there is a subalgebra C+ of End(AF′) and an isomorphism ofZℓ[ΓF′ ]-algebras

EndC+(H1(AF ,Zℓ)

) C+(Prim2k(XF ,Zℓ)(k)).

We subsequently writeLℓ for the ℓ-adic realizationH1(AF′ ⊗ F,Qℓ). The main result of the‘motivated’ theory of hyperkahlers, (the foundation of Andre’s results on the Tate and Shafarevichconjectures) is:

Theorem 4.2.8 (see Theorem 6.5.2 of [And96a]). For some finite extension F′/F, Prim2k(XF′)(k)is a direct factor (inMF) of End

(H1(AF′)

), and both Prim2k(XF′)(k) and H1(AF′ ) have connected

motivic Galois group.

Write ρA : GF′ → GL(LQ) andρV : GF → SO(VQ) for the motivic Galois (for the Betti realiza-tion) representations associated toH1(AF′ ) andPrim2k(X)(k). We will somewhat sloppily use thesame notationρA andρV for various restrictions of these representations (eg, toGF). Note that byCorollary 4.1.20, the image ofρA is the Mumford-Tate group ofKS(X), which is easily seen to becontained in GSpin(VQ) = x ∈ C+(VQ)× : xVQx−1 = VQ. More precisely:

Corollary 4.2.9.For some finite extension F′/F, ρA factors throughGSpin(VQ) and liftsρV|GF′ .A fortiori, ρA

ℓlifts ρV

ℓ|ΓF′ .

Proof. OverC (or F), the image ofρA is MT(AC), hence contained in GSpin(VQ). We cantherefore compare the two maps

GC(ρV,πρA)−−−−−−→ SO(VQ) × SO(VQ)→ GL

(C+(VQ)

) ×GL(C+(VQ)

),

whereπ denotes the projection GSpin(VQ)→ SO(VQ). Under the motivated isomorphism

C+(Prim2k(XC)(k)) End(H1(A)

)

the adjoint action ofρA on End(LQ) agrees with the action ofπ ρA onC+(VQ), so the two compo-sitionsGC → GL

(C+(VQ)

)above coincide. Now, ifm is odd, SO(VQ) → GL(C+(VQ)) is injective,

114

soρV = π ρA; that is, at least overC, ρA lifts ρV. If m is even, we deduce that the compositions

GC(ρV,πρA)−−−−−−→ SO(VQ) × SO(VQ)→ GL(∧2VQ) ×GL(∧2VQ)

agree.23 The kernel of∧2 : SO(VQ) → GL(∧2VQ) is ±1 (and central), so we see thatρV andπ ρA agree up to twisting by some characterχ : GC → ±1 ⊂ SO(VQ). But clearly this characterfactors throughGM

C, whereM = Prim2k(XC)(k) ⊕ H1(A), and by Theorem 4.2.8,GM

Cis connected.

Thereforeχ is trivial, andρA lifts ρV asGC-representations. By Lemma 4.1.21, the same holdsfor the correspondingGF-representations, and then as in Lemma 4.1.17 the same holdsover somefinite extensionF′/F.

This result does not always hold withF′ = F, and it is our task in the coming sections to achievea motivic descent overF itself. The Kuga-Satake variety is highly redundant, and itis technicallyconvenient to work with a smaller ‘spin’ abelian variety, many copies of which constitute theKuga-Satake variety. Since the rational Clifford algebraC+

Qmay not be split, this requires a finite

extension of scalarsE/Q, after which we can work in the isogeny categoryAV0F′ ,E of abelian

varieties overF′ with E-coefficients. We takeF′ as in the Corollary, and now to ease the notation,we write simplyA for AF′ .

Lemma 4.2.10.There exists a number field E and an abelian variety B/F′ with endomorphismsby E such that there is a decomposition in AV0

F′ ,E:

A⊗Q E B2nif m = 2n+ 1;

A⊗Q E B2n−1if m = 2n.

Theℓ-adic realization H1(BF,Qℓ) is isomorphic to the composite rspinρAℓ

as(E⊗Qℓ)[ΓF′ ]-modules,where as beforeρA

ℓdenotes the representationΓF′ → GSpin(Vℓ⊗E) obtained from Lℓ⊗E, and rspin

denotes either the spin (m odd) or sum of half-spin (m even) representations ofGSpin(Vℓ⊗E). Whenm= 2n, B decomposes in AV0

F′ ,E as B+ × B−, corresponding to the two half-spin representations.

Proof. Choose a number fieldE splittingC+(VQ). Then, lettingWE denote either the spin (modd) representation or the direct sumW+,E ⊕W−,E of the two irreducible half-spin representations(m even),C+(VE) is isomorphic as GSpin(VE)-representations toW2n

E , or to W2n−1

E,+ ⊕W2n−1

E,− . As E-algebra it is then isomorphic either to End(WE) M2n(E) or End(W+,E) ⊕ End(W−,E) M2n−1(E) ⊕M2n−1(E).24 Using the orthogonal idempotents inC+E C+(VE), we decompose the objectAF′ ⊗Q Eof AV0

F′ ,E into 2n (whenm is odd) or 2n−1 (whenm is even) copies of an abelian varietyB/F′ withcomplex multiplication byE.

We state a simple case of the main result of this section; for the proof, see§4.2.5. More generalversions will be proven in stages, depending on the complexity of the transcendental lattice ofVQ.

Theorem 4.2.11. Let (X, η) be a polarized variety over a number field F⊂ C for whichPrim2k(XC,Z)(k) satisfies axioms Ak, Bk, and B+k . Possibly enlarging F by a quadratic extension,we may assume as above that for allℓ, detVℓ = 1, so that theℓ-adic representationρV

ℓmapsΓF to

SO(Vℓ). We make the following hypothesis on the monodromy:

23The filtration onC+(VQ) given by the image ofV⊗≤2i is motivated, since O(V)-stable, soρV andπ ρA coincideon thei = 1 graded piece, which is just∧2(VQ).

24Note that the fieldE can be made explicit if we know the structure of the quadraticlattice VZ. See Example4.2.12 for a case whereE = Q.

115

• The transcendental lattice TQ hasEndQ−Hodge(TQ) = Q.• det(Tℓ) = 1.25

Then there exists:

• a finite extension E/Q• an object B of AV0F,E;• an Artin motive M inMF,E

• for each primeλ of E, a lift ρλ

GSpin(Vℓ ⊗ Eλ)

rspin// GLEλ(WEλ)

ΓF

ρλ66♥

♥♥

♥♥

♥♥

// SO(Vℓ) ⊂ SO(Vℓ ⊗ Eλ)

such that the composite rspinρλ is isomorphic to some number of copies26of Mλ⊗(H1(BF,Qℓ) ⊗

E⊗QℓEλ

)

as Eλ-linear ΓF-representations. In particular, the system of liftsρλλ is weakly compatible. It iseven ‘motivic’ in the sense of arising from a lift

GSpin(VE)

GF,E

ρ99s

ss

ss

ρ// SO(VE).

This theorem is an optimal (up to the O(Vℓ) ⊃ SO(Vℓ) distinction) arithmetic refinement of the(a priori highly transcendental) Kuga-Satake construction, showing the precise sense in which itdescends to the initial field of definitionF.

4.2.3. A simple case.To achieve the refined descent of Theorem 4.2.11, the basic idea isto apply our Galois-theoretic lifting results (which applyover F) to deduceΓF-invariance of theabelian varietyB that we know to exist overF′; Faltings’s isogeny theorem ([Fal83]) implies thatthis invariance is realized by actual isogenies; then we apply a generalization of a technique usedby Ribet ([Rib92]; our generalization is Proposition 4.2.27) to study elliptic curves overQ that areisogenous to all of theirΓQ-conjugates (so-called “Q-curves”). Ribet’s technique applies to ellipticcurves without complex multiplication, and we will have to keep track of monodromy groupsenough to reduce the descent problem to one for an absolutelysimple abelian variety. In somecases, a somewhat ‘softer’ argument than Ribet’s works– we give an example in Lemma 4.2.22–but in addition to being satisfyingly explicit, the Ribet method seems to be more robust.

But first we prove Theorem 4.2.11 in the simplest case, when the Hodge structureVQ is‘generic,’ dimVQ = 2n + 1 is odd, and the even Clifford algebraC+(VQ) is split overQ. Ourworking definition of ‘generic’ will be thatVQ contains no trivialQ-Hodge sub-structures (i.e., itis equal to its transcendental lattice), and that EndQ−Hodge(VQ) = Q.

