On Hecke Fields

This talk concerns Hecke fields of cohomological automorphic forms. These are the fields of definition
of the Hecke subquotients of the cohomology of algebraic local systems on locally symmetric varieties.
They are a large family of naturally occurring number fields and the purpose of this talk is to show
that in general they are precisely the  CM fields and totally real fields, and to deduce consequences.
The talk will explain why this is expected on the basis of standard conjectures,  and review historical
background.   The general result requires  Borel's Conjecture (Franke's Theorem), and for GL(N) may
be deduced using the work of Franke and Schwermer.