On the symmetric determinantal representations of the Fermat curves of prime degree

TL;DR

This paper proves that Fermat curves of prime degree cannot have their defining equations expressed as determinants of symmetric matrices with linear form entries over rationals, linking algebraic geometry and theta characteristics.

Contribution

It establishes a non-existence result connecting symmetric determinantal representations and theta characteristics for Fermat curves of prime degree.

Findings

Fermat curves of prime degree lack symmetric determinantal representations over rationals.

The proof uses relations between symmetric matrices and theta characteristics.

Results incorporate findings of Gross-Rohrlich on Jacobian torsion points.

Abstract

We prove that the defining equations of the Fermat curves of prime degree cannot be written as the determinant of symmetric matrices with entries in linear forms in three variables with rational coefficients. In the proof, we use a relation between symmetric matrices with entries in linear forms and non-effective theta characteristics on smooth plane curves. We also use some results of Gross-Rohrlich on the rational torsion points on the Jacobian varieties of the Fermat curves of prime degree.