L-indistinguishable p-adic automorphic forms Judith Ludwig University of Bonn, Mathematical Institute [email protected] Abstract We construct examples of L-indistinguishable p-adic automorphic eigenforms for an inner form of SL 2 using eigenvarieties and 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 still lacking a general definition of a p-adic automorphic representation we have good working definitions of p-adic automorphic forms in many situations. Given such a definition one can ask which aspects of the classical Langlands programme make sense for these p-adic automorphic forms and it is particularly interesting to ask about Langlands functoriality: If G and H are two connected reductive groups defined over a number field together with a classical Langlands transfer from G to H and a definition of p-adic automorphic forms, then is there a p-adic Langlands transfer? As in general one does not transfer single representations but rather sets 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 SL 2 , as proved in [3]. The pairs consist of a classical and a non-classical p-adic automorphic eigenform, which have the same system of Hecke eigenvalues and therefore give rise to the same Galois representation. In this sense these forms are L-indistinguishable. Although so far there is no definition of a global p-adic L-packet, our results 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 S B := {q, ∞}. Fix a prime p/ ∈ S B and a finite extension E/Q p . Let e G be the algebraic group over Q defined by the units B * and let G be the subgroup of elements of reduced norm one. For S a finite set of places, which includes p and S B , we have a Hecke algebra e H S := e H ur,S ⊗ E e A p for e G, which is the product of the spherical Hecke algebras at all places not in S and an Atkin-Lehner algebra at p. Let H S be the analogue for G. Tools • Classical transfer: There is a morphism of L-groups given by the natural projection L e G = GL 2 ( Q) → L G = PGL 2 ( Q), and as predicted by the Langlands functoriality conjectures, there is a corresponding transfer, which as- sociates to an automorphic representation e π of e G(A) an L-packet Π( e π ) of admissible representations of G(A). Formulas for the multiplicities m(π ) of π ∈ Π( e π ) in the automorphic spectrum of G(A) have been proved 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 points corresponding 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 e = ⊗e e l ∈ C ∞ c ( e G(A p f ), Q), such that e e l = 1 GL 2 (Z l ) for all but finitely many l, say for all l/ ∈ S (e e), gives rise to such an eigenvariety D(e e). There are morphisms e ψ : e H S (e e) →O(D(e e)) and e ω : D(e e) → f W , where the so called weight space f W = Hom cts ((Z * p ) 2 , G m ) is a rigid analytic space that interpolates the weights of classical automorphic representations. The set of points of D(e e) embeds D(e e)( Q p ) , → Hom( e H S (e e) , Q p ) × f W ( Q p ),z 7→ ( e ψ z , e ω (z )). Likewise, if e ∈ C ∞ c (G(A p f ), 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)( Q p ) , → Hom(H S (e) , Q p ) ×W ( Q p ),z 7→ (ψ z ,ω (z )), for the corresponding weight space W = Hom cts (Z * p , G m ). • p-adic transfer: The classical transfer can be interpolated to a p-adic transfer between eigenvarieties (see [2]). There are natural maps μ : f W→W and H S , → e H S for any finite set S . Let e e ∈ C ∞ c ( e G(A p f ), Q) be an idempotent. Then there exists an idempotent e ∈ C ∞ c (G(A p f ), Q) with S (e)= S (e e) =: S and a morphism ζ : D(e e) →D(e) compatible with the classical transfer and such that the following diagrams commute D(e e) e ω ζ // D(e) ω f W μ // W H S // e H S O(D(e)) ζ * // O(D(e 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 defined by z . Let π ( e θ ) be an algebraic automorphic representation of e G(A) associated with a Gr¨ oßencharacter e θ of an imaginary quadratic field in which p splits. Such a