Thomae formula for abelian covers of ℂℙ¹

Kopeliovich, Yaacov; Zemel, Shaul

doi:10.1090/tran/7764

Thomae formula for abelian covers of $\mathbb {CP}^{1}$
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Trans. Amer. Math. Soc. 372 (2019), 7025-7069 Request permission

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 $g\geq 1$, say, carries a finite set of $A$-invariant divisors of degree $g-1$ on $X$ that produce nonzero 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.

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