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

Yasuhiro Ishitsuka, Tetsushi Ito

Published 2014-12-29Version 1

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 with rational coefficients. In the proof, we use a relation between symmetric matrices with entries in linear forms and non-effective theta characteristics on plane curves, and the results of Gross-Rohrlich on the rational torsion points on the Jacobian varieties of the Fermat curves of prime degree.