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...

As automorphic L-functions or Artin L-functions, several classes of Lfunctions have Euler products and functional equations. In this paper we study the zeros of L-functions which have the Euler products and functional equations. We show that there exists some relation between the zeros of the Riemann zetafunction and the zeros of such L-functions. As a special case of our results, we find the r...

In previous papers [13], [2], [16], [3], [ l l ] , the theory of automorphic functions for some classical discontinuous groups F, such as the Siegel or Hilbert-Siegel modular groups, acting on certain bounded symmetric domains X, has been developed through the construction of a natural compactification of X/T , which is a normal analytic space, projectively embeddable by means of automorphic fo...

Euler equations for rotationally symmetrical motions of ideal fluid are considered. Basis of differential invariants for infinite-dimensional part of admitted group is calculated. The basis is used for construction of group foliation of Euler equations. Both automorphic and resolving systems are completed to involution. The resolving part of group foliation inherits finite-dimensional part of g...

The aim of the present note is to develop a unified theory of mean values of automorphic L-functions, albeit what we render here is less explicit and practical than we have desired. This is an outcome of the investigation commenced with the parts XII and XIV, where a framework was laid on the basis of the theory of automorphic representations, and a general approach to the mean values was envis...

Hecke operators play an important role in the theory of automorphic forms, and automorphic forms are closely linked to various cohomology groups. This paper is mostly a survey of Hecke operators acting on certain types of cohomology groups. The class of cohomology on which Hecke operators are introduced includes the group cohomology of discrete subgroups of a semisimple Lie group, the de Rham c...