Finding rational points on elliptic curves using 6-descent and 12-descent

TL;DR

This paper combines recent 3-descent and 4-descent methods to effectively find rational points on elliptic curves, confirming the predicted ranks for curves in a major database, and advancing computational techniques in number theory.

Contribution

It introduces a novel combination of 3-descent and 4-descent techniques to identify generators of the Mordell-Weil group on elliptic curves.

Findings

Confirmed the rank predictions for all elliptic curves of prime conductor in the database.

Demonstrated the effectiveness of combined descent methods for large height generators.

Enhanced computational approaches for analyzing elliptic curves.

Abstract

We explain how recent work on 3-descent and 4-descent for elliptic curves over Q can be combined to search for generators of the Mordell-Weil group of large height. As an application we show that every elliptic curve of prime conductor in the Stein-Watkins database has rank at least as large as predicted by the conjecture of Birch and Swinnerton-Dyer.