Let $G$ be a split semisimple group over a global function field $K$. Given a cuspidal automorphic representation $\Pi$ of $G$ satisfying a technical hypothesis, we prove that for almost all primes $\ell$, there is a cyclic base change lifting of $\Pi$ along any $\mathbb {Z}/\ell \mathbb {Z}$-extension of $K$. Our proof does not rely on any trace formulas; instead it is based on using modularity lifting theorems, together with a Smith theory argument, to obtain base change for residual representations. As an application, we also prove that for any split semisimple group $G$ over a local function field $F$, and almost all primes $\ell$, any irreducible admissible representation of $G(F)$ admits a base change along any $\mathbb {Z}/\ell \mathbb {Z}$-extension of $F$. Finally, we characterize local base change more explicitly for a class of toral representations considered in work of Chan and Oi.

We prove the compatibility of local and global Langlands correspondences for $\operatorname {GL}_n$ up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne [10] and Scholze [18]. More precisely, let $r_p(\pi )$ denote an n-dimensional p-adic representation of the Galois group of a CM field F attached to a regular algebraic cuspidal automorphic representation $\pi $ of $\operatorname {GL}_n(\mathbb {A}_F)$. We show that the restriction of $r_p(\pi )$ to the decomposition group of a place $v\nmid p$ of F corresponds up to semisimplification to $\operatorname {rec}(\pi _v)$, the image of $\pi _v$ under the local Langlands correspondence. Furthermore, we can show that the monodromy of the associated Weil-Deligne representation of $\left .r_p(\pi )\right |{}_{\operatorname {Gal}_{F_v}}$ is ‘more nilpotent’ than the monodromy of $\operatorname {rec}(\pi _v)$.

We prove the existence of $\mathrm {GSpin}_{2n}$-valued Galois representations corresponding to cohomological cuspidal automorphic representations of certain quasi-split forms of ${\mathrm {GSO}}_{2n}$ under the local hypotheses that there is a Steinberg component and that the archimedean parameters are regular for the standard representation. This is based on the cohomology of Shimura varieties of abelian type, of type $D^{\mathbb {H}}$, arising from forms of ${\mathrm {GSO}}_{2n}$. As an application, under similar hypotheses, we compute automorphic multiplicities, prove meromorphic continuation of (half) spin L-functions and improve on the construction of ${\mathrm {SO}}_{2n}$-valued Galois representations by removing the outer automorphism ambiguity.

We show that the completed Hecke algebra of $p$-adic modular forms is isomorphic to the completed Hecke algebra of continuous $p$-adic automorphic forms for the units of the quaternion algebra ramified at $p$ and $\infty$. This gives an affirmative answer to a question posed by Serre in a 1987 letter to Tate. The proof is geometric, and lifts a mod $p$ argument due to Serre: we evaluate modular forms by identifying a quaternionic double-coset with a fiber of the Hodge–Tate period map, and extend functions off of the double-coset using fake Hasse invariants. In particular, this gives a new proof, independent of the classical Jacquet–Langlands correspondence, that Galois representations can be attached to classical and $p$-adic quaternionic eigenforms.

In this article we explain how the results in our previous article on ‘algebraic Hecke characters and compatible systems of mod $p$ Galois representations over global fields’ allow one to attach a Hecke character to every cuspidal Drinfeld modular eigenform from its associated crystal that was constructed in earlier work of the author. On the technical side, we prove along the way a number of results on endomorphism rings of ${\it\tau}$-sheaves and crystals. These are needed to exhibit the close relation between Hecke operators as endomorphisms of crystals on the one side and Frobenius automorphisms acting on étale sheaves associated to crystals on the other. We also present some partial results on the ramification of Hecke characters associated to Drinfeld modular eigenforms. An important phenomenon absent from the case of classical modular forms is that ramification can also result from places of modular curves of good but non-ordinary reduction. In an appendix, jointly with Centeleghe we prove some basic results on $p$-adic Galois representations attached to $\text{GL}_{2}$-type cuspidal automorphic forms over global fields of characteristic $p$.

