Thomae formula for Abelian covers of the Sphere

Let $f\colon X\mapsto\mathbb{CP}^{1}$ be an Abelian cover of the sphere with $A$ as a group of automorphisms and $\lambda_1,\dots,\lambda_k$ be the ramification points of the cover. In this work we construct non-special divisors supported on $f^{-1}(\lambda_k)$. We evaluate theta functions on the images of these divisors in the Jacobian of $X$ and show that up to a certain constant not dependent on $\tau$, the period of the curve, these values are polynomials in $\lambda_1,\dots,\lambda_k.$ This work generalizes the work of Thomae for Hyperelliptic curves and Bershadsky and Radul for non-singular covers. (Joint with Shaul Zemel from Hebrew University).