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LECTURE: “THE ARITHMETIC OF THE LANGLANDS1
PROGRAM”2
JOHANNES ANSCHUTZ3
Contents4
1. Remarks on the lecture 25
2. A general introduction to the Langlands program 36
3. From modular forms to automorphic representations, part I 107
4. From modular forms to automorphic representations, part II 198
5. From modular forms to automorphic representations, part III 289
6. Langlands reciprocity for newforms 3810
7. Modular curves as moduli of elliptic curves (by Ben Heuer) 4611
8. The Eichler–Shimura relation (by Ben Heuer) 5312
9. Galois representations associated to newforms 6113
10. Galois representations for weight 1 forms (by Ben Heuer) 6914
11. The Langlands program for general groups, part I 7715
12. The Langlands program for general groups, part II 8316
References 9117
1
1. Remarks on the lecture18
• Aim: General introduction to the Langlands program19
• Focus: More on “picture”, proofs usually by reference. More “local, geomet-20
ric” than “global, analytic”.21
• Prerequisites: Vary a lot (algebraic number theory, algebraic groups, alge-22
braic geometry, harmonic analysis, representation theory,...). Usually detailed23
knowledge will not be necessary to understand the statements/picture, at least24
in some example.25
• Original plan: Yield background material for, now cancelled, trimester progam26
“The Arithmetic of the Langlands program” of the HIM, which was originally27
scheduled in 2020 (⇒ explains title of lecture)28
• Topics to be covered: Let’s see...29
• Hint: More analytic stuff on automorphic forms/Jacquet-Langlands in Edgar30
Assing’s lecture “Topics in Automorphic forms”31
• Disclaimer: I am by far not a person with serious knowledge/understanding32
of the Langlands program, thus I may oversimplify/overcomplicate things33
or miss subtelties. For definite/correct statements, one has to consult the34
references.35
• Reference: Getz/Hahn: “An introduction to automorphic representations36
with a view towards the trace formula” [GH19], more references during the37
course.38
• Style of lecture: I will explain this file via zoom (the layout is meant to be39
optimized for the use of the file in zoom). Any suggestion on improvement is40
welcome!41
• Video/Audio: Please turn off your video/audio (except for questions/comments).42
Do not record me!43
• Questions: via chat (in zoom), or orally (make me aware of you, e.g., raise44
hand via zoom, or speak), or by e-mail.45
• Problems with technic: This file will be uploaded on the webpage.46
Questions?47
2. A general introduction to the Langlands program48
Today:49
• Rough introduction to the Langlands program, and a first example of its50
arithmetic significance.51
Setup:52
• G reductive group over Q, most important example: G = GLn for some n ≥ 1.53
– an affine, algebraic group G/Q is reductive if it has no closed normal54
subgroups isomorphic to Ga.55
• Af := (xp)p ∈∏pQp | xp ∈ Zp for all but finitely many primes p 56
“ring of finite adeles”57
• A := Af × R58
“ring of adeles”59
Then:60
• A is a locally compact topological ring61
– On Af : unique ring topology such that∏pZp is open.62
– On A: product topology.63
• Q ⊆ A discrete subring (via diagonal embedding).64
• G(A) ∼= G(Af )×G(R) is a locally compact topological group65
– Choose an embedding G ⊆ Spec(Q[X1, . . . , Xm]) for some m ≥ 1, and66
take the subspace topology of induced embedding G(A) ⊆ Am, cf. [GH19,67
Theorem 2.2.1.].68
• G(Q) ⊆ G(A) is a discrete subgroup.69
Facts:70
• On any locally compact topological group H exists a right-invariant Haar71
measure, cf. [GH19, Section 3.2.].72
– unique up to a scalar in R>073
• On G(A) the right Haar measure is also left-invariant, and it descends to a74
G(A)-invariant measure on G(Q)\G(A), cf. [GH19, Lemma 3.5.4.], [GH19,75
Lemma 3.5.3].76
Definition 2.1. We set77
L2(G(Q)\G(A))
as the space of measurable functions f : G(Q)\G(A)→ C such that78 ∫G(Q)\G(A)
|f |2 <∞
where the integration is w.r.t. the G(A)-invariant measure on G(Q)\G(A).79
As usual: Two functions in the L2-space have to be identified if they agree outside80
a set of measure zero.81
Facts:82
• L2(G(Q)\G(A)) is a Hilbert space, i.e., it has an inner product.83
• the G(A)-action coming from right translation on it is unitary, i.e., preserves84
the inner product.85
Aim of the Langlands program (crude form):86
• Decompose the Hilbert space L2(G(Q)\G(A)) as a representation of G(A),87
according to arithmetic data.88
Questions?89
What does “decomposing” mean, roughly?90
• G := isomorphism classes of irreducible, unitary G(A)-representations.91
• Roughly, decomposing means to write (as a unitary G(A)-representation)92
L2(G(Q)\G(A)) “ ∼= ”⊕
[π]∈G
π⊕mπ
with mπ ∈ N ∪ ∞ multiplicity of π.93
• Looking for such a decomposition is too naive.94
Example: G = Gm = GL195
• In this case,96
G(Q)\G(A) ∼= Q×\A×
is a locally compact abelian group97
⇒ can use abstract Fourier theory to analyze L2(Q×\A×).98
Abstract Fourier theory (cf. [DE14, Chapter 3]):99
• A any locally compact abelian group100
• A = isom. classes of irreducible, unitary representations of A101
• A ∼= Homcts(A,S1) is again a locally compact abelian group.102
– via pointwise multiplication and the compact-open topology103
• A ∼= ˆA as topological groups via a 7→ (χ 7→ χ(a)).104
• A is discrete if and only if A is compact.105
“⇒” Choose⊕I
Z A, then A embeds into∏I
S1. “⇐” Use A ∼= ˆA and106
definition of compact-open topology.)107
Theorem 2.2 (Plancherel theorem, cf. [DE14, Chapter 3.4]). The Fourier transform108
F : L1(A) ∩ L2(A)→ L2(A)
defined by109
f 7→ (χ 7→∫A
f(x)χ(x)dµ(x))
extends to a unitary isomorphism110
F : L2(A) ∼= L2(A)
of Hilbert spaces.111
Fact:112
• F is an A-equivariant isomorphism where A acts on L2(A) via113
a · g(χ) := χ(a)g(χ)
for a ∈ A, g ∈ L2(A), χ ∈ A.114
Examples:115
• If A = S1, then116
Z ∼= A, n 7→ χn with χn(z) = zn, z ∈ S1.
Thus117
L2(S1) ∼=⊕n∈Z
Cχn
is the Hilbert space direct sum of the S1-equivariant subspaces118
Cχn ⊆ L2(S1).
Concretely: each L2-function f : S1 → C can uniquely be written as119
f =∑n∈Z
anχn, where an ∈ C,∑n∈Z|an|2 <∞.
⇒ We found a nice decomposition of the unitary S1-representation L2(S1)120
into irreducible subspaces.121
• If A = R, then R ∼= R via122
R→ R, x 7→ χx, where χx : R→ S1, y 7→ e2πixy.
The isomorphism123
F : L2(R) ∼= L2(R)
implies therefore that each f ∈ L2(R) can be written as124
f(x) =
∫R
g(y)χy(x)dy
for a unique function g ∈ L2(R). Thus,125
L2(R) ∼=∫R
χydy
is a Hilbert integral of representation. We cannot do better: χx /∈ L2(R)126
for any x ∈ R, and L2(R) contains no irreducible subrepresentation of R, cf.127
[GH19, Section 3.7].128
– Also no irreducible quotient, the statement about quotients in [GH19,129
Lemma 3.7.1] is wrong, I think.130
Back to G = Gm:131
• Factoring each n ∈ Q× into n = ±p1 . . . pk with pi prime implies132
Q×\A× ∼=∏p
Z×p × R>0.
• Let us ignore R>0 and look at133
L2(∏p
Z×p ) ∼=⊕χ
Cχ.
with χ :∏pZ×p → S1 all continuous characters of
∏pZ×p .134
• The characters χ are “arithmetic” data, namely135
Gal(Q(µ∞)/Q) ∼=∏p
Z×p ,
where Q(µ∞) =⋃n∈N
Q(e2πin ) is the cyclotomic extension of Q.136
• Thus the χα are (continuous) 1-dimensional Galois representations of Gal(Q/Q).137
– Namely: Gal(Q/Q)α
∏p prime
Z×pχ−→ S1 ⊆ C×138
• By Kronecker-Weber each 1-dimensional Galois representation139
σ : Gal(Q/Q)→ C×
is of the form σ = χ α for some χ.140
Theorem 2.3 (Kronecker-Weber). The field Q(µ∞) is the maximal abelian extension141
of Q, i.e., each finite Galois extension F/Q with abelian Galois group Gal(F/Q) is142
contained in Q(µ∞).143
The final decomposition for G = Gm:144
•L2(R>0Q×\A×) ∼=
⊕σ : Gal(Q/Q)→C×
Cχσ
with χσ characterised by χσ α = σ.145
• This (easy) example is prototypical for what one aims for in the Langlands146
program.147
Questions?148
Back to general G: Geometry of G(Q)\G(A)149
• There exists a central subgroup AG ⊆ G(R), with AG ∼= Rr>0, such that150
[G] := AGG(Q)\G(A)
has finite volume, cf. [GH19, Theorem 2.6.2.].151
– AG is the connected component of the maximal split subtorus of the152
center Z(G) of G, e.g., AGLn ⊆ GLn(R) is the group of scalar matrix153
with entries in R>0 while ASLn is trivial.154
• Tamagawa measure = some canonical measure on G(A), and155
τ(G) := vol([G]) = Tamagawa number of G.
Knowledge of τ(G) is arithmetically interesting. Cf. [Col11, Appendix B],156
[Lur, Lecture 1].157
– τ(SLn) = 1 ⇒ vol(SLn(R)/SLn(Z)) = ζ(2)ζ(3) . . . ζ(n), with ζ(s) =158
Riemann ζ-function159
• G(Q)\G(A) and [G] are profinite coverings of some real manifold, e.g.,160
GLn(Z)\(∏p
GLn(Zp)×GLn(R)) with diagonal action of GLn(Z)
which covers the real manifold161
GLn(Z)\GLn(R).
Similarly, [G] covers GLn(Z)R>0\GLn(R). Cf. [GH19, Section 2.6], [Del73,162
(0.1.4.1)].163
• K∞ ⊆ G(R) maximal compact connected subgroup ⇒ [G] is a principal K∞-164
bundle (or K∞-torsor) over [G]/K∞.165
– G = GLn ⇒ K∞ ∼= SOn(R)166
• G = GL2, then by Mobius transformations167
AG\GL2(R)/K∞∼−→ H±, g =
(a b
c d
)7→ g(i) :=
a · i+ b
c · i+ d
with168
H± := z ∈ C | Im(z) 6= 0
the upper/lower halfplane169
⇒ [GL2]/K∞ ∼= GL2(Z)\(GL2(∏p
Zp)×H±)
• K∞ ⊆ G(Af ) compact-open (“a level subgroup”)170
⇒ G(Q)\G(Af )/K∞ =
m∐i=1
G(Q)giRK∞
is finite (cf. [GH19, Theorem 2.6.1]) and171
[G]/K∞K∞ ∼=
m∐i=1
Γi\X
where172
X := AG\G(R)/K∞
is a real manifold, and173
Γi := G(Q) ∩G(R)giK∞g−1
i ⊆ G(R)
is a congruence subgroup, cf. [GH19, Section 2.6].174
– One usually has to assume that K∞ is sufficiently small, so that Γi acts175
freely on X, cf. [GH19, Definition 15.2].176
• G = GL2, then177
Γi\X ∼= Γ\H± (e.g., SL2(Z)\H)
with Γ ⊆ GL2(Z) a discrete subgroup containing178
Γ(m) := ker(GL2(Z)→ GL2(Z/m))
for some m ≥ 0. (later ⇒ relation to modular forms).179
• Usually dim(Γi\X) > 0⇒ geometry more complicated than in case G = Gm.180
• The Γi\X are interesting real manifolds, e.g., sometimes actually complex181
manifolds and canonically defined over number fields.182
• [G] compact (⇔ each Γi\X compact) if and only if any Gm ⊆ G lies in center,183
cf. [GH19, Theorem 2.6.2].184
– e.g., G the units in a non-split quaternion algebra over Q, i.e., units in a185
Q-algebra with presentation 〈x, y|x2 = a, y2 = b, xy = −yx〉, for suitable186
a, b ∈ Q×. Note that G(R) ∼= GL2(R) if ab < 0. Cf. [GS17, Chapter 1].187
More examples: Weil restrictions188
• F/Q finite extension, H reductive over F189
⇒ reductive group G := ResF/Q(H) over Q.190
– G represents the functor R 7→ H(R⊗QF ) on the category of Q-algebras,191
cf. [CGP15, Appendix A] or [GH19, Section 1.4].192
• G(Q)\G(A) = H(F )\H(AF ) with AF ring of adeles for F (⇒ no need to193
discuss general F ).194
• H = Gm,F ⇒ desired decomposition of L2([G]) is equivalent to class field195
theory for F .196
• F imaginary quadratic, H = GL2,F ⇒ G(R) ∼= GL2(C), K∞ ∼= U(n) unitary197
group and198
X = AG\G(R)/K∞
is the 3-dimensional hyperbolic space H3. The quotients Γi\X are arithmetic199
hyperbolic 3-manifolds, cf. [Thu82, Section 4 & 5].200
– H3 can be modelled on the quaternions q = x + y · i + z · j | x, y ∈201
R, z > 0 and
(a b
c d
)∈ GL2(C) acts as q 7→ (aq+ b)(cq+ d)−1, where202
C = x+ y · i | x, y ∈ R203
Questions?204
Definition 2.4 ([GH19, Definition 3.3]). Let G be a reductive group over Q. An auto-205
morphic representation (in the L2-sense) is an irreducibe unitary G(A)-representation206
π which is isomorphic to a subquotient of L2([G]).207
Facts:208
• G(A) acts on [G] = AGG(Q)\G(A) from the right ⇒ G(A) acts unitarily on209
L2([G])210
– After choice of a right-invariant Haar measure on G(A)...211
• Necessarily π is trivial on AG. Usually harmless: Schur’s lemma ⇒ can find212
for irreducible unitary G(A)-repr. π a character χ : G(R) → C× with χ ⊗ π213
trivial on AG. Cf. [GH19, Section 6.5].214
– For G = GLn take χ of the form g 7→ |det(g)|s) for some s ∈ R>0.215
• If [G] is not compact, L2([G]) will decompose into a “discrete” and a “con-216
tinuous” part, cf. [GH19, Section 10.4].217
• Recall G = isom. classes of irreducible unitary G(A)-representations.218
• G has a natural topology, the Fell topology, cf. [GH19, Section 3.8].219
– It generalizes the compact-open topology on A for A locally compact220
abelian.221
• Let π ∈ G, x ∈ π, and ϕ : π → C continuous C-linear. Then
fx,ϕ : G→ C, g 7→ ϕ(gx)
is a matrix coefficient of π. Roughly, two π, π′ ∈ G are close if their matrix222
coefficients agree for the compact-open topology.223
Theorem 2.5 ([GH19, Theorem 3.9.4]). There exists a measurable multiplicity func-
tion m : G→ 1, 2, . . . ,∞ and a measure µ on G such that
L2([G]) ∼=∫G
(⊕m(π)
π)dµ(π).
The m and µ are unique up changes on sets of measure 0.224
The discrete and continuous part of L2([G]):225
• We don’t need the exact meaning of the integral in 2.5, its occurence is due to226
the fact that some representations of G(A) “contribute” to L2([G]) without227
being subrepresentations.228
– Compare with the discussion of the Fourier transform for R.229
• Points π ∈ G with µ(π) > 0 appear (with multiplicitym(π)) as subrepresentations230
in L2([G]).231
• By Theorem 2.5232
L2([G]) ∼= L2disc([G])⊕ L2
cont([G])
where L2disc([G]) is the Hilbert sum of all irreducible, unitary subrepresentations233
of G(A), and L2cont([G]) its orthogonal complement, cf. [GH19, Section 9.1].234
• By definition, a closed, connected subgroup P ⊆ G is parabolic if G/P is235
proper over Spec(Q). The quotient of P by its unipotent radical is its Levi236
quotient M . Cf. [GH19, Section 1.9].237
– In GLn: take decomposition n = n1 + n2 + . . . + nk, P is subgroup of238
block upper triangular matrices in GLn with blocks of size n1, . . . , nk.239
The Levi quotient is M = block diagonal matrices. Up to conjugation240
all parabolics in GLn are of this form.241
• Langlands proved that the continuous part L2cont([G]) can be described via242
“inducing” the discrete parts L2disc([M ]) for all Levi quotients of parabolics243
P ⊆ G, P 6= G, cf. [GH19, Section 10.4]).244
– This is his theory of “Eisenstein series”, and it is a starting point of the245
Langlands program.246
• In this course, L2disc([G]) will be more important than L2([G]).247
• [G] is compact if and only G has no proper parabolics (defined over Q), this248
confirms that L2([G]) = L2disc([G]) in this case.249
Questions?250
3. From modular forms to automorphic representations, part I1
Today (and next time):2
• Construct automorphic representations for GL2 associated to modular forms3
• Keep some piece of paper ready!4
Last time:5
• G reductive over Q, A = Af × R ring of adeles6
• [G] = AGG(Q)\G(A) adelic quotient, has finite invariant volume7
• Langlands program: Decompose8
L2([G])
according to arithmetic data.9
• G = Gm ⇒10
L2(R>0Q×\A×) ∼=⊕
σ : Gal(Q/Q)→C×C χσ.
• Geometry of [G]:11
– K∞ ⊆ G(R) some maximal compact, connected12
– K = K∞ ⊆ G(Af ) some compact-open subgroup13
– Then14
[G]
K-torsor (in particular: a profinite covering)
[G]/K
K∞-torsor
a real manifold, at least for sufficiently small K
[G]/KK∞ =m∐i=1
Γi\X disjoint union of arithmetic manifolds
with Γi ⊆ G(R) some (discrete) congruence subgroups determined by K,15
and16
X := AG\G(R)/K∞.
• An irreducible unitary representation π of G(A) is automorphic if it is iso-17
morphic to a subquotient of L2([G]).18
• Examples:19
– G = Gm, then the automorphic representations of G(A) are the charac-20
ters of A× which are trivial on Q× · R>0 (by definition of [G])21
– [G] has finite volume22
⇒ the trivial representation (=constant functions on [G]) is automorphic,23
and in L2disc([G]).24
• Have the general orthogonal decomposition25
L2([G]) ∼= L2disc([G])⊕ L2
cont([G])
with L2disc([G]) the closure of the sum of all irreducible subrepresentations of26
G(A).27
• Langlands⇒ L2cont([G]) described via L2
disc([M ]) for M running through Levi28
quotients of parabolic subgroups P ⊆ G,P 6= G.29
Questions ?30
For G = GL2:31
• AGL2∼= R>0 embedded as scalar matrices into GL2(R)32
• We have33
GL2(Q)\GL2(A) ∼= GL2(Z)\(GL2(Z)×GL2(R)),
where34
Z := lim←−m
Z/m ∼=∏p
Zp.
• Indeed:35
– GL2(Af ) is the restricted product of the GL2(Qp) w.r.t. to the GL2(Zp),36
i.e.,37
GL2(Af ) = (Ap)p ∈∏p
GL2(Qp) | Ap ∈ GL2(Zp) for all but finitely many p
(cf. [GH19, Proposition 2.3.1]).38
– Q ⊆ Af is dense (use Chinese remainder theorem).39
– A×f = Q× · Z× (use prime factorization).40
– GL2(Zp),GL2(Qp) generated by elementary and diagonal matrices41
⇒ GL2(Af ) = GL2(Q)GL2(Z)42
⇒ GL2(Q)\GL2(Af ) ∼= GL2(Q) ∩GL2(Z)\GL2(Z) ∼= GL2(Z)\GL2(Z).43
⇒ GL2(Q)\GL2(A) ∼= GL2(Z)\(GL2(Z)×GL2(R))44
– The proof works for all n ≥ 1 and shows:45
GLn(Q)\GLn(A) ∼= GLn(Z)\(GLn(Z)×GLn(R)).
• Km = ker(GL2(Z)→ GL2(Z/m)) for m ≥ 1.46
– these groups are cofinal within all compact-open subgroups in GL2(Af ).47
– we have48
GL2(Q)\GL2(Af )/Km∼= GL2(Z)\GL2(Z)/Km
and GL2(Z)/Km∼= GL2(Z/m).49
– SL2(Z/m) is generated by elementary matrices: use the euclidean algo-50
rithm and the magic identity51 (a 0
0 a−1
)=
(1 a
0 1
)(1 0
−a−1 1
)(1 a
0 1
)(0 −1
1 0
)for any invertible element a in some ring R. Check!Check!52
⇒ SL2(Z)→ SL2(Z/m) surjective53
– We obtain54
GL2(Z)\GL2(Z)/Km∼= ±1\(Z/m)×
via the determinant.55
• K∞ = SO2(R) ⊆ GL2(R)56
• X = AGL2\GL2(R)/K∞ ∼= H± via57
AGL2\GL2(R)/K∞
∼−→ H±, g =
(a b
c d
)7→ g(i) :=
a · i+ b
c · i+ d
• In the end:58
GL2(Q)\(GL2(Af )/Km ×X) ∼=∐
±1\(Z/m)×
Γ(m)\H±
with59
Γ(m) := Km ∩GL2(Z) = ker(GL2(Z)→ GL2(Z/m)).
Finiteness of the volume of [G] for G = GL2: A direct argument60
• K∞Km compact61
⇒ It suffices to show that [GL2]/K∞Km has finite volume.62
⇒ Suffices to see that Γ(m)\H± has finite volume.63
• Up to scalar in R>0, the GL2(R)-invariant measure on H± is given by the64
volume form65
1
y2dx ∧ dy.
• Indeed:66
– Have to show67
g∗(1
y2dx ∧ dy) =
1
y2dx ∧ dy
for g ∈ GL2(R).68
– Write dx ∧ dy = i2dz ∧ dz, z = x+ iy.69
– Statement true for70
g =
(1 a
0 1
), a ∈ R.
– Statement true for AGL2.71
– Statement true for72
g =
(a 0
0 1
), a ∈ R×.
– Statement true for73
g =
(0 1
1 0
),
i.e., for z 7→ 1z = z
|z|2 .74
– From the magic identity we get invariance for lower triangular matrices,75
and thus for all of GL2(R). Check!Check!76
• A known fundamental domain for SL2(Z) on H is given by77
D := z ∈ H | |z| > 1, |Re(z)| < 1/2
and finiteness of the volume of GL2(Z)\H± follows from78 ∫D
1
y2dx ∧ dy <∞.
