# Algebraic Geometry Seminar

#### Local and Adelic Hecke Algebra Isomorphisms

**Speaker:**
Valentijn Karemaker, University of Pennsylvania

**Location:**
Warren Weaver Hall 201

**Date:**
Tuesday, January 31, 2017, 3:30 p.m.

**Synopsis:**

First, let K and L be number fields and let G be a linear algebraic group over Q. Suppose that there is a topological group isomorphism of points on G over the adeles of K and L, respectively. We establish conditions on the group G, related to the structure of its Borel groups, under which K and L have isomorphic adele rings. As a corollary, we show that when K and L are Galois over Q, an isomorphism of Hecke algebras for GL(n), which is an isometry in the L1-norm, implies that K and L are isomorphic as fields. This can be viewed as an analogue in the theory of automorphic representations of the theorem of Neukirch that the absolute Galois group of a number field determines the field if it is Galois over Q. Secondly, let K and L be non-archimedean local fields of characteristic zero. Analogous results to the number fields case still hold. However, we show that the Hecke algebra for GL(2) for any local field is Morita equivalent to the same complex algebra, determined by the Bernstein decomposition.