Shimura varieties are arithmetic quotients of locally symmetric spaces which are canonically defined over number fields. In this article, we discuss recent developments on the reciprocity law realized on cohomology groups of Shimura varieties which relate Galois representations and automorphic representations. Focus is put on the control of -adic families of Galois representations by -adic fami...

We state conjectures on the relationships between automorphic representations and Galois representations, and give evidence for them.

Rough exposition of the goal of this course. Hecke actions on Shimura varieties and their cohomology groups. The Matsushima formula in the classical setting; relative Lie algebra cohomology, spectral decomposition on L(Γ\G). The Matsushima formula in the adelic setting — automorphic representations appear in the Betti cohomology of Shimura varieties. Admissible (g, U)-modules and cohomological ...

In these notes, we shall attempt to make sense of the notions of semisimple and unipotent representations in the context of automorphic forms. Our goal is to formulate some conjectures, both local and global, which were originally motivated by the trace formula. Some of these conjectures were stated less generally in lectures [2] at the University of Maryland. The present paper is an update of ...

Abstract We propose a new method to construct rigid G -automorphic representations and $${\widehat{G}}$$ G ^ -local systems for reductive groups . The construction involves the notion of euphotic , proof rigidity geometry certain Hessenberg varieties.

We present a new method to estimate trilinear period for automorphic representations of SL2(R). The method is based on the uniqueness principle in representation theory. We show how to separate the exponentially decaying factor in the triple period from the essential automorphic factor which behaves polynomially. We also describe a general method which gives an estimate on the average of the au...

The method of L functions is one of the major methods for analyzing automorphic forms. For example, the Hecke Converse Theorem gives an equivalence via the Mellin transform between holomorphic modular forms on the upper half plane and certain L functions associated to Dirichlet series, which have analytic continuation and functional equation. The classical theory of automorphic forms on the gro...

We state a qualitative form of strong multiplicity one for GL1. We derive refinements of strong multiplicity one for automorphic representations arising from Eisenstein series associated to a Borel subgroup on GL(n), and for the cuspidal representations on GL(n) induced from idele class characters of cyclic extensions of prime degree. These results are in accordance with a conjecture of D. Rama...