On binary quartics and the Cassels-Tate pairing

TL;DR

This paper introduces a novel formula for the Cassels-Tate pairing on the 2-Selmer group of elliptic curves, leveraging binary quartic invariant theory and a special K3 surface, avoiding the need to solve conics.

Contribution

It provides a new explicit formula for the Cassels-Tate pairing that simplifies computations by eliminating the need to solve conics, using invariant theory and K3 surfaces.

Findings

New formula for Cassels-Tate pairing on 2-Selmer group

Avoids solving conics in the computation process

Utilizes invariant theory of binary quartics and K3 surfaces

Abstract

We use the invariant theory of binary quartics to give a new formula for the Cassels-Tate pairing on the $2$-Selmer group of an elliptic curve. Unlike earlier methods, our formula does not require us to solve any conics. An important role in our construction is played by a certain $K3$ surface defined by a $(2,2,2)$-form.