Splitting Brauer classes using the universal Albanese

TL;DR

This paper demonstrates that Brauer classes over fields can be split by torsors under abelian varieties, with specific conditions on the index influencing the splitting behavior via Albanese varieties and curves.

Contribution

It establishes new criteria for splitting Brauer classes using Albanese varieties and curves, highlighting the role of the index modulo 4 in the splitting process.

Findings

Brauer classes split over torsors under abelian varieties.

For indices not ≡ 2 mod 4, Albanese varieties of curves split the class.

When index ≡ 2 mod 4, adding a genus 1 factor enables splitting.

Abstract

We prove that every Brauer class over a field splits over a torsor under an abelian variety. If the index of the class is not congruent to 2 modulo 4, we show that the Albanese variety of any smooth curve of positive genus that splits the class also splits the class, and there exist many such curves splitting the class. We show that this can be false when the index is congruent to 2 modulo 4, but adding a single genus 1 factor to the Albanese suffices to split the class.