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 $g\ge 1$ 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.

50 pages, presentation improved and list of notation added

References in corpus (1)

• Thomae Formulae for General Fully Ramified $Z_n$ Curves

Cited by in corpus (3)

• General derivative Thomae formula for singular half-periods
• On Spaces Associated with Invariant Divisors on Galois Covers of Riemann Surfaces and Their Applications
• Thomae's Derivative Formulae for Trigonal Curves