• Finally, Γ(m) ⊆ GL2(Z) is of finite index79
⇒ Γ(m)\H± is of finite volume, as desired.80
Questions ?81
Modular forms classically (cf. [DS05]):82
• Γ ⊆ SL2(Z) congruence subgroup, later sufficiently small.83
• k ∈ Z84
• A function f : H± → C is a modular form of weight k for Γ if85
– f(γ · z) = (cz + d)kf(z) for γ =
(a b
c d
)∈ Γ86
– f is holomorphic87
– f is holomorphic at the cusps of Γ88
• “holomorphic at cusp”89
• Mk(Γ) = the C-vector space of modular forms of weight k for Γ.90
Let us call, for the moment, a function weakly modular if it satisfies the first two91
conditions.92
Holomorphic at cusps:93
• For each cusp σ a weakly modular form f for Γ has a Fourier expansion, i.e.,94
up to translating the cusp to ∞ there is an equality95
f(z) =∑n∈Z
anqn/h
for q = e2πiz, h the smallest periodicity of f , and an ∈ C, n ∈ Z.96
• f is holomorphic at the cusp, i.e., a modular form, if an = 0 for n < 0.97
• In this case, f is a cusp form if a0 = 0.98
• Let Sk(Γ) ⊆Mk(Γ) be the subspace of cusp forms.99
Modular forms as sections of line bundles on modular curves:100
• The embedding H± ⊆ P1C, z 7→ [z : 1] is GL2(R)-equivariant.101
• The GL2(C)-action on P1C lifts (naturally) to OP1
C(k):102
– Sufficient for k = −1103
– But OP1C(−1) is associated to the Gm-torsor104
A2C \ 0 → P1
C, (z1, z2) 7→ [z1 : z2],
which is naturally GL2(C)-equivariant.105
• We obtain a Γ-equivariant morphism106
OH±(k) := OP1C(k)×P1
CH±
H±
• Modding out Γ yields:107
ω⊗k := Γ\OH±(k)
Γ\H±,
with ω⊗k a holomorphic line bundle over Γ\H±108
• The space109
H0(Γ\H±, ω⊗k)
of holomorphic sections of ω⊗k identifies with weakly modular forms for Γ.110
• Indeed:111
– For k ∈ Z define the holomorphic GL2(R)-equivariant line bundle112
Lk := H± × C
over H± with GL2(R)-acting on the left by113
g · (z, λ) := (az + b
cz + d, (cz + d)−kλ).
for g =
(a b
c d
)∈ GL2(R).114
– The map115
L∗−1 → OH±(−1)∗ = H± ×P1CA2
C \ 0,(z, λ) 7→ (z, (λz, λ))
is a holomorphic isomorphism of GL2(R)-equivariant line bundles over Check!Check!116
H±. Here (−)∗ is the complement of the zero section in a line bundle.117
– By taking tensor powers:118
Lk ∼= OH±(k)
for all k ∈ Z, as GL2(R)-equivariant holomorphic vector bundles.119
– A holomorphic section120
H± → Lk, z 7→ (z, f(z))
is Γ-equivariant if and only if f(z) satisfies121
f(az + b
cz + d) = (cz + d)kf(z)
for
(a b
c d
)∈ Γ. Check!Check!122
– This shows that123
H0(Γ\H±, ω⊗k)
identifies with the space of weakly modular forms for Γ.124
• In particular, we have a natural embedding125
Mk(Γ) ⊆ H0(Γ\H±, ω⊗k)
with image defined by condition of being “holomorphic at the cusps”:126
– ω extends canonically to the canonical compactification Γ\H± of the127
algebraic curve Γ\H±128
– Mk(Γ) is the image of H0(Γ\H±, ω⊗k)129
⇒ In particular, Mk(Γ) is finite-dimensional over C. Moreover, ω is130
ample on Γ\H± which implies that Mk(Γ) = 0 for k < 0 and Mk(Γ) gets131
“big” for k 0.132
– Sk(Γ) ∼= H0(Γ\H±, ω⊗k(−D)) where D = Γ\H± \Γ\H± is the (reduced)133
divisor at infinity.134
Questions ?135
From sections of ω⊗k to functions on [G] = [GL2]:136
• Let K ⊆ GL2(Af ) compact-open and write137
GL2(Q)\(GL2(Af/K)×H±) ∼=m∐i=1
Γi\H±
• ω⊗k defines by pullback from H± a (complex) line bundle ω⊗k on138
GL2(Q)\(GL2(Af/K)×H±).
• The spaces Mk(Γi) of modular forms for Γi, i = 1, . . . ,m, embed into the139
space of holomorphic sections140
H0(GL2(Q)\(GL2(Af )/K ×H±), ω⊗k).
• Set141
Mk(K) :=
m⊕i=1
Mk(Γi).
• Then Mk(K) ⊆ H0(GL2(Q)\(GL2(Af )/K×H±), ω⊗k) is defined by condition142
of being holomorphic at all cusps.143
• ω⊗k defines by pullback a G(Af )-equivariant line bundle, again written ω⊗k,144
on145
GL2(Q)\(GL2(Af )×H±)
• Define146
H0(GL2(Q)\(GL2(Af )×H±), ω⊗k)
:= lim−→K
H0(GL2(Q)\(GL2(Af )/K ×H±), ω⊗k).
and147
Mk := lim−→K
Mk(K).
• The line bundle ω⊗k is GL2(R)-equivariantly trivial after pullback along148
AGL2\GL2(R)→ H±, g 7→ g(i).
Namely, for k = −1, there is the GL2(R)-equivariant section Check!Check!149
s : AGL2\GL2(R)→ AGL2\GL2(R)×H± OH±(−1) ∼= AGL2\GL2(R)×P1CA2
C \ 0,
g =
(a b
c d
)7→ (g, |det(g)|−1/2(ai+ b, ci+ d)).
• In particular, pullback of sections defines a morphism150
Φ: H0(GL2(Q)\(GL2(Af )×H±), ω⊗k)→ C∞([GL2])
which on the infinite component is induced by pullback along151
AGL2\GL2(R) 7→ H±, g 7→ g(i)
and the isomorphism152
AGL2\GL2(R)×H± Lk ∼= AGL2
\GL2(R)×H± OH±(k) ∼= AGL2\GL2(R)× C
where the last one is induced by the section s. One checks, by reducing to153
the case k = −1, that each section f : H± → Lk is send to the function154
g 7→ ϕf (g) := |det(g)|k/2(ci+ d)−kf(ai+ b
ci+ d)
for g =
(a b
c d
)∈ GL2(R). Indeed, for k = −1, the pullback of155
H± → L−1, z 7→ (z, f(z))
and the corresponding section156
AGL2\GL2(R)→ AGL2
\GL2(R)× C, g 7→ (g, ϕf (g))
are related, as sections of AGL2\GL2(R)×P1
CA2
C \ 0 by the formula Check!Check!157
f(g(i))(g(i), 1) = ϕf (g)|det(g)|1/2(ai+ b, ci+ d)
which forces158
ϕf (g) = |det(g)|−1/2(ci+ d)f(g(i))
as desired.159
• Note that the factor |det(g)|−1/2 is necessary only because by definition ele-160
ments of L2([GL2]) or C∞([GL2]) are constant along AGL2.161
We used the following general definition.162
Definition 3.1 (Smooth functions on [G]). Let G be a reductive group over Q. Then163
a function f : [G]→ C is smooth if it is stable under some K ⊆ G(Af ) compact-open164
and the resulting function165
f : [G]/K → C
is smooth, i.e., C∞. (Note that [G]/K is a real manifold.)166
Upshot:167
• We constructed a canonical morphism168
Mk ⊆ H0(GL2(Q)\(GL2(Af )×H±), ω⊗k)Φ−→ C∞([GL2]),
where Mk = lim−→K
Mk(K) and Mk(K) is a sum of spaces of modular forms169
m⊕i=1
Mk(Γi) for various congruence subgroups Γi ⊆ GL2(Z).170
• The image of Mk in H0(GL2(Q)\(GL2(Af ) × H±), ω⊗k) is defined by the171
condition of “holomorphicity at the cusps”.172
Questions ?173
When do modular forms give rise to functions in L2([GL2])?174
• More generally, pick a section f ∈ H0(GL2(Q)\(GL2(Af )×H±, ω⊗k).175
• Question: When is176
Φ(f) ∈ C∞([GL2])
in L2([GL2])?177
• There exists K ⊆ GL2(Af ) compact-open such that178
f ∈ H0(GL2(Q)\(GL2(Af )/K ×H±), ωωk),
and Φ(f) is L2 if and only if Φ(f) ∈ C∞([GL2]/K) ⊆ C∞([GL2]) (as K is179
compact).180
⇒ Reduce to question:181
For Γ ⊆ GL2(Z) some congruence subgroup, f ∈ H0(Γ\H±, ω⊗k). When is182
ΓAGL2\GL2(R)→ C, g 7→ ϕf (g) = |det(g)|k/2(ci+ d)−kf(g(i))
square-integrable, i.e., in L2?183
• Claim:184
|ϕf (g)|2 = |f(g(i))|2|Im(g(i))|k
for any g ∈ GL2(R).185
• Equivalently,186
|det(g)|−1/2|ci+ d| ?= |Im(g(i))|1/2
for g =
(a b
c d
)∈ GL2(R).187
• This is true as both sides are invariant under188
– K∞ = SO2(R)-invariant from the right,189
– the subgroup
(1 ∗0 1
)⊆ GL2(R) acting from the left,190
– the diagonal matrices in GL2(R) acting from the left. Check these!Check these!191
• Hence192 ∫ΓAGL2\GL2(R)
|ϕf (g)|2dg =
∫Γ\H±
|f(z)|2|y|k 1
y2dx ∧ dy
(Recall that we integrate over “the” invariant measure.)193
• Γ\H± has finitely many cusps ⇒ Reduce to local statement at cusps194
• Write f in Fourier expansion, wlog at ∞:195
f(z) =∑n∈Z
anqn/h.
• Then:196
|y|kf(z) = 2π|log(|q|)|k∑n∈Z
anqn/h
as y = log(|q|). Check!Check!197
• Now for 0 < r ≤ 1198 ∫0<|q|≤r
|log(|q|)|k−2∑n∈Z
anqn/h i
2dq ∧ dq <∞
if an = 0 for n ≤ 0, i.e., if f is cuspidal.199
Upshot:200
• The map Mk ⊆ H0(GL2(Q)\(GL2(Af ) × H±), ωk)Φ−→ C∞([GL2]) induces a201
canonical inclusion202
Φ: Sk → L2([GL2]),
where Sk := lim−→K
Sk(K).203
Observations:204
• GL2(Af ) acts on H0(GL2(Q)\(GL2(Af ) × H±), ωk), C∞([GL2]), L2([GL2]),205
Mk, Sk and all inclusions, in particular Φ, are equivariant.206
• E.g., for g ∈ GL2(Af ) and varying K ⊆ GL2(Af ) the isomorphism207
Mk(K)g−→Mk(g−1Kg) ⊆Mk
defines the action Mkg−→Mk as Mk = lim−→
K
Mk(K).208
• GL2(Af ) does not act on Mk(K), Sk(K), . . . for any fixed K ⊆ GL2(Af ).209
Next tasks:210
• Describe Sk as a GL2(Af )-representation.211
• Characterize the image of Sk in L2([GL2]).212
Questions ?213
4. From modular forms to automorphic representations, part II1
Today:2
• Continue construction of automorphic representations associated to cusp forms.3
Questions from last time:4
1) Can one classify GL2(R)-equivariant line bundles on H±?5
2) Similarly: Can one classify GL2(Z)-equivariant line bundles on H±?6
• For 1):7
– Assume G Lie group, H a closed subgroup. Then the category8
G-equivariant complex vector bundles on G/H
is equivalent to9
finite dimensional C-representations of H.
“⇒” Look at the representation of H at the coset 1 ·H ∈ G/H.10
“⇐” Pass to the quotient11
V := (G× V )/H
for the action h · (g, v) := (gh−1, hv).12
– Useful:13
σ : H → GL(V )
finite-dimensional C-representation of H. Then:14
Continuous resp. smooth sections15
G/H → V
identify with continuous resp. smooth functions16
f : G→ V
such that f(gh) = ρ(h)−1 · f(v). Check!Check!17
– In our case, G = GL2(R), H = AGL2K∞ with K∞ ∼= SO2(R). Then18
H ∼= C×
and the irreducible representations of H are given by19
z 7→ zpzq, p, q ∈ Z.
⇒ There exist more GL2(R)-equivariant line bundles on H±.20
– Only the ones for q = 0 define holomorphic line bundles.21
– The SL2(R)-equivariant line bundles on H are all isomorphic to ω⊗k for22
some k ∈ Z.23
– Potentially confusing: Exists modular forms of half-integral weight.24
– Explanation: These are not sections of an SL2(R)-equivariant line bundle25
on H.26
– But sections of an SL2(R)-equivariant line bundle.27
– Here: SL2(R)→ SL2(R), the “metaplectic group”, the degree 2-covering28
of SL2(R) (a non-algebraic Lie group).29
– Note: π1(SL2(R), 1) ∼= π1(S1, 1) ∼= Z.30
– More details: [Del73].31
• For 2):32
– M1,1 = stack of elliptic curves over Spec(C).33
– Analytic stack associated to M1,1 is the stacky quotient [GL2(Z)\H±].34
– Mumford:35
Pic(M1,1) ∼= Z/12,
generated by ω, with the relation ω⊗12 ∼= O induced by the cusp form36
∆ ∈ S12(SL2(Z)), cf. [Mum65].37
– Note: Does not imply Mk(Γ) ∼= Mk+12(Γ) for k ∈ Z, Γ ⊆ SL2(Z) con-38
gruence subgroup.39
– Reason: The canonical extensions of ω⊗k and ω⊗k+12 to the compactifi-40
cation of M1,1 differ.41
– I don’t know if all (holomorphic) line bundles on [GL2(Z)\H±] arise from42
an algebraic line bundle on M1,1, i.e., are algebraic.43
Last time:44
• We constructed a GL2(Af )-equivariant embedding45
Φ: Mk → C∞([GL2]),
inducing a GL2(Af )-equivariant embedding46
Φ: Sk → L2([GL2]).
• Here47
Mk = lim−→K⊆GL2(Af )
Mk(K)
with48
Mk(K) =
m⊕i=1
Mk(Γi)
a sum of spaces of modular forms for congruence subgroups Γi ⊆ GL2(Z).49
• Similarly:50
Sk = lim−→K
Sk(K)
with Sk(K) a sum of spaces of cusp forms for congruence subgroups.51
• Today: How can one characterize the images52
Φ(Mk) ⊆ C∞([GL2])
resp.53
Φ(Sk) ⊆ L2([GL2])?
.54
• Next time: How does Sk decompose as a GL2(Af )-representation?55
Recall the construction of Φ:56
• Consider GL2(R)-equivariant line bundle57
Lk → H±
associated to the representation z 7→ zk of C× = AGL2SO2(R).58
• By pullback: Obtain line bundle ω⊗k on GL2(Q)\(GL2(Af )×H±).59
• Smooth, holomorphic,... sections of ω⊗k identify with smooth, holomorphic,...60
functions61
f : GL2(Af )×H± → C, (g, z) 7→ f(g, z)
satisfying62
f(γg, γz) = (cz + d)kf(g, z)
for63
γ =
(a b
c d
)∈ GL2(Q).
• Define the function64
j : GL2(R)×H± → C×, (
(a b
c d
), z) 7→ cz + d.
Then65
j(gh, z) = j(g, hz)j(h, z)
for all g, h ∈ GL2(R), z ∈ H±. Check!Check!66
• For f : GL2(Af )×H± → C as above the function67
ϕf : GL2(Af )×GL2(R)→ C, (g, g∞) 7→ j(g∞, i)−kf(g, g∞i)
is GL2(Q)-equivariant, i.e., lies in C∞(GL2(Q)\GL2(A)).68
• This is not the function we denoted by Φ(f) last time: It is not invariant by69
AGL2!70
• This can be fixed by multiplying it with the function71
GL2(R)→ C×, g 7→ |det(g)|k/2.
• Thus72
Φ(f)(g, g∞) := |det(g∞)|k/2ϕf (g, g∞)
is the associated function in C∞([GL2]) from last time.73
• Recall74
C× ∼= AGL2SO2(R) =
(a −bb a
) ⊆ GL2(R).
• Then75
ϕf (g, g∞z) = z−kϕf (g, g∞)
for all z ∈ C× ⊆ GL2(R), (g, g∞) ∈ GL2(A). Check!Check!76
• We obtain that f 7→ ϕf identifies smooth sections77
GL2(Q)\GL2(Af )×H± → ω⊗k
with smooth functions78
ϕ : GL2(Q)\GL2(A)→ C
such that79
ϕ(g, g∞z) = z−kϕ(g, g∞)
for all (g, g∞) ∈ GL2(Af )×GL2(R), z ∈ C×.80
Questions?81
Upshot and next aims:82
• Describe the image of Mk in C∞(GL2(Q)\GL2(A)) under f 7→ ϕf by condi-83
tions soley on ϕ ∈ C∞(GL2(Q)\GL2(A)).84
• An additional condition will determine the image of Sk.85
⇒ Obtain a description of Sk ⊆ L2([GL2]).86
• We already know that an element ϕ in the image of Mk must satisfy87
ϕ(g, g∞z) = z−kϕ(g, g∞)
for z ∈ C× and (g, g∞) ∈ GL2(A) and that such ϕ descent to a smooth88
function89
fϕ : GL2(Af )×H± → C, (g, z) 7→ f(g, z)
satisfying modularity for GL2(Q).90
• Need to check:91
– When is fϕ holomorphic?92
– When is fϕ holomorphic at the cusps?93
When is fϕ holomorphic?94
• Reduces to: Given a smooth function95
f : H± → C.
Which condition on96
ϕf : GL2(R)→ C, g 7→ j(g, i)−kf(gi)
guarantees that f is holomorphic?97
The strategy:98
• The Lie algebra99
g := gl2(R) ∼= Mat2,2(R)
of GL2(R) acts on C∞(GL2(R)) by deriving the action of GL2(R) by right100
translations.101
• Make this action explicit on ϕf .102
• Then construct an element Y ∈ gC such that103
Y ∗ ϕf = 0 ⇔ f is holomorphic.
Infinitesimal actions (cf. [GH19, Section 4.2]):104
• H any Lie group with Lie algebra h := T1H (“tangent space at identity”)105
• V any representation of H, with V a Hausdorff topological C-vector space.106
• For X ∈ h, defined by some path γ : (−ε, ε) → H with γ(0) = 1, and v ∈ V107
consider108
(−ε, ε) \ 0 → V, t 7→ γ(t)v − vt
.
If the limit exists, we write109
X ∗ v := lim−→t→0
γ(t)v − vt
.
• An element v ∈ V is called smooth if110
X1 ∗ (. . . ∗ (Xn ∗ v)) . . .)
exists for all X1, . . . , Xn ∈ h.111
• Vsm ⊆ V the subspace of smooth vectors in V .112
• Vsm stable under H.113
• Vsm is a representation of h, equivalently a module under the enveloping114
algebra for h:115
U(h) :=⊕n≥0
h⊗n/〈X ⊗ Y − Y ⊗X − [X,Y ] | X,Y ∈ h〉.
• Here: [−,−] the Lie bracket of h.116
Questions?117
The explicit action of g = gl2(R) on ϕf :118
• Recall: f ∈ H± → C a smooth function.119
• Recall:120
ϕf : GL2(R)→ C, g 7→ j(g, i)−kf(gi)
with121
j(g, z) := cz + d ∈ C
for122
g =
(a b
c d
)∈ GL2(R), z ∈ H±.
• Recall:123
j(gh, z) = j(g, hz)j(h, z)
for g, h ∈ GL2(R), z ∈ H±.124
• We know125
ϕf (gz) = z−kϕf (g)
for z ∈ C×, g ∈ GL2(R)126
⇒ C ∼= Lie(C×) ⊆ g acts via the linear form127
C 7→ C, z 7→ −kz
on ϕf (g). Note that128
Lie C× = 〈
(1 0
0 1
),
(0 −1
1 0
)〉R
and the anti-diagonal matrix acts by multiplying with −ki. Check!Check!129
• Consider the subgroup130
U :=
(∗ ∗0 1
)⊆ GL2(R).
Then131
Lie(U) = 〈
(1 0
0 0
),
(0 1
0 0
)〉R.
• Define132
ϕf : GL2(R)×H± → C, (g, z) 7→ j(g, z)−kf(gz).
• For a fixed g ∈ GL2(R):133
∂ϕf∂z
(g, z) = det(g)j(g, z)−k−2 ∂f
∂z(gz).
Check! Needs that
z 7→ gz has ∂∂z -
derivative det(g)j(g,z)2 .
Check! Needs that
z 7→ gz has ∂∂z -
derivative det(g)j(g,z)2 .
134
• Thus, f is holomorphic if and only if135
∂ϕf∂z
(g, i) = 0
for all g ∈ GL2(R).136
• For any path γ : (−ε, ε)→ U with γ(0) = 1 we get137
ϕf (gγ(t)) = ϕf (gγ(t), i) = ϕf (g, γ(t)i)
because j(γ(t), i) = 1. Check!Check!138
• Set139
Xγ :=∂γ
∂t(0) ∈ Lie(U).
• Thus140
Xγ ∗ ϕf= ∂
∂t (ϕf (gγ(t))|t=0
=∂ϕf∂z (g, i) ∂∂t (γ(t) · i)|t=0 +
∂ϕf∂z (g, i) ∂∂t (γ(t) · i)|t=0
.
Check! (Use the
chain rule for
Wirtinger deriva-
tives.)
Check! (Use the
chain rule for
Wirtinger deriva-
tives.)
141
• Consider142
γ1 : (−ε, ε)→ U, t 7→
(1 + t 0
0 1
).
Then γ1(t) · i = (1 + t)i and thus143 (1 0
0 0
)∗ ϕf (g) = i
∂ϕf∂z
(g, i)− i∂ϕf∂z
(g, i).
• Consider144
γ2 : (−ε, ε)→ U, t 7→
(1 t
0 1
),
then γ1(t) · i = i+ t and thus145 (0 1
0 0
)∗ ϕf (g) =
∂ϕf∂z
(g, i) +∂ϕf∂z
(g, i).
• In particular,146
i
(1 0
0 0
)∗ ϕf (g) +
(0 1
0 0
)∗ ϕf (g) = 2
∂ϕf∂z
(g, i)
• Thus147 (i 1
0 0
)∈ gC.
would be a candidate for Y .148
• But we can do better:149
Y := 12
(−1 i
i 1
)
= 12
(1 0
0 1
)+ 1
2 i
(0 −1
1 0
)+ i
(i 1
0 0
)satisfies150
Y ∗ ϕf (g) = −k2ϕf (g) + i(−ki
2)ϕf (g)︸ ︷︷ ︸
=0
+2i∂ϕf∂z
(g, i) = 2i∂ϕf∂z
(g, i).
Check!Check!151
• Why better?152
– Set153
H :=
(0 −ii 0
), X :=
1
2
(−1 −i−i 1
).
– Then H,X, Y is an sl2-triple in gC.154
– Namely, in the basis155 (−i1
),
(i
1
),
it is given by156 (1 0
0 −1
),
(0 1
0 0
),
(0 0
1 0
).
Check!Check!157
Questions?158
When is fϕ holomorphic at the cusps?159
• Consider a function in Fourier expansion:160
f(z) =∑n∈Z
ane2πizn,
say on161
z ∈ C | |Re(z)| < 1, Im(z) > 1.
