I will begin by reviewing the notion of an algebraic Hecke character over a number field in some depth. This part of the talk should be accessible to a wide audience; the only prerequisites being basic algebraic number theory and some sheaf theory. The main goal of the talk will be to discuss the critical values of L-functions attached to algebraic Hecke characters. I will especially discuss certain signs that appear in reciprocity laws satisfied by these special values. Towards the end, I will talk about how this generalizes to L-functions for GL(n) over a CM field.

Introductory reading:

Read the first 5 sections of "On the arithmetic of Hecke characters": http://www.fields.utoronto.ca/sites/default/files/uploads/arithmetic_hec...