Example 4.2.12. IfX/F is a genericK3 surface, then these hypotheses are satisfied. TheK3-lattice H2(XC,Z) is an even unimodular lattice of rank 22 whose signature over R is (by Hodge

25Unlike the first hypothesis, this determinant condition is atechnicality; it can again be arranged after an (inde-pendent ofℓ) quadratic extension. Of course, in the generic case in which VQ = TQ, it is no additional hypothesis.

26Which can be made explicit.

116

theory) (3, 19). The classification of even unimodular lattices impliesit is isomorphic (overZ) to(−E8)⊕2 ⊕ U⊕3, whereE8 andU are theE8-lattice and the hyperbolic plane lattice. OverQ, theorthogonal complement of the ample classη is isomorphic to (−E8)⊕2 ⊕ U⊕2 ⊕ 〈−q(η)〉, whereqis the quadratic form and〈α〉 denotes the one-dimensional quadratic space with a generator whosesquare isα. SincePrim2(XC,Q) is odd-dimensional, the basic structure theory of Clifford algebras(see Chapter 9, especially Theorem 2.10, of [Sch85]) implies thatC+(Prim2(XC,Q) is a centralsimple algebra overQ, whose Brauer class is simply twice the Brauer class ofE8 plus twice the(trivial) Brauer class ofU. This is obviously the trivial class, soC+(VQ) is in this case isomorphicto M210(Q). More generally, this argument applies toVQ = Prim2(XC,Q)(1) for X a hyperkahlersatisfying:

• b2(X) > 3;• H2(XC,Z) is even;• the number of copies of theE8-lattice, i.e. the numberb2(X)−6

8 , is even.

Now, our ‘generic’ hypothesis implies that the Hodge group of VQ is the full SO(VQ) (moregenerally, see Zarhin’s result, quoted as Proposition 4.2.19, below); it follows without difficultythat the Mumford-Tate groupMT(AC) is the full GSpin(VQ), and End0(AC) = C+

Q. By our sec-

ond simplifying hypothesis, this Clifford algebra is isomorphic to a matrix algebraM2n(Q), and,writing WQ for the spin representation of GSpin(VQ), we have two isomorphisms of GSpin(VQ)-representations:

C+(VQ) W2n

Q ;

C+(VQ)ad End(WQ),

where GSpin(VQ) acts on the firstC+(VQ) by left-multiplication, and onC+(VQ)ad by conjugation(i.e., through the natural SO(VQ)-action).

We may certainly enlargeF′ to a finite extension over which all endomorphisms ofAC aredefined, and the complex multiplication byC+

Qthen gives an isogeny decompositionA ∼ B2n

,

whereB/F′ is an abelian variety with End0(B) = Q. We can take the spin representationWQto be equal toH1(BC,Q) (and, extending scalars and invoking comparison isomorphisms, we getidentifications with other cohomological realizations ofB– in particular, theℓ-adic realizationWλ =WQ ⊗Q Qℓ).

Proof of Theorem 4.2.11when VQ is generic. We sketch the argument, with some details post-poned until later sections– the goal here is to review Ribet’s method and outline the argument ina simple case. We will now apply the technique of [Rib92] in combination with our abstractGalois-theoretic lifting results. By Corollary 4.1.27, there exists a lift

ρℓ : ΓF → GSpin(VQℓ)

of ρVℓ, and we can normalize this lift so that in the spin representation its labeled Hodge-Tate

weights HTτ(rspinρℓ) are of ‘abelian variety’-type, i.e. 2n−1 zeroes and 2n−1 ones, for eachτ : F →Qℓ.

27 This normalization determines ˜ρℓ up to finite-order twist, and it implies that ˜ρℓ|ΓF′ is a finite-order twist ofρA

ℓ, since they both liftρV

ℓ|ΓF′ with the same Hodge-Tate data. We may therefore

replaceF′ by a finite extension and assume

ρAℓ = ρℓ|ΓF′ .

27For details, see Lemma 4.2.21.

117

Sinceρℓ begins life overF, we see thatρAℓ

is ΓF-conjugation invariant. The composition ofρAℓ

with the Clifford representation is 2n copies of theℓ-adic representationρBℓ

associated toB, soρBℓ

is alsoΓF-invariant. By Faltings’s theorem, for eachσ ∈ ΓF , there exists an isogenyµσ : σB→ B;we can and do arrange thatµσ is defined first for a (finite) system of representativesσi in ΓF forGal(F′/F), and then defined in general byµσih = µσi for all h ∈ ΓF′ . The collection ofµσ yields anobstruction class

[cB] ∈ H2(ΓF ,End0(B)×) = H2(ΓF ,Q×), 28

given by

cB(σ, τ) = µσ σµτ µ−1στ.

That is,cB measures the failure of the diagram

στBµστ

//

σµτ

B

σBµσ

==⑤⑤⑤⑤⑤⑤⑤⑤

to commute. Now, the classcB may be non-trivial, and the abelian varietyB may not descend(up to isogeny) toF.29 Nevertheless, Tate’s vanishing resultH2(ΓF ,Q

×) = 0 tells us that there is a

continous 1-cochainα : ΓF → Q×

whose coboundary equalscB, i.e. cB(σ, τ) = α(στ)α(τ)−1α(σ)−1.By continuity,α is locally constant with respect toΓF′′ for some finiteF′′/F′, and it takes valuesin some finite extensionQ(α) of Q. We now consider the restriction of scalars abelian varietyC := ResF′′/F(B).30 C is an abelian variety overF with endomorphism algebraR = End0(C),which asQ-vector space is isomorphic to

HomF′′

∏

σ∈Gal(F′′/F)

σB, B

=⊕

σ

Qµσ,

where again we use the fact that End0(BF) = Q. Write λσ for the element ofR corresponding toµσ under this isomorphism.

Lemma 4.2.13 (Lemma 6.4 of [Rib92]). The algebra structure ofR is given byλσλτ = cB(σ, τ)λστ,so there is aQ-algebra homomorphismα : R → Q(α) given by theQ-linear extension ofλσ 7→α(σ).

Since the isogeny category of abelian varieties overF is a semi-simple abelian category, wecan form the object

M = ResF′′/F(B) ⊗R,α Q(α).

We regardM as an object ofAV0F,Q(α), withQ(α)-rank (in the obvious sense) equal to theQ-rank of

B. Moreover, for any placeλ|ℓ of Q(α), theλ-adic realization

ρMλ : ΓF → GLQ(α)λ(Mλ)

28The local constancy of the isogeniesµσ implies this is a continuous cohomology class, withQ× equipped withthe discrete topology.

29That triviality of this class is equivalent to isogeny-descent is Theorem 8.2 of [Rib92].30HereB is really the base-change toF′′, but we omit this not to clutter the notation.

118

has projectivization isomorphic to the canonical projective descent toΓF of theΓF-invariant, ir-reducible representation ofΓF′′ on H1(BF,Q(α)λ).31 So, ρM

λ|ΓF′′ and ρB

ℓ|ΓF′′ rspin ρℓ|ΓF′′ are

isomorphic up to twist, hencerspin ρℓ andρMλ are twist-equivalent asΓF-representations. The

representationrspin: GSpin(VQ) → GL(WQ) is the identity on the center, so after identifyingρMλ

to a representation onWλ ⊗ Q(α)λ, we see that it factors through GSpin(Vℓ ⊗ Q(α)λ) as a lift ofρVℓ⊗Q(α)λ. The required motivic lift is the representation ofGF,Q(α) corresponding toH1(M);32 and

the variousρMλ

form a compatible system because they are formed from Tate modules of abelianvarieties.

4.2.4. Arithmetic descent: preliminary reduction. Now we proceed to a more general ar-gument, making first some preliminary reductions to the analogous lifting problem for the tran-scendental lattice. We must invoke Andre’s work on the Tateconjecture forX.

Theorem 4.2.14 (see Theorem 1.6.1 of [And96a]). Let (X, η) be a polarized variety over anumber field33 F satisfying Ak, Bk, and B+k . Then:

• Prim2k(XF ,Qℓ)(k) is a semi-simpleΓF-representation• the Galois invariants Prim2k(XF ,Qℓ)(k)ΓF are allQℓ-linear combinations of algebraic cy-

cles;

The Tate conjecture forH2k(X) then implies:

Lemma 4.2.15. There is an Artin motive Alg over F whose Betti realization isthe subspaceof VQ spanned by algebraic cycle classes, and whoseℓ-adic representation is the (Artin)ΓF-representation on VΓF′

ℓfor any F′/F large enough (and Galois) that all of these cycle classes are

defined over F′. The transcendental lattice T likewise descends to an object ofMF. In particular,there is an orthogonal decomposition

Vℓ = Algℓ ⊕ Tℓ

of ΓF-representations (not merelyΓF′ -representations).