• Recall162
|e2πiz| = e−2πy
with y = Im(z).163
• Then an = 0 for n < 0 if and only if164
|f(x+ iy)| ≤ CyN
for some C,N ∈ R>0 and y →∞.165
• By definition this means f is of moderate growth, or slowly increasing.166
Moderate growth for general reductive groups:167
• The moderate growth condition generalizes to arbitrary G reductive over Q.168
• For each embedding ρ : G → SLn,Q (not into GLn,Q) we obtain a norm169
|g| := supv
maxi,j=1,...,n
|ρ(g)i,jv|v
on G(A), where v runs through all primes and ∞.170
• A function ϕ : G(A)→ C is of moderate growth or slowly increasing if171
|ϕ(g)| ≤ C|g|N
for suitable C,N ∈ R>0, cf. [GH19, Definition 6.4.].172
• This definition does not depend on the embeddding ρ.173
Exercise:174
• Consider a (holomorphic) section175
f ∈ H0(GL2(Q)\(GL2(Af )×H±), ω⊗k).
• Then: Φ(f) ∈ C∞([GL2]) is of moderate growth176
⇔ f is holomorphic at the cusps177
• In other words, if and only if f lies in Mk.178
Upshot:179
• We can describe the image of180
Mk → C∞(GL2(Q)\GL2(A)), f 7→ ϕf .
• Namely, ϕ ∈ C∞(GL2(Q)\GL2(A)) lies in the image if and only if181
– ϕ(g, g∞z) = z−kϕ(g, g∞z) for (g, g∞) ∈ GL2(A), z ∈ C×.182
– ϕ is of moderate growth.183
– For each g ∈ GL2(Af )184
Y ∗ ϕ(g,−) = 0,
where Y ∈ gC = gl2(R)C is the element constructed before.185
• For k ≥ 1 we will see that there exists a unique, up to isomorphism, irreducible186
(gC,O2(R))-moduleDk−1 containing a non-zero element v, such that Y ∗v = 0,187
and C× acts on v via the character z 7→ z−k, cf. [Del73, Section 2.1].188
• Using Dk−1 one can rewrite our result as saying that189
Mk∼= Hom(gC,O2(R))(Dk−1, C
∞mg(GL2(Q)\GL2(A))),
where C∞mg(GL2(Q)\GL2(A)) ⊆ C∞(GL2(Q)\GL2(A)) denotes the subset of190
functions with moderate growth.191
A condition for cuspidality:192
• Let ϕ = ϕf ∈ C∞(GL2(Q)\GL2(A)) for some f ∈Mk.193
• Then f ∈Mk if and only if194
ϕB(g) :=
∫N(Q)\N(A)
ϕ(ng)dn = 0
for all g ∈ GL2(A), where195
N =
(1 ∗0 1
)is the unipotent radical in the standard Borel196
B =
(∗ ∗0 ∗
)and dn an N(A)-invariant measure on N(Q)\N(A).197
• Namely:198
– We may assume that f , and thus ϕ = ϕf , is invariant under some prin-199
cipal congruence subgroup K(m) ⊆ GL2(Af ).200
– Consider g = 1. Then using N(Q)\N(A) ∼= (mZ× R)/mZ:201 ∫N(Q)\N(A)
ϕf (n)dn
=∫
N(Q)\N(A)
j(n∞, i)−kf(n, n∞i)d(n, n∞)
=∫
R/mZ
∫mZ
f(a, i+ x)dadx
= µ(mZ)∫
R/mZf(1, i+ x)dx
= µ(mZ)a0
with µ(mZ) the volume of mZ and202
f(1, z) =
∞∑n=0
anqn/m
the Fourier expansion of the modular form f(1,−) for Γ(m) at the cusp203
∞.204
– Indeed: The crucial point is that205 ∫R/mZ
e2πi nmxdx = 0
if n > 0.206
– The condition ϕB(g) = 0 for all g ∈ GL2(A) is then equivalent to the207
vanishing of the constant Fourier coefficients at all cusps.208
– We leave the details as an exercise, cf. [Del73, Rappel 1.3.2.].209
• Note: Because ϕ is GL2(Q)-invariant for left translations, we can replace B210
by any conjugate gBg−1 with g ∈ GL2(Q).211
Questions?212
5. From modular forms to automorphic representations, part III1
Last time:2
• Let ϕ ∈ C∞(GL2(Q)\GL2(A)).3
• Then ϕ = ϕf for some f ∈Mk if and only if4
– ϕ(g, g∞z) = z−kϕ(g, g∞) for (g, g∞) ∈ GL2(A), z ∈ C×.5
– ϕ is of moderate growth.6
– For each g ∈ GL2(Af )7
Y ∗ ϕ(g,−) = 0,
where Y ∈ gC = gl2(R)C is a suitably constructed, natural element.8
• f ∈Mk is a cusp form if and only if for all B ⊆ GL2,Q proper parabolic, i.e.,9
Borel, with unipotent radical N ⊆ GL2,Q:10 ∫N(Q)\N(A)
ϕf (ng)dn = 0
for all g ∈ GL2(A).11
Today:12
• Describe Sk as a GL2(Af )-representation.13
• This will yield our main source of examples for automorphic representations.14
Observation:15
• Mk and Sk are smooth and admissible GL2(Af )-representations in the follow-16
ing sense.17
Definition 5.1. Let G be a locally profinite group (like GL2(Af )). A representation18
G on a C-vector space V is called smooth if19
V =⋃K⊆G
V K ,
where K runs through the compact-open subgroups of G and20
V K := v ∈ V | kv = v for all k ∈ K.
Equivalently, V is smooth if the action morphism21
G× V → V
is continuous with V carrying the discrete topology. Denote by22
Rep∞C G
the (abelian) category of smooth representations of G on C-vector spaces.23
Definition 5.2. Let G be a locally profinite group. A smooth representation V of G24
is called admissible if25
dimC V K <∞
for all K ⊆ G compact-open.26
Examples:27
• Mk =⋃
K⊆GL2(Af )
MKk with MK
k = Mk(K) is smooth and admissible because28
the Mk(K) are finite dimensional.29
• Similarly: Sk is an admissible representation of GL2(Af ).30
• ForG reductive over Q⇒ C∞(G(Q)\G(A)) is a smoothG(Af )-representation,31
but not admissible.32
• L2([G]) is not a smooth G(Af )-representation.33
The Hecke algebra (of a locally profinite group):34
• G a locally profinite group.35
– e.g., GL2(Af ), GLn(Qp), Gal(Q/Q), . . .36
• For K ⊆ G compact-open, exists a unique (left resp. right invariant) Haar37
measure µ on G with µ(K) = 1.38
• Concretely:39
– By translation invariance sufficient to define µ for K ′ ⊆ G compact-open40
subgroup.41
– K ′′ := K ′ ∩K compact-open ⇒ of finite index in K resp. K ′.42
– Set43
µ(K ′) :=[K ′ : K ′′]
[K : K ′′].
– This works and is also the unique possibility.44
– Note that µ takes actually values in Q.45
• Fix a left invariant Haar measure µ on G.46
• Define the “Hecke algebra”47
H(G) := C∞c (G)
of G as the set of locally constant, compactly supported functions G→ C.48
• H(G) is an algebra via convolution:49
f ∗ g(x) :=
∫G
f(xy)g(y−1)dy
for f, g ∈ H(G). Check associativ-
ity!
Check associativ-
ity!
50
• The multiplication is a purely algebraic operation:51
– Each f ∈ H(G) is a finite sum52
f =
m∑i=1
aiχgiKi
for some gi ∈ G, Ki ⊆ G compact-open subgroups.53
– Wlog: K1 = . . . = Km (by shrinking).54
– Let K ⊆ G be compact-open and g1, g2 ∈ G. Then55
g1Kg2K =
m∐j=1
hjK
for some hj ∈ G and56
χg1K ∗ χg2K =
m∑j=1
µ(K)χhjK .
Check!Check!57
• H(G) has a lot of idempotents:58
eK :=1
µ(K)χK
with K ⊆ G compact-open subgroups, χK the characteristic function of K,59
but in general no identity. Check!Check!60
Modules for the Hecke algebra (cf. [BRa]):61
• H(G) is a replacement for the group algebra C[G], which is better suited to62
study smooth representations of G.63
• If V ∈ Rep∞C G, then64
f ∗ v :=
∫G
f(h)hvdh
for f ∈ H(G), v ∈ V , defines an action of H(G) on V .65
• Concretely: If K ⊆ G is compact-open and sufficiently small, such that v ∈ V66
is fixed by K and f =m∑i=1
aiχgiK with ai ∈ C, gi ∈ G, then67
f ∗ v =
m∑i=1
aigiv.
• Assume K ⊆ G compact-open. Define the Hecke algebra relative to K68
H(G,K) := eKH(G)eK .
Then H(G,K) is an algebra with unit eK .69
• Let V be any smooth representation of G. Then70
V K = eK ∗ V
for any K ⊆ G compact-open. Check!Check!71
• In particular, H(G,K) ∼= C∞c (K\G/K) (K-biinvariant, compactly supported72
functions on G)73
Important property:74
• If V ∈ Rep∞C G is irreducible and V K 6= 0, then V K is irreducible as an75
H(G,K)-module.76
– Indeed: If M ⊆ V K is a non-trivial H(G,K)-submodule, then77
V = H(G) ∗M
by irreducibility and thus78
V K = eK(H(G) ∗M)) = eKH(G)eKM = M.
• Conversely, if V K is irreducible or zero for all K ⊆ G compact-open, then V79
is irreducible if V 6= 0. Check!Check!80
From irreducible, admissible GLn(Af )-representations to Hecke eigensys-81
tems:82
• V irreducible, admissible GLn(Af )-representation83
• Recall84
K(m) = ker(GLn(Z)→ GLn(Z/m)), m ≥ 0,
form basis of compact-open neighborhoods of the identity.85
• We know:86
– V K(m) is an irreducible H(GLn(Af ),K(m))-module for m 0.87
– Admissibility + Schur’s lemma ⇒ EndH(GLn(Af ),K(m))(VK(m)) ∼= C.88
• Set89
S :=⋂
m≥0, V K(m) 6=0
p prime dividing m,
90
ZS :=∏p/∈S
Zp
and91
ASf := Q⊗ ZS
the adeles away from S.92
• Note: For p /∈ S there exists 0 6= v ∈ V fixed by GLn(Zp) ⊆ GLn(Af ).93
• Define94
TS := H(GLn(ASf ),GLn(ZS)),
the “Hecke-algebra away from S”95
• Very important:96
TS is commutative!97
• Namely, consider98
σ : GLn(ASf )→ GLn(ASf ), g 7→ gtr.
Then:99
– σ preserves GLn(ZS).100
– σ(f1 ∗ f2) = σ(f2) ∗ σ(f1) for f1, f2 ∈ TS .101
– The cosets GLn(ZS)gGLn(ZS) with g ∈ GLn(Af ) a diagonal matrix span102
TS (by the elementary divisor theorem).103
– Conclusion: σ(f) = f for all f ∈ TS and thus104
f1 ∗ f2 = σ(f1 ∗ f2) = σ(f2) ∗ σ(f1) = f2 ∗ f1
as desired.105
• This argument is called Gelfand’s trick.106
• In fact, TS is central in H(GLn(Af ),K(m)) if all prime divisors of m lie in S.107
– If S1, S2 are two, possibly infinite, but disjoint sets of primes and gi ∈108
GLn(ASif ), then109
K(m)g1K(m)g2K(m) = K(m)g2K(m)g1K(m),
which implies the claim.110
• Claim: V defines a natural morphism111
σV : TS → C
of C-algebras.112
• Indeed:113
– For p /∈ S choose 0 6= v ∈ V fixed by GLn(Zp). Then V is fixed by K(m)114
for some m ≥ 0 with p - m.115
– V K(m) is irreducible, and thus H(GLn(Qp),GLn(Zp)) acts via a homo-116
morphism117
σV,p : H(GLn(Qp),GLn(Zp))→ EndH(GLn(Af ),K(m)(VK(m)) ∼= C,
which is independent of v.118
– The σV,p combine to σV .119
• We have not been very careful about Haar measures, or how the σV,p combine120
to σV .121
Definition 5.3. A Hecke eigensystem, or system of Hecke eigenvalues, is a maximal122
ideal in TS where S is a finite set of primes and TS := H(GLn(ASf ),GLn(ZS)).123
Observations:124
• TS is of countable dimension and thus each Hecke eigensystem has residue125
field C.126
• If S ⊆ S′ are two finite sets of primes, each Hecke eigensystem for S induces127
one for S′. ⇒ We call these Hecke eigensystems equivalent.128
• Each irreducible, admissible GLn(Af )-representation yields by the above con-129
struction a system of Hecke eigenvalues: V 7→ mV := ker(σV ).130
• A Hecke eigensystem for S is equivalently a collection of C-algebra homomor-131
phisms132
σp : H(GLn(Qp),GLn(Zp))→ C
for p /∈ S.133
A full description of H(GL2(Qp),GL2(Zp)):134
• Set G = GL2(Qp), K = GL2(Zp).135
• Normalize Haar measure such that µ(K) = 1.136
• Elementary divisor theorem: H(G,K) is free on the elements137
K
(pi 0
0 pj
)K
with (i, j) ∈ Z2,+ := (i, j) ∈ Z2 | i ≥ j.138
• Claim:139
f : C[X1, X±12 ]
'→ H(G,K)
via140
X1 7→ K
(p 0
0 1
)K,
X2 7→ K
(p 0
0 p
)K.
• Here we identify a double coset with its characteristic function.141
• Indeed:142
– The element143
K
(p 0
0 p
)K
is central in H(G,K) as144
KzK ·KgK = KgK ·KzK
for all g ∈ G, and z ∈ G in the center.145
– Moreover, let us check that each146
K
(pi 0
0 1
)K
with i ≥ 1 lies in the image of f .147
– Using induction we get that the n-fold product148
K
(p 0
0 1
)K · . . . ·K
(p 0
0 1
)K
agrees with the set of matrices in Mat2,2(Zp) of determinant of valuation149
n.150
– Thus, the product151
K
(p 0
0 1
)K · . . . ·K
(p 0
0 1
)K
is the characteristic function of this set.152
– As this characterstic function has the characteristic function of153
K
(pn 0
0 1
)K
as a summand with coefficient 1 we are done.154
– In particular, we see that H(G,K) must be commutative without using155
Gelfand’s trick.156
– Now it is not difficult to conclude that f must be an isomorphism. In-157
deed,158
Spec(H(G,K)) ⊆ Spec(C[X1, X±12 ])
defines a subscheme whose projection to Spec(C[X±12 ]) is free of infinite159
rank, but this is only possible for Spec(C[X1, X±12 ]) itself.160
Upshot:161
• Each irreducible, admissible GL2(Af )-representation V yields canonically a162
system of Hecke eigenvalues163
ap ∈ C, bp ∈ C×
for all primes p outside a finite set of primes S, which is naturally determined164
by V .165
• Concretely: If v ∈ GL2(Af ) is fixed by K(m) and p - m, then166
Tp(v) := GL2(Zp)
(p 0
0 1
)GL2(Zp) ∗ v = apv,
“the Hecke operator at p”, and167
Sp(v) := GL2(Zp)
(p 0
0 p
)GL2(Zp) ∗ v = bpv.
• Let Z ⊆ GL2 be the center, i.e., the scalar matrices.168
• Let ψ : A×f ∼= Z(Af ) → C× be the central character of V (exists by Schur’s169
lemma). Then170
bp = ψ((1, . . . , 1, p, 1, . . . , 1)
for p prime such that Z×p ⊆ A×f acts trivially on V .171
General results on the GL2(Af )-representation Sk:172
• The GL2(Af )-representation Sk decomposes into a direct sum173
Sk ∼=⊕i∈I
Vi
of irreducible, admissible representations Vi.174
• Indeed:175
– SkΦ⊆ L2([GL2]) and thus the C-vector space Sk is equipped with the176
non-degenerate GL2(Af )-equivariant hermitian pairing177
(·, ·) : Sk × Sk → C, (f, g) 7→∫
[GL2]
Φ(f)Φ(g)
– Each irreducible GL2(Af )-subquotient of Sk is already a direct summand178
(use orthogonal complements).179
– Using Zorn’s lemma each smooth non-zero GL2(Af )-representation V180
has an irreducible subquotient (Wlog: V generated by some v ∈ V .181
Then apply Zorn’s lemma to the set of submodules not containing v.)182
• Shalika/Piatetski-Shapiro: For each finite set S of primes and each systems183
of Hecke eigenvalues ap, bpp/∈S there exists at most one irreducible subrep-184
resentation V ⊆ Sk with this system of Hecke eigenvalues.185
– This is a special case of the ”strong multiplicty one theorem” for cus-186
pidal automorphic representations for GLn, cf. [GH19, Theorem 11.7.2]187
(maybe we’ll prove this later for n = 2).188
• Consequence: Assume that 0 6= f ∈ Sk is fixed by K(m), and an eigenvector189
for the Tp, Sp−operators for p - m. Then V := 〈f〉GL2(Af ) is irreducible.190
– Reason: Each v ∈ V yields a system of Hecke eigenvalues, which is191
equivalent to the one for f . Now apply strong multiplicity one.192
The decomposition of Sk indexed by newforms:193
• Atkin-Lehner/Casselman: For each irreducible GL2(Af )-subrepresentation194
V ⊆ Sk, there exists some m ∈ N and some 0 6= v ∈ V such that v is195
fixed by the subgroup196
K1(m) :=
(a b
c d
)| c ≡ 0, d ≡ 1 mod m ⊆ GL2(Z)
• Set197
Γ1(m) := GL2(Z) ∩K1(m).
• Remarks:198
– Our notation is different than in many sources, where Γ1(m) ⊆ SL2(Z).199
– The minimal m for which there exists such a non-zero v ∈ V is called200
the conductor of V . In this case, v is unique up to a scalar.201
– Maybe more details later, cf. [Del73].202
• The morphism203
Γ1(m)\H± '−→ GL2(Q)\(GL2(Af )/K1(m)×H±)
[z] 7→ [(1, z)]
is an isomorphism, compatible with ω⊗k.204
• In particular,205
Mk(Γ1(m)) ∼= Mk(K1(m)).
• Let f ∈Mk(K1(m)).206
• Let f ∈Mk(Γ1(m)) be the section corresponding to f .207
• Then208
f(g, z) = f(γ, z) = j(γ−1, z)−kf(γ−1z).
where g = γh with γ ∈ GL2(Q), h ∈ K1(m). Check!Check!209
• Fix p prime, p - m.210
• Tp, Sp act on Mk(K1(m)). Thus, by transport of structure on Mk(Γ1(m)).211
• Let us make Sp on Mk(Γ1(m)) explicit.212
• We have to find an expression213 (p 0
0 p
)= γh, with γ ∈ GL2(Q), h ∈ K1(m).
• Because p,m are prime there exists a, b ∈ Z, such that214
A :=
(a b
m p
)∈ GL2(Z).
• Then215
Adiag ·
(p−1 0
0 p−1
)diag
·
(p 0
0 p
)p
∈ K1(m),
where (−)diag denotes the diagonal embedding for GL2(Q) and (−)p the em-216
bedding at p.217
• In other words, we can take218
γ :=
(p 0
0 p
)A−1
• Thus Sp acts on Mk(Γ1(m)) via219
f 7→ (z 7→ j(γ−1, z)−kf(γ−1z)).
• The group220
Γ0(m) :=
(a b
c d
)| c ≡ 1 mod m
normalizes Γ1(m) and thus there is an action of Γ0(m)/Γ1(m) ∼= (Z/m)× via221
(B, f) 7→ j(B, z)−kf(Bz).
for B ∈ Γ0(m), f ∈Mk(Γ1(m)).222
• Thus, we obtain a decomposition223
Mk(Γ1(m)) =⊕
χ : (Z/m)×→C×Mk(Γ0(m), χ)
with Mk(Γ0(m), χ) the space of modular forms for Γ0(m) with nebentypus χ,224
i.e., modular forms for Γ1(m) satisfying225
f(az + b
cz + d)) = χ(d)(cz + d)kf(z) for
(a b
c d
)∈ Γ0(m).
• For each χ : (Z/m)× → C× there exists a unique character226
ψχ : Q×\A×/(1 +mZ) ∩ Z× → C×
such that ψχ(r) = r−k for r ∈ R>0.227
• We get that228
Mk(Γ0(m), χ) ∼= Mk(K1(m), ψχ)
where the RHS denotes the ψχ-eigenspace for the action of A× onMk(K1(m)).229
• Let us finish the description of Sp on Mk(Γ1(M)).230
• The action of Sp is given on Mk(Γ0(m), χ) ⊆Mk(Γ1(m)) by231
γ−1 =
(p−1 0
0 p−1
)A
where
(p−1 0
0 p−1
)acts by pk and A ∈ Γ0(m) by χ(p).232
• Let us describe now the action (by transport of structure) of Tp onMk(Γ0(m), χ).233
• We have234
GL2(Zp)
(p 0
0 1
)GL2(Zp) =
p−1∐j=0
(p j
0 1
)GL2(Zp)
∐(1 0
0 p
)GL2(Zp).
• We again have to find factorizations into235
γ · h, γ ∈ GL2(Q), h ∈ K1(m).
• We can take the factorization236 (p j
0 1
)p
=
(p j
0 1
)diag
· h
for j = 0, . . . , p− 1, with h ∈ K1(m), i.e.,237
γ−1 =
(p−1 −p−1j
0 1
).
• For238 (1 0
0 p
)we find, similar to the argument above,239
γ−1 =
(a b
m p
)·
(1 0
0 p−1
)=
(1 0
0 p−1
)·
(a p−1b
pm p
).
• Putting these together we obtain240
Tp(f) = pkχ(p)f(pz) +
p−1∑j=0
f(z − jp
).
• If f ∈Mk(Γ0(m), χ) is written in Fourier expansion241
f(z) =
∞∑n=0
anqn, q = e2πiz,
then242
Tp(f)(z) =
∞∑n=0
panpqn +
∞∑n=0
pkχ(p)anqpn.
• This follows from243
p−1∑j=0
e2πi−jnp =
p, p|n
0, otherwise.
Check!Check!244
• In particular, if f ∈Mk(Γ0(m), χ) is an eigenvector for Tp and a1 = 1, then245
Tp(f) = apf,
where246
Tp =1
pTp.
In other words, the Hecke eigenvalues are the Fourier coefficients of the nor-247
malized eigenform.248
Theorem 5.4. The GL2(Af )-representation Sk decomposes into irreducibles as249
Sk ∼=⊕f
〈f〉GL2(Af )
with f running through the set of (normalized) newforms for Γ0(N) with nebentypus250
χ for varying N and Dirichlet characters χ : (Z/N)× → C×.251
Newforms (roughly):252
• A newform is an eigenform realising the minimal N among all eigenforms253
with an equivalent system of Hecke eigenvalues.254
Proof. After the above preparation we can now quote [DS05, Proposition 5.8.4],255
[DS05, Theorem 5.8.2] which imply that for each system of Hecke eigenvalues ap-256
pearing in Sk there exists a unique normalized (i.e., in Fourier expansion a1 = 1)257
newform with equivalent system of Hecke eigenvalues. 258
Questions?259
6. Langlands reciprocity for newforms1
Last time:2
• We proved the decomposition3
Sk ∼=⊕f
〈f〉GL2(Af )
into irreducibles.4
• Here f is running through the set of (normalized) newforms of weight k for5
Γ0(N) with nebentypus χ for varying N and varying Dirichlet characters6
χ : (Z/N)× → C×.
