Explicit descent in the Picard group of a cyclic cover of the projective line

Brendan Creutz

Abstract

Given a curve \(X\) of the form \(y^p = h(x)\) over a number field, one can use descents to obtain explicit bounds on the Mordell–Weil rank of the Jacobian or to prove that the curve has no rational points. We show how, having performed such a descent, one can easily obtain additional information which may rule out the existence of rational divisors on \(X\) of degree prime to \(p\). This can yield sharper bounds on the Mordell–Weil rank by demonstrating the existence of nontrivial elements in the Shafarevich–Tate group. As an example we compute the Mordell–Weil rank of the Jacobian of a genus 4 curve over \(\mathbb{Q}\) by determining that the 3-primary part of the Shafarevich–Tate group is isomorphic to \(\mathbb{Z}/3 × \mathbb{Z}/3\)

Keywords: Descent, Mordell–Weil group, Shafarevich–Tate group.