Proof. Giving an Artin motive overF is equivalent to giving a representation ofΓF on a (finite-dimensional)Q-vector space. In our case, the space (Q-span) of cycles for homological equiva-lence

Zkhom(XF′ ) → H2k(XF ,Qℓ)(k),

or rather its intersection withVℓ, does the trick. Since the objectPrim2k(X)(k) ofMF is polarized,we can defineT inMF as the orthogonal complement ofAlg.

We will enlargeF′ as in the Lemma, so that all algebraic classes inPrim2k(XC,Q)(k) are alreadydefined overF′, and so that the motivic groupGT

F′ is connected (see Lemma 4.1.17). Note thatTQis an orthogonal Hodge structure of type (1,−1), (0, 0), (−1, 1), with h1,−1 = 1, so the Kuga-Satakeconstruction applies to it as well (see Variant 4.1.5 of [And96a]). Since the Kuga-Satake varietyassociated toVQ is simply an isogeny power of that associated toTQ, the latter, to be denotedA(T),also descends to some finite extensionF′/F.

31Under the homomorphismR → EndΓF′′ (⊕sigmaH1(B,Qℓ), the λσ permute the factors in the direct sum; the

projection viaR α−→ Q(α) collapses all the factors to a single copy with scalars extended toQ(α), i.e. toH1(B,Qℓ) ⊗QQ(α). This implies the claim about projective descents.

32For more details on how to check this carefully, see Corollary 4.2.18.33Faltings’s theorem works for finitely-generated extensions ofQ, so this does as well.

119

We introduce a little more notation. After extending scalars to a sufficiently large fieldE (omit-ted from the notation unless we want to emphasize it) we denote by (rspin,V,WV), (rspin,Alg,WAlg),and (rspin,T ,WT) the spin representations of these three spin groups; as before, by ‘the’ spin repre-sentation in theDn case we will mean the direct sum of the two half-spin representations.

Lemma 4.2.16.Suppose thatρVℓ

factors throughSO(Algℓ) × SO(Tℓ) → SO(Vℓ).34 If we have

found liftsρAlgℓ

andρTℓ

to GSpin(Algℓ⊗Qℓ) andGSpin(Tℓ⊗Qℓ) such that rspin,AlgρAlgℓ

and rspin,T ρTℓ

are motivic, thenρVℓ

has a lift ρℓ : ΓF → GSpin(Vℓ ⊗Qℓ) such that rspin,V ρℓ is also motivic. If theindividual lifts rspin,Alg ρAlg

ℓand rspin,T ρT

ℓbelong to compatible systems ofℓ-adic representations,

then the same holds for rspin,V ρℓ.35

Proof. The isomorphism ofgradedalgebras (for the graded tensor product⊗)

C(AlgQ)⊗C(TQ)∼−→ C(VQ)

induces an inclusionC+(AlgQ) ⊗C+(TQ) → C+(VQ), and then a map (not injective)

C+(AlgQ)× ×C+(TQ)× → C+(VQ)×,

which in turn induces a commutative diagram

GSpin(AlgQ) ×GSpin(TQ) //

GSpin(VQ)

SO(AlgQ) × SO(TQ) // SO(VQ).

As long asρVℓ

: ΓF → SO(Vℓ) factors through SO(Algℓ) × SO(Tℓ), this shows that we can liftAlgℓandTℓ in order to liftVℓ. We next want to understand the restriction to GSpin(AlgE)×GSpin(TE) ofthe spin representation of GSpin(VE); Just for this argument, we will ignore the similitude factor,i.e. work with weights of Spin rather than GSpin. We can writebases of the character lattices ofSO(AlgE), SO(TE), and SO(VE) as, respectively,χ1, . . . , χa, χa+1, . . . , χa+t, and

χ1, . . . , χa+t if either dim(T) or dim(Alg) is even;

χ1, . . . , χa+t+1 if both dim(T) and dim(Alg) are odd.

The set of weights of the spin representation ofso(AlgE) (and similarly for the other cases) is thenall 2a weights of the form36

a∑

i=1

±χi

2.

In the case where at least one of dim(T) and dim(Alg) is even, we see that the weights ofWAlg⊠WT

are precisely those ofWV|GSpin(AlgE)×GSpin(TE), and thereforeWV WAlg ⊠ WT as GSpin(AlgE) ×GSpin(TE)-representations. When both dim(T) and dim(Alg) are odd, weight-counting givesWV

(WAlg ⊠WT)⊕2.Thus, if we have found lifts ˜ρAlg

ℓand ρT

ℓ(to GSpin(Algℓ ⊗ Qℓ) and GSpin(Tℓ ⊗ Qℓ)) such that

rspin,Alg ρAlgℓ

and rspin,T ρTℓ

are motivic (respectively, belong to compatible systems ofℓ-adic

34At worst, ensuring this requires making a quadratic extension of F.35Recall that objects ofMF are not in general known to give rise to compatible systems.36The uniform description of the weights in the even and odd cases results from taking the sum of the two half-spin

representations.

120

representations), then the resulting lift to GSpin(Vℓ ⊗Qℓ), in its spin representation, is a direct sumof tensor products of motivic Galois representations (respectively, Galois representations belongingto compatible systems), hence is motivic.

Corollary 4.2.17. If detTℓ = 1 asΓF-representation, and if we can find a liftρTℓ

of ρTℓ

suchthat rspin,T ρT

ℓis motivic (respectively, belongs to a compatible system),then we can do the same

for ρVℓ.

Proof. SinceAlgℓ is an Artin representation, Tate’s vanishing result allowsus to lift to an Artinrepresentation ˜ρ

Algℓ

: ΓF → GSpin(Algℓ ⊗ Qℓ). The corollary follows.

Corollary 4.2.18.As in Corollary 4.2.17, assume thatdetTℓ = 1, and that for some numberfield E, and placeλ|ℓ of E, we can find a liftρT

λ: ΓF → GSpin(Tℓ ⊗Qℓ Eλ) such that rspin,T ρT

λis

theλ-adic realization of an object M ofMF,E whose base-change to some F′/F is one of the spindirect factors of the Kuga-Satake motive (with scalars extended to E) associated to TQ (see Lemma4.2.10). Then possibly enlarging E to a finite extension E′, there is a lifting of representations ofthe motivic Galois groupGF,E′ :

GSpin(VQ ⊗ E′)

GF,E′

77♣♣

♣♣

♣♣

ρV// SO(VQ ⊗ E′).

Proof. The Artin representationρAlgℓ

is definable overQ, and lifts to GSpin(AlgE1) after makingsome finite extensionE1/Q. The conclusion of the corollary will hold withE′ equal to the compos-ite E1E. We check it using the same principle as in the proof of Proposition 4.1.30: that is, we check‘geometrically’ (for the restriction toGF,E′) and for theλ-adic realization. By Corollary 4.2.9, themotivic representationρA(T) of GF , and by extension ofGF,E′ , factors through GSpin(TE′) and liftsρT : GF,E′ → SO(TE′). SinceH1(A(T)) ⊗Q E′ is just some number of copies37 of M, the same istrue ofρM. By assumption, theλ-adic realizationρM

λis justrspin ρT

λ, so this also factors through

GSpin(Tλ), lifting ρTλ. Using the sectionsλ : ΓF → GF,E′(E′λ), so thatGF,E′(E′λ) = sλ(ΓF) ·GF,E′(E

′λ),

we conclude as in Proposition 4.1.30 thatρM lifts ρT ⊗ E′.CombiningρM with the lift of ρAlg

ℓ, as in Lemma 4.2.16, we similarly find a lift to GSpin(VQ⊗Q

E′) of our givenρV : GF,E′ → SO(VQ ⊗Q E′).

4.2.5. Arithmetic descent: the generic case.To summarize, we have reduced the problemof finding motivic lifts of the motive (overF) Prim2k(X)(k) to the corresponding problem for thetranscendental latticeT. In this section we treat the ‘generic’ case in which the Hodge structureTQhas trivial endomorphism algebra, using a variant of Ribet’s method (which will return in§4.2.6)We isolate this case both to demonstrate a slightly different argument, and because the non-genericcases will require even deeper input, Andre’s proof of the Mumford-Tate conjecture in this context(see Theorem 4.2.25). The starting-point of the analysis ofthe motiveT is Zarhin’s calculation in[Zar83] of the Mumford-Tate group:

Proposition 4.2.19 (Zarhin).Let TQ be aQ-Hodge structure with orthogonal polarization andHodge numbers h1,−1 = 1, h0.0 > 0, and hp,q = 0 if |p− q| > 2. Moreover assume that TQ contains

372t for dimT odd; 2t−1 for dimT even.

