Computing the Cassels-Tate pairing on the 3-Selmer group of an elliptic curve

TL;DR

This paper extends Cassels' method to compute the Cassels-Tate pairing on 3-Selmer groups of elliptic curves, enabling better rank bounds through 3-descent in specific cases.

Contribution

It introduces a practical method for computing the Cassels-Tate pairing on 3-Selmer groups, expanding previous 2-Selmer techniques with necessary modifications.

Findings

Method is practical for small examples

Can improve upper bounds for elliptic curve ranks

Extends Cassels' approach to 3-Selmer groups

Abstract

We extend the method of Cassels for computing the Cassels-Tate pairing on the 2-Selmer group of an elliptic curve, to the case of 3-Selmer groups. This requires significant modifications to both the local and global parts of the calculation. Our method is practical in sufficiently small examples, and can be used to improve the upper bound for the rank of an elliptic curve obtained by 3-descent.