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

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

2012-04-26

arxiv.org/abs/1204.5803v1

Mathematics

Number Theory

Accepted to ANTS X

Scientific paper

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.

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