L-indistinguishable p-adic automorphic forms

Judith LudwigUniversity of Bonn, Mathematical Institute

jludwig@math.uni-bonn.de

Abstract

We construct examples of L-indistinguishable p-adic automorphic eigenforms for an inner form of SL2 using eigenvarietiesand a p-adic Labesse–Langlands transfer.

Introduction

In recent years a p-adic version of the Langlands programme has started to emerge. Although we are stilllacking a general definition of a p-adic automorphic representation we have good working definitions of p-adicautomorphic forms in many situations. Given such a definition one can ask which aspects of the classicalLanglands programme make sense for these p-adic automorphic forms and it is particularly interesting to askabout Langlands functoriality: If G and H are two connected reductive groups defined over a number fieldtogether with a classical Langlands transfer from G to H and a definition of p-adic automorphic forms, thenis there a p-adic Langlands transfer? As in general one does not transfer single representations but rathersets of such, called L-packets, a closely related question is: What does a p-adic L-packet look like?

Here we construct pairs of L-indistinguishable p-adic eigenforms for an inner form of SL2, asproved in [3]. The pairs consist of a classical and a non-classical p-adic automorphic eigenform, which havethe same system of Hecke eigenvalues and therefore give rise to the same Galois representation. In this sensethese forms are L-indistinguishable. Although so far there is no definition of a global p-adic L-packet, ourresults suggest that, for any future definition, such pairs should lie in the same L-packet.

Setup

Fix a prime q and let B/Q be the quaternion algebra ramified at SB := {q,∞}. Fix a prime p /∈ SB and

a finite extension E/Qp. Let G be the algebraic group over Q defined by the units B∗ and let G be thesubgroup of elements of reduced norm one. For S a finite set of places, which includes p and SB, we have aHecke algebra HS := Hur,S ⊗E Ap for G, which is the product of the spherical Hecke algebras at all placesnot in S and an Atkin-Lehner algebra at p. Let HS be the analogue for G.

Tools

•Classical transfer: There is a morphism of L-groups given by the natural projection

LG = GL2(Q)→ LG = PGL2(Q),

and as predicted by the Langlands functoriality conjectures, there is a corresponding transfer, which as-sociates to an automorphic representation π of G(A) an L-packet Π(π) of admissible representations ofG(A). Formulas for the multiplicities m(π) of π ∈ Π(π) in the automorphic spectrum of G(A) have beenproved by Labesse and Langlands in [1].

•Eigenvarieties: These are rigid analytic spaces parametrizing certain p-adic automorphic forms. Eigen-varieties are good tools to address the above questions as they often carry a Zariski-dense set of pointscorresponding to classical algebraic automorphic representations. By construction a point on an eigenva-riety gives rise to an (overconvergent) p-adic automorphic eigenform for some suitable Hecke algebra.

Any idempotent e = ⊗el ∈ C∞c (G(Apf ),Q), such that el = 1GL2(Zl) for all but finitely many l, say for all

l /∈ S(e), gives rise to such an eigenvariety D(e).

There are morphisms ψ : HS(e) → O(D(e)) and ω : D(e) → W , where the so called weight space

W = Homcts((Z∗p)2,Gm) is a rigid analytic space that interpolates the weights of classical automorphicrepresentations. The set of points of D(e) embeds

D(e)(Qp) ↪→ Hom(HS(e),Qp)× W(Qp), z 7→ (ψz, ω(z)).

Likewise, if e ∈ C∞c (G(Apf ),Q) is an idempotent with a set S(e) of bad places, we have an eigenvariety

D(e) for G. Again we have an embedding

D(e)(Qp) ↪→ Hom(HS(e),Qp)×W(Qp), z 7→ (ψz, ω(z)),

for the corresponding weight space W = Homcts(Z∗p,Gm).

• p-adic transfer: The classical transfer can be interpolated to a p-adic transfer between eigenvarieties(see [2]). There are natural maps µ : W → W andHS ↪→ HS for any finite set S. Let e ∈ C∞c (G(Apf ),Q)

be an idempotent. Then there exists an idempotent e ∈ C∞c (G(Apf ),Q) with S(e) = S(e) =: S and a

morphism ζ : D(e) → D(e) compatible with the classical transfer and such that the following diagramscommute

D(e)ω

��

ζ//D(e)

ω��

Wµ

//W

HS��

� � //HS��

O(D(e))ζ∗

//O(D(e))

.

Results

Definition. A point z on an eigenvariety is called classical, if there is a classical automorphic eigen-form in the corresponding space of p-adic forms, whose system of Hecke eigenvalues is that definedby z.

