Semi-integral Brauer-Manin obstruction and quadric orbifold pairs

TL;DR

This paper investigates semi-integral points on orbifolds related to quadric hypersurfaces, developing a new Brauer-Manin obstruction framework that bridges classical and integral versions, with results on local-global principles and quantitative failure analysis.

Contribution

It introduces a semi-integral Brauer-Manin obstruction for orbifolds, connecting classical and integral obstructions, and analyzes their effectiveness and limitations for quadric hypersurfaces.

Findings

Identifies conditions where local-global principles hold or fail.

Develops a semi-integral Brauer-Manin obstruction framework.

Quantifies the failure of the obstruction to explain integral points.

Abstract

We study local-global principles for two notions of semi-integral points, termed Campana points and Darmon points. In particular, we develop a semi-integral version of the Brauer-Manin obstruction interpolating between Manin's classical version for rational points and the integral version developed by Colliot-Th\'el\`ene and Xu. We determine the status of local-global principles, and obstructions to them, in two families of orbifolds naturally associated to quadric hypersurfaces. Further, we establish a quantitative result measuring the failure of the semi-integral Brauer-Manin obstruction to account for its integral counterpart for affine quadrics.