The local-global principle for symmetric determinantal representations of smooth plane curves in characteristic two

TL;DR

This paper explores the conditions under which smooth plane curves over global fields of characteristic two can be globally represented as determinants of symmetric matrices with linear entries, highlighting unique features of characteristic two.

Contribution

It establishes a local-global principle for symmetric determinantal representations of smooth plane curves specifically in characteristic two, utilizing Mumford's theory of theta characteristics.

Findings

A smooth plane curve over a global field of characteristic two admits a symmetric determinantal representation if and only if such a representation exists locally everywhere.

The result is unique to characteristic two, as analogous principles do not hold in other characteristics.

The work connects algebraic geometry, number theory, and the theory of theta characteristics in a novel way.

Abstract

We give an application of Mumford's theory of canonical theta characteristics to a Diophantine problem in characteristic two. We prove that a smooth plane curve over a global field of characteristic two is defined by the determinant of a symmetric matrix with entries in linear forms in three variables if and only if such a symmetric determinantal representation exists everywhere locally. It is a special feature in characteristic two because analogous results are not true in other characteristics.