Adjoint Selmer groups of automorphic Galois representations of unitary type

TL;DR

This paper proves the vanishing of the Bloch--Kato adjoint Selmer group for certain automorphic Galois representations, providing key insights into their arithmetic properties and implications for related automorphic forms and elliptic curves.

Contribution

It establishes the vanishing of the adjoint Selmer group for automorphic Galois representations of unitary type under mild hypotheses, advancing understanding of their deformation theory.

Findings

Vanishing of the adjoint Selmer group for automorphic Galois representations of unitary type.

Results apply to non-CM Hilbert modular forms and elliptic curves over totally real fields.

Provides new tools for studying automorphic Galois representations and their deformations.

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.