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

29 pages. Major revision: completely different methods, an earlier error corrected, paper reorganized. Comments are still welcome!

References in corpus (2)

• Twisted sheaves and the period-index problem
• Boundedness of families of canonically polarized manifolds: A higher dimensional analogue of Shafarevich's conjecture

Cited by in corpus (12)

• Twisted sheaves and the period-index problem
• Applications of patching to quadratic forms and central simple algebras
• Refinements to patching and applications to field invariants
• Cohomological obstruction theory for Brauer classes and the period-index problem
• On the behaviour of Brauer $p$-dimensions under finitely-generated field extensions
• On Brauer $p$-dimensions and index-exponent relations over finitely-generated field extensions
• Colliot-Thélène's conjecture and finiteness of u-invariants
• Weierstrass preparation and algebraic invariants
• Embedding problems of division algebras
• Patching subfields of division algebras
• Arithmetic aspects of moduli of sheaves on curves
• Lower bounds and infinity criterion for Brauer $p$-dimensions of finitely-generated field extensions