Generalized period-index problem with an application to quadratic forms

TL;DR

This paper establishes an upper bound for the index of finite subgroups in the Brauer group of function fields over discretely valued fields, with applications to quadratic forms and their Witt indices.

Contribution

It provides a new upper bound for subgroup indices in the Brauer group of certain function fields, linking algebraic invariants to quadratic form properties.

Findings

Upper bound for the index of finite subgroups in the Brauer group.

Application to bounding degrees of extensions for quadratic forms.

Connection between arithmetic invariants and Witt index maximization.

Abstract

Let $F$ be the function field of a curve over a complete discretely valued field. Let $\ell$ be a prime not equal to the characteristic of the residue field. Given a finite subgroup $B$ in the $\ell$ torsion part of the Brauer group ${}_{\ell}Br(F)$, we define the index of $B$ as the minimum of the degrees of field extensions which split all elements in $B$. In this manuscript, we give an upper bound for the index of any finite subgroup $B$ in terms of arithmetic invariants of $F$. As a simple application of our result, given a quadratic form $q/F$, where $F$ is the function field of a curve over an $n$-local field, we provide an upper bound to the minimum of degrees of field extensions $L/F$ so that the Witt index of $q\otimes L$ becomes the largest possible.