Yuval Flicker

• The Ohio State University

We compute explicitly the cardinality of a set of Galois-invariant isomorphism classes of irreducible rank two \(\overline{\mathbb {Q}}_\ell \)-smooth sheaves on \(X-S\), where X is a smooth projective absolutely irreducible curve of genus g over a finite field \(\mathbb {F}_q\) and S is a reduced divisor, with pre-specified tamely ramified ramific...

We compute the cardinality of a set of Galois-invariant isomorphism classes of irreducible rank two \(\overline{{\mathbb {Q}}}_\ell \)-smooth sheaves on \(X_1-S_1\), where \(X_1\) is a smooth projective absolutely irreducible curve of genus g over a finite field \({\mathbb {F}}_q\) and \(S_1\) is a reduced divisor, with pre-specified tamely ramifie...

We compute the cardinality of a set of Galois-invariant isomorphism classes of irreducible rank two Q¯ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\mathbb...

Schur duality is an equivalence, for d\leq n , between the category of finite-dimensional representations over \mathbb{C} of the symmetric group S_d on d letters, and the category of finite-dimensional representations over \mathbb{C} of \operatorname{GL}(n,\mathbb{C}) whose irreducible subquotients are subquotients of \smash{\overline{\mathbb{E}}^{...

We give a new proof of the well-known classification of finite subgroups of \({\text {SL}}(2,\mathbb C)\), that generalizes to the three dimensional case of \({\text {SL}}(3,\mathbb C)\); recall the geometric proof, based on study of the motions of the Platonic solids, that does not seem generalizable to higher \({\text {SL}}(n)\) nor to other fiel...

We determine the linearly reductive finite subgroup schemes G of SL(3,F), namely those all of whose modules are completely reducible, where F is an algebraically closed field of positive characteristic, up to conjugation. This extends work of M. Hashimoto from SL(2,F) to SL(3,F).

We attempt to give a complete exposition of the classification of the finite subgroups of \({\text {SL}}(3,\overline{F})\), where \(\overline{F}\) is a separably closed field of characteristic not dividing the order of the finite group, in contemporary language, completing some of the proofs of Blichfeldt from the early twentieth century. The class...

International audience We develop a modular version of a super analogue of Schur's duality by means of supergroups, rather than Lie superalgebras, in preparation for a geometric analogue.

The Schur-Weyl duality, which started as the study of the commuting actions of the symmetric group Sd and GL(n, ℂ) on V⊗d where V = ℂn, was extended by Drinfeld and Jimbo to the context of the finite Iwahori-Hecke algebra Hd(q2) and quantum algebras Uq(gl(n)), on using universal R-matrices, which solve the Yang-Baxter equation. There were two exten...

The Schur-Weyl duality, which started as the study of the commuting actions of the symmetric group $S_d$ and $\mathrm{GL}(n,\mathbb{C})$ on $V^{\otimes d}$ where $V=\mathbb{C}^n$, was extended by Drinfeld and Jimbo to the context of the finite Iwahori-Hecke algebra $H_d(q^2)$ and quantum algebras $U_q(\mathrm{gl}(n))$, on using universal $R$-matric...

here has recently been renewed interest in the trace formula–in particular, that of the initial case of GL(2)–due to counting applications in the function field case. For these applications, one needs a very precise form of the trace formula, with all terms computed explicitly. Our aim in this work is to compute the trace formula for GL(2) over a n...

The character tr p of an irreducible admissible representation p of the group G(F) of F-points of a reductive connected linear algebraic group G over a local non-Archimedean field F has been shown by Harish-Chandra to be locally constant on the regular set and locally integrable, that is, representable by a function with such properties, when the c...

Let G be a reductive connected algebraic group defined over a number field F. Fix a minimal parabolic subgroup P 0 and a Levi component \(M_{P_{0}}\) of P 0, both defined over F. In this chapter we work only with standard parabolic subgroups of G, that is, parabolic subgroups P, defined over F, which contain P 0. We shall refer from now on to such...

Let G be a reductive connected linear algebraic group over a field F. Let M be a fixed Levi subgroup of G over F.

The well-known Poisson summation formula applies to a lattice \(\Gamma \) in \(\mathbb{R}\) and a function \(f \in C_{c}^{\infty }(\mathbb{R})\).

