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

TL;DR

The paper develops a method to enhance descent techniques on cyclic covers of the projective line, providing sharper bounds on Mordell-Weil ranks and insights into the Shafarevich-Tate group, exemplified by a genus 4 curve.

Contribution

It introduces a way to extract additional information from descents on cyclic covers, improving bounds on ranks and understanding of the Shafarevich-Tate group.

Findings

Computed the Mordell-Weil rank of a genus 4 Jacobian over Q.

Determined the 3-primary part of the Shafarevich-Tate group as Z/3 x Z/3.

Demonstrated the method's effectiveness in ruling out rational divisors of certain degrees.

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 Q by determining that the 3-primary part of the Shafarevich-Tate group is isomorphic to Z/3 x Z/3.