121

no copies of the trivialQ-Hodge structure. Then ET = EndQ−Hodge(TQ) is a totally real or CM field,and TQ is a simple MT(TQ)-module. There is a non-degenerate ET-hermitian38 pairing

〈·, ·〉 : TQ × TQ → ET

such that

MT(TQ) = Aut(TQ, 〈·, ·〉E) ⊂ SO(TQ).

For the rest of this section, we assume that EndQ−Hodge(TQ) = Q; the Mumford-Tate groupMT(TQ) is then the full SO(TQ).

Lemma 4.2.20. AssumeEndQ−Hodge(TQ) = Q. Then the Mumford-Tate group, and thereforethe motivic Galois group, of the Kuga-Satake variety A(T) is equal toGSpin(TQ). Consequently,End0(A(T)) = C+

Q C+(TQ).

Proof. Easy.

We now choose a number fieldE/Q splittingC+(TQ), and consider the decomposition inAV0F′ ,E

A(T) ∼ B(T)2tif dim(TQ) = 2t + 1;

A(T) ∼ B(T)2t−1 ∼ (B+(T) × B−(T))2t−1if dim(TQ) = 2t,

as in Lemma 4.2.10. We saw in Corollary 4.2.9 thatρA(T) factors through GSpin(TQ) and liftsρT ; viewing ρA(T) in GSpin(TQ), we then have the relationrspin (ρA(T) ⊗ E) = ρB(T), taking theBetti realization ofB(T) to be our model for the spin representation. We letB0 equalB(T) inthe odd case andB+(T) in the even case. Similarly, we letr0 denoterspin or one of the half-spinrepresentations (which we may assume corresponds toB+(T)). We also for convenience fix anembeddingE → Qℓ.

Lemma 4.2.21. (Without any assumption onEndQ−Hodge(TQ)) There exists a liftρTℓ

: ΓF →GSpin(Tℓ ⊗ Qℓ) of ρT

ℓand a finite extension F′′/F′ such that

rspin ρTℓ |ΓF′′ H1(B(T)F,Qℓ) ⊗E Qℓ|ΓF′′ .

Proof. As in the arguments of§3.2, we first choose a liftΓF → GSpin(Tℓ ⊗ Qℓ) with finite-order Clifford norm. In the root datum notation of§2.8, the Hodge-Tate cocharactersµτ of ρT

ℓ, for

all τ : F → Qℓ, are (conjugate to)λ1, and the finite-order Clifford norm lifts have Sen operators(up to conjugacy) corresponding to the ‘co-characters’39 µτ = λ1. We can modify this initial lift bytwisting byλ0ω′, whereω′ : ΓF → Q

×ℓ is a Galois character whose square differs from the inverse

ω−1 of the cyclotomic character by a finite-order twist (suchω′ exists for anyF). This gives a newlift ρT

ℓwhose composition with the Clifford norm (which, recall, is 2χ0 in our notation) isω−1,

up to a finite-order twist, and whoseτ-labeled Hodge-Tate weights in the spin representation are1 and 0 with equal multiplicity (compare the argument of§4.1.1). Thenrspin ρT

ℓ|ΓF′ differs from

H1(B(T)F ,Qℓ)⊗E Qℓ by a finite-order twist, hence they are isomorphic after someadditional finitebase-changeF′′/F′.

38In the case of totally realET , this means symmetric.39That is, in the the co-character group tensored withQ.

122

Lemma 4.2.22. There is a factor M ofResF′′/F(B0) (viewed as an object of AV0F,E), having

endomorphisms by a finite extension E′/E, and an embedding E′ → Qℓ extending our fixed E→Qℓ such that the associatedℓ-adic realization Mℓ40 is isomorphic asQℓ[ΓF ]-representation to r0 ρTℓ.

Proof. By the (E-linear) Tate conjecture,

EndAV0F,E

(ResF′′/F(B0)

) ⊗E Qℓ EndQℓ[ΓF ]

(IndF

F′′(H1(B0,F,Qℓ) ⊗E Qℓ)

).

The Galois representation being induced isr0 ρTℓ, and inside this endomorphism ring we can

considerHomQℓ[ΓF ]

(r0 ρT

ℓ , IndFF′′ (r0 ρT

ℓ )),

which by Frobenius reciprocity is just

EndQℓ[ΓF′′ ]

(r0 ρT

ℓ

)= Qℓ,

since EndAV0F′′,E

(B0) = E. In other words, there is a uniqueQℓ-line in EndAV0F,E

(ResF′′/F(B0)

) ⊗E Qℓ

consisting of projectors onto ther0 ρTℓ-isotypic piece of theℓ-adic representation. Decomposing

the semi-simpleE-algebra EndAV0F,E

(ResF′′/F(B0)

)into simple factors, we see that this line lives in

a unique simple component (tensored withQℓ), which itself must be just a finite field extensionE′

of E (else ther0 ρTℓ-isotypic piece would have multiplicity greater than 1); itthen corresponds

to exactly one of the simple factors ofE′ ⊗E Qℓ, i.e. a particular embeddingE′ → Qℓ. Wecan therefore takeM to be the abelian variety corresponding to this factorE′; M has complexmultiplication by E′, and via this specified embeddingE′ → Qℓ, the ℓ-adic realizationMℓ isisomorphic tor0 ρT

ℓ.

Remark 4.2.23. This is at its core the same proof as given in§4.2.3; the latter proof is probablymore transparent, but this one is somewhat ‘softer.’ I don’tthink it translates as well to the moregeneral context of§4.2.6, however.

Corollary 4.2.24.Theorem 4.2.11 holds for the motive Prim2k(X)(k) over F.

Proof. When dim(TQ) is odd, we are done, by the previous lemma and Corollaries 4.2.17 and4.2.18. When dim(TQ) is even, we take the outputM in AV0

F,E′ of the previous lemma, view itinMF,E′ , and form the twisted dualM∨(−1). Here there are two cases: if dimTQ is not divisibleby four, this object corresponds to the composition of the other half-spin representation with ˜ρT

ℓ,41

and we can now apply the earlier corollaries toM ⊕ M∨(−1). If dimTQ is divisible by four (sothe half-spin representations are self-dual), apply Lemma4.2.22 to the composition of ˜ρT

ℓwith the

otherhalf-spin representation as well; so instead of a single motive M, we now have two motivesM+ andM−

42 such that theℓ-adic realization (M+⊕M−)ℓ is isomorphic torspin ρTℓ. Then as before,

we may apply Corollary 4.2.18.

40Via E′ → Qℓ: that is,Mℓ := H1(MF ,Qℓ) ⊗E′ Qℓ.41The two half-spin representations of GSpin2n have highest weights−χ0 +

12(

∑n−1i=1 χi + χn) (for r0) and−χ0 +

12(

∑n−1i=1 χi −χn) (for the other half-spin representation), so the lowest weight of r∨0 ⊗ (−2χ0) is−χ0− 1

2(∑n

i=1 χi), which,whenn is even, is visibly the lowest weight of the other half-spin representation.

42Enlarge the coefficient fields ofM+ andM−, viewed as subfields ofQℓ via the respective embeddings producedby Lemma 4.2.22, to some common over-field.

123

4.2.6. Non-generic cases:dim(TQ) odd. To study the non-generic case EndQ−Hodge(TQ) , Q,we have to understand theℓ-adic algebraic monodromy groups of the representationsρT

ℓ, i.e. the

ℓ-adic analogue of Zarhin’s result. Andre has announced ([And96a, Theorem 1.6.1]) a proof of theMumford-Tate conjecture in this context; the arguments of [And96a, §7.4] are not quite complete,but will be completed in a forthcoming paper ([Moo]) of Ben Moonen, in the course of generalizingsome of Andre’s work. We therefore accept as proven the following:

Theorem 4.2.25 (Andre-Moonen).Let (X, η) be a polarized variety over a number field43 F

satisfying Ak, Bk, and B+k . Then the inclusionρℓ(ΓF)Zar

→ GVF ⊗Q Qℓ of the algebraic monodromy

group into theℓ-adic motivic Galois group of V= Prim2k(X)(k) is an isomorphism on connectedcomponents of the identity.

Recall that over the fieldF′, we may assume the groupsρℓ(ΓF′)Zar

andGVF′ are connected,

and therefore isomorphic. Combining Theorem 6.5.1 of [And96a] with Andre’s result that Hodgecycles on abelian varieties are motivated, and with Zarhin’s description ([Zar83]) of the Mumford-Tate group of the transcendental latticeT2k(XC,Q)(k), we obtain (see Corollary 1.5.2 of [And96a]):

Corollary 4.2.26.The semisimpleQ-algebra ET := EndGTF′

(T) is a totally real or CM field,and there is a natural ET-hermitian pairing〈·, ·〉ET : T×T → ET . The motivic group (which equalsthe Mumford-Tate group, and equals, after⊗QQℓ, theℓ-adic algebraic monodromy group)GT

F′ isthen isomorphic to the full orthogonal (ET totally real) or unitary (ET CM) group

Aut(T, 〈·, ·〉ET ) → SO(TQ).

