Thomae type formulae for singular Z_N curves

TL;DR

This paper provides a straightforward, rigorous proof of Thomae type formulae specifically for singular Z_N curves, employing classical variational methods rooted in Riemann, Thomae, and Fuchs's work.

Contribution

It offers the first elementary and rigorous derivation of Thomae formulae for singular Z_N curves using traditional variational techniques.

Findings

Rigorous proof of Thomae formulae for singular Z_N curves

Application of classical variational methods to algebraic curves

Clarification of the relationship between singular curves and Thomae formulae

Abstract

We give an elementary and rigorous proof of the Thomae type formula for singular $Z_N$ curves. To derive the Thomae formula we use the traditional variational method which goes back to Riemann, Thomae and Fuchs.