• To each irreducible, admissible GL2(Af )-representation π we associated a7
system of Hecke eigenvalues8
ap(π), bp(π)p/∈S
for S := p prime with πGLn(Zp) = 0. In fact, ap(π) resp. bp(π) is the eigen-9
value of the double coset10
GL2(Zp)
(p 0
0 1
)GL2(Zp)
resp.11
GL2(Zp)
(p 0
0 p
)GL2(Zp)
when acting on a non-zero vector fixed by GL2(Zp).12
• If π = 〈f〉GL2(Af ) for a normalized newform f ∈ Sk(Γ0(N), χ), then S =13
prime divisors of N and14
ap(π) = pap, bp(π) = pkχ(p)
if p /∈ S, where f(q) =∞∑n=1
anqn is the Fourier expansion of f .15
Today:16
• Traces of Frobenii for `-adic Galois representations.17
• Statement of Langlands program for Sk.18
Traces of Frobenii for `-adic representations:19
• ` some prime.20
• Q` = lim−→E/Q`
E is a topological field via colimit topology.21
• Let W be a finite dimensional Q`-vector space.22
• Let σ : GQ := Gal(Q/Q)→ GL(W ) be a continuous representation (an “`-adic23
Galois representation”)24
• Recall: For each prime p there exists an embedding25
GQp := Gal(Qp/Qp)→ GQ,
well-defined up to conjugacy in GQ.26
• GQp sits in the exact sequence:27
1→ Ip → GQp → Gal(Fp/Fp) = (Frobgeomp )Z → 1
with Ip the inertia subgroup, and28
Frobgeomp : Fp → Fp, x 7→ x1/p
the geometric Frobenius at p (= inverse of usual Frobenius).29
• Let S be a finite set of primes. We say σ is unramified outside S if σ(Ip) = 130
for all p /∈ S.31
• If σ unramified outside S and n := dimQ`W , we can associate to σ a system32
c1,p, . . . , cn,pp/∈S
of elements c1,p, . . . , cn,p ∈ Q`. Namely,33
ci,p := Tr(σ(Frobgeomp )|ΛiW )
• Note: The ci,p are well-defined because σ(Ip) = 1.34
• Note: The ci,p = Tr(σ(Frobgeomp )|ΛiW ) depend only on p as the trace is35
invariant under conjugation.36
• If W is semisimple, then σ is determined by the37
c1,p, . . . , cn,pp/∈S
for each finite set of primes S such that W is unramified outside S, cf. [Ser97,38
I-10].39
• In fact, the collection c1,pp/∈S is sufficient (+ W semisimple and unramified40
outside S).41
• Indeed:42
– Consider Λ := Q`[GQ,S ], where GQ,S is the quotient of GQ by the closure43
of the subgroup generated by the Ip, p /∈ S.44
– Assume W,W ′ are two semisimple, continuous GQ,S-representations with45
Tr(Frobgeomp |W ) = Tr(Frobgeom
p |W ′)
for all p /∈ S.46
– By Chebotarev density and continuity this implies47
Tr(λ|W ) = Tr(λ|W ′),
for all λ ∈ Λ.48
– For irreducible, pairwise non-isomorphic Λ-modules W1, . . . ,Wm and49
each i = 1, . . . ,m, there exists µi ∈ Λ, such that50
µi = 1
on Wi, but51
µi = 0
on Wj , j 6= i (this is a version of the Chinese remainder theorem).52
– Write53
W =
m⊕i
W⊕nii , W ′ =
m⊕i=1
W⊕n′ii
in isotypic components withW1, . . . ,Wm irreducible, pairwise non-isomorphic,54
and ni, n′i ∈ N, possibly zero.55
– Then56
ni dimQ`Wi = Tr(µi|W ) = Tr(µi|W ′) = n′i dimQ`Wi
for all i = 1, . . . ,m, i.e., ni = n′i, i = 1, . . . ,m and W ∼= W ′ as desired.57
• This statement can be seen as an analog of “strong multiplicity one” for58
cuspidal automorphic representations for GLn mentioned last time.59
Arithmetic significance of the traces of Frobenii:60
• Let σ : GQ → GL(W ) be an `-adic representation, unramified outside S.61
• The traces62
c1,p := Tr(σ(Frobgeomp )|W )
encode significant arithmetic information.63
• For example, assume F/Q is Galois with Gal(F/Q) ∼= S3 and64
σ : GQ Gal(F/Q)→ GL2(Q`)
is inflated from the unique irreducible, 2-dimensional representation of S3.65
Set66
S := p ramified in F
and for p /∈ S67
FrobgeomF,p
as the image of Frobgeomp ∈ GQ,S in Gal(F/Q) ∼= S3.68
• Then:69
∗ Tr(Frobgeomp |W ) = 2⇔ Frobgeom
F,p ∈ 1 ⇔ p splits completely in OF ,70
∗ Tr(Frobgeomp |W ) = 0⇔ Frobgeom
F,p ∈ (1, 2), (1, 3), (2, 3) ⇔ p splits into three distinct primes in OF ,71
∗ Tr(Frobgeomp |W ) = −1⇔ Frobgeom
F,p ∈ (1, 2, 3), (1, 3, 2) ⇔ p splits into two distinct primes in OF .72
• Thus: Knowledge of traces of Frobenii for all `-adic Galois representations73
(with finite image) implies knowledge of the decomposition of unramified74
primes in all finite extensions of Q.75
• Other examples of `-adic representations:76
Hiet(XQ,Q`)
for proper, smooth schemes X over Q, and i ≥ 0.77
• TheGQ-action is induced by functoriality ofHiet(−,Q`) from the (right) action78
of GQ on79
XQ = X ×Spec(Q) Spec(Q).
• Proper, smooth base change + Grothendieck-Lefschetz trace formula ⇒ If X80
has good reduction at p and p 6= `, then81
2dim(X)∑i=0
(−1)iTr(Frobgeomp |Hi
et(XQ,Q`)) = ]X (Fp),
where X → Spec(Z(p)) is a proper, smooth model of X at p.82
• The last example motivates why we took the geometric Frobenius and not83
the arithmetic Frobenius Frobarithp = (Frobgeom
p )−1.84
Questions?85
The whole point of Langlands reciprocity:86
• Traces of Frobenii are supposed to match Hecke eigenvalues!87
• Thus, arithmetic information is expected to be encoded in automorphic in-88
formation, which might be more concrete.89
• Example (Hecke):90
– Let F be the splitting field of x3 − x− 1.91
– Then F is the Hilbert class field of Q(√−23), and Galois over Q with92
Galois group S3.93
– Consider as before the associated 2-dimensional, irreducible Galois rep-94
resentation σ : GQ → Gal(F/Q) → GL2(Q`).95
– Hecke proved that there exists a normalized newform f(q) =∞∑i=1
anqn
96
such that97
ap = Tr(σ(Frobgeomp ))
for all primes p with p - 23.98
– One can make f more explicit: Considering ramification, f must lie in99
S1(Γ0(23), χ) with χ : (Z/23)× → C× the quadratic character determin-100
ing101
Q(√−23) ⊆ Q(ζ23),
and people knowing modular forms will tell us that we must have102
f(q) = q
∞∏i=1
(1− qi)(1− q23i).
– We can now calculate the developement of f (or look it up in some103
database like LMFDB):104
f(q) = q − q2 − q3 + . . .+ q58 + 2q59 + . . .
and deduce how primes decompose in F , e.g., 59 splits completely!105
– This example was posted by Matthew Emerton on mathoverflow. More106
insightful posts by him can be found via his webpage: http://www.107
math.uchicago.edu/~emerton/ . Find and read
them!
Find and read
them!
108
• Fermat’s theorem was proved by Wiles/Taylor using the following strategy109
(initiated by Frey, and complemented by Serre, Ribet):110
– Assume up + vp + wp = 0 with u, v, w ∈ Q, uvw 6= 0 and p ≥ 3.111
– Consider the elliptic curve E over Q with (affine) Weierstraß equation112
y2 = x(x+ up)(x− vp)
(after possibly manipulating u, v, w a bit).113
– Consider the Galois representation114
σ : GQ → GL(H1et(EQ,Q`)) ∼= GL2(Q`).
– Show that σ is modular, i.e., there exists a normalized newform f1(q) =115∞∑n=1
anqn of weight 2 such that116
Tr(σ(Frobgeom)) = ap
for almost all primes p.117
– Using a theorem of Ribet, one concludes that f1 must be congruent to a118
normalized newform f2 in119
S2(Γ0(2)),
i.e., the Fourier coefficients of f1, f2 are algebraic integers and congruent120
modulo some prime.121
– But S2(Γ0(2)) = 0, and thus f2 cannot exist.122
– This yields the desired contradiction.123
– For more details, cf. [Wil95], [Rib90].124
Langlands reciprocity for newforms:125
• Fix a prime ` and an isomorphism126
ι : Q` ∼= C.
• For newforms the Langlands program combined with the Fontaine-Mazur127
conjecture predicts a bijection128
LL: Amod'−→ Gmod
from the set129
Amod := irreducible GL2(Af )-subrepresentations π ⊆⊕k≥1
Sk,
to a certain set130
Gmod
consisting of irreducible, 2-dimensional Galois representations σ : GQ → GL2(Q`)131
• LL should satisfy that for all π ∈ Amod with σ := LL(π) we have132
ι(Tr(σ(Frobarithp ))) = ap(π), ι(det(σ(Frobarith
p ))) = bp(π)
for p outside some specified finite set S of primes. In a previous ver-
sion I considered
the geometric
Frobenius, which
was wrong.
In a previous ver-
sion I considered
the geometric
Frobenius, which
was wrong.
133
• Here, the eigenvalues ap(π) resp. bp(π) are used and not the ap(π), bp(π).134
• Note: Such a bijection LL is uniquely determined (by the respective multi-135
plicity one theorems), if it exists.136
• How is S specified? Let us call a prime p unramified for σ resp. π if137
σ(Ip) = 1
resp.138
πGL2(Zp) 6= 0.
• Then a prime p 6= ` is conjecturally unramified for π if and only if it is139
unramified for σ := LL(π).140
• The matching of Hecke eigenvalues states then more precisely that σ(Frobarithp )141
should have characteristic polynomial142
X2 − ι−1(ap(π))X + ι−1(bp(π))
for each unramified prime p 6= ` for π.143
• It is the next aim of the course to indicate how the map LL can be constructed,144
i.e., how to associate Galois representations to normalized newforms. For this145
we follow results of Deligne, cf. [Del71b], and Deligne/Serre, cf. [DS74].146
The set Gmod:147
• The set Gmod was defined by Fontaine/Mazur in relation with their remarkable148
conjecture on geometric Galois representations, cf. [FM95].149
• Let σ : GQ → GL2(Q`) be an irreducible, 2-dimensional `-adic representation.150
• Then by definition σ ∈ Gmod if and only if151
– σ unramified at almost all p, i.e., σ(Ip) = 1 for p outside some finite set152
of primes S.153
– σ is odd, i.e., for each complex conjugation c ∈ Gal(C/R) ⊆ GQ we have154
det(σ(c)) = −1.
– The restriction155
σ` := σ|GQ`: GQ` → GL2(Q`)
is de Rham with Hodge-Tate weights 0, ω for some ω ∈ N (note that156
ω = 0 is allowed).157
• If σ = LL(π) with π ⊆ Sk, then conjecturally ω = k − 1.158
• Using the work of many people (Wiles, Kisin, Emerton, Skinner-Wiles, Pan,. . . )159
any σ ∈ Gmod with distinct Hodge-Tate weights is modular if ` ≥ 5, cf. [Pan19,160
Theorem 1.0.4.].161
• If σ ∈ Gmod has finite image, then σ is modular, cf. [PS16], [GH19, Section162
13.4.] (and the references therein).163
• Conjecturally: HT-weights 0 ⇔ image of σ finite.164
de Rham representations:165
• p prime (the previous `).166
• K/Qp a discretely valued, non-archimedean extension (e.g., K/Qp finite).167
• σ : GK := Gal(K/K)→ GLm(Qp) a continuous representation.168
• Then σ has coefficients in some finite extension E/Qp (by compactness of GK169
as Qp carries the colimit topology here).170
• We obtain a Qp-linear representation, a “local p-adic Galois representation”,171
ρ : GK → GLm(E) ⊆ GLn(Qp),
where n := m · dimQpE.172
• Upshot: Let V be a finite dimensional Qp-vector space and ρ : GK → GL(V )173
a continuous representation.174
• Then V is called de Rham if175
dimK(BdR ⊗Qp V )GK = dimQp V
for a certain field extension BdR of K with GK-action, cf. [BC09, Section 6].176
• Namely: BdR is Fontaine’s field of p-adic periods, cf. [Fon94], [BC09, Defini-177
tion 4.4.7]. Abstractly,178
BdR∼= CK((t)),
where CK is the completion of K for its p-adic valuation (this topology is179
coarser than the colimit topology on K).180
• BdR is a discretely valued field, with residue field CK , the deduced decreasing181
filtration Fil•BdR is GK-stable and as CK-semilinear GK-representations182
gr•BdR∼=⊕j∈Z
CK(j),
where CK(j) := CK ⊗Zp Zp(j), with183
Zp(1) := Tpµp∞(CK)
the p-adic Galois representation associated to the cyclotomic character184
χcyc : GK → Z×p ∼= Aut(µp∞(CK)).
• Important Theorem of Tate, cf. [BC09, Theorem 2.2.7]: Let χ : GK → Q×p be185
a character. Then186
H0(GK ,CK(χ)) =
K, if χ|IK has finite image
0, otherwise ,
here IK ⊆ GK is the inertia subgroup.187
• In particular:188
H0(GK ,CK(j)) = 0, j 6= 0
as χjcyc|IKhas infinite image if j 6= 0.189
• Conclusion: If dimC V = 1, then V is de Rham if and only if V is isomorphic190
to χ ·χjcyc for some character χ : GK → Q×p with IK of finite order, and j ∈ Z191
⇒ There exists many representations, which are not de Rham: (χp−1cyc )a with192
a ∈ Zp \ Z.193
• Facts:194
– ρ : GK → GL(V ) is de Rham if ρ|GK′ is de Rham for some finite extension195
K ′ of K, cf. [BC09, Proposition 6.3.8].196
– de Rham representations are stable under subquotients, duals and tensor197
products, cf. [BC09, Section 6.1].198
• Let V be a de Rham representation of dimension n. Then199
V ⊗Qp CK ∼= CK(j1)⊕ . . .⊕ CK(jn)
as CK-semilinear GK-representations, where j1, . . . , jn ∈ Z. The unordered200
collection (j1, . . . , jn) is called the collection of Hodge-Tate weights for V .201
Strategy for constructing LL (roughly):202
• Let π ∈ Amod, with associated system of Hecke eigenvalues203
ap, bpp/∈S .
• Assume for simplicity that π ⊆ S2, i.e., associated to a weight 2 newform.204
• Then we will construct a GL2(Af )-equivariant embedding205
S2 → H1(GL2(Q)\(GL2(Af )×H±),C) := lim−→K
H1(GL2(Q)\(GL2(Af )/K ×H±),C)
of S2 into cohomology.206
• The given isomorphism ι : Q` ∼= C yields an isomorphism207
H1(GL2(Q)\(GL2(Af )×H±),C) ∼= H1(GL2(Q)\(GL2(Af )×H±),Q`)
• For K ⊆ GL2(Af ) compact-open, the complex manifold208
GL2(Q)\(GL2(Af )/K ×H±)
is actually algebraic, i.e., given by XK(C) for some quasi-projective scheme209
XK → Spec(C).210
• The etale comparison theorem implies211
H1(GL2(Q)\(GL2(Af )×H±),Q`) ∼= H1et(XK ,Q`).
• Now the miracle happens: For K ⊆ GL2(Af )212
the quasi-projective scheme XK → Spec(C) is canonically defined over Spec(Q)!213
In other words, XK∼= XK ×Spec(Q) Spec(C) for (compatible) quasi-projective214
schemes XK → Spec(Q).215
• In particular,216
H1et(XK ,Q`) ∼= H1
et(XK,Q,Q`)
by invariance of etale cohomology under change of algebraically closed base217
fields. But the RHS carries an action of GQ!218
• Look now at the GQ ×GL2(Af )-module219
H1et(XQ,Q`) := lim−→
K
H1et(XK,Q,Q`).
• Then we will check that the GQ-module220
LL(π) := HomGL2(Af )(π,H1et(XQ,Q`))
is 2-dimensional and that we can express the traces of Frobenii by Hecke221
eigenvalues.222
• Replacing Q` by some local system a similar strategy works for weight k ≥ 2.223
• However, this does not help for k = 1. Here one has to use that a weight 1224
modular form can be congruent to some modular form of weight k ≥ 2, and225
use the previous case.226
Questions?227
7. Modular curves as moduli of elliptic curves (by Ben Heuer)1
Last time:2
• First rough sketch of construction of3
LL: Amod'−→ Gmod
Amod :=irreducible GL2(Af )-subrepresentations π ⊆⊕k≥1
Sk,
Gmod :=σ : GQ → GL2(Q`), unramified outside finite set S, odd, de Rham
• Crucial point for construction: The complex manifold4
GL2(Q)\(GL2(Af )/K ×H±)
(1) is algebraic over Spec(C),5
(2) admits canonical model XK over Spec(Q).6
This gives rise to the Galois action on l-adic etale cohomology H1et(XK,Q,Q`).7
Today:8
• Explain why the algebraic model XK exists.9
• Reinterpret notions defined before in algebro-geometric terms: modular forms,10
compactification, q-expansions, adelic Hecke action, ...11
Reminder on elliptic curves Reference: [Silverman: The Arithmetic...]12
• Let K be any field. We have the following equivalent definitions:13
Definition 7.1. An elliptic curve over K is a14
– connected smooth projective curve E|K of genus 1 with point O ∈ E(K).15
– connected smooth projective algebraic group E|K of dimension 1.16
– non-singular plane cubic curve, defined by a Weierstraß equation17
E : y2 = x3 + ax+ b, a, b ∈ K
(if charK 6∈ 2, 3, otherwise more terms are required).18
• Fact: The group scheme is automatically commutative (!).19
• Fact: The non-singularity can be expressed as a condition on the discriminant:20
E non-singular ⇔ ∆(a, b) = −16(4a3 + 27b2) 6= 0.
• Fact: Given E, the Weierstraß equation is not unique. But the j-invariant21
j(E) := j(a, b) = 17284a3
4a3 + 27b2
is independent of choice of Weierstraß equation.22
• Fact: If K is algebraically closed, we have23
E ∼= E′ ⇔ j(E) = j(E′).
• More generally, we can replace Spec(K) by any base scheme S.24
Definition 7.2. An elliptic curve over S is a proper smooth curve E → S25
with geometrically connected fibres of genus 1, together with a point 0 : S → E.26
• This can be thought of as family of elliptic curves parametrised by S.27
• Again, this automatically has a group structure:28
Theorem (Abel). There is a unique way to endow an elliptic curve E → S29
with the structure of a commutative S-group scheme s.t. 0 = identity section.30
• Zariski-locally on S, one can describe E in terms of a Weierstraß equation.31
Definition 7.3. A morphism of elliptic curves E → E′ over S is a homomor-32
phism of S-group schemes. An isogeny is an fppf-surjective homomorphism33
with finite flat kernel.34
Example 7.4. For any N ∈ Z, the group scheme structure gives the mor-35
phism “multiplication by N”, which is denoted by [N ] : E → E.36
• Fact: [N ] is finite flat of rank N2. It is finite etale if N is invertible on S.37
• In particular, [N ] is an isogeny.38
Definition 7.5. E[N ] := ker[N ] ⊆ E, the N -torsion subgroup. This is a39
finite flat closed subgroup scheme. It is finite etale if N is invertible on S.40
• More generally, for any finite flat subgroup scheme D ⊆ E[N ] of exponent N ,41
∃ unique elliptic curve E/D with isogeny ϕ : E → E/D with kernel D.42
• [N ] induces a dual isogeny ϕ∨ : E/D → E with kernel E[N ]/D.43
Elliptic curves over K = C Reference: [Diamond–Shurman: A first course...]44
• Let E|C be an elliptic curve. Then E(C) has the structure of a compact45
complex Lie group. As such, it can be canonically uniformised:46
• Let LieE be the tangent space at 0 ∈ E(C). Then LieE ∼= C and there is an47
isomorphism of complex Lie groups48
E(C) = LieE/H1(E,Z) ∼= C/Λ, where Λ ∼= Z2.
• For any N ∈ Z, we have49
E(C)[N ] = ( 1NΛ)/Λ ⊆ C/Λ.
• Note: E(C)[N ] = ( 1NZ/Z)2 = (Z/N)2 is of rank N2, as claimed.50
• Upshot: E(C) is a 1-dimensional complex torus: A quotient of C by a51
lattice Z2 ∼= Λ ⊆ C such that Λ⊗Z R = C.52
• morphisms of complex tori := morphisms of complex Lie groups53
Theorem. There is an equivalence of categories54
elliptic curves over C → 1-dim complex tori
• Essential surjectivity: Let Λ ⊆ C as above. Define the Weierstraß ℘-function55
℘(z,Λ) =1
z2+∑w∈Λw 6=0
(1
z2− 1
(z − w)2
)
• Then ℘(z) = ℘(z,Λ) is holomorphic on C\Λ with complex derivative56
℘′(z) = −2∑w∈Λ
1
(z − w)3.
• This is clearly Λ-periodic ℘ and ℘′ are meromorphic functions on C/Λ.57
Theorem. The functions ℘ and ℘′ are related by a Weierstraß equation
EΛ : ℘′2 =4(℘3 + g2(Λ)℘+ g3(Λ))
for explicit g2(Λ), g3(Λ) ∈ C. We thus get an isomorphism of Lie groups58
(℘, ℘′) : C/Λ ∼−→ EΛ(C).
• Sending C/Λ 7→ EΛ defines a quasi-inverse59
1-dim complex tori → elliptic curves over C.
Complex moduli spaces of elliptic curves60
• We can now parametrise complex elliptic curves up to isomorphism61
• For this, we just have to determine when two lattices Λ1,Λ2 satisfy62
C/Λ1∼= C/Λ2.
• Fact: Any homomorphism of complex tori is multiplication by a ∈ C.63
• ⇒ this happens iff Λ1 = aΛ2 for some a ∈ C×.64
• Can now uniformise as follows: Write Λ = w1Z⊕ w2Z.65
• The condition Λ⊗Z R = C implies w1/w2 ∈ C \ R = H±.66
• ⇒ For every complex torus E, there is τ ∈ H± such that67
E ∼= Eτ := C/Λτ , Λτ = Z⊕ τZ
• Finally, for τ, τ ′ ∈ H±, we have68
Z + τZ = Z + τ ′Z⇔ ∃γ =(a bc d
)∈ GL2(Z) : γτ =
aτ + b
cτ + d= τ ′
• This shows: τ 7→ Eτ defines69
GL2(Z)\H± ∼= complex tori/ ∼∼= elliptic curves over C/ ∼
This gives our space from earlier,70
SL2(Z)\H = GL2(Z)\H± = GL2(Q)\(GL2(Af )/K ×H±) for K = GL2(Z)
an interpretation as a “moduli space” (so far, in a weak sense):71
SL2(Z)\H = elliptic curves over C/ ∼
• C algebraically closed⇒ j-invariant defines bijection (in fact biholomorphism)72
j : SL2(Z)\H ∼−→ C.
