Vol. 1, No. 1, 2013

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

Brendan Creutz

Vol. 1 (2013), No. 1, 295–315

Given a curve X of the form yp = 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 by determining that the 3-primary part of the Shafarevich-Tate group is isomorphic to 3 × 3.

abelian variety, Mordell-Weil group, explicit descent
Mathematical Subject Classification 2010
Primary: 11G10
Secondary: 11Y50
Published: 14 November 2013
Brendan Creutz
School of Mathematics and Statistics
University of Sydney
Sydney, NSW 2006