Moduli of Galois representations and generalized class groups

Abstract

This lecture describes some recent progress on establishing the conjectural relationships between Galois representations and automorphic forms, together with the number-theoretic motivation for these results. This includes equidistribution theorems like the Sato–Tate conjecture concerning the number of points on reductions of elliptic curves modulo primes. The connection to these equidistribution statements goes back to Dirichlet’s work on primes in arithmetic progressions, in which the class number formula for real quadratic fields plays an important role (at least from one point of view). We discuss how results on ‘generalized class groups’ defined using Galois cohomology have been crucial to more recent developments.