Let π(θ) be an algebraic automorphic representation of G(A) associated with a Großencharacter θ of animaginary quadratic field in which p splits. Such a representation gives rise to two points on D(e) for a suit-

able idempotent e ∈ C∞c (G(Apf ),Q), which can be distinguished by their slope, i.e., by the p-adic valuation

of the eigenvalue of a certain Hecke operator Up ∈ Ap. Namely only one point has valuation zero.Let x be the point of non-zero slope and consider its image in Hom(HS,Qp)×W(Qp) under the composite

of the maps

D(e)(Qp) //

φ++VVVVVVVVVVVVVVVVVVVVVV

Hom(HS,Qp)× W(Qp)��

Hom(HS,Qp)×W(Qp),

which we denote by φ.

Theorem. There exist automorphic representations π(θ) of G(A) as above together with idempotents

e ∈ C∞c (G(Apf ),Q) and e1, e2 ∈ C∞c (G(Apf ),Q), such that the image φ(x) of x lifts to a non-classical

point on the eigenvariety D(e1) and to a classical point on D(e2).

Sketch of proof. Fix π(θ) such that π(θ)l is unramified for all l 6= q, the L-packet Π(π(θ)q) = {τ1, τ2}defined by π(θ)q is of size two and such that precisely one of the representations

π1 :=⊗l 6=q

π0l ⊗ τ1 ⊗ π∞, π2 :=

⊗l 6=q

π0l ⊗ τ2 ⊗ π∞,

say π2, is automorphic. Here π0l denotes the unique member of the local L-packet Π(π(θ)l), which has a

non-zero fixed vector under SL2(Zl).For all l 6= q define el := eGL2(Zl) and let eq be the special idempotent attached to the Bernstein com-

ponent defined by the supercuspidal representation π(θ)q. Define e = ⊗lel ∈ C∞c (G(Apf ),Q) and let

S = S(e) = {p, q}.Let eq,1 (resp. eq,2) ∈ C∞c (G(Qq),Q) be the special idempotent associated with τ1 (resp. τ2) and define

e1 :=⊗l 6=q,p

eSL2(Zl) ⊗ eq,1 and e2 :=⊗l 6=q,p

eSL2(Zl) ⊗ eq,2 ∈ C∞c (G(Apf ),Q).

This setup does the job: Firstly, for i = 1, 2 there exists a unique element πi in the L-packet Π(π(θ)) suchthat ei(πi)

pf 6= 0. As π2 is automorphic φ(x) lifts to D(e2). In order to show that φ(x) also lifts to D(e1),

we use the p-adic transfer D(e) → D(e) for a suitable idempotent e. We choose e so that we also have aclosed immersion D(e1) ↪→ D(e). The transfer allows us to lift φ(x) to a point y on D(e). To see that ylies in D(e1) the crucial point is that in a neighbourhood of x we can find many points associated with

automorphic representations of G(A) that do not come from a Großencharacter. These give rise to stableL-packets of G and therefore, via the p-adic transfer, to points on D(e) that all lie in D(e1). A geometricargument implies that y ∈ D(e1) as well.

Consequences

Corollary. Let D(e) be an eigenvariety for G and assume z ∈ D(e)(Qp) is a point whose system of

Hecke eigenvalues ψz comes from a classical automorphic representation π of G(A). Then z is notnecessarily classical.

For an idempotent e and a point κ ∈ W(Qp) let M(e, κ) be the space of p-adic automorphic forms ofweight κ and tame level e. It is a HS(e)-module and by construction a point on D(e) of weight κ gives rise

to a character of HS(e) occurring in M(e, κ).

An eigenvariety D(e) carries a family of Galois representations which at a classical point z specializes tothe Galois representation associated with the classical automorphic representation giving rise to z.

Corollary. Define ϕ := ψx|HSand let n = µ(ω(x)) ∈ W(E). Then there exists an overconvergent

non-classical p-adic automorphic eigenform f ∈ M(e1, n)ϕ and a classical automorphic eigenformg ∈M(e2, n)ϕ. The Galois representations ρf and ρg associated with f and g agree. In this sense thetwo forms f and g are L-indistinguishable.

References

[1] J.-P. Labesse and R. P. Langlands. L-indistinguishability for SL(2). Canad. J. Math., 31(4):726–785,1979.

[2] Judith Ludwig. A p-adic Labesse–Langlands transfer. 2014. arXiv:1412.4140.

[3] Judith Ludwig. L-Indistinguishability on Eigenvarieties. 2015. arXiv:1508.06187.