We express the cohomology of the basic Newton stratum of some unitary Shimura varieties associated to division algebras in terms of automorphic representations of the group in the Shimura datum. 2010 Mathematics Subject Classification: Primary 11G18 ; Secondary 11F70, 11F72.

We prove a natural analogue of the Sato-Tate conjecture for modular forms of weight 2 or 3 whose associated automorphic representations are a twist of the Steinberg representation at some finite place. 2000 Mathematics Subject Classification: 11F33.

We prove a natural analogue of the Sato-Tate conjecture for modular forms of weight 2 or 3 whose associated automorphic representations are a twist of the Steinberg representation at some finite place.

The “Galois representations” of the title are modular representations ρ of the Galois groups of a number field F , and the ”automorphic realization” of the title refers to obtaining these representations as constituents of Galois representations attached to automorphic representations of general linear groups over F . The present article refines the moduli-theoretic arguments of [HST] to show t...

We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor–Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology. In practice, this imposes several restrictions – one must be in a Shimura variety settin...

We report briefly on an endoscopic classification of representations by focusing on one aspect of the problem, the question of embedded Hecke eigenvalues. 1. The problem for G By “eigenvalue”, we mean the family of unramified Hecke eigenvalues of an automorphic representation. The question is whether there are any eigenvalues for the discrete spectrum that are also eigenvalues for the continuou...

Conjectures concerning the relations between motives and automorphic representations of GL(n) over number elds have been made by Clozel C1]. They make precise part of the programme indicated by Langlands in La]. Once a motive is attached to an automorphic representation, one can derive a compatible series of-adic representations of Gal(Q=Q) attached to it in the usual way. In C2], Clozel has gi...

In the first part of this paper we study minimal representations of simply connected simple split groups G of type Dk or Ek over local nonarchimedian fields. Our main result is an explicit formula for the spherical vectors in these representations. In the case of groups over R and C, such a formula was obtained recently in [8]. We also use our techniques to study the structure of the space of s...

Let G be a connected, real, semisimple Lie group contained in its complexification GC, and let K be a maximal compact subgroup of G. We construct a KC-G double coset domain in GC, and we show that the action of G on the K-finite vectors of any irreducible unitary representation of G has a holomorphic extension to this domain. For the resultant holomorphic extension of K-finite matrix coefficien...

This is an attempt to formulate a geometric Langlands conjecture for G = GSpin2n+1 and GSp2n, n ≥ 1. Namely, let Ǧ be the Langlands dual group (over Q̄l), let X be a smooth projective connected curve. Given a Ǧ-local system E on X , assuming some irreducibility type conditions on VE for some representations V of Ǧ, we propose a conjectural construction of a distinguished E-Hecke automorphic shea...