Galois deformations and arithmetic geometry of Shimura varieties

Abstract

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 ℓ-adic families of Galois representations by ℓ-adic families of automorphic representations. Arithmetic geometrical ideas and methods on Shimura varieties are used for this purpose. A geometrical realization of the Jacquet–Langlands correspondence is discussed as an example.