Tenth Algorithmic Number Theory Symposium ANTSX

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 MordellWeil 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 MordellWeil rank by demonstrating the existence of nontrivial elements in the ShafarevichTate group. As an example we compute the MordellWeil rank of the Jacobian of a genus 4 curve over Q by determining that the$3primary part of the ShafarevichTate group is isomorphic to Z/3 x Z/3.
Files available: paper (PDF)
© 201112 Kiran S. Kedlaya (with thanks to Pierrick Gaudry and Emmanuel ThomÃ©)
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