Ramified descent and transcendental Brauer–Manin obstruction

Demeio, Julian Lawrence

doi:10.2140/ant.2026.20.419

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Ramified descent and transcendental Brauer–Manin obstruction

Julian Lawrence Demeio

Vol. 20 (2026), No. 3, 419–443

DOI: 10.2140/ant.2026.20.419

Abstract

Given a smooth geometrically connected variety $X$ defined over a number field $K$ and an étale torsor $V \to U$ over a Zariski-open $U$ of $X$ , we investigate the problem of which adelic points of $X$ can be approximated by adelic points that lift to a (twist of a) $V$ . The question has long been investigated in the literature when $U = X$ , but less so in the general case. We introduce a Brauer–Manin obstruction to the problem, and provide an example where this obstruction is nontrivial and purely transcendental. This answers in the negative a question posed by Harari at a 2019 workshop. Our example is also an explicit example of a nontrivial transcendental Brauer–Manin obstruction on a smooth compactification of a quotient ${SL}_{n} ∕ G$ , with $G$ constant metabelian.

Keywords

rational point, Brauer–Manin obstruction, ramified cover

Mathematical Subject Classification

Primary: 14G05, 14G12

Secondary: 11G35

Milestones

Received: 11 February 2023

Revised: 19 March 2025

Accepted: 2 May 2025

Published: 24 March 2026

Authors

Julian Lawrence Demeio
Department of Mathematics University of Bath Bath United Kingdom

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Vol. 20, No. 3, 2026