Before continuing, we formulate a variant of Ribet’s methodwith coefficients:

Proposition 4.2.27.Let F′/F be an extension of number fields, and let E/Q be a finite exten-sion. Suppose we are given an object B of AV0

F′ ,E such that for some embedding E→ Qℓ, the

associatedℓ-adic realization Bℓ = H1(BF,Qℓ) ⊗E Qℓ satisfies the invariance conditionσBℓ Bℓ

for all σ ∈ ΓF . Further assume thatEndAV0F,E

(BF) = E.44 Then there exists a finite extension E′/E,

an extension E′ → Qℓ of the embedding E→ Qℓ, a number field F′′/F′, and an object M ofAV0

F,E′ such that (H1(MF ,Qℓ) ⊗E′ Qℓ

)|ΓF′′ Bℓ|ΓF′′ .

That is, B, up to twist, has a motivic descent to F.

Proof. This is proven as on page 117, using theE-linear variant of Faltings’ theorem:

HomAV0F′,E

(σB, B) ⊗E Qℓ∼−→ HomQℓ[ΓF′ ]

(H1(σBF,Qℓ) ⊗E Qℓ,H

1(BF,Qℓ) ⊗E Qℓ

).

In the notation of the earlier proof,M = ResF′′/F(B) ⊗R,α Q(α) is the required motivic descent, toan isogeny abelian variety overF with Q(α)-multiplication (so in the statement of the proposition,E′ → Qℓ isQ(α) ⊂ Q → Qℓ, extending the initialE → Qℓ).

We now assume that dimQ TQ is odd, say of the formcd whered = 2d0 + 1 = [ET : Q]. Inparticular,ET is totally real. Let ˜ρT

ℓandB(T) be as in Lemma 4.2.21, and replaceF′ by a large

enough extension to satisfy the conclusion of that lemma, and such that (ρTℓ(ΓF′))Zar is connected.

43Or, again, a finitely-generated extension ofQ.44This necessarily holds forB in AV0

F′ ,E as well.

124

Recall thatB(T) is an object ofAV0F′ ,E for some finite extensionE/Q large enough to splitC+(TQ).

Corollary 4.2.26 implies that

(ρTℓ⊗ Qℓ)(ΓF′ )

Zar

d∏

1

SOc(Qℓ) ⊂ SOcd(Qℓ).

Restricting the spin representationWcd of socd(Qℓ) to∏d

1 soc(Qℓ), we obtain (via a weight calcula-tion as in Lemma 4.2.16, and writingWc for the spin representation ofsoc)

Wcd|∏ soc (⊠d1Wc)

2d0.

The Lie algebra of the lift ˜ρTℓ

is one copy of the additive groupga times the Lie algebra ofρTℓ⊗Qℓ,

and thisga acts by scalars in the spin representation, sorspinρTℓ|ΓF′ has an analogous decomposition

as 2d0 copies of some (absolutely, Lie) irreducible representationW′ of ΓF′ . For a suitable enlarge-mentE′ of E (and extensionE′ → Qℓ), andF′′ of F′, we can realizeW′ asH1(B(T)′,Qℓ)⊗E′Qℓ foran objectB(T)′ of AV0

F′′ ,E′ .45 Then EndAV0

F′′,E′(B(T)′) = E′, and the isomorphismrspin ρT

ℓ (W′)2d0

impliesΓF-conjugation invariance ofW′. Thus we can apply Ribet’s method to deduce:

Lemma 4.2.28.There exists a finite extension E′′/E′, an embedding E′′ → Qℓ extending thegiven E′ → Qℓ, and an object M of AV0F,E′′ such that

(H1(MF ,Qℓ) ⊗E′′ Qℓ|ΓF′′′

)W′|ΓF′′′

for some still further finite extension F′′′/F′′.

Let Mℓ denote the associatedℓ-adic realization (viaE′′ → Qℓ). Sincerspin ρTℓ

is Lie-isotypic,andMℓ is a descent toΓF of its unique (after finite restriction) Lie-irreducible constituent, Corollary3.4.2 shows that there is an Artin representationω of ΓF such that

rspin ρTℓ Mℓ ⊗ ω.

Possibly enlarging the field of coefficients yet again, we deduce:

Theorem 4.2.29. SupposedimTQ is odd, and thatdetTℓ = 1 (as ΓF-representation). Thenthere exists

• a number fieldE and an embeddingE → Qℓ;• an objectM ofMF,E that is a tensor product of an Artin motive and (the image of) an

object of AV0F,E

;

• and a lift ρTℓ

: ΓF → GSpin(Tℓ ⊗ Qℓ) of ρTℓ;

such that rspin ρTℓ

is isomorphic to theℓ-adic realization (viaE → Qℓ) of M. Moreover:

• ρTℓ

lives in a weakly-compatible system of lifts;• ρT

ℓactually arises from a lifting of representations of the motivic Galois groupGF,E, as in

Theorem 4.2.11.• The same conclusions hold with the motive V= Prim2k(X)(k) in place of its transcenden-

tal lattice T (and, again, a possible enlargement ofE).

45More precisely, we need the decomposition ofWcd|∏ soc to be defined overE′; the first claim then follows fromFaltings. The extensionF′′/F′ is needed to decomposeB(T)F ⊗E E′ over some finite extension.

125

Proof. The number fieldE is the composite (inside the ambientQℓ) of the fieldE′′ and thefield needed to define the Artin representationω. To conclude the proof of the theorem, we makethree observations:

• MF,E is Tannakian (note that we already know thatM46 andω are motivic);• M andω both give rise to compatible systems ofℓ-adic representations;• Corollary 4.2.18 applies to lift the representations of motivic Galois groups.

4.2.7. Non-generic cases:dimTQ even. We do not fully treat the case of even-rank transcen-dental lattice, but here give a couple examples, describingthe ‘shape’ of the Galois representationsin light of Proposition 3.4.1.

First, continue to assumeET is totally real. Let dimTQ = 2n = cd, with d = [ET : Q]. For anyN, denote byW2N,± (for each choice of±) the two half-spin representations ofso2N, and continueto writeW2N+1 for the spin representation ofso2N+1.

Lemma 4.2.30.When c is even, the restriction Wcd,+|∏ soc is given by

Wcd,+|∏d1 soc

⊕

ǫ=(ǫi )∈±d∏ǫi=1

⊠di=1Wc,ǫi ,

where the indexing set ranges over all choices of signs with− occurring an even number of times.This is a direct sum of distinct Lie-irreducible representations.

When c is odd, so d= 2d0 is even,

Wcd,+|∏d1 soc 2d0−1

⊠d1 Wc,

a single Lie-irreducible representation occurring with multiplicity 2d0−1.

Now, recall (Lemma 4.2.21) that after a sufficient base-changeF′/F, we can find an abelianvarietyB+(T) (with coefficients in a number fieldE, embedded inQℓ; we may by extending scalarsassumeE is large enough that the above decomposition of spin representations is defined overE),and a liftρT

ℓsuch thatr+ ρT

ℓ|ΓF′ is isomorphic toH1(B+(T),Qℓ) ⊗E Qℓ. We assumeF′ sufficiently

large that this Galois representation is a sum of Lie-irreducible representations.

Proposition 4.2.31.Suppose c is even. Then motivic lifting holds forρT.

Proof. Sincec is even, the previous lemma shows thatr+ ρTℓ

is Lie-multiplicity-free, henceis a direct sum of inductions of non-conjugate, Lie-irreducible Galois representations. Ifπ0(ρT

ℓ) is

trivial, in which case ˜ρTℓ

also has connected monodromy group,47 then no inductions occur in thisdecomposition, so each factor⊠d

1Wc,ǫi in Lemma 4.2.30 corresponds to a Lie-irreducible factor ofr+ ρT

ℓ, asΓF-representation. Each of these factors is, overF′, of the form

H1(Bǫ,Qℓ) ⊗E Qℓ,

whereBǫ is an object ofAV0F′ ,E with endomorphism algebra justE itself (the usual application of

Faltings’ theorem, using the Lie-multiplicity-free property, and the fact that the spin representation

46Precisely,M ⊗E′′ E.47Since it contains the center of GSpin; the sort of example this avoids isρ f ,ℓ⊗ω

1−k2 , wheref is a classical modular

form of odd weightk.

