On the vanishing of adjoint Bloch--Kato Selmer groups of irreducible automorphic Galois representations

Abstract

Let $\rho$ be the $p$-adic Galois representation attached to a cuspidal, regular algebraic, polarizable automorphic representation of $GL_n$. Assuming only that $\rho$ satisfies an irreducibility condition, we prove the vanishing of the adjoint Bloch--Kato Selmer group attached to $\rho$. This generalizes previous work of the author and James Newton.