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

TL;DR

This paper proves the vanishing of the adjoint Bloch--Kato Selmer group for a broad class of irreducible automorphic Galois representations, extending previous results in the field.

Contribution

It establishes the vanishing under minimal assumptions, broadening the scope of known cases for automorphic Galois representations.

Findings

Vanishing of the adjoint Bloch--Kato Selmer group for irreducible automorphic Galois representations

Generalization of previous vanishing results to more general automorphic cases

Relies on minimal irreducibility assumptions without additional conditions

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.