A Note on the Automorphic Langlands Group

Journal of the Institute of Mathematics of Jussieu, 2012

We prove new automorphy lifting theorems for essentially conjugate self-dual Galois representations into GLn. Existing theorems require that the residual representation have ‘big’ image, in a certain technical sense. Our theorems are based on a strengthening of the Taylor–Wiles method which allows one to weaken this hypothesis.

Inventiones Mathematicae, 1994

The Langlands philosophy contemplates the relation between automorphic representations and Galois representations. A particularly interesting case is that of the non-selfdual automorphic representations of GL 3 . Clozel conjectured that the L-functions of certain of these are equal to L-functions of Galois representations.

Japanese Journal of Mathematics, 2020

The functoriality conjecture of Langlands, as stated in [33], offers a general organizational scheme for all automorphic representations. This conjecture can be roughly stated as follows: if H and G are reductive groups defined over a global field k, ξ : L H → L G a homomorphism between their Langlands dual groups, then it is possible to attach to each automorphic representation π H = ⊗ v π H v of H a packet of automorphic representations π = ⊗ v π v of G. In the formula π = ⊗ v π v , local components π v are irreducible admissible representations of G(k v), k v being the completion of k at a place v ∈ |k|. The automorphic functoriality is expected to be of local nature in the sense that it should be compatible with the local functoriality principle that consists in a map from the set of packets of irreducible admissible representations of H(k v) to the set of packets of irreducible admissible representations of G(k v). Since unramified representations of H(k v) at an unramified nonarchimedean place v are parametrized by certain semi-simple conjugacy classes in L H, the local functoriality is required to be compatible with the transfer of conjugacy classes from L H to L G. A crucial part of the functoriality program, known as endoscopy, is now almost completely understood thanks to cumulative efforts of many mathematicians over the last 40 years, see [5]. Its main ingredients include in particular the construction of the general trace formula, mainly due to Arthur [4] and the proof of the transfer conjecture and the fundamental lemma, see [54] and [39]. The endoscopic case is concerned with the case where L H is essentially the centralizer of a semi-simple element in L G. In spite of this severe restriction, the endoscopic case is crucial in the general picture for it gives a sense to the concept of packet, both in local and automorphic settings, along with global multiplicity formulae [31]. The endoscopic program also produced the stable trace formula expected to be an essential tool for future works on the conjecture of functoriality. In [30], Langlands proposed a conjectural limiting form of the stable trace formula aiming at isolating the part of the automorphic spectrum of G that comes from a smaller group H by functoriality. These limiting forms of trace formulae are designed to have a spectral expansion involving logarithmic

Journal of the European Mathematical Society, 2000

Mathematical Research Letters, 1995

Twenty-five years ago R. Langlands proposed [L] a "fantastic generalization" of Artin-Hasse reciprocity law in the classical class field theory. He conjectured the existence of a correspondence between automorphic irreducible infinite-dimensional representations of a reductive group G over a global number field on the one hand, and (roughly speaking) finite dimensional representations of the Galois group of the field, on the other hand. Following an earlier idea of A. Weil, Langlands' conjecture was reinterpreted by V. Drinfeld [Dr 3] and G. Laumon [La] in purely geometric terms. In the complex geometry setup the number field gets replaced by a Riemann surface X, the Galois group of the field gets replaced by the fundamental group, π 1 (X), and an automorphic representation gets replaced by a perverse sheaf, cf. [BBD], on the moduli space of algebraic principal G-bundles on X. The purpose of this note is to formulate some results and conjectures related to Langlands reciprocity for algebraic surfaces, as opposed to algebraic curves. That would amount, in number theory, to a non-abelian higher dimensional generalization of the class field theory, the subject that remains a total mystery at the moment. A key step of our geometric approach is a construction of Hecke operators for vector bundles on an algebraic surface. The main point of this paper is that in certain cases the corresponding algebra of Hecke operators turns out to be a homomorphic image of the quantum toroidal algebra. The latter is a quantization, in the spirit of Drinfeld [Dr 1] and Jimbo [Ji], of the universal enveloping algebra of the universal central extension of a "double-loop" Lie algebra. This yields, in particular, a new geometric construction of affine quantum groups of types A (1) , D (1) and E (1) in terms of Hecke operators for an elliptic surface.

