Thomae's formula for \(Z_n\) curves (Q628355)

A nonsingular curve \(Z_n\) is an \(n\)-sheeted branched cover of the sphere, branched over \(nr\) distinct points \(\lambda_i\) with branching order at each branch point \(n-1\), and whose equation is given by \[ w^n=\prod_{i=0}^{nr-1}(z-\lambda_i) \] where \(r\) is a positive integer greater than 1. In the paper under review, the authors show how to construct integral divisors \(\Delta\) of degree \(g\) whose support lies in the branch set and which are non-special. A divisor is said to be non-special if there are no non-zero holomorphic differentials whose divisors are multiples of it. The authors use these non-special divisors to build identities satisfied by the theta functions with rational characteristics associated with the \(Z_n\) curves. The work originates in work done in the nineteenth century by J. Thomae, who considered the case \(n=2\). Thomae's work was already extended by Bershadsky-Radul and by Nakayashiki. According to the authors, the present proof is more transparent than the one given in previous works.