John Bergdall
University of Arkansas
Title
On the distribution of modular points on deformation rings
Abstract
The goal of this talk is to discuss new results on a local-global phenomenon in the Langlands program on automorphic forms and Galois representations. Spaces of modular forms are parametrized by weights, and those same weights also parametrize local deformation rings that are defined by conditions in p-adic Hodge theory. In the mid-2000's, M. Kisin's work on modularity showed that components of local deformation rings always support modular points. The purpose of this talk is to describe an enhancement of Kisin's result, by giving a specific method for counting how many points a given component has, in terms of that component's special fiber geometry. This is joint work with Chengyang Bao (Imperial College, London) and Brandon Levin (Rice University, Houston).