• Via the covering map H± → GL2(Z)\H±, we get from the above:
H± =complex tori with ordered basis α : Z2 ∼−→ Λ/ ∼
=elliptic curves E over C with α : Z2 ∼−→ H1(E,Z)/ ∼
• What about moduli interpretations of other levels? Recall for N ∈ N, we had73
the level K = K(N),74
XK := GL2(Q)\(GL2(Af )/K ×H±) =∐
Γ(N)\H±
where75
Γ(N) = γ ∈ GL2(Z)|γ ≡ ( 1 00 1 ) mod N.
• These are precisely the automorphisms of Λ that preserve the subgroup76
EΛ[N ] = 1NΛ/Λ ⊆ C/Λ.
Consequently, we have the more general statement
Γ(N)\H± =complex tori with ordered basis α : (Z/N)2 ∼−→ 1NΛ/Λ/ ∼
=elliptic curves E over C with α : (Z/N)2 ∼−→ E(C)[N ]/ ∼ .
• In between GL2(Z) and Γ(N), we had the groups77
Γ1(N) = γ ∈ GL2(Z)|γ ≡ ( ∗ ∗0 1 ) mod N,78
Γ0(N) = γ ∈ GL2(Z)|γ ≡ ( ∗ ∗0 ∗ ) mod N.
• The modular curves associated to these have moduli interpretations
Γ1(N)\H± =elliptic curves E|C with point Q ∈ E(C)[N ] of exact order N/ ∼
Γ0(N)\H± =elliptic curves E|C with cyclic subgroup C ⊆ E(C)[N ] of rank N/ ∼ .
Motivating Example: The Legendre family79
• Consider the complex manifold U = P1\0, 1,∞ in the variable λ.80
• Over U , we have a complex family of elliptic curves:81
• The Legendre family is cut out of P2U → U by the Weierstraß equation82
Eλ : Y 2 = X(X − 1)(X − λ).
Proposition 7.6. (1) We have Eλ[2] = O, (0, 0), (1, 0), (λ, 0). In partic-83
ular, there is a natural isomorphism Eλ[2] = (Z/2Z)2.84
(2) For any elliptic curve E over C together with a trivialisation85
α : (Z/2Z)2 ∼−→ E[2],
there is a unique point x ∈ U such that E = (Eλ)x.86
• Upshot: Eλ → U is universal elliptic curve with level Γ(2), in some sense.87
• This is all algebraic! Can make sense of Eλ → U as morphism of schemes C.88
• Even better: These schemes are already defined over Q.89
• Can elaborate this argument to show that90
Γ(2)\H± =: XΓ(2)(C) = P1\0, 1,∞.
• NB: In particular, the compactification is X∗Γ(2) = P1.91
• This shows that S2(Γ0(2)) = 0, which we used for Fermat’s Last Theorem.92
Idea:93
Get algebraic model of Γ(N)\H± over Q by passing from C to moduli of elliptic
curves over general Q-schemes S.
• Question: Is there a scheme representing the functor94
PSL2(Z) : S 7→ elliptic curves over S/ ∼
on schemes over Q? This would be a scheme X → Q with C-points SL2(Z)\H.95
• Answer: No. An easy way to see this is the phenomenon of twists:96
• Can have two non-isomorphic E 6∼= E′ over Q that become isomorphic over a97
quadratic extension K. In particular, j(E) = j′(E).98
• Clearly, two Q-points of X agree iff they do over K. So X cannot exist!99
• Same problem for the Legendre family above: This does not represent100
S 7→ elliptic curves E over S with α : (Z/2Z)2 ∼−→ E[2]/ ∼
because this fails to account for quadratic twists.101
• Fact: Twists of elliptic curves E over a field K correspond to 1-cocycles102
GK|K → Aut(E).
Quadratic twists come from the involution ([−1] : E → E) ∈ Aut(E).103
• Slogan: Nontrivial automorphisms are bad for representability104
• Upshot: In order to get a representable functor, need additional data.105
Moduli problems of elliptic curves [Katz–Mazur: Arithmetic moduli...]106
• Consider the moduli functors on schemes over Z[ 1N ]:
PΓ0(N) : S 7→ (E|S,C ⊆ E[N ] cyclic subgroup scheme of rank N)/ ∼,
PΓ1(N) : S 7→ (E|S,Q ∈ E[N ](S) point of exact order N)/ ∼,
PΓ(N) : S 7→ (E|S, α : (Z/NZ)2 ∼−→ E[N ] isom of group schemes)/ ∼ .
• We can also form “mixed moduli problems” like107
PΓ1(N)∩Γ0(p) := PΓ1(N) ×PSL2(Z)PΓ0(p) : S 7→ (E,Q,C)/ ∼
• Crucial point: Let (E,C) ∈ PΓ0(N)(S). The automorphism [−1] : E → E108
sends C 7→ C. In particular, [−1] ∈ Aut(E,C).109
• In contrast, let (E,Q) ∈ PΓ1(N)(S), then [−1] : E → E sends Q ∈ E[N ] to110
−Q ∈ E[N ], which is different if N ≥ 3. In particular, [−1] 6∈ Aut(E,Q).111
• Similarly for PΓ(N)(S). This explains the problem with the Legendre family:112
N = 2 is too small to deal with the automorphism [−1]! We need N ≥ 3.113
• Upshot: The moduli problems PΓ(N) and PΓ1(N) are “rigid” (:= have no114
non-trivial automorphisms) for N ≥ 3, in contrast to PΓ0(N) and PΓ(2).115
Theorem. Let N ≥ 3 and p any prime.
(1) The moduli problems PΓ(N) and PΓ1(N) are each representable by smooth
affine curves XΓ(N) and XΓ1(N) over Z[ 1N ].
(2) For any n ∈ N, the moduli problem PΓ(N)∩Γ0(pn) is representable by a flat
affine curve XΓ1(N)∩Γ0(pn) over Z[ 1N ] that is smooth over Z[ 1
pN ].
• Remark: In order to represent PSL2(Z), one can pass from schemes to the116
bigger category of stacks. Get “moduli stack of elliptic curves” M1,1.117
• Between varying level structures Γ ⊆ Γ′ ⊆ Γ(N), have forgetful morphisms118
XΓ′ → XΓ.
These are all finite etale over Z[ 1N ].119
• GL2(Z/NZ) acts from the right on PΓ(N) by precomposition120
α 7→ α γ.
Alternatively, this is a left action via precomposition with γ∨ = γ−1 det γ.121
• This induces a GL2(Z/NZ)-action on XΓ(N).122
• Similarly, there is a natural (Z/NZ)×-action on XΓ1(N).123
• We can then define a modular curve124
XΓ0(N) := XΓ1(N)/(Z/NZ)×.
• This does not represent P(Γ0(N)), but it is “as close as possible”.125
• In particular, XΓ0(N)(K) = P(Γ0(N))(K) for algebraically closed K.126
Compactification127
• The j-invariant associated to Weierstraß equations defines a finite flat function128
j : X → A1.
• By normalisation in A1 → P1, can define129
j : X∗ → P1,
still a finite flat morphism. Think of X∗ as X plus a finite divisor of points.130
• Fact: X∗ is smooth and proper.131
• Reason: This can be seen using the Tate curve over Z[ 1N ][[q]].132
• X∗(C) is the smooth compactification X∗C of Γ\H± mentioned earlier.133
• Remark: X∗ is a moduli space of “generalised elliptic curves”.134
Geometric Modular forms References: [Katz], [Loeffler: lectures notes]135
• Fix N ≥ 3. Let X = XΓ1(N).136
• Goal: Geometric reinterpretation of modular forms as sections of sheaves on137
X:138
Definition 7.7. Let ω := e∗Ω1E|X where e : X → E is the identity section.139
• Since E → X is smooth of dimension 1, this is a line bundle.140
• Fact: This extends uniquely to a line bundle ω on X∗. Reason: The universal141
elliptic curve extends to a group scheme E∗ → X∗, take e∗Ω1E∗|X∗ .142
Proposition 7.8. The analytification of ω⊗k is naturally isomorphic to the143
sheaf ωk of modular forms of weight k on X∗(C).144
Sketch of proof. Recall: H± is moduli space of E|C with α : Z2 ∼−→ H1(E,Z).145
Integration∫α(1,0)
defines an isomorphism e∗Ω1E|C
∼−→ C.146
Get canonical trivialisation of ω over H±. Check GL2(Z)-action coincides. 147
• In particular, the sheaf ω already has an algebraic model over Z[ 1N ].148
Definition 7.9. For any Z[ 1N ]-algebra R, can define149
Mk(Γ1(N), R) = Γ(X∗Γ1(N) ×Spec(Z[1/N ]) Spec(R), ω⊗k),
the R-module of modular forms of weight k with coefficients in R.150
• Fact: This is a finite free Z[ 1N ]-module.151
• Fact: Using Tate curves, get q-expansions in R[[q]]. Can define cusp forms.152
• Fact: If R,S are flat Z[ 1N ]-algebras (e.g. any Q-algebras), then153
Mk(Γ1(N), R)⊗R S = Mk(Γ1(N), S).
• Upshot: Modular forms are manifestly objects of algebraic geometry!154
8. The Eichler–Shimura relation (by Ben Heuer)1
Last time:2
• There is a GL2(Z)-equivariant bijection.3
H± → elliptic curves E over C with α : Z2∼=−→ H1(E,Z)/ ∼
τ 7→ (Eτ := C/(Z + Zτ), α : Z2 → Z + Zτ, (1, 0) 7→ 1, (0, 1) 7→ τ)
• Corollary:4
GL2(Z)\(GL2(Z)×H±) ∼= elliptic curves E over C with α : Z2 ∼−→ TE/ ∼,
where TE := lim←−N∈N
E[N ] is the full adelic Tate module of E.5
• This can be defined over Q!6
• Set X → Spec(Q) as the moduli scheme representing the functor7
Q−Sch→ Sets, S 7→ elliptic curves E over S with α : Z2 ∼−→ TE/ ∼ .
Here, TE = lim←−N
E[N ] is an inverse limit of finite, etale group schemes over S8
• −1 ∈ GL2(Z) acts trivially on X.9
• For K ⊆ GL2(Z) compact-open with K ⊆ K1(3) (note in particular, −1 /∈ K)10
have smooth quasi-projective curve11
XK = X/K.
• E.g.: If K = K(N), N ≥ 3, then12
XK(N)
parametrizes elliptic curves E together with trivialization (Z/N)2 ∼−→ E[N ].13
• Other examples, K = K1(N) Z/N → E[N ] , K = K0(N) subgroups.14
Next goal:15
• Use modular curves to reinterpret Hecke operators geometrically.16
• From geometry of modular curves, deduce fundamental relation between Ga-17
lois action and Hecke action: The Eichler–Shimura relation.18
The adelic action:19
• Clearly, GL2(Z) acts on X, by precomposition on α : Z2 ∼= TE.20
• But: Want GL2(Af )-action to get Hecke operators21
⇒ work with elliptic curves up to quasi-isogeny.22
• Recall: Af = Z⊗Z Q and23
GL2(Z)\(GL2(Z)×H±) ∼= GL2(Q)\(GL2(Af )×H±).
• On elliptic curves can interpret this as follows.24
• Recall that the LHS geometrizes to X, which represents the functors
P : Q−Sch→Sets, S 7→(E|S, α : Z2 ∼−→ TE)/isomorphism
• Let V E = TE ⊗Z Q be the rational adelic Tate module, and define
P ′ : Q−Sch→Sets, S 7→(E|S, α : A2f∼−→ V E)/quasi-isogeny
Definition 8.1. Recall: An isogeny is a surjective homomorphism E → E′25
with finite flat kernel. A quasi-isogeny is an element of26
E → E′ isogeny ⊗Z Q.
A p-quasi-isogeny is an element of27
E → E′ isogeny of degree a power of p ⊗Z Z[ 1p ].
• Then P ∼= P ′. Indeed:28
– Clearly have natural transformation P → P ′. Sketch for inverse:29
– Take (E,α : A2f∼−→ V E) ∈ P ′(S). Multiply by isogeny N : E → E until30
α−1 restricts to31
TE → Z2.
Reduce both sides mod M ∈ N. Let D be the kernel of the induced map32
E[M ]→ (Z/M)2,
this stabilises for M 0. The dual isogeny E/D → E induces33
α : Z2 ∼−→ T (E/D).
– This defines unique representative in P (S). Thus P (S) P ′(S).34
• This shows that the natural GL2(Z)-action extends to a GL2(Af )-action.35
• BThis does not fix the projection to any XK because it changes E.36
• Example:(p 00 1
)acts by sending E 7→ E/D where D ⊆ E[p] is the subgroup37
generated by the image of α(1, 0) under TpE E[p].38
• Fact: All of this still works after compactification.39
Geometric interpretation of Hecke operators40
• Let V = C∞(GL2(Af ) × H±) be the smooth GL2(Af )-representation of41
smooth functions42
f : GL2(Af )×H± → C, (g, z) 7→ f(g, z),
i.e., f is fixed under some compact-open subgroup in GL2(Af ) and z 7→ f(g, z)43
is smooth for any g ∈ GL2(Af ).44
• Fix a prime p.45
• Let ϕ ∈ H(GL2(Qp)) be any element of the Hecke algebra at p, i.e., a locally46
constant function ϕ : GL2(Qp)→ C with compact support.47
• Recall that for f ∈ V we have48
ϕ ∗ f(g, z) =
∫GL2(Qp)
ϕ(h)f(gh, z)dh,
where we chose the Haar measure on GL2(Qp) with volume 1 on Kp.49
• The integral above can be rewritten as follows:50
• Consider the following diagram:51
GL2(Af )×H± ×GL2(Qp)
GL2(Af )×H± GL2(Af )×H±,
q2 q1
q1(g, z, h) =(g, z),
q2(g, z, h) =(gh, z).
• Such a diagram is called a correspondence.52
• We also have the projection to the third factor:53
π : GL2(Af )×H± ×GL2(Qp)→ GL2(Qp)
Lemma 8.2. For f : GL2(Af )×H± → C in V we have54
ϕ ∗ f = q1,!(q∗2(f) · π∗(ϕ)),
where (−)∗ means the pullback of a function and (−)! integrating along the55
fiber (which makes sense because ϕ has compact support).56
• Proof. The fiber of q1 over (g, z) ∈ GL2(Af )×H± is57
q−11 (g, z) = (g, z, h)| h ∈ GL2(Qp) ∼= GL2(Qp)
and q∗2(f) · π∗(ϕ) is the function58
(g, z, h) 7→ f(gh, z)ϕ(h).
Applying q1,!, the integral over this fibre, gives59 ∫GL2(Qp)
ϕ(h)f(gh, z)dh = ϕ ∗ f(g, z).
Note that this is well-defined because ϕ has compact support. 60
Passage to finite level61
• Let N be coprime to p and let K := K1(N) ⊆ GL2(Af ). Then Kp = GL2(Zp).62
• Write Kp for the prime-to-p-part of K, i.e. K = KpKp and Kp ∩Kp = 1.63
• Assume that f is fixed under K, i.e., arises by pullback from a function64
f : GL2(Af )/K ×H± → C.
• Assume ϕ ∈ H(GL2(Qp),Kp), i.e., that ϕ is Kp-biinvariant.65
• ϕ is the pullback of a function with finite support66
ϕ : Kp\GL2(Qp)/Kp → C.
• In this case, ϕ ∗ f is again K-invariant, i.e., the pullback of a function67
ϕ ∗ f : GL2(Af )/K ×H± → C.
• We can change the above diagram (while retaining notation) to:68
(GL2(Af )/Kp ×H±)×Kp GL2(Qp)/K
GL2(Af )/K ×H± GL2(Af )/K ×H±.
q2 q1
Here − ×Kp − is the contracted product (i.e., the quotient for the action69
k · (g, z, h) := (gk−1, z, kh)) and70
q2([g, z, h]) = [gh, z], q1([g, z, h]) = [g, z],
where square brackets indicate equivalence classes.71
• We again have a projection
π : (GL2(Af )/Kp ×H±)×Kp GL2(Qp)/Kp →Kp\GL2(Qp)/Kp,
[g, z, h] 7→[h].
• Next, mod out by GL2(Q) on the left: Recall this gives a complex manifold72
XΓ1(N)(C) = GL2(Q)\(GL2(Af )/K ×H±)
• We then need to replace functions f by sections in ωk, k ∈ Z on this.73
• More precisely, we obtain74
(1)
GL2(Q)\(GL2(Af )/Kp ×H±)×Kp GL2(Qp)/Kp
XΓ1(N)(C) XΓ1(N)(C).
q2 q1
and for f ∈ H0(XΓ1(N)(C), ωk), i.e., a weakly modular form of level K1(N),75
ϕ ∗ f = q1,!(q∗2(f) · π∗(ϕ)).
• To make this precise, use: We have a canonical isomorphism76
q∗2(ωk) ∼= q∗1(ωk)
since ωk is defined via pullback of a GL2(Q)-equivariant line bundle on H±.77
• Thus can regard q∗2(f) · π∗(ϕ) ∈ q∗2(ωk) as section of q∗1(ωk).78
• Sum up along the fibers to obtain the section79
q1,!(q∗2(f) · π∗(ϕ)) ∈ Γ(XΓ1(N)(C), ωk).
Interpretation in terms of moduli80
• On elliptic curves, (1) corresponds to a Hecke correspondence:81
(ι : E1 99K E2, α)
(E2, α′) (E1, α)
q2 q1 where
– (E1, α) is an elliptic curve over C with a Γ1(N)-level structure82
α : Z/NZ → E1[N ]
– Equivalently, this is the datum of a point Q := α(1) ∈ E1[N ](C).83
– (ι : E1 99K E2, α) is the data of elliptic curves E1, E2 over C with α a84
Γ1(N)-level structure on E1, and ι a p-quasi-isogeny:85
– q1 : (ι : E1 99K E2, α) 7→ (E1, α).86
– q2 : (ι : E1 99K E2, α) 7→ (E2, α′) where α′ is the composition87
Z/NZ E1[N ] E2[N ]α∼ι
(use: ι is an isomorphism on N -torsion as (p,N) = 1). Equivalently, this88
sends Q to the image Q′ in E2[N ].89
– The map π sends (ι : E1 99K E2, α) to the class of the isomorphism90
Tp(ι) : TpE1 ⊗Zp Qp∼−→ TpE2 ⊗Zp Qp.
More concretely, choose any bases of TpE1 resp. TpE2. Then Tp(ι) is rep-91
resented by a matrix A ∈ GL2(Qp). The class of A ∈ Kp\GL2(Qp)/Kp92
is independent of the chosen basis.93
• Problem: Functor of p-quasi-isogenies not representable (only by ind-scheme).94
• However: Mainly interested in ϕ given by the characteristic functions for95
Kp
(p 0
0 p
)Kp ↔ Sp,
96
Kp
(p 0
0 1
)Kp ↔ Tp.
• Then term ϕ(h) in defining integral is supported on this double coset.97
• Hecke correspondences are supported on subspaces represented by schemes!98
• For Sp, consider tuples (ι : E1 99K E2, α) of the form99
([p] : E → E,α)
i.e. E1 = E2 and ι : E1 → E2 is multiplication [p].100
• For Tp, instead need to consider101
(ι : E1 → E2, α)
where ι : E1 → E2 is an isogeny of degree p.102
• Equivalently, this is the datum of a cyclic subgroup ker ι ⊆ E1 of rank p.103
• Last time: These are now representable by schemes, even over Spec(Q)!104
Hecke action on modular curves – perspective of algebraic geometry105
• We now pass to the finite level K = K1(N) for some p - N .106
• Corresponding modular curve XΓ1(N) over Z[ 1N ] represents pairs (E,Q) over107
Z[ 1N ]-schemes S of an elliptic curve E over S with a point108
Q ∈ E[N ](S).
• We start with 〈p〉 := Sp. By the above, just need to multiply Q by p.109
• For this, use action of (Z/NZ)× on XΓ1(N).110
• By the above, the correspondence for Sp now becomes111
XΓ1(N)
XΓ1(N) XΓ1(N).
p·∼
id
• Fact: For any a ∈ (Z/NZ)×, have canonical isomorphism a∗ω = ω.112
• Since the right map is an isomorphism, the integration over fibers is trivial.113
• Everything extends to compactifications.114
• Putting everything together, we have proved:115
Proposition 8.3. Consider the operator defined as the composition116
Mk(Γ1(N),Z[ 1N ]) = H0(X∗Γ1(N), ω
k)p∗−→ H0(X∗Γ1(N), p
∗ωk) = Mk(Γ1(N),Z[ 1N ]).
Then its base-change to C is the Hecke operator Sp.117
• For Tp, instead of isogenies (ι : E → E′, α), parametrise triples (E,D,Q),
where D = ker ι finite cyclic subgroup of rank p. Get maps of moduli functors
π1 :(E,D,Q)7→(E,Q),
π2 :(E,D,Q)7→(E/D,Q′), Q′ := Q+D = image of Q under E[N ]→ (E/D)[N ]
• These induce a Hecke correspondence118
XΓ0(p)∩Γ1(N)
XΓ1(N) XΓ1(N).
π2 π1
• Fact: π1 and π2 extend to compactifications X∗Γ0(p)∩Γ1(N) → X∗Γ1(N).119
• Fact: π1 and π2 are finite flat of degree p+ 1, even on compactifications.120
• Fact: There are canonical isomorphisms π∗1ω = π∗2ω121
• Since π1 has finite fibers, the integral occurring in the definition of the Hecke122
operator becomes a discrete sum, which we can interpret as a trace map123
Trπ : π1,!π∗1ω → ω
• Combining all this, we have proved:124
Proposition 8.4. The base-change to C of the operator125
H0(X∗Γ1(N), ω⊗k)
π∗2−→ H0(X∗Γ1(N)∩Γ0(p), ω⊗k)
Trπ1−−−→ H0(X∗Γ1(N), ω⊗k).
is the Hecke operator Tp.126
• Note: This is all defined already over Q (even over Z[ 1N ])! Consequence:127
Corollary 8.5. The eigenvalues of Sp, Tp on Mk(Γ1(N),C) are algebraic.128
• To give a more explicit description of Tp, reinterpret in terms of divisors:129
• Let Div(XΓ1(N)) = set of divisors on X (≈ formal sums of points).130
• We can then reinterpret 〈p〉 := Sp and Tp as operators on Div(XΓ1(N)) :
Sp : Div(XΓ1(N))→ Div(XΓ1(N)), [E,Q] 7→ [E, p ·Q],
Tp : Div(XΓ1(N))→ Div(XΓ1(N)), [E,Q] 7→∑
D⊆E[p]
[E/D,Q+D],
where E is an elliptic curve, and Q ∈ E[N ] a point of order N .131
Modular curves in characteristic p132
• Let E be an elliptic curve over a scheme S of characteristic p.133
• Then E[N ] is etale for all p - N .134
• The morphism [p] : E → E factors into135
EF−→ E(p) V−→ E,
the Frobenius and Verschiebung isogeny. Here E(p) = E ×S,F S.136
• ⇒ always have kerF ⊆ E[p]. This is a connected, i.e. kerF (S) = 0.137
Definition 8.6. Over S = Fp, there are two cases:138
(1) E(Fp) = Z/pZ. Then E is called ordinary, and E[p] = µp × Z/pZ.139
(2) E(Fp) = 0. Then E is called supersingular, and E[p] is connected.140
• B “supersingular” is not about smoothness. It just means “really special”:141
• Fact: There are only finitely many isomorphism classes of supersingular el-142
liptic curves over Fp. All others are ordinary.143
• Recall: XΓ1(N) is defined over Z[ 1N ].144
• Can therefore reduce to Fp to get modular curve XΓ1(N),Fp → Spec(Fp).145
• Similarly for XΓ1(N)∩Γ0(p) representing (E,D,Q) with D ⊆ E[p] of rank p.146
• Deligne–Rapoport: The fibre XΓ1(N)∩Γ0(p),Fp → Spec(Fp) is of the form147
148
id F
D = µp
D = Z/pZXΓ1(N)∩Γ0(p),Fp
XΓ1(N),Fp149
• Two copies of XΓ1(N),Fp , corresponding to D = µp or D = Z/pZ as subgroup150
of E[p], with transversal intersections at supersingular points (black).151
• Projection is identity (deg=1) on one copy, and Frobenius (deg=p) on other.152
Eichler–Shimura relation [Diamond–Shurman], [Conrad: Appendix to Serre’s...]153
• Eichler–Shimura relation expresses reduction Tp mod p in terms of Frobenius:154
• Base change to algebraic closure Fp.155
• Consider morphism Div0(X∗Γ1(N),Fp
)→ Pic0(X∗Γ1(N),Fp
)156
• Fact: Pic0(X∗Γ1(N),Fp
) is points of a group scheme over Fp. In particular,157
multiplication by p factors through Frobenius F : there is V = F∨ s.t.158
p = V F = F V
Theorem. (Eichler–Shimura relation) In Pic0(X∗Γ1(N),Fp
), we have159
Tp = F + 〈p〉V.