Let F be a number field. The results of this chapter also hold, we believe, for the function field of a smooth projective absolutely irreducible curve over a finite field, once the invariant trace formula—established for number fields in Chapters 3–5—is established for such a function field.

Let F u be a local non-Archimedean field, G(F u ) the multiplicative group of a division algebra D u central of rank n over F u , and \(G'(F_{u}) =\mathop{ \mathrm{GL}}\nolimits (n,F_{u})\).

The first example of a trace formula is the classical Poisson summation formula

The number of rank two ℚl-local systems, or ℚl-smooth sheaves, on (X — {u}) ⊗𝔽q 𝔽, where X is a smooth projective absolutely irreducible curve over 𝔽q, 𝔽 an algebraic closure of 𝔽q and u is a closed point of X, with principal unipotent monodromy at u, and fixed by Gal(𝔽/𝔽q), is computed. It is expressed as the trace of the Frob...

Let 𝔽q be a finite field with q elements, 𝔽 = 𝔽̅q an algebraic closure of 𝔽q, X1 an absolutely irreducible projective smooth curve over 𝔽q, and S1 a finite set of closed points of X1. Let N1 be the cardinality of S1. Erasing the index 1 indicates extension of scalars to 𝔽. Replacing it by m indicates extension of scalars to 𝔽qm ⊂ 𝔽. Let Fr be the F...

These are purely expository notes of Opdam’s analysis [O1] of the trace form τ(f) = f(e) on the Hecke algebra H = C c (I\G/I) of compactly supported functions f on a connected reductive split p-adic group G which are biinvariant under an Iwahori subgroup I, extending Macdonald’s work. We attempt to give details of the proofs, and choose notations w...

We write out and prove the trace formula for a convolution operator on the space of cusp forms on GL(2) over the function field F of a smooth projective absolutely irreducible curve over a finite field. The proof - which follows Drinfeld - is complete and all terms in the formula are explicitly computed. The structure of the homogeneous space GL(2,...

Let X-1 be a curve of genus g, projective and smooth over F-q. Let S-1 subset of X-1 be a reduced divisor consisting of N-1 closed points of X-1. Let (X, S) be obtained from (X-1, S-1) by extension of scalars to an algebraic closure F of F-q. Fix a prime l not dividing q. The pullback by the Frobenius endomorphism Fr of X induces a permutation Fr*...

In this chapter we compute the orbital integrals of a certain spherical function, which is introduced in Definition 9.1. We give two methods of computation. That of Prop. 9.9 is natural; it is based on representation theoretic techniques, as presented, e.g., in [BD, BZ, Bo, C, F2, FK1, K1, K2]. That of Prop. 9.12 is elementary. It is due to Drinfel...

The definition in Chap. 3of elliptic modules as A−structures on the additive group \({\mathbb{G}}_{a,K}\) over a field K over A has a natural generalization in which the field K, that is, the scheme SpecK, is replaced by an arbitrary scheme S over A and \({\mathbb{G}}_{a,K}\) is replaced by an invertible (locally free rank one) sheaf \(\mathbb{G}\)...

Let p be a prime number, d a positive integer, q=p d , \({\mathbb{F}}_{q}\) a field of q elements, C an absolutely irreducible smooth projective curve defined over \({\mathbb{F}}_{q}\), and F the function field \({\mathbb{F}}_{q}(C)\) of C over \({\mathbb{F}}_{q}\), that is, the field of rational functions on C over \({\mathbb{F}}_{q}\). At each pl...

In the proof of Theorem 10.8 we use the Grothendieck fixed-point formula of Theorem 6.6, which applies to the cohomology \({H}_{c}^{i}({\overline{X}}_{v}, \mathbb{L}(\rho ))\) of the geometric fiber \({\overline{X}}_{v} = {X}_{v} {\otimes }_{{\mathbb{F}}_{v}}{\overline{\mathbb{F}}}_{v}\) of the special fiber \({X}_{v} = {M}_{r,I} {\otimes }_{A}{\ma...

In this Sect. we summarize some properties of the étale cohomology groups with compact support needed for our study of the action of the Hecke operators and the Galois group on them. This is a rather selective summary, and not a complete exposition. For an introductory textbook to the subject see. The shorter exposition of , Arcata, Rapport, is ver...

