Loïc Merel

Abstract:   There is a longstanding interest among number theorists for values at the integers of the Riemann zeta function (and its generalizations called L-functions). The first important discoveries are possibly Euler's formulas (\zeta(2)=\pi^2/6, etc) and Dirichlet's class number formula.

Since then many L-functions attached to various arithmetic-geometric objects (representations of Galois groups, Dirichlet characters, elliptic curves, modular forms, etc.) have been identified. I'll partially explain how the concept has been unified.

Bloch and Kato, after the work of many other people, proposed a formula predicting the values at integers of L-functions that extends many conjectures proposed before (by Artin, Beilinson, Birch, Deligne, Stark, Swinnerton-Dyer, etc.). Their prediction has been confirmed in a few of the most primitive cases. I will not fully discuss the conjecture; instead, based on examples, I will explain a sense in which the formula of Bloch and Kato is not yet satisfactory. I will also suggest a way that leads slightly beyond the conjecture of Bloch and Kato.