Speaker: Joel Bellaiche (Columbia)
Title: Extensions of Galois representations
Abstract: A great number of results and open questions in algebraic number theory can be reformulated, and thus unified, in terms of existence, or non-existence, of extensions with prescribed properties between given irreducible representations of an absolute Galois group.
In the first part of this talk, I want to explain how two old and classical diophantine problems (solving the Pell-Fermat equation, and counting rational points on elliptic curves) can be reformulated in term of the existence of suitable extensions of Galois representations, and to formulate the general conjecture, due to Bloch and Kato, that predicts the dimension of the space of extensions between two general Galois representations.
In the second part, I want to show how such extensions can be constructed using families of automorphic forms.