# Period-index bounds for arithmetic threefolds

## TL;DR

This paper establishes a new upper bound on the index of Brauer classes over certain fields, showing it divides the fourth power of the period, advancing understanding of the period-index problem in algebraic geometry.

## Contribution

It proves the first uniform period-index bounds for Brauer classes over fields of transcendence degree 2 over p-adic fields, using Gabber's alterations and deformation theory.

## Key findings

- Index divides the fourth power of the period for classes with period prime to 6p.
- Provides the first uniform bounds over such fields.
- Advances the period-index conjecture in algebraic geometry.

## Abstract

The standard period-index conjecture for the Brauer group of a field of transcendence degree 2 over a $p$-adic field predicts that the index divides the cube of the period. Using Gabber's theory of prime-to-$\ell$ alterations and the deformation theory of twisted sheaves, we prove that the index divides the fourth power of the period for every Brauer class whose period is prime to $6p$, giving the first uniform period-index bounds over such fields.