|Joint IAS/Princeton University Number Theory Seminar|
|Topic:||Adjoint Selmer groups for polarized automorphic Galois representations|
|Affiliation:||University of Illinois, Urbana-Champaign|
|Date:||Thursday, October 15|
|Time/Room:||4:30pm - 5:30pm/S-101|
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.