Congruences between automorphic forms and lower bounds on Selmer group

Columbia University, New York NY

Investigators

Abstract

The study of the absolute Galois group of a number field F or, from a Tannakian point of view, of its abelian category of continuous finite dimensional representations (let us say over a p-adic field) has long been recognized as one of important challenge in pure mathematics. Among those representations, the ones that are geometric, in the sense of Fontaine and Mazur, are especially of arithmetic significance, and their category should be equivalent to the (still conjectural) category of mixed motives over F. It is important to understand the first Ext groups in that category (higher Ext groups should be zero) and Bloch and Kato have made precise conjectures relating the dimension of those groups to the order of L-functions at integers values of the variable. The project aims to construct as much extensions as possible in those Ext groups (hopefully as much as predicted by the conjecture, in the case corresponding to the center of the functional equation of the L-function) using p-adic deformations of non-tempered automorphic forms. An important step should be the study of the local geometry of the "moduli space of p-adic automorphic forms" called Eigenvarieties around the non-tempered automorphic forms. Many old problems in arithmetic, some of them going back as far as Diophantes, as well as some new ones, fit well in the framework of Galois theory: they often can be translated into questions about existence, or non-existence, of certain Galois representations (that is representations of the absolute Galois group G of the field Q of rational numbers, or of some open subgroups of G) with prescribed properties. And then, sometimes, they can be proven, as was Fermat's Last Theorem by Wiles. The study of Galois representations splits up into two parts : finding irreducible Galois representations, and then determining extensions between them. Even if the first problem is far from being solved, precise conjectures about the second one were made by Bloch and Kato. The projects aims to give partial answers to those conjectures, by constructing some interesting extensions. The method uses the theory of automorphic forms, which was once quite a different topic, but which is now strongly tied to the theory of Galois representations by the Langland's program. The idea is that one can obtain interesting extensions of Galois representations by looking at (p-adic) deformations of some very special automorphic forms, the so-called non tempered forms. The more deformations there are, the more extensions one should be able to construct. Those deformations are encoded in the geometry of a (p-adic) variety, known as the Eigenvariety, and developing tools to study that geometry is an important part in the project.