Higher descents on an elliptic curve with a rational 2-torsion point

TL;DR

This paper discusses advanced descent techniques on elliptic curves with rational 2-torsion points, improving bounds for the Mordell-Weil group rank and enabling practical 8-descent computations for curves with large discriminants.

Contribution

It extends existing descent methods by incorporating rational 2-torsion points, particularly enabling practical 8-descent on elliptic curves over ield  with large discriminants.

Findings

Enhanced 8-descent method for elliptic curves with full rational 2-torsion.

Practical implementation for curves with large discriminants.

Improved bounds for Mordell-Weil group rank.

Abstract

Let $E$ be an elliptic curve over a number field $K$. Descent calculations on $E$ can be used to find upper bounds for the rank of the Mordell-Weil group, and to compute covering curves that assist in the search for generators of this group. The general method of 4-descent, developed in the PhD theses of Siksek, Womack and Stamminger, has been implemented in Magma (when $K={\mathbb Q}$) and works well for elliptic curves with sufficiently small discriminant. By extending work of Bremner and Cassels, we describe the improvements that can be made when $E$ has a rational 2-torsion point. In particular, when $E$ has full rational 2-torsion, we describe a method for 8-descent that is practical for elliptic curves $E/{\mathbb Q}$ with large discriminant.