Witt Group of Nondyadic Curves

TL;DR

This paper computes the derived Witt groups of smooth proper curves over nondyadic local fields with characteristic not equal to 2, expanding understanding beyond hyperelliptic cases and exploring Theta characteristics.

Contribution

It provides the first comprehensive calculation of derived Witt groups for non-Archimedean curves beyond hyperelliptic cases, using reduction techniques and studying Theta characteristics.

Findings

Derived Witt groups computed for nondyadic local fields

Established conditions for the existence of Theta characteristics

Extended known results beyond hyperelliptic curves

Abstract

Witt group of real algebraic curves has been studied since Knebusch in 1970s. But few results are known if the base field is non-Archimedean except the hyperelliptic case by works of Parimala, Arason et al.. In this paper, we compute the derived Witt groups of smooth proper curves over nondyadic local fields with $char\neq2$ by reduction, with a general study of the existence of Theta characteristics.