Second p descents on elliptic curves

TL;DR

This paper introduces an algorithm for computing specific elements in the Shafarevich-Tate group of elliptic curves, enabling practical 9-descents over the rationals, which advances the computational techniques in number theory.

Contribution

It presents a novel algorithm for second p-descents on elliptic curves, providing explicit models and practical methods for higher descents.

Findings

Algorithm computes elements D with pD = C in the Shafarevich-Tate group

Enables practical 9-descents on elliptic curves over rationals

Provides explicit models for these elements as curves in projective space

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 9-descents on elliptic curves over the rationals.