Thomae formula for Abelian covers of $\mathbb{CP}^{1}$

TL;DR

This paper generalizes Thomae's classical formula to Abelian covers of the complex projective line, relating theta constants, divisors, and branching data in a way that remains constant across moduli spaces.

Contribution

It extends Thomae's formula to a broad class of Abelian covers of olds, providing a unified approach to relate theta constants and branching data.

Findings

Derived a quotient involving theta constants and divisors that is moduli-invariant.

Generalized classical Thomae formula to Abelian covers with fixed Galois group.

Established a polynomial relation in branching values for these covers.

Abstract

Abelian covers of $\mathbb{CP}^{1}$, with fixed Galois group $A$, are classified, as a first step, by a discrete set of parameters. Any such cover $X$, of genus $g\geq1$ say, carries a finite set of $A$-invariant divisors of degree $g-1$ on $X$ that produce non-zero theta constants on $X$. We show how to define a quotient involving a power of the theta constant on $X$ that is associated with such a divisor $\Delta$, some polynomial in the branching values, and a fixed determinant on $X$ that does not depend on $\Delta$, such that the quotient is constant on the moduli space of $A$-covers with the given discrete parameters. This generalizes the classical formula of Thomae, as well as all of its known extensions by various authors.