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

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.