Thomae formula for abelian covers of ℂℙ¹

Abstract

Abelian covers of C P 1 \mathbb {CP}^{1} , with fixed Galois group A A , are classified, as a first step, by a discrete set of parameters. Any such cover X X , of genus g ≥ 1 g\geq 1 , say, carries a finite set of A A -invariant divisors of degree g − 1 g-1 on X X that produce nonzero theta constants on X X . We show how to define a quotient involving a power of the theta constant on X X that is associated with such a divisor Δ \Delta , some polynomial in the branching values, and a fixed determinant on X X that does not depend on Δ \Delta , such that the quotient is constant on the moduli space of A 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.