Non vanishing divisors for general cyclic covers and their Thomae formula

Abstract. Let X be a general cyclic cover of CP1 ramified at m points, λ1...λm. 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. 1.