Number Theory Seminar

Mathematicians have observed close and fruitful connections between congruences for modular forms and Galois representations. In this direction, Serre made a rather definitive conjecture (for cuspidal eigenforms of weight at least 2) that was ultimately proven by Khare and Wintenberger. There has been substantial work on generalizing Serre's conjecture beyond GL(2). We will survey recent work on Herzig's generalization of the weight part of Serre's conjecture focusing on the case of GSp(4). The proofs rely on new results about Galois deformation rings obtained from a theory of local models. This talk is based on joint work with B. Le Hung, B. Levin, and S. Morra and joint work in progress with B. Le Hung and H. Lee.