Modul: MAT770 Oberseminar: Algebraische Geometrie

## The Iwasawa Main Conjecture for modular forms.

Dr. Olivier Fouquet talk

Speaker invited by: Prof. Dr. Joseph Ayoub

**Date:** 29.10.18 **Time:** 13.15 - 14.45 **Room:** Y27H25

Since Dirichlet’s Class Formula and the Birch and Swinnerton-Dyer Conjectures, it is known (or conjectured) that values at integers of L-functions of geometric objects can be computed in terms of arithmetical and cohomological invariants. The Main Conjectures of Iwasawa theory are generalizations of that philosophy which aims at predicting not only the values of L-functions but also their p-adic variations as the underlying geometric objects moves in a p-adic family (for instance the family of twists by Dirichlet characters, a p-adic family of p-adic modular forms…). I will explain the statement and meaning of these conjectures and then present a joint work with Xin Wan in which we establish these conjectures for modular forms with an irreducible residual Galois representations.