In this article, we first define under Langlands' functoriality conjecture a rather general class M of L-functions which are both automorphic and elements of the Selberg class. Then we define the automorphism group Aut(M) of the considered class of L-functions, and derive from it and field automorphisms of C the automorphism group of any element of M. Finally, we give a new approach to the Riemann Hypothesis through Hadamard-Weierstrass factorization theorem, whose very meaning is that an L-function and its multiset of non-trivial zeroes are actually the same object. AMS MSC 2010 : 11M26

Annali della Scuola normale superiore di Pisa. Classe di scienze, 2020

By making use of Langlands functoriality between GSp(4) and GL(4), we show that the images of the Galois representations attached to "genuine" globally generic automorphic representations of GSp(4) are "large" for a set of primes of density one. Moreover, by using the notion of (n, p)-groups (introduced by Khare, Larsen and Savin) and generic Langlands functoriality from SO(5) to GL(4) we construct automorphic representations of GSp(4) such that the compatible system attached to them has large image for all primes. 2010 Mathematics Subject Classification. 11F80, 11F12, 11F46.

We prove that a strengthened form of the local Langlands conjecture is valid throughout the principal series of any connected split reductive $p$-adic group. The method of proof is to establish the presence of a very simple geometric structure, in both the smooth dual and the Langlands parameters. We prove that this geometric structure is present, in the same way, for the general linear group, including all of its inner forms. With these results as evidence, we give a detailed formulation of a general geometric structure conjecture.

arXiv: Number Theory, 2020

This communication is an introduction to the Langlands Program and to ($G$-) shtukas (over algebraic curves) over function fields. Modular curves and Drinfeld (elliptic) modules and shtukas are used in coding theory. From this point of view the communication is concerned with mathematical models and methods of coding theory. For a connected reductive group $ G $ over a global field $ K $, the Langlands correspondence relates automorphic forms on $ G $ and global Langlands parameters, i.e. conjugacy classes of homomorphisms from the Galois group $ {\mathcal Gal} ({\overline K} / K) $ to the dual Langlands group $ \hat G ({\overline {\mathbb Q}}_p) $. In the case of fields of algebraic numbers, the application and development of elements of the Langlands program made it possible to strengthen the Wiles theorem on the Shimura-Taniyama-Weil hypothesis and to prove the Sato-Tate hypothesis. In this review article, we first present results on Langlands program and related representation o...

arXiv (Cornell University), 2020

The purpose of this paper is to survey some of the important results on Langlands program, global fields, D-shtukas and finite shtukas which have influenced the development of algebra and number theory. It is intended to be selective rather than exhaustive, as befits the occasion of the 80-th birthday of Yakovlev, 75-th birthday of Vostokov and 75-th birthday of Lurie. Under assumptions on ground fields results on Langlands program have been proved and discussed by Langlands, Jacquet, Shafarevich, Parshin, Drinfeld, Lafforgue and others. This communication is an introduction to the Langlands Program, global fields and to D-shtukas and finite shtukas (over algebraic curves) over function fields. At first recall that linear algebraic groups found important applications in the Langlands program. Namely, for a connected reductive group G over a global field K, the Langlands correspondence relates automorphic forms on G and global Langlands parameters, i.e. conjugacy classes of homomorphisms from the Galois group Gal(K/K) to the dual Langlands groupĜ(Q p). In the case of fields of algebraic numbers, the application and development of elements of the Langlands program made it possible to strengthen the Wiles theorem on the Shimura-Taniyama-Weil hypothesis and to prove the Sato-Tate hypothesis. V. Drinfeld and L. Lafforgue have investigated the case of functional global fields of characteristic p > 0 (V. Drinfeld for G = GL 2 and L. Lafforgue for G = GL r , r is an arbitrary positive integer). They have proved in these cases the Langlands correspondence. Under the process of these investigations, V. Drinfeld introduced the concept of a F-bundle, or shtuka, which was used by both authors in the proof for functional global fields of characteristic p > 0 of the studied cases of the existence of the Langlands correspondence. Along with the use of shtukas developed and used by V. Drinfeld and L. Lafforge, other constructions related to approaches to the Langlands program in the functional case were introduced. G. Anderson has introduced the concept of a t-motive. U. Hartl, his colleagues and students have introduced and have explored the concepts of finite, local and global G-shtukas. In this review article, we first present results on Langlands program and related representation over algebraic number fields. Then we briefly present approaches by U. Hartl, his colleagues and students to the study of D-shtukas and finite shtukas. These approaches and our discussion relate to the Langlands program as well as to the internal development of the theory of G-shtukas.