On the Thomae formula for $Z_N$ curves

TL;DR

This paper provides a rigorous, elementary proof of the Thomae formula for ${f Z}_N$ curves, utilizing a variational approach and explicit algebraic expressions, advancing understanding of theta functions and algebraic curves.

Contribution

It offers a new proof of the Thomae formula for ${f Z}_N$ curves using a classical variational method instead of Laplacian determinants.

Findings

Explicit algebraic expression of chiral Szeg"{o} kernels.

Proof of vanishing of derivatives of theta functions with ${f Z}_N$ invariant characteristics.

Validation of the Thomae formula for ${f Z}_N$ curves.

Abstract

We shall give an elementary and rigorous proof of the Thomae formula for ${\bf Z}_N$ curves which was discovered by Bershadsky and Radul. Instead of using the determinant of the Laplacian we use the traditional variational method which goes back to Riemann, Thomae, Fuchs. In the proof we made explicit the algebraic expression of the chiral Szeg\"{o} kernels and proves the vanishing of zero values of derivatives of theta functions with ${\bf Z}_N$ invariant $1/2N$ characteristics.