Period and index in the Brauer group of an arithmetic surface (with an appendix by Daniel Krashen)

TL;DR

This paper introduces novel methods to analyze the period and index in the Brauer group of arithmetic surfaces using stacks, leading to new bounds and insights into the period-index conjecture and related arithmetic problems.

Contribution

It presents two new splitting techniques for Brauer classes on surfaces via stacks, resulting in new bounds and connections to the Hasse principle and moduli spaces.

Findings

New bounds on period-index relation for classes on curves over higher local fields

Relation between Hasse principle for moduli spaces and period-index problem

Sharp local period-index bounds proven by Krashen

Abstract

In this paper we introduce two new ways to split ramification of Brauer classes on surfaces using stacks. Each splitting method gives rise to a new moduli space of twisted stacky vector bundles. By studying the structure of these spaces we prove new results on the standard period-index conjecture. The first yields new bounds on the period-index relation for classes on curves over higher local fields, while the second can be used to relate the Hasse principle for forms of moduli spaces of stable vector bundles on pointed curves over global fields to the period-index problem for Brauer groups of arithmetic surfaces. We include an appendix by Daniel Krashen showing that the local period-index bounds are sharp.