Brauer classes not split by genus one curves

TL;DR

The paper demonstrates that certain Brauer classes over a field cannot be split by any genus one curve, answering a question posed by Clark and Saltman.

Contribution

It provides the first example of Brauer classes not split by genus one curves, resolving an open problem in the field.

Findings

Existence of Brauer classes not split by genus one curves

Answers a question of Clark and Saltman

Advances understanding of Brauer groups and genus one curves

Abstract

We show that there exist Brauer classes over a field $F$ which are not split by any genus one curve over $F$. This answers a question of Clark and Saltman.