Vanishing thetanulls for some dihedral and cyclic coverings of Riemann surfaces

Abstract

Let Wg → Wz be a ramified p-sheeted covering of Riemann surfaces of genus g and z, (z > 0) where p is an odd prime. Assume that the Galois group is either dihedral or cyclic. Assume, moreover, that the covering is full; that is, there us an integral divisor E, of degree 2r on Wz which lifts to be canonical on Wg. Then g = rp + 1, where r ≥ 1. Clearly, Wg admits 22z half-canonical linear series of dimension at least r − z arising from divisors on Wz whose double is E. Theorem 1 Of these 22z half-canonical linear series uz (= 2z−1 (2z−1)) have dimension at least r − z + 1. Theorem 2 Let Wg (g = 3r + 1, r ≥ 3) admit four half canonical linear series, three of dimension r − 1, and one of dimension r, whose sum is bi-canonical, where the half-canonical linear series of dimension r is unique. Then Wg is a full elliptic-trigonal Riemann surface. (This characterizes the cases z = 1, p = 3, g ≥ 10)