126

decomposition holds overE). By ΓF-invariance of each factor, we can apply Proposition 4.2.27toproduce an objectMǫ of AV0

F,Eǫ, for some finite extensionEǫ/E insideQℓ, with Mǫ,ℓ|ΓF′ isomorphic

to a finite-order twist ofH1(Bǫ ,Qℓ)⊗EǫQℓ. By Lie-irreducibility, some finite-order twist (a characterof Gal(F′/F), in fact, and, again, we may have to enlargeEǫ) M′

ǫ of Mǫ hasℓ-adic realizationisomorphic to the corresponding factor ofr+ ρT

ℓ. Inside the ambientQℓ, we take the compositum

E′ of the variousEǫ, extending scalars on eachM′ǫ. Then the object⊕ǫM′

ǫ ofMF,E′ satisfies thehypotheses of Corollary 4.2.18, so we deduce the existence of the desired motivic lift.

We sketch the case of non-connected monodromy. Take the (motivic) Lie-irreducible factorsof r+ ρT

ℓ|ΓF′ , and partition them intoΓF-orbits. Fix a representativeWi of each orbit, and consider

the stabilizerΓFi in ΓF of Wi. Arguing as in Proposition 4.2.31, we can apply Ribet’s method todescend eachWi to an objectMi ofMFi ,Ei for some extensionEi of E insideQℓ. Moreover, twistingyields anM′

i whoseℓ-adic realization is isomorphic to a factor ofr+ ρTℓ.48 Then we can induce

(the representation of motivic Galois groups) fromFi to F to obtain our motives overF. Thiscompletes the case of evenc (andET totally real).

Having demonstrated the available techniques in a couple ofquite different situations (namely,where the lifts range between the extremes of being Lie-multiplicity-free and Lie-isotypic), westop here, remarking only that whenET is CM, the analysis must begin not from Lemma 4.2.30 butfrom the restrictionWcd,+|∏ glc, the product ranging over pairs of complex-conjugate embeddingsET → Qℓ. This restriction is given by:

Lemma 4.2.32. Suppose d= 2d0 is even, with notation otherwise as above. We denote irre-ducible representations ofglc by W(r), where W is an irreducible representation ofslc, and (r)indicates that the restriction to the centerga ⊂ glc is multiplication by r. Then, letting Vi denotethe standard representation of the ith copy ofslc

Wcd,+|∏d01 glc

⊕

i1,...,id0:∑i j∈2Z

∧i1V∗1(c2− i1) ⊠ . . . ⊠ ∧id0V∗d0

(c2− id0).

Remark 4.2.33. • Note that the representations occurring here do not necessarily extendto representations of GLc, sincec may be odd; they do extend on the (connected) double-cover of GLc.• In particular, we see that whenET is CM, r+ ρT

ℓis Lie-multiplicity-free. This suggests

proceeding as in Proposition 4.2.31, although we will stop here.

4.3. Towards a generalized Kuga-Satake theory

4.3.1. A conjecture. It is fair to assume that one could establish a motivic lifting result forthe remaining hyperkahler cases. More important, these lifting results clamor for generalization.Motivated by the Fontaine-Mazur conjecture, Theorem 3.2.10, Proposition 4.1.30, and Theorem4.2.29, we are led to the following much more ambitious conjecture:

Conjecture 4.3.1. Let F and E be number fields, and let H′ ։ H be a surjection, with centraltorus kernel, of linear algebraic groups over E. Suppose we are given a homomorphismρ : GF,E →

48The isomorphisms HomΓF′ (M1,V) = HomΓFi(IndFi

F′ (Mi),V) = HomΓFi(Mi ⊗ Qℓ[Gal(F′/Fi)],V), with V =

r+ ρTℓ, imply this claim, using the fact thatMi is Lie-irreducible and after finite restriction occurs withmultiplicity

one inr+ ρTℓ.

127

H. Then if F is imaginary, there is a finite extension E′/E and a homomorphismρ : GF,E′ → H′E′lifting ρ ⊗E E′. If F is totally real, then such a lift exists if and only if theHodge number parityobstruction of Corollary 3.2.8 vanishes.

To indicate the scope of this conjecture, letX/F be any smooth projective variety, and con-sider for anyk ≤ dimX the motiveH2k(X)(k) (or Prim2k(X)(k), having chosen an ample linebundle). This gives rise to an orthogonal representation ofGF, and the conjecture in this case (forGSpin→ SO, or the variant with the full orthogonal group in place of SO) amounts to a generaliza-tion of the Kuga-Satake construction toarbitrary orthogonally-polarized motivic (overF) Hodgestructures. For other choices ofH′ andH (for instance, GLn → PGLn, where necessarily we willhave coefficient field larger thanQ), the conjectured generalization is even more mysterious.49

Note the role of Lemma 4.1.31 in building our confidence in this conjecture. LetH′ → H bea morphism of groups, say overQ, as in the conjecture. One way of formulating the Fontaine-Mazur conjecture is thatGF ⊗Q Qℓ should be isomorphic to the Tannakian group for the Tannakiancategory Repg,ss

Qℓ(ΓF) of semi-simple geometricQℓ-representations ofΓF . In particular, a geometric

lift ρℓ : ΓF → H′(Qℓ) of ρℓ : ΓF → H(Qℓ) should arise from an algebraic homomorphism ofQℓ-groupsGF ⊗ Qℓ → H′

Qℓ. Lemma 4.1.31 then tells us that ifρ arises from someGF ⊗ Q→ H, then

we can find a lift ˜ρ : GF ⊗ Q→ H′ of homomorphisms ofQ-groups.Now that we know to look for such a thing, we conclude by givingone more example in which

it is easy to construct a ‘generalized Kuga-Satake motive.’

4.3.2. Motivic lifting: abelian varieties. If A/C is an abelian surface, thenH2(A,Q) ∧2H1(A,Q) has Hodge numbersh2,0 = 1 andh1,−1 = 4. The classical Kuga-Satake construc-tion then associates toA another abelian variety,KS(A), with rational cohomologyC+(H2(A,Q))(or the analogue withPrim2(A,Q), if we have fixed a polarization), and a theorem of Morrison([Mor85]) asserts thatKS(A) is isogenous toA8 (or A4 if we work with Prim2(A,Q)). We nowshow that this construction can be generalized to abelian varieties of any dimension.

Let F be any subfield ofC, and letA/F be an abelian variety of dimensiong with a fixedpolarization. Consider the algebraic representation

ρ : GF → GSp(H1(AC,Q)) GSp2g,

or, for the theory overC, its restriction toGC. We will composeρ with a homomorphism GSp2g→GSpinN for suitableN to produce the generalized Kuga-Satake motive. LetW = H1(AC,Q), andlet re1+e2 : Sp(V) → GL(Ve1+e2), in the weight notation of§2.8, denote the irreducible representa-tion of Sp(V) obtained as the complement of the trivial representation in ∧2(W). This absolutelyirreducible representation, defined overQ, has image in SO(Ve1+e2), where the quadratic form isinduced from the pairing canonically induced on∧2(W), and which coincides with the inducedpolarization onPrim2(AC,Q). Since Sp(W) is simply-connected, there is a lift to an algebraichomomorphism ˜r : Sp(W) → Spin(Ve1+e2).

50 The dimension ofVe1+e2 is(2g2

)− 1, which is odd

49Compare Lemma 4.1.33.50OverC this follows from standard Lie theory, and such a homomorphism descends toQ (see Lemma 4.1.31);

since the map of Lie algebrassp(W)→ so(Ve1+e2) is defined overQ, we see that the map overQ is ΓQ-invariant, hencedescends to a morphism overQ.

128

(= 2n+ 1) if g is even, and even (= 2n) if g is odd. In either case, we have the representationrn

of Spin(Ve1+e2)Q having highest weight n =∑n

i=1 χi

2 .51

Lemma 4.3.2. Let c denote the non-trivial (central) element of the kernelof Spin(Ve1+e2) →SO(Ve1+e2). If g ≡ 2, 3 (mod 4), thenr(−1) = c, and if g≡ 0, 1 (mod 4), thenr(−1) = 1.

Proof. In all cases,re1+e2(−1) = 1, so ˜r(−1) equals either 1 orc. Sincec acts as−1 in any of thespin representations (c is the element−1 of the Clifford algebra), it suffices to computern r(−1).The weights ofVe1+e2 are

±(ei + ej)1≤i< j≤g ∪ ei − eji, j ,

except with one copy of the weight zero deleted (so that zero has multiplicityg− 1 rather thang).It follows thatrn r has a weight equal to

12

∑

1≤i< j≤g

(ei + ej) +∑

1≤i< j≤g

(ei − ej)

=g∑

i=1

(g− i)ei .

