On torsion of superelliptic Jacobians

Abstract

We give a structure theorem for the m-torsion of the Jacobian of a general superelliptic curve y m = F (x). We study existence of torsion on curves of the form y q = x p − x + a over finite fields of characteristic p. We apply those results to bound from below the Mordell-Weil ranks of Jacobians of certain superelliptic curves over Q.