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.