Thomae formula for abelian covers of CP¹

Abelian covers of CP¹, with fixed Galois group A, are classified, as a first step, by a discrete set of parameters. Any such cover X, of genus g ≥ 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 Δ, some polynomial in the branching values, and a fixed determinant on X that does not depend on Δ, 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.