We prove a simple form of the converse theorem for GL(n) over a function field F, “simple” referring to a cuspidal component. Thus a generic admissible irreducible representation π of the adèle group GL(\(n, \mathbb{A}\)) with cuspidal components at a finite nonempty set S of places of F whose product L-function L(t, π ×π′) is a polynomial in t and...

We shall now describe each isogeny class in \({M}_{r,v}({\overline{\mathbb{F}}}_{p})\) and the action of the Frobenius on it. The group \(G({\mathbb{A}}_{f})\) acts transitively on the isogeny class, and our task is to find the stabilizer of an element in the class, in order to describe the isogeny class as a homogeneous space.

Let \(F = {\mathbb{F}}_{q}(C)\) be the field of functions on a smooth projective absolutely irreducible curve C over \({\mathbb{F}}_{q}\), \(\mathbb{A}\) its ring of adèles, \(\overline{F}\) a separable algebraic closure of F, G = GL(r), and ∞ a fixed place of F, as in Chap. 2. This section concerns the higher reciprocity law, which parametrizes th...

Definition 2.5 of an elliptic module over a field extension of F ∞ is purely algebraic. So it has a natural generalization defining elliptic modules over any field over A.

The group \(GL(r, {\mathbb{A}}_{f})\) acts (by Prop.4.15) on the moduli scheme \({M}_{r} = Spec{A}_{r} =\mathop{ \lim }_\longleftarrow {M}_{r,I}\) constructed in Theorem 4.10. The central group F × acts trivially. In this section we construct a covering scheme \({\widetilde{M}}_{r}\) of M r for which the action of \(GL(r, {\mathbb{A}}_{f})\) extend...

The purpose of this section is to prove Ramanujan’s conjecture for cuspidal representations π of \(GL(r, \mathbb{A})\) over a function field, which have a cuspidal component, namely that all unramified components of such a π are tempered, namely that all of their Hecke eigenvalues have absolute value one. This is deduced from a form of the trace fo...

The main tool which is applied in Part IV is a comparison of the “arithmetic” fixed point formula with the “analytic” trace formula. To carry out this comparison we need to describe the arithmetic data, which is the cardinality of the set of points on the fiber M r,v at v of the moduli scheme M r , over finite field extensions of \({\mathbb{F}}_{v}...

Let E/F be a quadratic extension of p-adic fields, p ≠ 2. Let $x \mapsto \overline{x}$ be the involution of E over F. The representation π of GL(3, E) normalizedly induced from the trivial representation of the maximal parabolic subgroup is invariant under the involution $\sigma(g) = J{}^t\overline{g}^{-1}J$. We compute — by purely local means — th...

The Saito–Kurokawa lifting of automorphic representations from PGL(2) to the projective symplectic group of similitudes PGSp(4) of genus 2 is studied using the Fourier summation formula (an instance of the "relative trace formula"), thus characterizing the image as the representations with a nonzero period for the special orthogonal group SO(4, E/F...

The tame subgroup It of the Iwahori subgroup I and the tame Hecke algebra Ht = Cc(It\G/It) are introduced. It is shown that the tame algebra has a presentation by means of generators and relations, similar to that of the Iwahori-Hecke algebra H = Cc(I\G/I). From this it is deduced that each of the generators of the tame algebra is invertible. This...

The construction of eigenvarieties due to Chenevier is extended to the Hilbert case, that is, to unitary groups over a totally real field F that are anisotropic at each archimedean place. This permits us to ask about the relationship of the eigenvarieties that we construct for two totally real fields, one being a cyclic extension of the other. Bibl...

A work of Sorensen is rewritten here to include nontrivial types at the infinite places. This extends results of K. Ribet and R. Taylor on level-raising for algebraic modular forms on D^{\times}, where D is a definite quaternion algebra over a totally real field F. This is done for any automorphic representations \pi of an arbitrary reductive group...

This paper has been withdrawn by the author as it has already been submitted under the title "Twisted character of a small Representation of GL(4)".

It is shown that the only possible pole of the twisted tensor L-functions in Re(s)\geq 1 is located at s=1 for all quadratic extensions of global fields.

We provide a purely local computation of the (elliptic) twisted (by "transpose-inverse") character of the representation \pi=I(\1) of PGL(3) over a p-adic field induced from the trivial representation of the maximal parabolic subgroup. This computation is independent of the theory of the symmetric square lifting of [IV] of automorphic and admissibl...

