Thomae formula for $2$ Abelian covers of $\mathbb{CP}^1$

Abstract

Let $X$ be an Abelian cover $\mathbb{CP}^{1}$ ramified at $mr$ points, $\lambda_1...\lambda_{mr}.$ 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 Riemann theta function doesn't vanish on their image in $J(X).$ We obtain a Thomae formula similar to the formulas [BR],[Na],[Z] ,[EG] and [Ko]. We show that up to a certain determinant of the non standard periods of $X$, the value of the Riemann theta function at these divisors raised to a high enough power is a polynomial in the branch point of the curve $X.$ Our approach is based on a refinement of Accola's results and Nakayashiki's approach explained in [Na] for Abelian covers.