General cyclic covers and their Thomae formula

TL;DR

This paper generalizes Thomae's formula for cyclic covers of the projective line, expressing theta function values at specific divisors as polynomials in branch points, extending previous results to more general singular covers.

Contribution

It introduces a new class of divisors on cyclic covers and derives a polynomial expression for theta functions at these divisors, broadening the scope of Thomae-type formulas.

Findings

Theta function values are polynomial in branch points.

Generalization of Thomae's formula to singular cyclic covers.

Extension of Accola and Nakayashiki's methods to broader cases.

Abstract

Let $X$ be a general cyclic cover of $\mathbb{CP}^{1}$ ramified at $m$ points, $\lambda_1...\lambda_m.$ we define a class of non positive divisors on $X$ of degree $g-1$ supported in the pre images of the branch points on $X$, such that the the standard theta function doesn't vanish on their image in $J(X).$ These divisors generalize the divisors introduced in [BR] and [Na]. Generalizing the results of [BR],[Na] and [EG] we show that up to a certain determinant of the non standard periods of $X$, the value of the theta functions at these divisors is a polynomial in the branch point of the curve $X.$ Our treatment is based on a generalization of Accola's results of the 3 cyclic sheeted cover [Ac1] and a straightforward generalization of Nakayashiki's approach explained in [Na] in the non singular case for any singular cyclic cover.