This paper proves two results on the field of rationality $\mathbb{Q}({\it\pi})$ for an automorphic representation ${\it\pi}$, which is the subfield of $\mathbb{C}$ fixed under the subgroup of $\text{Aut}(\mathbb{C})$ stabilizing the isomorphism class of the finite part of ${\it\pi}$. For general linear groups and classical groups, our first main result is the finiteness of the set of discrete automorphic representations ${\it\pi}$ such that ${\it\pi}$ is unramified away from a fixed finite set of places, ${\it\pi}_{\infty }$ has a fixed infinitesimal character, and $[\mathbb{Q}({\it\pi}):\mathbb{Q}]$ is bounded. The second main result is that for classical groups, $[\mathbb{Q}({\it\pi}):\mathbb{Q}]$ grows to infinity in a family of automorphic representations in level aspect whose infinite components are discrete series in a fixed $L$-packet under mild conditions.

Cet article est une contribution à la fois au calcul du nombre de fibrés de Hitchin sur une courbe projective et à l’explicitation de la partie nilpotente de la formule des traces d’Arthur-Selberg pour une fonction test très simple. Le lien entre les deux questions a été établi dans [Chaudouard, Sur le comptage des fibrés de Hitchin. À paraître aux actes de la conférence en l’honneur de Gérard Laumon]. On décompose cette partie nilpotente en une somme d’intégrales adéliques indexées par les orbites nilpotentes. Pour les orbites de type «régulières par blocs», on explicite complètement ces intégrales en termes de la fonction zêta de la courbe.

In this paper we generalize the work of Harris–Soudry–Taylor and construct the compatible systems of two-dimensional Galois representations attached to cuspidal automorphic representations of cohomological type on ${\rm GL}_2$ over a CM field with a suitable condition on their central characters. We also prove a local-global compatibility statement, up to semi-simplification.

We give an explicit construction of the cusp eigenforms on an elliptic curve defined over a finite field, using the theory of Hall algebras and the Langlands correspondence for function fields and ${\mathrm{GL} }_{n} $. As a consequence we obtain a description of the Hall algebra of an elliptic curve as an infinite tensor product of simpler algebras. We prove that all these algebras are specializations of a universal spherical Hall algebra (as defined and studied by Burban and Schiffmann  [On the Hall algebra of an elliptic curve I, Preprint (2005), arXiv:math/0505148 [math.AG]] and Schiffmann and Vasserot [The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials, Compositio Math. 147 (2011), 188–234]).

We prove modularity lifting theorems for ℓ-adic Galois representations of any dimension satisfying a unitary type condition and a Fontaine–Laffaille condition at ℓ. This extends the results of Clozel, Harris and Taylor, and the subsequent work by Taylor. The proof uses the Taylor–Wiles method, as improved by Diamond, Fujiwara, Kisin and Taylor, applied to Hecke algebras of unitary groups, and results of Labesse on stable base change and descent from unitary groups to GLn.

The local Langlands conjectures imply that to every generic supercuspidal irreducible representation of G2 over a p-adic field, one can associate a generic supercuspidal irreducible representation of either PGSp6 or PGL3. We prove this conjectural dichotomy, demonstrating a precise correspondence between certain representations of G2 and other representations of PGSp6 and PGL3. This correspondence arises from theta correspondences in E6 and E7, analysis of Shalika functionals, and spin L-functions. Our main result reduces the conjectural Langlands parameterization of generic supercuspidal irreducible representations of G2 to a single conjecture about the parameterization for PGSp 6.

This work is the geometric part of our proof of the weighted fundamental lemma, which is an extension of Ngô Bao Châu’s proof of the Langlands–Shelstad fundamental lemma. Ngô’s approach is based on a study of the elliptic part of the Hichin fibration. The total space of this fibration is the algebraic stack of Hitchin bundles and its base space is the affine space of ‘characteristic polynomials’. Over the elliptic set, the Hitchin fibration is proper and the number of points of its fibers over a finite field can be expressed in terms of orbital integrals. In this paper, we study the Hitchin fibration over an open set larger than the elliptic set, namely the ‘generically regular semi-simple set’. The fibers are in general neither of finite type nor separated. By analogy with Arthur’s truncation, we introduce the substack of ξ-stable Hitchin bundles. We show that it is a Deligne–Mumford stack, smooth over the base field and proper over the base space of ‘characteristic polynomials’. Moreover, the number of points of the ξ-stable fibers over a finite field can be expressed as a sum of weighted orbital integrals, which appear in the Arthur–Selberg traceformula.

