Irreducibility of Galois representations

יום ד', 02/01/2019 - 10:30

According to the Langlands philosophy, there should be a close relationship between automorphic representations and Galois representations. When are these Galois representations irreducible?

In the 1970s, Ribet proved that the p-adic Galois representation attached to a modular form f is irreducible if and only if f is cuspidal. More generally, it is conjectured that the p-adic Galois representation associated to any cuspidal automorphic representation of GL(n) is irreducible. 

The goal of this talk is to provide an overview of this conjecture, focusing on the special case of Galois representations attached to low weight, genus 2 Siegel modular forms. These two-dimensional analogues of weight 1 modular forms are, conjecturally, the automorphic objects that correspond to abelian surfaces.