Joint IAS/PU Number Theory
On the Distribution of Modular Points on Deformation Rings
The goal of this talk is to discuss a new 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 parametrize local deformation rings defined by conditions in p-adic Hodge theory. In the mid-2000's, M. Kisin's work on modularity showed that modular points can be found on each and every component of one of these local deformation rings. The purpose of this talk is to describe an extension 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).