Models for Galois deformation rings

Workshop on Motives, Galois Representations and Cohomology Around the Langlands Program
Topic:Models for Galois deformation rings
Speaker:Brandon Levin
Affiliation:University of Chicago
Date:Thursday, November 9
Time/Room:4:00pm - 5:00pm/S-101
Video Link:https://video.ias.edu/MotivesGaloisRepsandCohomology/2017/1109-BrandonLevin

Abstract: An important input into modularity lifting theorems is an understanding of the geometry of Galois deformation rings, especially local deformation rings with p-adic Hodge theory conditions at $\ell=p$.  Outside of a few cases (ordinary, Fontaine-Laffaille...), the detailed structure of these deformation rings is very mysterious.  I will introduce models which in generic situations conjecturally have the same singularities as those of a class of potentially crystalline deformation rings. This will ultimately have applications to the weight part of Serre's conjecture, the Breuil-Mèzard conjecture, and modularity lifting in dimension $> 3$ where relatively little is known. This is joint work with Daniel Le, Bao V. Le Hung, and Stefano Morra.