Let F be a number field or a p-adic field of odd residual characteristic. Let E be a quadratic extension of F, and F' an odd degree cyclic field extension of F. We establish a base-change functorial lifting of automorphic (respectively, admissible) representations from the unitary group U(3, E/F) to the unitary group U(3, F' E/F'). As a consequence...

We compute by a purely local method the elliptic, twisted by transpose-inverse, character \chi_\pi of the representation \pi=I_{(3,1)}(1_3) of PGL(4,F) normalizedly induced from the trivial representation of the maximal parabolic subgroup of type (3,1), where F is a p-adic field. Put C=(GL(2,F)xGL(2,F))'/F^x (F^x embeds diagonally, prime means equa...

We compute by a purely local method the (elliptic) θ-twisted character χπY of the representation πY = I(3, 1)(13 × χY) of G = GL(4, F), where F is a p-adic field, p ≠ 2, and Y is an unramified quadratic extension of F; χY is the nontrivial character of F×/NY/FY×. The representation πY is normalizedly induced from $\big(\begin{array}{cc} m_3\quad\as...

Publications on automorphic representations of the group U(3) assumed the validity of multiplicity one theorem since I claimed it in 1982. But the argument, published 1988, was based on a misinterpretation of a claim of Gelbart and Piatetski-Shapiro, SLN 1041 (1984), Prop. 2.4(i): ``L^2_{0,1} has multiplicity 1'', as meaning that each irreducible i...

In this talk, I report my recent joint work with K. Konno on non-tempered automorphic representations on low rank groups (KK). We obtain a fairly complete classification of such automorphic representations for the quasisplit unitary groups in four variables. 1 CAP forms The term CAP in the title is a short hand for the phrase "Cuspidal but Associat...

The area of automorphic representations is a natural continuation of studies in the 19th and 20th centuries on number theory and modular forms. A guiding principle is a reciprocity law relating infinite dimensional automorphic representations with finite dimensional Galois representations. Simple relations on the Galois side reflect deep relations...

We announce results of [F1] on automorphic forms on $\mathrm{SO}(4)$. An initial result is the proof by means of the trace formula that the functorial product of two automorphic representations $\pi_1$ and $\pi_2$ of the adèle group $\mathrm{GL}(2, \mathbf{A}_F)$ whose central characters $\omega_1$, $\omega_2$ satisfy $\omega_1 \omega_2 = 1$, exist...

We announce results of [F1] on automorphic forms on SO(4). An initial result is the proof by means of the trace formula that the functorial product of two automorphic representations π1 and π2 of the adèle group GL(2, AF) whose central characters ω1, ω2 satisfy ω1ω 2 = 1, exists as an automorphic representation π1 Squared times π2 of PGL(4, AF). Th...

The theory of lifting of automorphic and admissible representa- tions is developed in a new case of great classical interest: Siegel automorphic forms. The self-contragredient representations of PGL(4) are determined as lifts of representations of either symplectic PGSp(2) or orthogonal SO(4) rank two split groups. Our approach to the lifting uses...

We provide a purely local computation of the (elliptic) twisted (by “transpose-inverse”) character of the representationπ=I(1) of PGL(3) over ap-adic field induced from the trivial representation of the maximal parabolic subgroup. This computation is independent of the theory of the symmetric square lifting of [IV] of automorphic and admissible rep...

A local-global principle is shown to hold for all conjugacy classes of any inner form of GL(n), SL(n), U(n), SU(n), and for all semisimple conjugacy classes in any inner form of Sp(n), over fieldsk with vcd(k)≤1. Over number fields such a principle is known to hold for any inner form of GL(n) and U(n), and for the split forms of Sp(n), O(n), as wel...

The torsor P_s=Hom(H_{\DR},H_s) under the motivic Galois group G_s=Aut H_s of the Tannakian category M_k generated by one-motives related by absolute Hodge cycles over a field k with an embedding s into the complex numbers is shown to be determined by its global projection [P_s\to (P_s)/(G_s)^0] to a Gal(\ov k/k)-torsor, and by its localizations (P...

This paper starts by introducing a bi-periodic summation formula for automorphic forms on a group G ( E ), with periods by a subgroup G ( F ), where E / F is a quadratic extension of number fields. The split case, where E = F ⊕ F , is that of the standard trace formula. Then it introduces a notion of stable bi-conjugacy, and stabilizes the geometri...

