Adjoint Selmer groups of automorphic Galois representations of unitary type

Abstract

Let $\rho$ be the $p$-adic Galois representation attached to a cuspidal, regular algebraic automorphic representation of ${\mathrm{GL}}_n$ of unitary type. Under very mild hypotheses on $\rho$, we prove the vanishing of the (Bloch–Kato) adjoint Selmer group of $\rho$. We obtain definitive results for the adjoint Selmer groups associated to non-CM Hilbert modular forms and elliptic curves over totally real fields.