Arithmetic aspects of moduli of sheaves on curves

TL;DR

This paper explores the arithmetic properties of moduli spaces of stable vector bundles on curves, linking the period-index problem for Brauer classes to the Hasse principle for rational points, thus advancing understanding of rationality issues.

Contribution

It establishes a connection between the period-index problem and the Hasse principle for moduli spaces, refining classical results and providing new insights into their arithmetic behavior.

Findings

Connection between period-index problem and Hasse principle clarified

Refinement of classical results by Artin and Tate achieved

Insights into rational points on etale forms of moduli spaces obtained

Abstract

We describe recent work on the arithmetic properties of moduli spaces of stable vector bundles and stable parabolic bundles on a curve over a global field. In particular, we describe a connection between the period-index problem for Brauer classes over the function field of the curve and the Hasse principle for rational points on etale forms of such moduli spaces, refining classical results of Artin and Tate. Submitted to proceedings of a Clay workshop on vector bundles on curves in honor of Newstead.