In Boyer [Monodromy of perverse sheaves on vanishing cycles on some Shimura varieties, Invent. Math. 177 (2009), 239–280 (in French)], a sheaf version of the monodromy-weight conjecture for some unitary Shimura varieties was proved by giving explicitly the monodromy filtration of the complex of vanishing cycles in terms of local systems introduced in Harris and Taylor [The geometry and cohomology of some simple Shimura varieties (Princeton University Press, Princeton, NJ, 2001)]. The main result of this paper is the cohomological version of the monodromy-weight conjecture for these Shimura varieties, which we prove by means of an explicit description of the groups of cohomology in terms of automorphic representations and the local Langlands correspondence.

Let X be a smooth projective curve. We consider the dual reductive pair , over X, where H splits on an étale two-sheeted covering . Let BunG (respectively, BunH) be the stack of G-torsors (respectively, H-torsors) on X. We study the functors FG and FH between the derived categories D(BunG) and D(BunH), which are analogs of the classical theta-lifting operators in the framework of the geometric Langlands program. Assume n=m=1 and H nonsplit, that is, with connected. We establish the geometric Langlands functoriality for this pair. Namely, we show that FG :D(BunH)→D(BunG) commutes with Hecke operators with respect to the corresponding map of Langlands L-groups LH→LG. As an application, we calculate Waldspurger periods of cuspidal automorphic sheaves on BunGL2 and Bessel periods of theta-lifts from to . Based on these calculations, we give three conjectural constructions of certain automorphic sheaves on (one of them makes sense for -modules only).

Let $F$ be a $p$-adic local field and let $A_{i}^{\times }$ be the unit group of a central simple $F$-algebra ${{A}_{i}}$ of reduced degree $n\,>\,1\,(i\,=\,1,\,2)$. Let ${{\mathcal{R}}^{2}}\left( A_{i}^{\times } \right)$ denote the set of irreducible discrete series representations of $A_{i}^{\times }$. The “Abstract Matching Theorem” asserts the existence of a bijection, the “Jacquet-Langlands” map, $J{{L}_{{{A}_{2}},{{A}_{1}}}}\,:\,{{R}^{2}}\left( A_{1}^{\times } \right)\,\to \,{{R}^{2}}\left( A_{2}^{\times } \right)$ which, up to known sign, preserves character values for regular elliptic elements. This paper addresses the question of explicitly describing the map 𝒥ℒ, but only for “level zero” representations. We prove that the restriction $J{{L}_{{{A}_{2}},{{A}_{1}}}}\,:\,R_{0}^{2}\left( A_{1}^{\times } \right)\,\to \,R_{0}^{2}\left( A_{2}^{\times } \right)$ is a bijection of level zero discrete series (Proposition 3.2) and we give a parameterization of the set of unramified twist classes of level zero discrete series which does not depend upon the algebra ${{A}_{i}}$ and is invariant under $J{{L}_{{{A}_{2}},{{A}_{1}}}}$ (Theorem 4.1).

Langlands has conjectured the existence of a universal group, an extension of the absolute Galois group, which would play a fundamental role in the classification of automorphic representations. We shall describe a possible candidate for this group. We shall also describe a possible candidate for the complexification of Grothendieck's motivic Galois group.

In this paper, we study the restriction of irreducible representations of the group SO2n+1 × SO2m to a spherical subgroup.

The object of this paper is to prove certain p-adic orbital integral identities needed in order to accomplish the symmetric square transfer via the twisted Arthur trace formula. Only §5 of this article contains original material, the rest of it is due to R. Langlands. Very briefly, we reduce the problem of proving certain orbital integral identities for “matching” functions in the respective Hecke algebras to two counting problems on the buildings. We give Langlands’ solution of one of these problems in the case of the unit elements of the respective Hecke algebras and §5 provides the solution to the other one, again, in the unit element case. The main results assume p ≠ 2.