## Adjoint Selmer groups for polarized automorphic Galois representations

Patrick Allen

University of Illinois, Urbana-Champaign

October 15, 2015

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.