Computing the Cassels-Tate pairing on 3-isogeny Selmer groups via cubic norm equations

TL;DR

This paper introduces a new method for computing the Cassels-Tate pairing on 3-isogeny Selmer groups of elliptic curves, enhancing rank bounds and providing an efficient algorithm for solving cubic norm equations.

Contribution

It presents a novel algorithm for cubic norm equations that bypasses S-unit computations and improves the rank bounds for elliptic curves using 3-isogeny descent.

Findings

Enhanced upper bound on elliptic curve rank from 3-isogeny descent

Exact rank determination for specific elliptic curves with torsion subgroup of order 3

Efficient cubic norm equation solving algorithm avoiding S-unit computations

Abstract

We explain a method for computing the Cassels-Tate pairing on the 3-isogeny Selmer groups of an elliptic curve. This improves the upper bound on the rank of the elliptic curve coming from a descent by 3-isogeny, to that coming from a full 3-descent. One ingredient of our work is a new algorithm for solving cubic norm equations, that avoids the need for any S-unit computations. As an application, we show that the elliptic curves with torsion subgroup of order 3 and rank at least 13, found by Eroshkin, have rank exactly 13.