Moduli of metaplectic bundles on curves and Theta-sheaves (original) (raw)

A Note on the Automorphic Langlands Group

Canadian Mathematical Bulletin, 2002

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.

On Langlands program, related representation and GGG-shtukas

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 ,theLanglandscorrespondencerelatesautomorphicformson, the Langlands correspondence relates automorphic forms on ,theLanglandscorrespondencerelatesautomorphicformson 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...

The Tannakian formalism and the Langlands conjectures

Algebra & Number Theory, 2014

Let H be a connected reductive group over an algebraically closed field of characteristic zero, and let Γ be an abstract group. In this note we show that every homomorphism of Grothendieck semirings φ : K + 0 [Γ] → K + 0 [H], which maps irreducible representations to irreducible, comes from a group homomorphism ρ : Γ → H(K). We also connect this result with the Langlands conjectures.

On Weyl modules for the symplectic group

A rich information can be found in the literature on Weyl modules for Sp(2n, F), but the most important contributions to this topic mainly enlighten the algebraic side of the matter. In this paper we try a more geometric approach. In particular, our approach enables us to obtain a sufficient condition for a module as above to be uniserial and a geometric description of its composition series when our condition is satisfied. Our result can be applied to a number of cases. For instance, it implies that the module hosting the Grassmann embedding of the dual polar space associated to Sp(2n, F) is uniserial.

The Langlands parameter of a simple supercuspidal representation: symplectic groups

The Ramanujan Journal

Let π be a simple supercuspidal representation of the symplectic group Sp 2l (F), over a p-adic field F. In this work, we explicitly compute the Rankin-Selberg γ-factor of rank-1 twists of π. We then completely determine the Langlands parameter of π, if p = 2. In the case that F = Q 2 , we give a conjectural description of the functorial lift of π, with which, using a recent work of Bushnell and Henniart, one can obtain its Langlands parameter.

Kloosterman sheaves for reductive groups

Annals of Mathematics, 2013

Deligne constructed a remarkable local system on È 1 − {0, ∞} attached to a family of Kloosterman sums. Katz calculated its monodromy and asked whether there are Kloosterman sheaves for general reductive groups and which automorphic forms should be attached to these local systems under the Langlands correspondence.

Arithmetic invariants of discrete Langlands parameters

Duke Mathematical Journal, 2010

For general groups, one invariant of a discrete series representation π is its formal degree. Recently, Hiraga, Ichino, and Ikeda have formulated a conjecture for the formal degree of a discrete series representation, in terms of the L-function and ε-factor of the adjoint representation of the L-group.