F13.97> Fr(g) = (Fr(g i;j )), ~ Fr(g) = t Fr(g) Gamma1 , where t signifies transpose, and ~ Fr w (g) = w ~ Fr(g)w. Put ~ G 0 = G 0 Theta F 0 Theta . The unitary group U = U(F ) is fg 2 G 0 ; g = ~ Fr w (g)g, and the group ~ U = ~ U(F ) of unitary similitudes is f(g; ) 2 ~ G 0 ; (g; ) = ~ Fr w (g; )g, where ~ Fr w (g; ) = ( ~ Fr w

Matching of (twisted) orbital integrals of corresponding spherical functions on a reductive p-adic group G and its (twisted) endoscopic group H is a prerequisite to lifting representations from H to G by means of a comparison of trace formulae. Kottwitz-Shelstad [KS] conjectured the precise form that; the matching takes. This matching statement is...

"January 1999, volume 137, number 655 (fourth of 6 numbers)" Incluye bibliografía

. A theorem of Grothendieck asserts that over a perfect field k of cohomological dimension one, all non-abelian H 2 -cohomology sets of algebraic groups are trivial. The purpose of this paper is to establish a formally real generalization of this theorem. The generalization --- to the context of perfect fields of virtual cohomological dimension one...

. The fundamental lemma in the theory of automorphic forms is proven for the (quasi-split) unitary group U(3) in three variables associated with a quadratic extension of p-adic fields, and its endoscopic group U(2), by means of a new, elementary technique. This lemma is a prerequisite for an application of the trace formula to classify the automorp...

A theorem of Grothendieck asserts that over a perfect field k of cohomological dimension one, all non-abelian H^2-cohomology sets of algebraic groups are trivial. The purpose of this paper is to establish a formally real generalization of this theorem. The generalization -- to the context of perfect fields of virtual cohomological dimension one --...

this paper is to compare the notion of being GL(2; A )-distinguished with the notion (defined below) of being distinguished with respect to another subgroup of GL(2; A E ). Using a "relative trace formula", Jacquet and Lai [JL] carried out such comparisons in certain cases. To extend their results, one could either develop an extensive theory of or...

for Zentralblatt. Let E=F be a (separable) quadratic extension of global fields with charF 6= 2, and A X ; A Theta X the X-adeles, X-ideles. Put G = GL(2). A cuspidal G(A E )-module ß 0 is called distinguished if there is a form OE 2 ß 0 with R G(F )Z(A F )nG(A F ) OE(h)dh 6= 0: These ß 0 are characterized in the author's paper "Twisted tensors and...

E v is a field. Under our assumption that D is unsplit by E, the algebra D E is division, central over E of degree d = deg D. Let H be a simple algebra of degree 2 central over F . The multiplicative group G of Mm (D H ), where D H = D Theta F H, is an algebraic F -group, which is an inner form of the split group G sp = GL(2n)=F , n = md. Put G = G...

Introduction. Langlands' principle of functoriality [B] conjectures that there is a parametrization of the set RepF (G) of admissible [BZ] or automorphic [BJ] representations of a reductive group G over a local or global field F , by admissible homomorphisms ae : WF ! GoWF . Here WF is a form of the Weil group [T] of F , and G is the connected (com...

. A twisted analogue of Kazhdan's decomposition of compact elements into a commuting product of topologically unipotent and absolutely semi-simple elements, is developed and used to give a direct and elementary proof of the Langlands' fundamental lemma for the symmetric square lifting from SL(2) to PGL(3) and the unit element of the Hecke algebra....

F72.42> , where G # is the semi-direct product Go ! oe ? of G with the group ! oe ? generated by oe . When ß is irreducible then S is uniquely determined up to an ` th root i of unity in C . Let M (G) be the category of G -modules. An element E of M (G) is called finitely generated if for any filtered system of proper subobjects E i in E , the subo...

. In the first Section of this paper we obtain an asymptotic expansion near semi simple elements, of orbital integrals ¯ ~ x ( ~ f) of C 1 c -functions ~ f on symmetric spaces G=H. Here G is a reductive p-adic group, and H is the group of fixed points of an involution oe on G. This extends the germ expansion of Shalika [Sh] and Vigneras [V] in the...