• Note: There is a version for X∗Γ0(N),Fp
, which famously reads160
Tp = F + V
We get a second important variant by multiplying by F and using FF∨ = p.161
F 2 − TpF + 〈p〉p = 0.
Proof. (Sketch) Let E be an ordinary elliptic curve over Zp = W (Fp). Let C ⊆ E[p]162
be the “canonical subgroup”:= generated by kernel of163
E[p](Zp)→ E[p](Fp).
• Key fact: For D ⊆ E[p] cyclic subgroup scheme of rank p,164
(1) the isogeny E → E/D reduces to F : E → E(p) ⇔ D = C.165
(2) the isogeny E → E/D reduces to V : E → E(p−1) ⇔ D 6= C.166
• Here E(p−1)
:= E ×Fp,F−1 Fp = base-change along inverse of Frobenius.167
• F sends Q ∈ E[N ] to base-change Q(p) ∈ E(p)[N ] = E[N ](p).168
• Since V F = p on E[N ], this implies: V sends Q to pQ(p−1) ∈ E(p−1)[N ].169
• Recall: There are p+ 1 different subgroups D of E[p] over W (Fp).170
• Thus Tp([E,Q]) =∑D⊆E[p][E/D,Q+D] in Div(XΓ1(N)) reduces to171
[E(p), Q(p)] + p[E(p−1), pQ(p−1)] in Div(XΓ1(N),Fp).
• The first summand is F [E,Q].172
• By definition, 〈p〉[E(p−1), Q(p−1)] = [E(p−1), pQ(p−1)].173
• Now pass to Pic0(X∗Γ1(N),Fp
). Here: p = V F . We therefore have174
p[E(p−1), Q(p−1)] = pF−1[E,Q] = V [E,Q].
(technically, we need to work with [E,Q]− [E′, Q′] to be in Div0). 175
9. Galois representations associated to newforms1
Exams:2
• Write me to arrange an oral exam on 31.7. (second week after end of lectures).3
Next aim:4
• Realize newforms in cohomology.5
• Finish construction of Galois representations associated to newforms of weight6
k = 2.7
• Give hints on how to proceed in the case that k ≥ 2.8
De Rham cohomology:9
• X a real manifold10
• C∞(X) = space of smooth functions f : X → C.11
• Ai(X) = space of smooth i-forms on X, i ≥ 0.12
• de Rham complex of X:13
A•(X) : A0(X)=C∞(X)
d−→ A1(X)d−→ A2(X)→ . . .
with d the exterior derivative.14
• de Rham cohomology of X:15
H∗dR(X) := H∗(A•(X))
• de Rham comparison isomorphism:16
H∗dR(X) ∼= H∗(X,C)
with RHS the sheaf cohomology of the constant sheaf C on X.17
• Moreover,18
H∗(X,C) ∼= H∗sing(X,C).
with RHS = singular cohomology of X with values in C.19
• If Γ is a discrete group acting properly discontinuously and freely on X, then20
A•(Γ\X) ∼= A•(X)Γ = (A0(X)Γ
=C∞(X)Γ
d−→ A1(X)Γ d−→ A2(X)Γ → . . .).
• Finally, we set21
H∗dR(GL2(Q)\(GL2(Af )×H±)) := lim−→K
H∗dR(GL2(Q)\(GL2(Af )/K ×H±))
and22
H∗(GL2(Q)\(GL2(Af )×H±),C) := lim−→K
H∗(GL2(Q)\(GL2(Af )/K ×H±),C),
where the colimit is over all compact-open subgroups K ⊆ GL2(Af ), and the23
transition maps are induced by pullback.24
• Then, for i ≥ 0, the (GL2(Af )-equivariant) de Rham comparison25
HidR(GL2(Q)\(GL2(Af )×H±)) ∼= Hi(GL2(Q)\(GL2(Af )×H±),C)
holds by passing to the colimit.26
Upshot:27
• Can construct classes in H∗(X,C) using (closed) differential forms.28
Cohomology classes associated to modular forms of weight 2:29
• Note that30
GL2(Q)\(GL2(Af )×H±) ∼= GL2(Q)+\(GL2(Af )×H),
where GL2(Q)+ ⊆ GL2(Q) is the subgroup of elements of positive determi-31
nant.32
• Pick f ∈ H0(GL2(Q)+\(GL2(Af )×H), ω⊗k), k ∈ Z.33
• We view f as a function34
f : GL2(Af )×H→ C, (g, z) 7→ f(g, z)
satisfying35
f(γg, γz) = (cz + d)kf(g, z)
for γ ∈
(a b
c d
)∈ GL2(Q)+.36
• Assume that f is of weight k = 2.37
• Then the differential form f(g, z)dz on GL2(Af )×H is closed and satisfies38
γ∗(f(g, z)dz) = f(γg, γz)γ∗dz = det(γ)f(g, z)dz
for γ ∈ GL2(Q)+.39
• Indeed:40
– Closedness follows from holomorphicity as41
d(f(g, z)dz) =∂
∂zf(g, z)dz ∧ dz − ∂
∂zf(z)dz ∧ dz = 0.
– If γ =
(a b
c d
)∈ GL2(Q), then42
γ∗(dz) =det(γ)
(cz + d)2dz.
• Let43
| − | : Q×\A× → R>0
be the adelic norm, i.e.,44
|(x2, x3, . . . , x∞)| :=∏p
|xp|p · |x∞|∞
with | − |p : Qp → R≥0, x 7→ p−vp(x) the p-adic norm, and | − |∞ : R → R≥045
the real norm.46
• | − | defines the character47
χno := | − | det
of GL2(A), which is trivial on GL2(Q).48
• Thus, for49
f(g, z) := χno((g, 1))f(g, z)
(with (g, 1) ∈ GL2(A) ∼= GL2(Af )×GL2(R)), the form50
ηf := f(g, z)dz
on GL2(Af )×H is GL2(Q)+-equivariant.51
• Get a map, which is not GL2(Af )-equivariant,52
α : M2 → H1dR(GL2(Q)+\(GL2(Af )×H)) ∼= H1(GL2(Q)+\(GL2(Af )×H),C)
by sending f to [ηf ].53
• For a smooth GL2(Af )-representation V and a smooth character54
χ : GL2(Af )→ C×
define V (χ) := V ⊗C χ.55
• Then56
α : M2(χno)→ H1(GL2(Q)+\(GL2(Af )×H),C)
is GL2(Af )-equivariant.57
• Let K ⊆ GL2(Af ) be compact-open, and set58
XK := GL2(Q)+\(GL2(Af )/K ×H).
with canonical compactification59
X∗K := GL2(Q)+\(GL2(Af )×H∗)
where H∗ := H ∪ P1(Q) (equipped with Satake topology).60
• If f ∈ S2(K), then ηf extends to a holomorphic differential form on X∗K .61
– Indeed: If q = e2πiz, then62
dq = 2πiqdz,
i.e.,63
dz =1
2πiqdq.
Now express f(g, z) at each cusp in Fourier expansion, i.e., as a function64
of q.65
• The diagram66
H1dR(X∗K)
'
// H1dR(XK)
'
H1(X∗K ,C) // H1(XK ,C)
commutes.67
• Moreover, the morphism68
H1c (XK ,C)→ H1(X∗K ,C) = H1
c (X∗K ,C)
is surjective as its cokernel embeds into H1(cusps,C) = 0.69
• Thus, α(S2(K)) lies in the interior cohomology H1(XK ,C) ∼= H1(X∗K ,C) of70
XK , which by definition is the image of H1c (XK ,C)→ H1(XK ,C).71
• Note that the above discussion applies similarly to antiholomorphic modular72
forms, i.e., complex conjugates of modular forms.73
• This yields the map74
α : S2(K)→ H1(XK ,C), f 7→ [χnofdz],
where S2(K) denotes the C-linear space of antiholomorphic modular forms.75
• We can pass to infinite level and obtain the morphism76
α⊕ α : S2 ⊕ S2 → H1(GL2(Q)\(GL2(Af )×H±),C)
with (hopefully) self-explaining notation.77
Theorem 9.1 (Eichler–Shimura). The map78
α⊕ α : S2(χno)⊕ S2(χno)→ H1(GL2(Q)\(GL2(Af )×H±),C)
is a GL2(Af )-equivariant isomorphism.79
• The statement is equivalent to the analogous statement for all (sufficiently80
small) compact-open subgroup K ⊆ GL2(Af ).81
• Thus, fix some K ⊆ GL2(Af ) compact-open.82
• The proof will exploit the ∪-product pairing83
H1c (XK ,C)×H1(XK ,C)→ H2
c (XK ,C)integrate−−−−−→ C,
which under the de Rham comparison is induced from the pairing84
(η1, η2) 7→∫XK
η1 ∧ η2
of differential forms, where η1 has compact support.85
• We will use the following observation:86
– H∗c (XK ,C) ∼= H∗(A•c(XK)), where A•c denotes differential forms with87
compact supports.88
– The canonical map89
(2) A•c(XK)→ A•rd(XK)
is a quasi-isomorphism, where the RHS denotes differentials forms of90
rapid decay, cf. [Bor80, Theorem 2].91
• For f ∈ S2 the form ηf is rapidly decreasing.92
– Use that |e2πinz| = e−2πnIm(z) decreases rapidly if Im(z)→∞ and n ≥ 1.93
• Thus α lifts naturally to a map94
α : S2(K)→ H1c (XK ,C).
• Consider f ∈ S2(K), g ∈ S2(K). Then:95
∗ α(f) ∪ α(f) = 0 as dz ∧ dz = 0.96
∗ Similarly for g.97
∗ Using Stokes’ theorem, one proves98
tr(α(f) ∪ α(g)) =
∫XK
ηf ∧ ηg.
∗ This implies that the Poincare pairing induces (up to a scalar) the Pet-99
terson scalar product aka L2-pairing 〈−,−〉 on S2.100
• We can now prove that β := α⊕ α is injective.101
• Indeed:102
– If β(f + g) = 0, then103
β(f + g) ∪ α(f) = 〈f, f〉 = 0,
i.e., f = 0. Similarly, g = 0.104
• Now we check H1(XK ,C) = 2dim(S2(K)).105
⇒ Proof of the Eichler-Shimura isomorphism is finished.106
• Namely (we use lower case latters to denote dimensions):107
∗ h1(XK ,C)) = h1(X∗K ,C) = 2 · h0(X∗K ,C)− χtop(X∗K)108
∗ Here: χtop is the topological Euler characteristic.109
∗ On the other hand:
dim(S2(K))
= h0(X∗K ,Ω1X∗K
)RR= deg(Ω1
X∗K) + χhol(X
∗K) + h1(X∗K ,Ω
1X∗K
)
= −χhol(X∗K) + h0(X∗K ,C)
∗ Here: χhol is the holomorphic Euler characteristic.110
∗ χtop(X∗K) = 2χhol(X∗K)111
Galois representations associated to newforms:112
• Recall that in the last two lectures we introduced a scheme113
X → Spec(Q)
with GL2(Af )-action such that naturally114
X(C) ∼= GL2(Q)\(GL2(Af )×H±).
• ForK ⊆ GL2(Af ) compact-open (plus sufficiently small), get a quasi-projective,115
smooth curve116
XK → Spec(Q)
with117
XK(C) ∼= GL2(Q)\(GL2(Af )/K ×H±).
(note clash in notation with previous section, where XK was equal to RHS).118
• Fix a prime `.119
• For K ⊆ GL2(Af ) can consider the interior etale cohomology120
H1et(XK,Q,Q`) := Im(H1
c,et(XK,Q,Q`)→ H1et(XK,Q,Q`)) ∼= H1
et(X∗K,Q,Q`).
• In the limit, we get121
H1et(XQ,Q`) := lim−→
K
H1et(XK,Q,Q`)
• By naturality of H1c,et, H
1et this space has a natural action of122
Gal(Q/Q)×GL2(Af ),
i.e., an action of Gal(Q/Q) and an action of GL2(Af ), and these two actions123
commute.124
• Fix an isomorphism ι : Q` ∼= C.125
• Using etale comparison theorems (and ι) we get the isomorphisms126
H1et(XQ,Q`) ∼= H1
et(XC,Q`) ∼= H1(X(C),Q`)ι∼= H1(X(C),C).
• These isomorphisms are GL2(Af )-equivariant.127
• Let π ⊆ S2 be an irreducible GL2(Af )-representation.128
• Using ι, we will view each smooth GL2(Af )-representation over C as a smooth129
representation over Q`.130
• The space131
ρπ := HomGL2(Af )(π(χno), H1et(XQ,Q`))
is two-dimensional by the Eichler-Shimura isomorphism132
H1(X(C),C) ∼= S2(χno)⊕ S2(χno)
and because S2 is multiplicity free as a GL2(Af )-representation.133
• In summary, as a Gal(Q/Q)×GL2(Af )-representation134
(3) H1et(XQ,Q`) ∼=
⊕π
ρπ ⊗Q` π(χno),
where the sum is running over all irreducible GL2(Af )-representations π ⊆ S2135
(note that there is a natural morphism from the RHS to the LHS).136
The Eichler–Shimura relation in etale cohomology:137
B We need to show that Hecke eigenvalues match with traces of Frobenii. B138
• Recall: To π ⊆ Sk with system of Hecke eigenvalues139
ap(π), bp(π) = χ(p)pk−1p/∈S ,
we want to attach a 2-dimensional `-adic representation140
ρπ : Gal(Q/Q)→ GL2(Q`),
such that for p /∈ S each arithmetic Frobenius Frobarithp at p has characteristic141
polynomial Confession: I
stated the Lang-
lands reciprocity
for newforms
wrongly!
Confession: I
stated the Lang-
lands reciprocity
for newforms
wrongly!
142
X2 − ap(π)X + χ(p)pk−1
(we omit ι, ρπ in the following), i.e.,143
(Frobarithp )2 − ap(π)Frobarith
p + χ(p)pk−1 = 0.
• By passing to K1(N)-invariants in (3), we have (for N ≥ 3)144
H1et(X
∗Γ1(N),Q,Q`) ∼=
⊕π
ρπ ⊗Q` π(χno)K1(N)
(note XK1(N)∼= XΓ1(N)).145
• Let p be a prime, p - `N .146
• Fix a place of Q over p. This determines an algebraic closure Fp of Fp.147
• Recall: For an abelian variety A over a field L, and ` a prime, we write148
T`A = lim←−A[`n](L), V`A = T`A⊗Z` Q`
for the `-adic Tate module resp. the rationalized `-adic Tate module.149
Lemma 9.2. Let p be prime, p - `N . Then we have150
H1et(X
∗Γ1(N),Q,Q`) = V`Pic0(X∗
Γ1(N),Fp)∨,
where (−)∨ denotes the Q`-dual.151
Proof. The Kummer sequence 1→ µ`n → Gm`n−→ Gm → 1 implies152
H1et(X
∗Γ1(N),Q,Q`(1)) ∼= V`Pic0(X∗
Γ1(N),Q).
The Weil pairing yields a canonical isomorphism153
V`Pic0(X∗Γ1(N),Q)∨ ∼= V`Pic0(X∗
Γ1(N),Q)(−1).
Finally, because ` 6= p154
V`Pic0(X∗Γ1(N),Q) ∼= V`Pic0(X∗
Γ1(N),Qp) ∼= V`Pic0(X∗
Γ1(N),Fp)
by lifting torsion points. 155
• In particular, the Galois representation156
H1et(X
∗Γ1(N),Q,Q`)
is unramified at p.157
• On V`Pic0(X∗Γ1(N),Fp
), we have the Eichler–Shimura relation, which we recall158
reads159
F 2 − TpF + pSp = 0
with F the arithmetic Frobenius.160
• Last time it was written Tp, but the action of the double coset161
GL2(Zp)
(p 0
0 1
)GL2(Zp)
was considered, which yields Tp.162
• Moreover, instead of Sp, which corresponds to163
GL2(Zp)
(p 0
0 p
)GL2(Zp),
it was written 〈p〉.164
Proposition 9.3. Let p be a prime with p - `N . Then on H1et(X
∗Γ1(N),Q,Q`), we have165
(Frobgeomp )2 − TpFrobgeom
p + pSp = 0.
• For a normalized newform f ∈ S2(Γ0(N), χ) we define finally166
ρf := (ρπ)∨
with π the irreducible GL2(Af )-representation generated by f , and167
ρπ := HomGL2(Af )(π(χno), H1et(XQ,Q`)).
Some analysis of ρf , cf. [Rib77]:168
• Write the (normalized) newform f ∈ S2(Γ0(N), χ) in Fourier expansion169
f(q) =∑n≥1
anqn.
Then for p - `N , Frobarithp has characteristic polynomial170
X2 − apX + χ(p)p
on ρf .171
• Indeed:172
– By the (dual of the) Eichler–Shimura relation173
(Frobarithp )2 − TpFrobarith
p + pSp
on ρf .174
– Recall that Tp has eigenvalue pap on πK1(N), while Sp has eigenvalue175
χ(p)p2.176
– We had to twist the GL2(Af )-action on S2 by χno = |det(−)|. Thus Tp177
has eigenvalue ap on178
ρπ ⊗Q` π(χno)K1(N) ⊆ H1et(X
∗Γ1(N),Q,Q`).
while Sp has eigenvalue 1p2χ(p)p2 = χ(p).179
• The determinant of ρf is180
ψ · χcyc
where181
χcyc : GQ → Q`×
denotes the cyclotomic character, and182
ψ : Q×R>0\A× → Q`× ι∼= C×
the (finite order) adelic character determined by χ.183
• Indeed:184
– Use Chebotarev and check that both sides agree on arithmetic Frobenius185
elements for p - `N .186
• In particular, ρf is odd, i.e., the determinant of each complex conjugation is187
−1.188
– This uses that for all k ≥ 1 non-triviality of Sk(Γ0(N), χ) implies χ(−1) =189
(−1)k.190
• ρf is irreducible, cf. [Rib77, Theorem (2.3)].191
• Because, ρf is realized in the etale cohomology of a proper, smooth scheme192
over Q, the de Rham comparison theorem in `-adic Hodge theory implies that193
ρf |GQ`is de Rham.194
Outline of the construction in weights ≥ 2:195
• Instead of196
H1et(XQ,Q`)
use197
H1et(XQ,L
k),
with the local system198
Lk := Symk−1R1f∗(Q`)
for f : E → X the universal elliptic curve.199
• Problem: Lk does not extend to a local system on the compactification. Thus200
one needs an additional argument to justify base change to Fp.201
• Use similar strategy to obtain analogs of the Eichler–Shimura relation/isomorphism.202
• Also replace the de Rham comparison/etale comparsion theorems by their203
versions with coefficients.204
10. Galois representations for weight 1 forms (by Ben Heuer)1
Last time: Constructed Galois representations attached to newforms in weight ≥ 2:2
Theorem 10.1 (Deligne ’71). Let N ≥ 3 and k ≥ 2. Let ε : (Z/NZ)× → C× be a3
Dirichlet character. Let f ∈ Mk(Γ0(N), ε) be a newform of weight k. Let K be the4
number field generated by the ap(f) and ε. Then for any place λ of K over a prime5
` - N , there is a continuous, irreducible Galois representation6
ρ : GQ → GL2(Kλ),
unramified outside N , such that the characteristic polynomial of Frobp is7
X2 − apX + ε(p)pk−1 for all p - N`.
Galois reps associated to weight 1 modular forms8
• Goal of today: prove the following Theorem9
Theorem 10.2 (Deligne–Serre ’73). Let N ≥ 3. Let ε : (Z/NZ)× → C× be a
Dirichlet character. Let f ∈M1(Γ0(N), ε) be a newform of weight 1. Then there
is a continuous, odd, irreducible Galois representation
ρ : GQ → GL2(C),
unramified outside N , such that the characteristic polynomial of Frobp is
X2 − apX + ε(p) for all p - N.
• This finishes the construction of newforms Galois reps.10
• Note that ρ has image in GL2(C), i.e. is an Artin representation, in contrast11
to the ones for weight ≥ 2, which were `-adic. Why is that?12
– Easy fact: Any Artin representation has finite image.13
– In fact, ρ is defined already over a number field, so could embed into p-14
adic field Kλ as before. But using C is a way of signaling “finite image”.15
– On the Galois side, see that ρ can only be finite for k = 1 since the16
determinant is required to be det ρ = εχk−1cycl , and χcycl has infinite image.17
• If ε is even, then M1(Γ0(N), ε) = 0. Hence ε is odd, which implies ρ odd.18
First sketch of construction19
Compared to last time, the proof will be less geometry, more Galois representations:20
• Reduce f mod p. This gives a “mod p modular form”21
• Use congruences of modular forms to show there is a mod p Hecke eigenform22
f ′ with same eigenvalues but higher weight ≥ 2.23
• Lift f ′ back to characteristic 0 using “Deligne–Serre lifting”24
• Attach Galois representation ρf ′ using weight ≥ 2 construction25
• Reduce Galois representation mod `26
• Lift Galois representation from F` to C.27
Introduction to mod p modular forms28
• Let N ≥ 3. Let S be any Z[ 1N ]-scheme S.29
• Recall: We had defined modular forms over S of weight k by30
Mk(Γ1(N);S) := Γ(X∗Γ1(N),S , ωk).
• ω is the ample line bundle of relative differentials on universal elliptic curve31
• Can define q-expansions: Let S = Spec(A), then have the Tate curve T (q)32
over Z((q))⊗Z A. Over Spec(Z((q))⊗Z A), have canonical isomorphism33
ωGm = ωT (q)
• bundle ωGm is trivial: the canonical differential dxx on Gm is invertible section34
• Via this trivialisation, get for every cusp ofX∗Γ1(N),A an associated q-expansion35
Mk(Γ1(N);S)→ A[[q]].