In particular,rn r(−1) is multiplication by (−1)g(g−1)/2, and the lemma follows.

Corollary 4.3.3. ˜r : Sp(W)→ Spin(Ve1+e2) extends to an algebraic homomorphismGSp(W)→GSpin(Ve1+e2). If g ≡ 2, 3 (mod 4), then this map can be chosen so the Clifford norm coincideswith the symplectic multiplier; if g≡ 0, 1 (mod 4), then this map can be chosen to factor throughSpin(Ve1+e2). The composition

GFρ−→ GSp(W)

r−→ GSpin(Ve1+e2) → GL(C+(Ve1+e2))

defines the generalized Kuga-Satake lift of A.

Proof. This follows immediately from Lemma 4.3.2 and the identifications:

Gm× Sp(W)〈(−1,−1)〉

∼−→ GSp(W)

Gm× Spin(Ve1+e2)

〈(−1, c)〉∼−→ GSpin(Ve1+e2).

Wheng ≡ 2, 3 (mod 4), we take the mapGm→ Gm to be the identity, and wheng ≡ 0, 1 (mod 4),we take it to be trivial.

Remark 4.3.4. • Repeating the above arguments with∧2(W) in place ofVe1+e2, we cansimilarly construct liftsGF → GSpin(∧2(W)).• Wheng = 2, this recovers the classical construction. In that case, the compositionrn r

is the identity, and, decomposingC+(Ve1+e2) as 4 copies ofrn (as GSpin(Ve1+e2) represen-tation), the identification (up to isogeny)KS(A) ∼ A4 is nearly a tautology.

The motivic formalism now tells us thatrn r ρ is (the Betti realization of) an object ofMF.Since theℓ-adic realizations ofρ form a weakly52 compatible system, the same is true for theℓ-adicrealizations of this Kuga-Satake motive. In this case, however, we can say more, and will realizethis explicitly (and unconditionally) as a Grothendieck motive. The first step is to compute the

51We use the common fundamental weight notation here. This is the spin representation in the odd case; one ofthe half-spin representations in the even case.

52In fact, strictly.

129

plethysmrn r; we will do this, but first mention an equivalent, structurally appealing plethysm.First we treat theDn case, that is whend = 2g = dimH1(X) satisfies 2n =

(d2

); this amounts

to dimX being even. LetV = H2(X) = ∧2W. We use the following (common) notation forfundamental weights ofDn:

• i = χ1 + . . . + χi, andri = ∧iV, for i = 1, . . . , n− 2;• n−1 =

χ1+...+χn−1−χn

2 , n =∑n

1 χi

2 , andrn−1, rn are the two half-spin representations.As representations ofso2n, we have the following identity:

2n⊕

i=0

∧i(V) =(rn−1 ⊕ rn

)⊗2.

The plethysm problem that we expect to solve, then, is to describe a representationrKS(X) of Sp(W)such that

2n⊕

i=0

∧i(∧2W) = (rKS(X))⊗2.

(More ambitiously, we could attempt this with∧2k(W) instead.) Similarly, in theBn case, we havefundamental weights i =

∑i1 χi for i = 1, . . . , n− 1, andn =

∑χi

2 , corresponding, respectivelyto the wedge powers∧iV of the standard representation (r1) and the spin representation. Asrepresentations ofso2n+1, we have the identity

n⊕

i=0

∧i(V) = r⊗2n,

so in this case we want a representationrKS of Sp(W) satisfyingn⊕

i=0

∧i(∧2(W)) = r⊗2KS.

Proposition 4.3.5. Writingωi (i = 1, . . . , g) for the usual fundamental weights53 of Cg, then inall cases (g odd or even) we have

∧•(∧2(W)) 2g(Vω1+...+ωg−1)⊗2,

as Sp(W)-representations. Here∧• denotes the full exterior algebra. This is deduced from theabove discussion and the two calculations:

• (even) AsSp(W)-representations,

(rn−1 ⊕ rn) r 2g/2Vω1+...+ωg−1.

• (odd) AsSp(W)-representations,

rn r 2(g−1)/2Vω1+...+ωg−1.

Proof. We treat the even case, the odd case being essentially identical. There areg2−gnon-zeroweights in∧2(W), and the weight zero occurs with multiplicityg. The weights of (rn−1 ⊕ rn) rare sums of plus or minus anyn = g2− g

2 weights of∧2(W), then the total divided by two. It followsthat the highest weight of (rn−1 ⊕ rn) r is, as previously computed,

∑g−1i=1 ωi, but moreover that

it occurs with multiplicity 2g/2 (here g2 = n − (g2 − g); we can choose the weight+0 or −0 this

53ωi = e1 + . . . + ei in the standard coordinate system

130

many times). It follows that (rn−1 ⊕ rn) r contains 2g/2Vω1+...+ωg−1; by dimension count (see thefollowing Lemma 4.3.6), they are isomorphic.

Lemma 4.3.6. With the above notation, the dimension of the irreducibleSp(W)-representationVω1+...+ωg−1 is 2g(g−1).

Proof. Simplifying the Weyl dimension formula, we find

dimVω1+...+ωg−1 = 2|Φ+ |

∏

1≤i≤ j≤g

2g+ 1− (i + j)2g+ 2− (i + j)

,

where the number|Φ+| of positive roots is 2g2. The product telescopes: fix ani, and the correspond-

ing product overj is equal to12. The lemma follows.

Corollary 4.3.7. Let F be any subfield ofC, and let X/F be an abelian variety of any dimen-sion g, giving rise to a representation

ρH1(X) : GF → GSp(H1(X,Q)).

Then the Kuga-Satake lift (Corollary 4.3.3 and remark following) of the representation

GF → SO(H2(X,Q)(1))

can be explicitly realized in the spin (or sum of half-spin) representation as2⌊g2⌋ copies of the com-

position rω1+...+ωg−1 ρH1(X) with the highest weightω1+ . . .+ωg−1 representation ofSp(H1(X,Q)).54

Let us call this object ofMF the Kuga-Satake motive KS(X). Then KS(X) is in fact a Grothendieckmotive, for either numerical or homological equivalence.

Proof. It remains to check thatKS(X) can be cut out by algebraic cycles. We start with theexplicit description (due to Weyl; see§17.3 of [FH91]) of the representationVω1+···+ωg−1. From nowon, abbreviateλ =

∑g−11 ωi, andr =

∑g−11 i (the length ofλ); in fact, what follows applies to an

arbitrary partitionλ. ThenVλ =W〈r〉 ∩ Sλ(W)

as Sp(W)-representation. We have to explain this notation:Sλ(W) denotes the Schur functor asso-ciated to the partitionλ, which explicitly is equal to the image of the Young symmetrizercλ actingon W⊗r ; andW〈r〉 is the subspace ofW⊗r given by intersecting the kernels of all the contractions(1 ≤ p < q ≤ r)

cp,q : W⊗r −→W⊗(r−2)

v1 ⊗ . . . ⊗ vr 7→ 〈vp, vq〉 · v1 ⊗ . . . ⊗ vp ⊗ . . . ⊗ vq ⊗ . . . ⊗ vr ,

where〈·, ·〉 represents the symplectic form onW. The result will now follow from some basic factsabout algebraic cycles and, crucially, the Lefschetz and K¨unneth55Standard Conjectures for abelianvarieties (due to Lieberman; for a proof, see [Kle68]). Fix a polarization ofX, giving rise to aLefschetz operatorLX. Recall Jannsen’s fundamental result ([Jan92]) that numerical motives forma semi-simple abelian category. This and the Kunneth Standard Conjecture for abelian varietiesimply that the category of numerical motives generated by abelian varieties overF is a semi-simpleTannakian category. We deduce that the following are (numerical) sub-motives ofH(Xr ):

54Extended to GSp on the center by the prescription of Corollary 4.3.3. Throughout this argument we will ignorethese extra scalars to simplify the notation.

55We will write πiX for the algebraic cycle onX × X inducingH(X)։ H i(X) → H(X).

131

• M = (X, π1X, 0), i.e. the object corresponding toH1(X);

• The kernel of each contractionH1(X)⊗r → H1(X)⊗(r−2). For notational simplicity, takep = 1, q = 2. If we were working in cohomology, we would compute the kernel of c1,2 as

ker(H1(X) ⊗ H1(X)

〈·,·〉−−→ Q(−1))⊗ H1(X)⊗r−2,

where recall that the polarization〈·, ·〉 is defined by〈w1,w2〉 = Ld−1X (w1∪w2). Now,W⊗2 =

Sym2(W)⊕∧2(W), and cup-product kills Sym2(W), while mapping∧2(W) isomorphicallyto H2(X). The kernel ofc1,2 is therefore isomorphic to

(Sym2(H1(X)) ⊕ Prim2(X)

)⊗ H1(X)⊗r−2.