It is shown that the only possible pole of the twisted tensor L-functions in Re(s) 1 is located at s = 1 for all quadratic extensions of global fields. 0. Introduction. Let E be a quadratic separable field extension of a global field F . Denote by AE , A F the corresponding rings of adeles. Put G n for GL n and Z n for its center. Then Z n (AE ) is...

The notion of a period of a cusp form on GL(2,D()), with respect to the diagonal subgroup D()X × D()X, is defined. Here D is a simple algebra over a global field F with a ring of adeles. For Dx = GL(1), the period is the value at 1/2 of the L-function of the cusp form on GL(2, ). A cuspidal representation is called cyclic if it contains a cusp form...

this paper. At the archimedean places the L-factors are the associated L-factors of the representation of the Weil groups which parametrize ß v (and so r(ß v

Ideas underlying the proof of the "simple" trace formula are used to show the following. Let F be a global field, and A its ring of adeles. Let π be a cuspidal representation of GL(n, A) which has a supercuspidal component, and ω a unitary character of Ax/Fx. Let S0 be a complex number such that for every separable extension E of F of degree n, the...

In this paper we prove the existence of the symmetric-square lifting of admissible and of automorphic representations from the group SL(2) to the group PGL(3). Complete local results are obtained, relating the character of an SL(2)-packet with the twisted character of self-contragredient PGL(3)-modules. Our global results relate packets of cuspidal...

We define the symmetric square lifting for admissible and automorphic representations, from the group H = H0 = SL(2), to the group G = PGL(3), and derive its basic properties. This lifting is defined by means of Shintani character relations. The definition is suggested by the computation of orbital integrals (stable and unstable) in our On the symm...

We define the symmetric square lifting for admissible and automorphic representations, from the group H = H0 = SL(2), to the group G = PGL(3), and derive its basic properties. This lifting is defined by means of Shintani character relations.The definition is suggested by the computation of orbital integrals (stable and unstable) in our On the symme...

We compute a trace formula—for a test function on PGL(3,A)— twisted by the outer automorphism The resulting formula is then compared with trace formulae for H = H00 = SL(2) and H1 = PGL(2), and matching functions f0 and f1 thereof. We obtain a trace formula identity which plays a key role in our study of the symmetric square lifting from H(A) to G(...

We develop Drinfeld's theory of elliptic modules and their moduli schemes to establish the correspondence of irreducible Galois representations and cuspidal automorphic representations – of GL(r) over a function field – which have a cuspidal local component, on realizing it in thé etale cohomology with compact support of the geometric fiber of the...

> [H] of ?0u , and ? ?u that of ? u .By virtue of [K2], it suffices to prove this for F u of characteristic zero. In fact, all of ourarguments hold also in the positive characteristic case, except for the reference [K1] to theorthonormality relations for characters used in the proof of Proposition 5. These relationsare known to follow once the loca...

This chapter provides an overview of the geometric Ramanujan conjecture and Drinfeld reciprocity law. The proof of the purity conjecture and the deduction of the Drinfeld explicit reciprocity law from Deligne's conjecture are based on a new form of the Selberg trace formula for a test function with at least one supercusp component. This new trace f...

The Selberg trace formula is of unquestionable value for the study of automorphic forms and related objects. In principal it is a simple and natural formula, generalizing the Poisson summation formula, relating traces of convolution operators with orbital integrals. This paper is motivated by the belief that such a fundamental and natural relation...

We prove the fundamental lemma for spherical functions with respect to the natural (induction) lifting fromPGL(2) toPGL(3) which appears as the unstable counterpart of the stable symmetric-square lifting fromSL(2) toPGL(3) (see [IV] for an introduction to this project, and [VI] for the final results). Thus spherical functions onPGL(2) andPGL(3) whi...

Soit L(s,r(π),V)=Π v der[1−q v − sr(t(π v ))] −1 (v¬∈V) le produit eulerien attache a une representation cuspidale (irreductible, automorphe, unitaire) Π du groupe adelique GL(n,A E ), ou E/F est une extension quadratique de corps globaux et r est la representation «tenseur tordue» de Ĝ=[GL(n,C)×GL(n,C)]×Gal(E/F) sur C n ⊗C n . La fonction L(s,rcπ)...