Theorem 10.3 (Katz’ q-expansion principle). This is injective for all k at each cusp.36
• Now apply this all to S = Spec(F) where F is a finite field of characteristic p:37
The resulting modular forms are called mod p modular forms.38
• The following fact perhaps helps you get a feeling for mod p modular forms:39
Lemma 10.4. For weight k ≥ 2, any mod p modular form is the reduction40
of a modular form over a number field modulo p. In particular, in this case41
Mk(Γ1(N);F) = Mk(Γ1(N);Z[ 1N ])⊗Z F.
Proof. Using Riemann–Roch, one can show that H1(X∗Γ1(N), ω⊗k) = 0. 42
• However, it is often better to have a more intrinsic definition. For example:43
Example of mod p modular form: The Hasse invariant44
• Recall: For any elliptic curve in characteristic p, have Verschiebung isogeny45
V : E(p) → E.
• In particular, have this for the universal elliptic curve.46
• On relative differentials, this defines a pullback map in the other direction47
V ∗ : ω → ω⊗p.
• Tensoring with ω−1 yields48
V ∗ : O → ω⊗(p−1) V ∗ ∈ Γ(ω⊗(p−1)) = Mp−1(Γ1(N);F).
• This is the Hasse invariant, denoted by Ha.49
• It is a mod p modular form of weight p− 1. In fact: an eigenform of level 1.50
Proposition 10.5. The Hasse invariant has constant q-expansion = 1 ∈ Fp[[q]].51
Proof. Sketch: Use ωT (q) = ωGm : On Gm have F = [p] and thus V = 1. 52
• Note: 1 ∈M0(Γ1(N);Fp) is also a mod p modular form with q-expansion 1.53
• This does not contradict the q-expansion principle as the weights are different.54
Upshot: In characteristic p, we have a new phenomenon:55
• Modular forms of different weight can have same q-expansion!56
• E.g. for any modular form f of weight k, any n ∈ N, have modular form57
Han · f of weight k + n(p− 1) with same q-expansion.58
• In particular, this preserves eigenforms a system of eigenvalues for Hecke59
operators can correspond to eigenforms of different weights.60
• BThe multiplication map61
Mk(Γ1(N);S)·Ha−−→Mk+p−1(Γ1(N);S)
is injective (q-expansion principle) but not surjective!62
• Reason: Ha has zeros, precisely on supersingular points of X∗Γ1(N)(Fp)63
• This is one motivation for removing those points p-adic modular forms.64
Eisenstein series65
• There is a more classical perspective on this that we want to at least mention:66
Definition 10.6.67
For any even k ≥ 4, let68
Gk :=∑
(n,m)∈Z2
(n,m) 6=(0,0)
1
(m+ nτ)k.
This is a modular eigenform of weight k and level 1 with q-expansion69
2ζ(k)(1−∞∑n=0
4
Bkσk−1(n)qn)
where σk−1(n) =∑d|n d
k−1 and Bk is the k-th Bernoulli number.70
Define the normalised Eisenstein series as71
Ek :=1
2ζ(k)Gk = 1− 4
Bk
∞∑n=0
σk−1(n)qn ∈ Z(p)[[q]]
Lemma 10.7. Assume p ≥ 5, then in terms of q-expansions, we have72
Ep−1 ≡ 1 mod p.
Proof. This follows from Kummer’s congruences for Bernoulli numbers. 73
Upshot: For p ≥ 5, the reduction mod p of the Eisenstein series Ep−1 is = Ha:74
Ep−1 ≡ Ha mod p “Deligne’s congruence”
Strategy (Deligne–Serre):75
• Given eigenform of weight 1, reduce mod p.76
• Multiply with Ha so it becomes eigenform of weight ≥ 2.77
• Use construction from last lecture in this case.78
Galois reps attached to mod p modular forms79
Theorem 10.8. Let k ≥ 1 and let f ∈ Mk(Γ1(N), ε;Fp) be a mod p cuspidal eigen-80
form. Let Ff be the (finite) field generated over Fp by the ap(f) and ε. Then there is81
a semi-simple Galois representation82
ρf : GQ → GL2(Ff ),
unramified outside N` such that for all ` - Np,83
charpoly(Frobp|ρf ) = X2 − apX + ε(p)pk−1.
Step 1: Reduce to case of k ≥ 2: If k = 1, multiply with Ha. This does not change84
the q-expansion, hence is still an eigenform. But its weight is ≥ 2.85
Step 2: Lift the Hecke eigensystem to characteristic 0. This is always possible in86
weight ≥ 2, as we shall discuss next.87
Deligne–Serre lifting lemma88
• Let R be a Z[ 1N ]-algebra without zero-divisors. Let89
TR ⊆ End(Mk(Γ1(N);R))
be the Hecke algebra: R-subalgebra generated by the Hecke operators Tl, Sl.90
• A Hecke eigensystem is a ring homomorphism91
ψ : TR → R.
The archetypal example of a Hecke eigensystem is:92
• Let g be an eigenform in Mk(Γ1(N);R), this gives rise to a Hecke eigensystem93
ψg : TR → R, T 7→ aT s.t. T (g) = aT g.
Lemma 10.9. Assume that R = K is a field. Then any Hecke eigensystem94
ψ : TK → K
comes from a unique Hecke eigenform.95
Proof. • TK is an Artinian K-algebra.96
• After passing to extension, it is the product of Artinian local rings Ti.97
• Since TK acts faithfully on M := Mk(Γ1(N);K), the submodule Mi := Ti ·M98
is non-zero (commutative algebra fact).99
• Take any eigenvector in Mi. Then this has eigensystem ψ.100
• Uniqueness: The Hecke eigenvalues determine the q-expansion. 101
• Let K be a number field, p a place over p - N . Let Fp be the residue field.102
• Write OK,(p) for the valuation subring of K of p-integral elements.103
Lemma 10.10 (Deligne–Serre Lifting Lemma). Let k ≥ 2 and let g ∈Mk(Γ1(N);Fp)104
be an eigenform. Then there is a finite extension K ′|K, an extension p′|p and an105
eigenform g ∈Mk(Γ1(N);OK′,(p′)) such that g reduces to g mod p′.106
Proof. • Consider the ring homomorphism107
TOK → TFp
ψg−−→ Fp.
• This corresponds to a maximal ideal m. Since108
Spec(TOK )→ Spec(OK)
is locally free, can find prime ideal q ⊆ m such that q ∩ O = 0.109
• This defines a non-zero prime ideal of TK , corresponding to Hecke eigensystem110
TK → K ′.
• By Lemma 10.9, this corresponds to an eigenform.111
• Use uniqueness in Lemma 10.9 to see that this reduces to g. 112
Proof of Theorem 10.8113
• Let f be mod p eigenform of weight ≥ 2, defined over extension Ff of Fp.114
• Lift to eigenform f over some number field K. Let p be a place over p.115
• Associated to this, have Galois representation116
ρf : GQ → GL2(Kp).
• Since image is compact, can always find isomorphic representation of the form117
ρf : GQ → GL2(OK,p).
• Reduce this mod p to get representation118
ρf : GQ → GL2(Fp).
• This might not yet be semi-simple Define ρf as semi-simplification119
ρf := ρssf
(:=direct sum of Jordan–Holder factors). Same trace and determinant as ρf .120
• Since ker ρf ⊆ ker ρf , this is unramified outside of pN .121
• Remains to show: ρf already defined over Ff : For this, STP:122
ρf = σ ρf for all σ ∈ Gal(Fp|Ff ).
• Certainly, all characteristic polynomials of Frob` for ` - pN ,123
X2 − a`X + ε(`),
are preserved by σ, since a`, ε(`) ∈ Ff by definition. Now use:124
Theorem 10.11 (Brauer–Nesbitt). Let F be a perfect field. Let G be a finite125
group. Let V,W be two semi-simple finite dimensional representations of G126
over F . Then V ∼= W if and only if127
charpoly(g|V ) = charpoly(g|W ) for all g ∈ G.
• This finishes the construction of f ρf . 128
Serre’s conjecture129
• A quick digression before we continue with weight 1 forms:130
• In 73’, Serre gave a conjecture that described exactly which mod p Galois131
representations arise in this way. It became known as132
Conjecture 10.12 (Serre’s Modularity Conjecture, or Serre’s Conjecture). Let ρ :133
GQ → GL2(Fp) be a continuous Galois representation that is odd and irreducible.134
Then there is a mod p eigenform g such that135
ρ = ρg.
In 1987, Serre published an article refining this conjecture:136
• He predicted the optimal level of g (in terms of the Artin conductor of ρ)137
• He predicted the optimal weight (recipe in terms of ramification of ρ at p)138
• He showed that his conjecture would imply Fermat’s Last Theorem.139
• Serre’s Conjecture is now a theorem by Khare–Wintenberger (2008).140
Lifting ρf to C141
• Back to our earlier setup: f ∈M1(Γ0(N), ε) eigenform142
• K number field generated by eigenvalues and ε.143
• Let L := set of primes of Q which decompose completely in K. This is an144
infinite set by Chebotarev.145
• Let ` ∈ L and λ a place over ` in K. Let OK,(λ) ⊆ K be λ-integral elements.146
• BNow apply previous section with role of p played by varying ` ∈ L.147
• For any such λ, consider reduction f of f mod λ. This is a mod ` eigenform.148
• Theorem 10.8 associates a semi-simple Galois representation149
ρf,` : GQ → GL2(F`)
(over F` as λ completely split) with charpoly of Frobp for any p - `N given by150
charpoly(Frobp|ρf,`) = X2 − apX + ε(p).
• Zeros of this are n-th units roots, which we may assume are all in OK (by the151
next lemma) and thus in Fl. Thus152
= (X − a)(X − b) for some a, b ∈ F×` .
Definition 10.13. Let G` := im(ρf,`) ⊆ GL2(F`). This is a finite group.153
Lemma 10.14. If f is of weight 1, there is A ∈ N such that |G`| ≤ A for all ` ∈ L.154
Proof. After using an estimate on the coefficients of weight 1 modular forms this155
becomes purely group theoretical: use classification of subgroups of GL2(F`). We156
shall not discuss it, but if you want to see it: [DS74, Lemma 8.4]. 157
• Enlarge K so that it contains all n-th unit roots for n ≤ A. This shrinks L.158
• Also exclude the primes ≤ A from L. Then L is still infinite.159
Lemma 10.15. Let G be a finite group of order prime to `. Then any representation160
ρ : G→ GL2(F`) lifts to a representation161
ρ : G→ GL2(OK,(λ)).
Proof. Step 1: Lift to GL2(Z`):162
• STP: any G→ GL2(Z/`nZ) lifts to G→ GL2(Z/`n+1Z).163
• For this, consider the short exact sequence164
1→ 1 + `nM2(F`)→ GL2(Z/`n+1Z)→ GL2(Z/`nZ)→ 1
• The first group is isomorphic to M2(F`) via 1 + `nx 7→ x.165
• Now use non-abelian group cohomology: Consider the left-action166
G acts on Map(G,GL2(Z/`nZ) via g · ϕ := ϕ(g)ϕ(g)−1.
• Then167
Hom(G,GL2(Z/`nZ)) = Map(G,GL2(Z/`nZ))G.
• Apply Map to the above short exact sequence, then G gives a long exact168
sequence of non-abelian group cohomology169
· · · → Hom(G,GL2(Z/`n+1))→ Hom(G,GL2(Z/`n))→ H1(G,Map(G,M2(F`)))
• Fact in group cohomology: H1(G,A) = 0 if |G|, |A| finite and coprime.170
Step 2: Already defined over number field171
• Since G is finite, ρ is semi-simple.172
• Traces must be sums of roots of unity of order ≤ |G| ≤ A contained in K.173
• Brauer–Nesbitt ensures that ρ is already defined over K.174
• Finally, K ∩ Z` = OK,(λ) 175
Back to prove of Theorem 10.2 (Galois representation for weight 1 forms)176
• Apply this to ρg : G` → GL2(F`), get G` → GL2(OK,(λ))177
• Compose with GQ → G` to get Galois representation178
ρg,` : GQ → GL2(OK,(λ)).
Proposition 10.16. The lift ρg,` : GQ → GL2(OK,(λ)) is such that for all p - `N ,179
charpoly(Frobp|ρg,`) = X2 − apX + ε(p).
The key is to vary `: Recall that we get ρg,` for all ` ∈ L.180
Definition 10.17. Let Y be the set of polynomials of the form (X − α)(X − β) for181
α, β ∈ OK,(λ) roots of unity of order ≤ |A|. This is a finite set.182
• Note: Given n ≤ A, ` ∈ L, and n-th root of unity x ∈ F×` , there is exactly183
one lift to root of unity in K (existence: n ≤ A, uniqueness: ` ≥ A).184
• Consequence: If F,G ∈ Y , then F ≡ G mod λ implies F = G.185
• Idea: Apply this to F = X2 − apX + ε(p) and G = charpoly(Frobp|ρg,`)186
Lemma 10.18. X2 − apX + ε(p) ∈ Y .187
Proof. For all λ over a prime in L, there is F ∈ Y for which188
charpoly(Frobp|ρg,`) ≡ F mod λ
since ρg,` has finite image. Thus also189
X2 − apX + ε(p) ≡ F mod λ.
Since Y is finite, there is F for which this holds for infinitely many λ. Thus190
X2 − apX + ε(p) = F ∈ Y
Proof of Proposition. • As G` finite of order ≤ A, have191
charpoly(Frobp|ρg,`) ∈ Y
• charpoly(Frobp|ρg,`) ≡ charpoly(Frobp|ρg,`) ≡ X2 − apX + ε(p) mod λ192
• Thus both sides are in Y , which implies193
charpoly(Frobp|ρg,`) = X2 − apX + ε(p).
Remains to prove:194
• ρ is odd: This is simply because ε is odd and c has order 2.195
• ρ is irreducible: Use complex analytic estimate due to Rankin 196
This finishes the construction of eigenforms Galois representations!197
Summary of the weight 1 constructioneigenforms
of weight 1
eigenforms of
weight ≥ 2
`-adic Galois
representations
complex Galois
representations
mod ` eigenforms
of weight 1
mod ` eigenforms
of weight ≥ 2
mod ` Galois
representations
reduce
congruence Deligne’s
construction
reduce
·Ha
DS-lifting
mod ` construction
Theorem 10.8
lift
11. The Langlands program for general groups, part I1
Next aims:2
• More or less precise statements (for parts of) Langlands program for GLn, or3
even general G reductive over Q.4
Langlands(-Clozel-Fontaine-Mazur) reciprocity for GLn,F :5
• F arbitrary number field6
• ` some prime. Fix an isomorphism C ∼= Q`.7
• Then for any n ≥ 1, there exists a (unique) bijection between8
i) the set of L-algebraic cuspidal automorphic representations of GLn(AF ),9
ii) the set of (isomorphism classes) of irreducible continuous representations10
Gal(F/F )→ GLn(Q`) which are almost everywhere unramified, and de11
Rham at places dividing `,12
such that the bijection matches Satake parameters with eigenvalues of Frobe-13
nius elements.14
Comments:15
• This is only a part of the Langlands program for G = ResF/QGLn,F .16
• This does not describe all automorphic representations for GLn,F , nor all17
Galois representations.18
• Not known for F = Q, n ≥ 2, e.g., there exists L-algebraic cuspidal automor-19
phic representations of GL2(A) (associated to Maassforms), which should cor-20
respond to even irreducible, 2-dimensional `-adic representations of Gal(Q/Q)21
(with finite image).22
• Langlands-Tunnell: If the image is solvable, automorphicity of the Galois rep-23
resentation is known, cf. [GH19, Theorem 13.4.5.]. This does not cover almost24
everywhere unramified 2-dimensional, irreducible Galois representations with25
image SL2(F5).26
• Unicity follows from the (strong) multiplicity one theorems on both sides.27
• Today: Explain “L-algebraic”.28
Automorphic forms for general groups:29
• G an arbitrary reductive group over Q.30
• Will introduce a convenient (=more algebraic) replacement for L2([G]).31
• Recall for G = GL2 = GL2,Q, k ∈ Z, the embedding32
Φ: Mk → C∞(GL2(A)), f 7→ ϕf .
• Recall the description of the image of Φ(Mk) resp. Φ(Sk), namely:33
– For k ∈ Z an element ϕ ∈ C∞(GL2(A)) lies in the image of34
Φ: Mk → C∞(GL2(A))
if and only if35
∗ ϕ(g, g∞z) = z−kϕ(g, g∞) for (g, g∞) ∈ GL2(A), z ∈ C× ⊆ GL2(R).36
∗ ϕ is of moderate growth (implies the vanishing of negative Fourier37
coefficients).38
∗ For each g ∈ GL2(Af )39
Y ∗ ϕ(g,−) = 0,
where Y ∈ gC = gl2(R)C is a suitably constructed, natural element40
(implies holomorphicity).41
∗ ϕ(γg) = ϕ(g) for all γ ∈ GL2(Q).42
– f is moreover cuspidal if and only if43 ∫N(Q)\N(A)
ϕf (ng)dn = 0
for all unipotent radicals N in proper parabolics P ⊆ G (defined over44
Q).45
• Fix a maximal compact (usually not connected) subgroup K∞ of G(R), e.g.,46
O2(R) = SO2(R) ∪ sSO2(R) ⊆ GL2(R) with s =
(−1 0
0 1
).47
Definition 11.1 (cf. [GH19, Definition 6.5]). Let G be a reductive group over Q. An48
(adelic) automorphic form for G is a smooth function ϕ : G(A)→ C such that49
1) ϕ is of moderate growth (was defined in Section 4, not important for this50
lecture).51
2) ϕ is right K∞-finite, i.e., the functions g 7→ ϕ(gk) for k ∈ K∞ ⊆ G(A) span52
a finite dimensional vector space.53
3) ϕ is killed by an ideal in Z(gC) of finite codimension, where gC denotes the54
(complexified) Lie algebra of G(R), and Z(gC) the center of the enveloping55
algebra U(gC) of gC.56
4) ϕ is left G(Q)-invariant, i.e., ϕ(γg) = ϕ(g) for all γ ∈ G(Q).57
The Casimir element for GL2:58
• Let us check that 3) is indeed verified for ϕf : C∞(GL2(A))→ C with f ∈Mk.59
• Namely, the element Y was not in Z(gl2,C).60
• Set61
H :=
(0 −ii 0
), X :=
1
2
(−1 −i−i 1
), Y :=
1
2
(−1 i
i 1
), Z :=
(1 0
0 1
).
• Then62
∗H ∗ ϕf = −kϕf
as z · ϕf = z−kϕf for z ∈ C×.63
∗ Y ∗ ϕf = 0.64
• [X,Y ] = XY − Y X = H (because H,X, Y form an sl2-triple)65
• Define the Casimir operator66
∆ :=1
4(H2 + 2XY + 2Y X).
Then ∆ ∈ Z(gl2,C).67
• In fact, Z(gl2,C) is generated, as a C-algebra, by ∆ and Z ∈ gl2,C (this follows68
from the Harish-Chandra isomorphism, cf. [GH19, Theorem 4.6.1]).69
• One calculates (using Y ∗ ϕf = 0)70
∆ ∗ ϕf =1
4(H2 − 2XY + 2Y X) ∗ ϕf =
1
4(H2 − 2H) ∗ ϕf =
1
4(k2 − 2k)ϕf ,
thus ϕf is indeed killed by the ideal 〈∆ − 14 (k2 − 2k), Z + k〉 ⊆ Z(gl2,C) of71
finite codimension.72
More on automorphic forms:73
• G arbitrary reductive over Q.74
• Set A(G) as the space of automorphic forms, and A([G]) as the subspace of75
automorphic forms, which are invariant under AG (acting via left or right76
translations on G(A)).77
• Unfortunately, the spaces A([G]) and A(G) are not stable under G(R), be-78
cause K∞-finiteness need not be preserved.79
More on automorphic forms (continued):80
• Clearly, G(Af ) acts on A(G) via ϕ 7→ (g 7→ ϕ(gh)) for h ∈ G(Af ).81
• Similarly, K∞, even AGK∞, acts on A(G)82
• Less clear, but true (cf. [GH19, Proposition 4.4.2.]): gC acts on A(G).83
⇒ U(gC), in particular Z(gC) ⊆ U(gC), acts on A(G) by differential operators.84
• The gC and K∞-action make A(G) into a (gC,K∞)-module.85
Definition 11.2 (cf. [GH19, Definition 4.5.]). A (gC,K∞)-module is a C-vector space86
V , which is simultaneously a gC- and K∞-module, such that87
1) V is a union of finite dimensional K∞-stable subspaces.88
2) For X ∈ k := Lie(K∞)C and ϕ ∈ V we have89
X ∗ ϕ =∂
∂t(exp(tX)ϕ)t=0.
3) For h ∈ K∞, X ∈ gC and ϕ ∈ V we have90
h(X ∗ (h−1ϕ) = (Ad(h)(X)) ∗ ϕ.
Comments:91
• We required no topology on V , thus (gC,K∞)-modules are much more alge-92
braic than G(R)-representations, and thus easier to handle.93
• In 2) the limit is taken for the natural topology of a finite dimensional K∞-94
stable subspace of V containing ϕ.95
• For each representation of G(R) on some Hilbert space C-vector space, the96
space of smooth and K∞-finite vectors is naturally a (gC,K∞)-module, cf.97
[GH19, Proposition 4.4.2.], and on unitary representations one does not loose98
any informations, cf. [GH19, Theorem 4.4.4.], namely: isomorphism classes of99
irreducible unitary representations of G(R) inject into isomorphism classes of100
irreducible (gC,K∞)-modules.101
• Moreover, the (gC,K∞)-modules obtained in this way are admissible, i.e., for102
each (finite dimensional) irreducible representation σ of K∞, the σ-isotypic103
component is finite dimensional, cf. [GH19, Theorem 4.4.1.].104
• Version of Schur’s lemma⇒ On each irreducible, admissible (gC,K∞)-module105
π the center Z(gC) of U(gC) acts via some morphism106
ωπ : Z(gC)→ C
of C-algebras, called the “infinitesimal character of π”.107
• The Harish-Chandra isomorphism, cf. [GH19, Theorem 4.6.1.], furnishes108
Z(gC) ∼= U(tC)W
for each Cartan subalgebra t ⊆ g and W the corresponding Weyl group. Note109
U(tC) ∼= Sym•tC because tC is an abelian Lie algebra.110
• As W is finite, C-algebra morphisms111
Z(gC) ∼= U(tC)W → C
correpond bijectively to W -orbits of C-algebra homomorphisms112
U(tC) ∼= Sym•tC → C.
Moreover,113
HomC−alg(U(tC),C) ∼= HomC−lin(tC,C) =: t∨C .
U(tC)W → U(tC) is finite.114
• Thus, the infinitesimal character defines a canonical W -orbit of elements in115
the “weight space” t∨C .116
• Now assume that t = Lie(T ) for some maximal torus T ⊆ GR. Then W -117
equivariantly118
X∗(TC) ⊆ t∨C .
• An irreducible, admissible (gC,K∞)-module π is called L-algebraic if the W -119
orbit in t∨C corresponding to the infinitesimal character ωπ lies in the finite120
free Z-module X∗(TC) ⊆ t∨C , cf. [BG10, Definition 2.3.1.], [Tho, Definition 93].121
(gl2,C,O2(R))-modules generated by modular forms:122
• Let f ∈ Mk. We will analyze the (gl2,C,O2(R))-module V ⊆ A(GL2) gener-123
ated by ϕ := ϕf .124
• Recall125
Y ∗ ϕ = 0, z · ϕ = z−kϕ for z ∈ C×.
