Geometric deformations of orthogonal and symplectic Galois representations

Joint IAS/Princeton University Number Theory Seminar
Topic:Geometric deformations of orthogonal and symplectic Galois representations
Speaker:Jeremy Booher
Affiliation:Stanford University
Date:Thursday, November 19
Time/Room:4:30pm - 5:30pm/S-101
Video Link:https://video.ias.edu/puias/2015/1119-Booher

For a representation of the absolute Galois group of the rationals over a finite field of characteristic $p$, we would like to know if there exists a lift to characteristic zero with nice properties. In particular, we would like it to be geometric in the sense of the Fontaine-Mazur conjecture: ramified at finitely many primes and potentially semistable at $p$. For two-dimensional representations, Ramakrishna proved that under technical assumptions, odd representations admit geometric lifts. We generalize this to higher dimensional orthogonal and symplectic representations. The key ingredient is a smooth local deformation condition obtained by analysing unipotent orbits and their centralizers in the relative situation, not just over fields.