p-adic L-functions and Galois cohomology

Brandeis University, Waltham MA

Investigators

Abstract

The project focuses on the conjectural relations between certain arithmetic invariants attached to Galois representations, in particular the cohomological invariants called Selmer groups, and the order of vanishing at integers of either the L-functions, or the p-adic L-functions attached to those Galois representations. The main objective is to prove that the rank of the Selmer group of a Galois representation attached to an automorphic form for a unitary group is at least equal to the order of vanishing at 0 of the corresponding p-adic L-function (one actually expects that the equality holds). The strategy consists in a study of the geometry of the eigenvariety, which is the universal family of automorphic forms, at a point attached to the given Galois representation, and in the construction of a family of p-adic L-functions on that eigenvariety. The project has an other aspect, which consists in reformulating and generalizing the conjectures relating p-adic L-functions and Galois cohomology. While the automorphic methods used in this project are very promising, it is not expected that they alone will solve the vast array of conjectures and questions concerning the relations between p-adic L-functions and Galois cohomology. Many tools and ideas from various parts of mathematics will be needed to do so. One of the aims of the PI in reformulating those conjectures is to separate more clearly what part can be done with which methods or combination of methods, and to foster a greater involvement of mathematicians in other areas (e.g., theory of transcendence) in the work on those conjectures.