Irreducibility of Hurwitz spaces of covering of an elliptic curve of prime degree with one point of total ramification

Abstract

Let Y be an elliptic curve, p a prime number and WH_{p,n}(Y) the Hurwitz space that parametrizes equivalence classes of p-sheeted branched coverings of Y , with n branch points, n − 1 of which are points of simple ramification and one of total ramification.In this paper, we prove that WH_{p,n}(Y) is irreducible if n − 1≥ 2 p.