Second $p$-descents on elliptic curves

Brendan Creutz

Abstract

Let $p$ be a prime and let $C$ be a genus one curve over a number field $k$ representing an element of order dividing $p$ in the Shafarevich-Tate group of its Jacobian. We describe an algorithm which computes the set of $D$ in the Shafarevich-Tate group such that $pD=C$ and obtains explicit models for these $D$ as curves in projective space. This leads to a practical algorithm for performing explicit 9-descents on elliptic curves over $\mathbb{Q}$

Keywords: elliptic curves, descent, Shafarevich-Tate group.

AMS Subject Classification: Primary 11G05; secondary 11Y50.