Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 5

Special Issue in honor of Professor Benedict Gross’s 70th birthday

Guest Editors: Zhiwei Yun, Shouwu Zhang, and Wei Zhang

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

Pages: 2159 – 2202

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n5.a5


Jack A. Thorne (Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, United Kingdom)


Let $\rho$ be the $p$-adic Galois representation attached to a cuspidal, regular algebraic, polarizable automorphic representation of $\mathrm{GL}_n$. 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.

The author’s work received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 714405).

Received 10 March 2021

Received revised 2 February 2022

Accepted 6 March 2022

Published 12 January 2023