No transcendental Brauer-Manin obstructions on abelian varieties

TL;DR

This paper proves that for torsors under abelian varieties over number fields, the Brauer-Manin obstruction to rational points is fully explained by locally constant Brauer classes, removing previous finiteness assumptions.

Contribution

It establishes unconditional results showing no transcendental Brauer-Manin obstructions exist on abelian varieties, extending prior work that required finiteness of Tate-Shafarevich groups.

Findings

Any adelic point orthogonal to algebraic Brauer group is orthogonal to all Brauer classes.

If a Brauer-Manin obstruction exists, it is due to locally constant classes.

Results hold unconditionally, without assumptions on Tate-Shafarevich groups.

Abstract

Suppose $X$ is a torsor under an abelian variety $A$ over a number field. We show that any adelic point of $X$ that is orthogonal to the algebraic Brauer group of $X$ is orthogonal to the whole Brauer group of $X$. We also show that if there is a Brauer-Manin obstruction to the existence of rational points on $X$, then there is already an obstruction coming from the locally constant Brauer classes. These results had previously been established under the assumption that $A$ has finite Tate-Shafarevich group. Our results are unconditional.