representation gives rise to two points on D(e e) for a suit- able idempotent e e ∈ C ∞ c ( e G(A p f ), Q), which can be distinguished by their slope, i.e., by the p-adic valuation of the eigenvalue of a certain Hecke operator U p ∈ e A p . Namely only one point has valuation zero. Let e x be the point of non-zero slope and consider its image in Hom(H S , Q p ) ×W ( Q p ) under the composite of the maps D(e e)( Q p ) // φ ++ V V V V V V V V V V V V V V V V V V V V V V Hom( e H S , Q p ) × f W ( Q p ) Hom(H S , Q p ) ×W ( Q p ), which we denote by φ. Theorem. There exist automorphic representations π ( e θ ) of e G(A) as above together with idempotents e e ∈ C ∞ c ( e G(A p f ), Q) and e 1 ,e 2 ∈ C ∞ c (G(A p f ), Q), such that the image φ( e x) of e x lifts to a non-classical point on the eigenvariety D(e 1 ) and to a classical point on D(e 2 ). Sketch of proof. Fix π ( e θ ) such that π ( e θ ) l is unramified for all l 6= q , the L-packet Π(π ( e θ ) q )= {τ 1 ,τ 2 } defined by π ( e θ ) q is of size two and such that precisely one of the representations π 1 := O l 6=q π 0 l ⊗ τ 1 ⊗ π ∞ , π 2 := O l 6=q π 0 l ⊗ τ 2 ⊗ π ∞ , say π 2 , is automorphic. Here π 0 l denotes the unique member of the local L-packet Π(π ( e θ ) l ), which has a non-zero fixed vector under SL 2 (Z l ). For all l 6= q define e e l := e GL 2 (Z l ) and let e e q be the special idempotent attached to the Bernstein com- ponent defined by the supercuspidal representation π ( e θ ) q . Define e e = ⊗ l e e l ∈ C ∞ c ( e G(A p f ), Q) and let S = S (e e)= {p, q }. Let e q,1 (resp. e q,2 ) ∈ C ∞ c (G(Q q ), Q) be the special idempotent associated with τ 1 (resp. τ 2 ) and define e 1 := O l 6=q,p e SL 2 (Z l ) ⊗ e q,1 and e 2 := O l 6=q,p e SL 2 (Z l ) ⊗ e q,2 ∈ C ∞ c (G(A p f ), Q). This setup does the job: Firstly, for i =1, 2 there exists a unique element π i in the L-packet Π(π ( e θ )) such that e i (π i ) p f 6= 0. As π 2 is automorphic φ( e x) lifts to D(e 2 ). In order to show that φ( e x) also lifts to D(e 1 ), we use the p-adic transfer D(e e) →D(e) for a suitable idempotent e. We choose e so that we also have a closed immersion D(e 1 ) , →D(e). The transfer allows us to lift φ( e x) to a point y on D(e). To see that y lies in D(e 1 ) the crucial point is that in a neighbourhood of e x we can find many points associated with automorphic representations of e G(A) that do not come from a Gr¨oßencharacter. These give rise to stable L-packets of G and therefore, via the p-adic transfer, to points on D(e) that all lie in D(e 1 ). A geometric argument implies that y ∈D(e 1 ) as well. Consequences Corollary. Let D(e) be an eigenvariety for G and assume z ∈D(e)( Q p ) is a point whose system of Hecke eigenvalues ψ z comes from a classical automorphic representation e π of e G(A). Then z is not necessarily classical. For an idempotent e and a point κ ∈W ( Q p ) let M (e, κ) be the space of p-adic automorphic forms of weight κ and tame level e. It is a H S (e) -module and by construction a point on D(e) of weight κ gives rise to a character of H S (e) occurring in M (e, κ). An eigenvariety D(e) carries a family of Galois representations which at a classical point z specializes to the Galois representation associated with the classical automorphic representation giving rise to z . Corollary. Define ϕ := e ψ e x | H S and let n = μ( e ω ( e x)) ∈W (E ). Then there exists an overconvergent non-classical p-adic automorphic eigenform f ∈ M (e 1 ,n) ϕ and a classical automorphic eigenform g ∈ M (e 2 ,n) ϕ . The Galois representations ρ f and ρ g associated with f and g agree. In this sense the two 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.
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L-indistinguishable p-adic automorphic forms
Judith LudwigUniversity of Bonn, Mathematical Institute
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.
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