Kempf.04.04

Abstract: Shimura varieties are arithmetic quotients of locally symmetric spaces which are canonically defined over number fields. In this lecture, we discuss recent developments on the (non-abelian) reciprocity law realized on cohomology groups of Shimura varieties. The reciprocity law relates Galois representations and automorphic representations (generalizing the classical class field theory of Takagi-Artin, and formulated by Langlands representation theoretically), and plays a central role in number theory. A focus is put on the control of $\ell$-adic families of Galois representations by $\ell$-adic families of automorphic representations, and I hope to include the recent new direction (applications to algebraic number theory).