7th Feb: Diego Chicharro

Title: L-functions and Deligne's conjecture 

Abstract: Special values of L-functions is one of the most striking topics in number theory - even more so because the general picture, except in a few simple cases, remains at most conjectural. One of the classical examples that is well-understood is the Riemann zeta function, that is, the L-function of Spec(Q). At even positive integers, it evaluates to a rational multiple of a power of pi, and is a rational number at odd negative integers. Another important example in dimension 1 are elliptic curves, whose L-functions L(E,s) at s=1 are rational multiples of some “period” when the rank is zero, as predicted by the BSD conjecture. In this talk, I will try to explain a conjecture of Deligne that predicts the special values of L-functions of “critical” varieties (or more generally, “critical” motives), and in particular we will show how it generalises the aforementioned examples.