Adjoint Selmer groups for polarized automorphic Galois representations

Joint IAS/Princeton University Number Theory Seminar
Topic:Adjoint Selmer groups for polarized automorphic Galois representations
Speaker:Patrick Allen
Affiliation:University of Illinois, Urbana-Champaign
Date:Thursday, October 15
Time/Room:4:30pm - 5:30pm/S-101
Video Link:https://video.ias.edu/jointnts/2015/1015-PatrickAllen

Given the $p$-adic Galois representation associated to a regular algebraic polarized cuspidal automorphic representation, one naturally obtains a pure weight zero representation called its adjoint representation. Because it has weight zero, a conjecture of Bloch and Kato says that the only de Rham extension of the trivial representation by this adjoint representation is the split extension. We will discuss a proof of this case of their conjecture, under an assumption on the residual representation. This is done by using the Taylor-Wiles patching method, Kisin's technique of analyzing the generic fibre of deformation rings, and a characterization of smooth closed points in the generic fibre of certain local deformation rings.