The d-primary Brauer-Manin obstruction for Curves

TL;DR

This paper investigates the role of the d-primary Brauer group in obstructing rational points on curves over global fields, providing examples that challenge existing conjectures and heuristics.

Contribution

It introduces new examples illustrating the complexity of d-primary Brauer-Manin obstructions and refutes a stronger version of Poonen's conjecture.

Findings

Existence of infinitely many coprime d with d-primary obstructions

An example where the odd part of the Brauer group does not obstruct rational points

Counterexamples to a stronger form of Poonen's conjecture

Abstract

For a curve over a global field we consider for which integers d the d-primary part of the Brauer group can obstruct the existence of rational points. We give examples showing it is possible that there is a d-primary obstruction for infinitely many relatively coprime d, and also an example where the odd part of the Brauer group does not obstruct although there is a Brauer-Manin obstruction. These examples demonstrate that a slightly stronger form of a conjecture of Poonen is false, despite being supported by the same heuristic.