Number Theory Seminar

Speaker: Adrien Morin (Bordeaux)

Title: Special values of L-functions for Z-constructible sheaves in dimension 1

Abstract: Weil-étale cohomology is a conjectural cohomology theory for arithmetic schemes that is better behaved than étale cohomology and has conjectural links to special values of zeta functions. In this talk, I will explain how one can define in dimension 1 the Weil-étale cohomology with compact support of a Z-constructible sheaf and link it with the special value at s=0 of the corresponding L-function. There are three special cases of interest: one obtains a cohomological formula for the special value at s=0 of the spectrum of an order in a number field, generalizing the analytic class number formula, a formula for the special value at s=0 of an Artin L-function associated to a rational representation of the Galois group of a global field, and finally the formula for a constructible sheaf recovers Tate's formula for the Euler characteristic of a number field.