• It is not difficult to see that there exists a unique, up to isomorphism, (gl2,C,O2(R))-126
module D′k−1 (the dual of Dk−1 in [Del73, Section 2.1]) generated by such an127
element ϕ, and that D′k−1∼= V is irreducible.128
• In particular, it only depends on k, not f .129
• We obtain, as was mentioned some time ago,130
Mk∼= Hom(gl2,C,O2(R))(D
′k−1,A(GL2)),
(note: the space H(GA) in [Del73, Scholie 2.1.3.] agrees with our A(GL2)131
only up to inversion on GL2(A)).132
• We calulated133
∆ ∗ ϕ =1
4(k2 − 2k)ϕ, Z ∗ ϕ = −kϕ,
and this determines the infinitesimal character ωD′k−1.134
• The W -orbit of C-algebra homomorphisms U(tC) ∼= C[Z,H]→ C correspond-135
ing to ωD′k−1under the Harish-Chandra isomorphism is given by the homo-136
morphisms137
Z 7→ −k, H 7→ ±(k − 1).
• The lattice X∗(T ) ⊆ 〈Z,H〉∨C is given by C-linear maps138
Z 7→ a, H 7→ b
with a, b ∈ Z and a ≡ b mod 2.139
• Thus, D′k−1 is not L-algebraic (it is C-algebraic in the sense of [BG10]).140
• However,141
D′k−1 ⊗C |det|1/2+a
is L-algebraic for any a ∈ Z because (the (gl2,C,O2(R))-module associated142
with) |det|1/2 has infinitesimal character corresponding to143
Z 7→ 1, H 7→ 0.
Upshot:144
• For any G there exists the space A(G) of (adelic) automorphic forms, which145
for GL2 naturally contains the ϕf with f ∈Mk.146
• The space A(G) carries a (gC,K∞) × G(Af )-action, and is a more algebraic147
replacement for L2([G]) with its G(A)-action.148
A new notion of automorphic representations:149
• We redefine the notion of an automorphic representation using A(G) instead150
of L2([G]).151
• Namely, an automorphic representation for G is an irreducible G(Af ) ×152
(gC,K∞)-subquotient of A(G), cf. [GH19, Definition 6.8.].153
• The former automorphic representations will from now on be called “L2-154
automorphic representations”.155
• For informations how both notions relate, cf. [GH19, Section 6.5.].156
Flath’s theorem (cf. [GH19, Theorem 5.7.1.]):157
• For almost all primes p the reductive group GQp := G ×Spec(Q) Spec(Qp) is158
“unramified”, i.e., extends to a reductive group scheme159
Gp → Spec(Zp).
• Equivalently, GQp is quasi-split (=contains a Borel subgroup defined over Qp)160
and split(=contains a maximal and split torus) over an unramified extension.161
• Then162
G(A) =
′∏p
(G(Qp),Gp(Zp))×G(R),
where Gp(Zp) ⊆ G(Qp) is a compact-open subgroup (cf. [GH19, Proposition163
2.3.1.]), called “hyperspecial”.164
• Flath’s theorem (cf. [GH19, Section 5.7.] and [Fla79]) states that irreducibe,165
admissible G(Af ) × (gC,K∞)-modules π decompose (uniquely) into a “re-166
stricted tensor product”167
π ∼=′⊗
p prime
πp ⊗ π∞
of irreducible, smooth (even admissible) GQp -representations πp and an irre-168
ducible, admissible (gC,K∞)-module π∞.169
• Moreover, for almost all primes p the representation πp is unramified, i.e., the170
group GQp is unramified and πGp(Zp)p 6= 0, where Gp is a reductive model of171
GQp .172
• Important, cf. [GH19, Theorem 7.5.1.]: H(G(Qp),Gp(Zp)) is again commuta-173
tive! (⇒ If πGp(Zp)p 6= 0, then its dimension is one.)174
Upshot:175
• Thus irreducible, admissible G(Af ) × (gC,K∞)-modules are a collection of176
“local data”.177
• Being automorphic puts strong relations among these local data.178
• For almost all primes p get homomorphism179
χp : H(G(Qp),Gp(Zp))→ C,
similarly to case for GL2 ⇒ this will yield analog of a “system of Hecke180
eigenvalues”.181
• For π set182
πf :=
′⊗p
πp,
thus183
π ∼= πf ⊗ π∞.
Definition 11.3 (L-algebraic automorphic representation). An automorphic repre-184
sentation185
π ∼= πf ⊗ π∞is L-algebraic if the irreducible, admissible (gC,K∞)-module π∞ is L-algebraic, cf.186
[BG10, Definition 3.1.1.].187
Examples:188
• An automorphic representation π ⊆ A(GL2) which is generated by a modular189
form is not L-algebraic, but the twist190
π ⊗C |det|1/2+aadelic
is for any a ∈ Z.191
• A continuous character ψ : Q×\A× → C is L-algebraic if and only if192
ψ = χ| − |kadelic
for some k ∈ Z, | − |adelic the adelic norm and χ : Q×\A× → C a character of193
finite order.194
12. The Langlands program for general groups, part II1
Last time:2
• F arbitrary number field3
• The Langlands(-Clozel-Fontaine-Mazur) reciprocity conjecture for GLn,F says:4
– ` some prime. Fix an isomorphism C ∼= Q`.5
– Then for any n ≥ 1, there exists a (unique) bijection between6
i) the set of L-algebraic cuspidal automorphic representations of GLn(AF ),7
ii) the set of (isomorphism classes) of irreducible continuous repre-8
sentations Gal(F/F ) → GLn(Q`) which are almost everywhere9
unramified, and de Rham at places dividing `,10
such that the bijection matches Satake parameters with eigenvalues of11
Frobenius elements.12
• Introduced space of (adelic) automorphic forms A(G) for any G.13
• Flath’s theorem⇒ automorphic representations π factorize: π ∼=⊗′
p πp⊗π∞.14
• Explained “L-algebraic” (=infinitesimal character of π∞ is “integral”).15
Today:16
• Introduce “cuspidal” automorphic representations.17
• Explain Satake parameters.18
• Discuss some expectations of the Langlands program for general G.19
Definition 12.1 ([GH19, Definition 9.2]). Let G/Q be reductive. We call ϕ ∈ L2([G])20
resp. ϕ ∈ A(G) cuspidal if21 ∫N(Q)\N(A)
ϕ(ng)dn = 0
for all unipotent radicals N of proper parabolic subgroups P ⊆ G (defined over Q),22
and almost all g ∈ G(A).23
Cuspidal automorphic representations:24
• Set L2cusp([G]) resp. Acusp(G) resp. Acusp([G]) as the subspaces of cuspidal25
elements.26
• There is an embedding with dense image27
Acusp([G]) ⊆ L2cusp([G]),
i.e., cuspidal automorphic forms satisfy a growth condition strong enough to28
make them L2 (they are “rapidly decreasing”), cf. [GH19, Section 6.5.].29
• An L2-automorphic representation of G(A) is cuspidal if it is isomorphic to a30
subquotient (actually subrepresentation) of L2cusp([G]).31
• Similarly, an automorphic representation is cuspidal if it occurs as a subquo-32
tient of Acusp(G).33
• Gelfand, Piatetski-Shapiro: As a unitary G(A)-representation34
L2cusp([G]) ∼=
⊕π∈G(A)
mππ
with each mπ finite, i.e., mπ ∈ N ∪ 0, cf. [GH19, Corollary 9.1.2].35
• In particular, L2cusp([G]) ⊂ L2
disc([G]) (note: the trivial representation occurs36
in L2disc([G]) \ L2
cusp([G])).37
• From38
L2cusp([G]) ∼=
⊕π∈G(A)
mππ
on deduces the decomposition (into irreducible, admissible G(Af )×(gC,K∞)-39
modules)40
Acusp([G]) ∼=⊕
π∈G(A)
mππK∞−finite,
where πK∞−finite ⊆ π denotes the (dense) subspace of K∞-finite vectors, cf.41
[GH19, Theorem 4.4.4.].42
• Thus, the decomposition of L2cusp([G]) is “the same as” the decomposition of43
Acusp([G]).44
• Piatetski-Shapiro, Shalika, cf. [GH19, Theorem 11.3.4]: F number field, G =45
ResF/QGLn,F . Then the G(A)-representation on L2cusp([G]) is multiplicity46
free (“multiplicity one”).47
• “Strong multiplicity one”: Assume π, π′ are cuspidal automorphic represen-48
tations for GLn,F . If πp ∼= π′p for almost all primes p, then π ∼= π′, cf. [GH19,49
11.7.2].50
• Not true without cuspidality, cf. [BG10, page 38].51
• Moeglin, Waldspurger, cf. [GH19, Theorem 10.7.1.], [MW89]: G = ResF/QGLn,F .52
Then L2disc([G]) is known, up to parametrizing cuspidal automorphic repre-53
sentations of GLd,F for d|n, and turns out to be multiplicity free.54
Satake parameters:55
• Describe πp if π is unramified at p, i.e.,56
πGp(Zp)p 6= 0
with Gp → Spec(Zp) a reductive model of GQp (which exists for almost all57
primes p).58
• Thus, describe C-algebra homomorphisms59
H(G(Qp),K)→ C
for K := Gp(Zp).60
• Langlands/Satake, cf. [GH19, Theorem 7.5.1., Corollary 7.5.2.], [BG10, Sec-61
tion 2.1.]: There exists62
– a “natural” algebraic group G over Q,63
– an automorphism Frp : G(Q) ∼= G(Q),64
– and a bijection65
HomC−alg(H(G(Qp),K),C)1:1↔
Frobenius semisimple G(C)− conjugacy classes in G(C) o FrZp.
• The group GC has the dual root datum as G. Thus for example, cf. [Bor79]:66
– if G = GLn, then G = GLn,C,67
– if G = SLn, then G = PGLn,C,68
– if G = SO2n, then G = SO2n,C,69
– if G = SO2n+1, then G = Sp2n,C,70
– if G = GSp2n, then G = GSpin2n+1,C.71
• If GQp is split, then the automorphism Frp is trivial, and Frobenius semisimple72
conjugacy classes are just conjugacy classes of semisimple elements in G(C).73
• If G = GLn, then semisimple conjugacy classes are uniquely determined by74
their characteristic polynomials.75
• Let π =⊗′
p πp⊗π∞ be an automorphic representation of G. Then we obtain76
the following analog of a system of Hecke eigenvalues:77
– a finite set S of primes, such that πp is unramified for p /∈ S,78
– for each p /∈ S a Frobenius semisimple conjugacy class cp(π) in G(C) o79
FrZp .80
• For GLn the eigenvalues of (each element in) cp(π) are the Satake parameters81
of π.82
• Concretely, if the coset83
GLn(Zp)Diag(p, . . . , p︸ ︷︷ ︸i−times
, 1, . . . , 1︸ ︷︷ ︸(n−i)−times
)GLn(Zp),
has eigenvalue ap,i on πGLn(Zp)p , then the elements in cp(π) have characteristic84
polynomial85
Xn − p(1−n)
2 ap,1Xn−1 + . . .+ (−1)ip
i(i−n)2 ap,iX
i + . . .+ (−1)nap,n,
cf. [GH19, Section 7.2.].86
• Even more concrete, for π generated by a newform f =∞∑i=1
anqn ∈ Sk(Γ0(N), χ)87
the cp(π), p - N, have characteristic polynomial88
X2 − p−1/2papX + χ(p)pk.
Recall that π ⊗C |det|1/2adelic is L-algebraic. For π ⊗C |det|1/2adelic we obtain the89
polynomial90
X2 − apX + χ(p)pk−1,
which (after choosing an isomorphism Q` ∼= C) was the characteristic poly-91
nomial of an arithmetic Frobenius at p.92
The L-group:93
• G reductive over Q94
• ` a prime.95
• With a little work (cf. [GH19, Section 7.3.]) the group Gal(Q/Q) acts on the96
reductive group G over Q (the action is trivial if G is split).97
• Set98
LG := G(Q`) o Gal(Q/Q).
• An L-parameter is by definition a G(Q`)-conjugacy class of continuous homo-99
morphisms100
Gal(Q/Q)→ LG,
whose projection to Gal(Q/Q) is the identity.101
• If F/Q is finite and G = ResF/QGLn,F , then an L-parameter identifies with102
an isomorphism class of an n-dimensional `-adic Galois representation of103
Gal(F/F ).104
The Buzzard–Gee conjecture for L-algebraic automorphic representations:105
106
• Let ` be a prime.107
• Fix an isomorphism ι : C ∼= Q`.108
• Let π be an L-algebraic automorphic representation of G.109
• Then Buzzard–Gee (cf. [BG10, Conjecture 3.2.2.]) conjecture that there exists110
an L-parameter111
ρπ : Gal(Q/Q)→ LG
such that (in particular)112
– if p 6= ` is unramified for π, then each arithmetic Frobenius Frobp ∈113
Gal(Qp/Qp) maps to the conjugacy class of114
ι(cp(π)) ∈ G(Q`) o FrobZp
under115
Gal(Qp/Qp)→ G(Q`) o Gal(Qp/Qp) G(Q`) o FrobZp ,
– for each continuous representation LG → GLN (Q`), which is algebraic116
on G(Q`), the composition117
Gal(Q`/Q`)ρ−→ LG→ GLN (Q`)
is de Rham.118
• This generalizes the association of Galois representations to (a twist of) the119
automorphic representation associated to some newform.120
• Contrary to the case of GLn the L-parameter ρπ is not conjectured to be121
unique, cf. [BG10, Remark 3.2.4.].122
• Moreover, we may chose the same L-parameter ρ for different L-algebraic123
automorphic representations π.124
• For GLn the Buzzard–Gee conjecture predicts in addition to the previous125
Langlands-Clozel-Fontaine-Mazur conjecture the existence of Galois repre-126
sentations attached to possibly non-cuspidal (L-algebraic) automorphic rep-127
resentations. These Galois representations are then no longer conjectured to128
be irreducible.129
The Langlands group L:130
• Conjecturally, there should exist a (very big) locally compact group L, the131
(global) Langlands group, with (at least) the following properties, cf. [GH19,132
Section 12.6.], [Art02], [LR87]:133
– There exists a canonical surjection134
L Gal(Q/Q)
(in fact, LWQ with WQ the global Weil group).135
– For each n ≥ 0 the isomorphism classes Irrn(L) of continuous irreducible136
n-dimensional C-representations of L are naturally in bijection with the137
set Cspn of cuspidal automorphic representations of GLn(A).138
– Fix an isomorphism Q` ∼= C. Then there is an injection of the set Ggeom139
of isomorphism classes of irreducible, almost everywhere unramified `-140
adic Galois representations Gal(Q/Q) which are de Rham at ` to the set141
Irr of isomorphism classes of irreducible representations of L, with image142
corresponding to L-algebraic cuspidal automorphic representations.143
– For each place v of Q there is an injection144
Lv → L
of the local Langlands groups, which is well-defined up to conjugation in145
L. Here:146
∗ Lv ∼= WR, the non-split extension of C× by Gal(C/R), if v =∞.147
∗ Lv ∼= WQp × SU2(R) if v = p prime and148
WQp∼= IQp o Z ⊆ Gal(Qp/Qp) ∼= IQp o Z
is the subgroup of elements mapping to integral powers of Frobe-149
nius.150
– If π ∼=⊗′
v πv ∈ Cspn corresponds to a continuous representation ρπ : L →151
GLn(C), then πv should correspond to ρπ |Lv under the (conjectural) local152
Langlands correspondence, cf. [GH19, Section 12.5.]. Next semester Pe-
ter is giving a lec-
ture on this ⇒Follow it!
Next semester Pe-
ter is giving a lec-
ture on this ⇒Follow it!
153
• If this conditions hold, then necessarily Lab ∼= Q×\A×. Note that by global154
class field theory there exists a surjection Q×\A× → Gal(Q/Q).155
• The existence of L is currently completely out of reach, even more conjec-156
turally L should surject onto the “motivic Galois group” Gmot over C, cf.157
[Art02], [LR87].158
The global Langlands correspondence for general G (very, very rough):159
• No precision is claimed, and all statements are close to being empty!160
• For precise statements one should consult [GH19, Chapter 12] or [Art94].161
• In order to parametrize non L-algebraic automorphic representations (at least162
those relevant for the decomposition of L2disc([G])) one should replace the163
previous Galois version of L-parameters by (certain) L-parameters164
L → LG
(actually in the Weil form WQ n G(C) of the L-group).165
• Then one hopes to construct a surjective map (cf. [GH19, Conjecture 12.6.2.])166
(certain) automorphic representations LL−−→ (certain) L− parameters.
• The fibers of LL are called L-packets .167
• A precise construction of a local Langlands correspondence (cf. [GH19, Con-168
jecture 12.5.1.]) should yield a parametrization of the L-packets, cf. [GH19,169
(12.23)].170
• For each π ⊆ L2disc([G]) its multiplicity should be computable via the L-171
parameter LL(π) and the precise parametrizations of the L-packets, cf. [GH19,172
Conjecture 12.6.3.].173
• However, for non-tempered representations the above should not be reason-174
able and one should consider Arthur parameters175
L × SL2(C)→ LG
instead of L-parameters, cf. [Art94].176
Important stuff, which was not mentioned in the lecture:177
• L-functions, [Bor79], [GH19, Chapter 11, Section 12.7.],178
• Functoriality, [GH19, Section 12.6.],179
• The trace formula, [Art05], [GH19, Section 18],180
• The Langlands program for function fields,181
• ...182
A glimpse on Shimura varieties:183
• Let K∞ ⊆ G(R) be a maximal (connected) compact subgroup.184
• Let K ⊆ G(Af ) be a (sufficiently small) compact-open subgroup.185
• Recall that186
–
[G]→ [G]/K = G(Q)AG\G(A)/K
is a profinite covering of a real manifold187
–
[G]/K → XK := [G]/KK∞ = G(Q)AG\G(A)/KK∞
is a K∞-bundle over XK , which is a disjoint union of arithmetic mani-188
folds (=quotients of a symmetric spaces by arithmetic subgroups)189
• Note that implicitly XK depends on K∞.190
• Let us set (just to simplify some notations later)191
X := lim←−K
XK .
Stuff we encountered for G = GL2:192
(1) The upper/lower halfplane H± ∼= G(R)/K∞, which is naturally a complex193
manifold194
(2) The holomorphic embedding195
H± → P1C
(3) The G(R)-equivariant vector bundles196
ωk
on H±, or by pullback on XK for K ⊆ G(Af ) compact-open.197
(4) For k ∈ Z the space of modular forms198
Mk ⊆ H0(X, ωk),
where the RHS denotes holomorphic sections.199
(5) The canonical compactification X∗K of XK , with the extension of ωk on it.200
(6) A scheme X → Spec(Q), whose C-valued points are naturally isomorphic to201
X.202
(7) The Gal(Q/Q)×G(Af )-representation203
H1et(XQ,L)
associated to certain G(Af )-equivariant Q`-local systems L on X , e.g., L ∼=204
Q`.205
(8) The Eichler–Shimura isomorphism relating Sk for k ≥ 2, to certain206
H1et(XQ,L).
(9) The Eichler–Shimura relation expressing a relation of the Gal(Q/Q)-action207
and the G(Af )-action on208
H1et(XQ,L),
which was a consequence of the mod p geometry of some modular curve.209
(10) A map210
LL: Amod → Gmod
with (conjecturally) describable image.211
For general G there are unfortunately no analogs - but wait: there exists212
G giving rise to Shimura varieties!213
• Already (1) fails for general G: The space214
G(R)/K∞
need not be complex manifold! (E.g., if G = GL3 , then the real dimension215
dim(GL3(R)/AGL3(R)SO3(R)) = 3(3+1)2 − 1 = 5 is odd)216
• In particular, everything related to holomorphicity has no analog for such G,217
e.g., 2, 3, 4, 6, 7,...218
• That is bad news!219
• Good news: For groups G giving rise to Shimura data (like ResF/QGL2,F for220
F/Q totally real, or GSp2n,... , cf. [Lan17], [Mil05], [Del71a], [Del79]), the221
real manifold222
G(R)/K∞AG
is a complex manifold.223
• A Shimura datum is a pair (G,X) with G a reductive group G over Q, and X224
a G(R)-conjugacy classes of morphisms h : S := ResC/RGm → GR such that225
1) For each h ∈ X, the characters of S(R) = C× acting on Lie(GR)C (via226
Ad h) are either z 7→ zz , z 7→ 1, or z 7→ z
z .227
2) For each h ∈ X, the group g ∈ G(C) | h(i)gh(i)−1 = g is compact228
modulo its center.229
3) The adjoint group Gad has no factor, defined over Q, for which the230
projection of h ∈ X is trivial.231
• E.g., take G = GL2, and X as the conjugacy class of the usual embedding232
C× → GL2(R). Note that X ∼= H±.233
• In general, for h ∈ X the stabilizer Kh of h in G(R) is compact modulo234
the center of G(R) (this follows from condition 2)) and X ∼= G(R)/Kh(∼=235
G(R)/AGK∞).236
• For K ⊆ G(Af ) recall237
XK := G(Q)\(G(Af )/K ×X).
Other notation: ShK(G,X) the Shimura variety of levelK attached to (G,X).238
• We get:239
(1) X is a disjoint union of hermitian symmetric domains, cf. [Mil05], [Del79].240
(2) To each h ∈ X is naturally attached a parabolic subgroup Ph ⊆ G(C)241
(cf. [CS17, Section 2.1]), and sending h→ Ph defines an open embedding242
π : X ∼= G(R)/Kh → F l ∼= G(C)/Ph
of X into a flag variety (this generalizes the embedding H± → P1(C)). In243
particular, X carries a natural complex structure. More is true (Baily-244
Borel): XK is naturally a quasi-projective variety, cf. [Lan17, Theorem245
2.4.1.].246
(3) The pullback of G(C)-equivariant vector bundles on F l gives rise to247
G(Q)-equivariant vector bundles on G(Af )×X. By descent one one ob-248
tains automorphic vector bundles E on XK for any K ⊆ G(Af ) compact-249
open, i.e., analogs of the ωk, k ∈ Z, [Har88]. Note that G(C)-equivariant250
vector bundles on F l are equivalent to (algebraic) representations of Ph.251
(4), (5) If dimC(XK) > 1, there do exist compactifications XK , but these are252
no longer canonical. However, it is possible to define subspaces in the253
cohomology of H∗(XK , E)), which generalize Mk for k ∈ Z, cf. [Har88].254
(6) The space X (=inverse limit of complex manifolds) arises again as the255
C-points of a scheme X → Spec(E), the canonical model, defined over256
some number field (the “reflex field of the Shimura data”), cf. [Lan17,257
Theorem 2.4.3.].258
(8),(9),(10) Having the canonical model, one can attempt (sometimes sucessfully)259
to obtain analogs of points (18),(19),(20) and finally relate (certain)260
automorphic representations to (variants of) Galois representations, cf.261
[Har88], [BRb]. Needless to say that everything (compactifications, Eichler–262
Shimura isomorphism/relation, integral models, mod p geometry, interior263
vs intersection cohomology,...) is much more complicated!264
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