Explicit descent for Jacobians of prime power cyclic covers of the projective line

Abstract

The Jacobian of a cyclic cover of the projective line is isogenous to a product of abelian subvarieties, one for each positive divisor of the degree of the cover. In this article, we show how to compute a Selmer group that bounds the Mordell-Weil rank for each abelian subvariety corresponding to a non-trivial prime power divisor of the degree. In the case that the Chabauty condition holds for that abelian subvariety, we show how to bound the number of rational points on the curve.