To realize the analogous numerical motive, we can define Sym2(M) as

(X × X,12

(1+ (12)) · (π1X × π1

X), 0).

For the projector, we have taken the product of two commutingidempotents, the firstone being the Young symmetrizer associated to the Schur functor Sym2. There is also aprojectorp2 onto primitive cohomology (see 1.4.4, 2.3, and 2A11 of [Kle68]), so (X, p2, 0)realizesPrim2(X) as a numerical motive. Finally, since direct sums and tensor productsexist in our category, we can therefore describe the kernel of any cp,q as a numericalmotiveMp,q.• In general, the Young symmetrizercλ is not quite an idempotent. Writec2

λ= x−1cλ,56

makingxcλ an idempotent algebraic correspondence onXr . cλ commutes with ther-foldKunneth projector (π1

X)r (this is easily checked at the level of cohomological correspon-dences, soa fortiori holds for numerical equivalence), so

(Xr , xcλ · (π1X)r , 0)

defines a numerical motive, denotedSλ(H1(X)).• We would like to conclude the proof by intersecting the objectsMp,q (p < q) andSλ(H1(X)).

This is possible for numerical motives since the category isabelian. Finally, since all ofthe cycles considered in the proof are cycles on (disjoint unions of) abelian varieties incharacteristic zero, where numerical and homological equivalence coincide,57 we also de-duce the existence of a homological motive corresponding toKS(X).

Remark 4.3.8. A weight calculation yields the Hodge numbers ofKS(X).

4.3.3. Coda. Identifying among all rational Hodge structures those thatare motivic is one ofthe fundamental problems of complex algebraic geometry, and this generalized Kuga-Satake theorywould systematically construct newmotivicHodge structures from old, in a way not achievable bysimply playing the Tannakian game. I hope that investigation of these phenomena will provide astimulating testing-ground for thinking about Hodge theory in non-classical weights.

It is also tempting to ask what should be true if we replaceF by other fields, especially finitely-generated subfields ofC. Our motivic descent in the hyperkahler case works as written, except

56x depends onλ, but we have fixed aλ.57In the presence of the Standard Conjecture of Hodge type, theLefschetz conjecture implies ‘num= hom’: see

Proposition 5.1 of [Kle94].

132

for the critical absence of Tate’s basic vanishing result; in other contexts, it might be hoped thata similar descent works, conditional on the relevant cases of the Tate conjecture. I suspect onlya qualitativepotential lifting result will hold in this generality– that would suffice to imply theanalogous lifting conjecture forC itself. But another clearly important question to ask is: what, ifany, is the analogue of Tate’s vanishing theorem when the number fieldF is replaced by any fieldfinitely-generated overQ?

133

Index of symbols

(·)D, with argument a topological group, 14A: short-hand for the abelian varietyAF′ , 115A(T), 119A∗hom(X), 97Ak, 111AF′ : Andre’s descent of the classical Kuga-Satake

abelian variety, 114Alg: space of algebraic cycle classes as an object of

MF , 119B: ‘spin representation’ abelian variety, 115B(T), 122B+, 115B−, 115B0, 122Bk, 111B+k , 111C(VZ), 113C∗hom(X,Y), 97C+, 113C+(VZ), 113C•mot(X,Y)E, 100CF , 13Fcm, 14G, 1, 36, 57, 66G∨, LG, 13G∨sc, 67H, 47H′, 47H∗B, 97KS(X), 114KS(X), for an abelian varietyX, 131L(λ), 84LZ, 113Lℓ, 114MT(A), 102Prim2k(X)(k), 111S, 66T: transcendental lattice as an object ofMF , 119VE, 112VQ, 112VZ, 113Vℓ, 112

W〈r〉, 131WE, 113WV, WAlg, WT : spin representations associated to the

various orthogonal spacesV, Alg, T, 120Wn, for some odd integern, 126W+,E, 113W−,E, 113Wn,+, Wn,−, for some even integern, 126BdR, 21BHT, 21DdR, 21DHT, 21Q(π f ), 35Qcm, 14Θρ,ι, 23ΓF , 13ιℓ, 13ι∗∞,ℓ(τ), ι

∗(τ), 13ι∞, 13Sλ, 131GM

F , 101LF , 65MF , 100TF , 65TF,E, 109GSpin, 51HTτ, 22, 33HTτ(ρ), 31Spin, 51µv, νv, 2µιv , νιv, 2πi

X, 98recv, 13ρA, 114ρMℓ

, 113ρV, 114ρM, for a motiveM: the associated motivic Galois

representation, 113∼s, 73∼w, 13, 73∼ew, 73∼w,∞, 14, 73

135

G, 1, 57, 66T, 57Z, 57, 66τ∗ℓ,∞(ι), τ∗(ι), 13θρ,ι, 69ω, 58(·), with argument a field, 14σπ, 35cλ, 131mv, 25r0, 122rspin,V, rspin,Alg, rspin,T : homomorphisms giving the

spin representations associated to the variousorthogonal spacesV, Alg, T, 120

rspin, 113tv, 25, 14

136

Index of terms and concepts

L-packetarchimedean, 61unramified, 61

W-algebraic automorphic representation, 2, 36W-arithmetic, 37Q-curves, 116ℓ-adic Hodge theory propertyP, 5

Adler-Prasad conjecture, 3algebraic correspondence, 97archimedean purity lemma of Clozel, 38Artin motive, 101automorphic Langlands group, 65

C-algebraic automorphic representation, 2central character of an automorphic representation, 27central character of an automorphic representation–

how to compute, 58CM coefficients, 9CM descent prototype, 26CM field, 14compatible system ofℓ-adic representations valued in

linear algebraic group, 20correspondence between automorphic representations

and Galois representations, 13

de Rham Galois representation, 21

equivalence of automorphic representations andℓ-adic representations, 73

fiber functor, 95fiber of a functorial transfer

Tate’s lifting problem, 61fibers of a functorial transfer

GL2 ×GL2 tensor product, 41Fontaine-Mazur conjecture, 11Fontaine-Mazur-Langlands conjecture, 11Fontaine-Mazur-Tate conjecture, 11

Galois lifting problem, 4, 47geometric Galois representation, 11Grunwald-Wang special case, 59

Hodge symmetry, 105Hodge-Tate Galois representation, 21Hodge-Tate-Sen weights, 23hyperkahler variety, 111

image of a functorial transfer, 30cyclic automorphic induction, 42Tate’s lifting problem, 59

infinity-type of a Hecke character, 25infinity-type of an automorphic representation, 35infinity-types of automorphic representations:

questions about, 47infinity-types of Hecke characters, all possible, 29

K3 surface, 111Kunneth Standard Conjecture, 98Kuga-Satake construction, 114Kuga-Satake motive associated to an abelian variety,

131

L-algebraic automorphic representation, 2labeled Hodge-Tate weights, 22, 31, 33labeled Hodge-Tate weights of a Hodge-Tate

representation, 23Lie irreducible, 82Lie multiplicity free, 84Lie-multiplictiy free, or LMF, 9LMF, 84local reciprocity map, 13

mixed-parity Hilbert modular form, 36mixed-parity Hilbert modular representation, 43modified commutativity constraint forMhom

F , 98motivated correspondences, 100motivated cycles, 99motive for homological equivalence, 97motives for motivated cycles, 100motivic Galois group of an objectM ofMF , 101Mumford-Tate conjecture for hyperkahler varieties,

124Mumford-Tate group of aQ-Hodge structure, 102

neutral Tannakian category, 95

137

polarized variety, 111potential automorphy theorem, 50primitive cohomology, as a motive, 111pure weakly compatible system, 13

Ramanujan conjecture (archimedean), 37regular automorphic representation for the group GLn,

35regular filtered module, 34rigid tensor category, 95root data for Spin and GSpin groups, 52

Sen operator, 23Standard Conjectures, 11

Taniyama group, 65, 109Tate conjecture for hyperkahler varieties, 119Tate’s theorem, 15tensor category, 94transcendental lattice, or subspace, of the Hodge

structurePrim2k(XC,Q)(k), 113typeA Hecke character, 2, 25typeA0 Hecke character, 2, 25

weak equivalence of automorphic representations, 6weak transfer of automorphic representations, 61weakly compatible system ofλ-adic representations

valued in a reductive group, 13weakly compatible system ofλ-adic, orℓ-adic,

representations, 12

138

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