Computing the Cassels-Tate pairing on the 2-Selmer group of a genus 2 Jacobian

TL;DR

This paper presents a practical method for computing the Cassels-Tate pairing on the 2-Selmer group of genus 2 Jacobians, aiding in determining their rational point ranks, with successful application to the LMFDB database.

Contribution

It introduces a new computational approach for the Cassels-Tate pairing on genus 2 Jacobians that remains practical regardless of Galois action complexities.

Findings

Successfully computed ranks for all genus 2 Jacobians in LMFDB

Method is practical and does not require restrictive Galois conditions

Enabled unconditional rank determination for numerous cases

Abstract

We describe a method for computing the Cassels-Tate pairing on the 2-Selmer group of the Jacobian of a genus 2 curve. This can be used to improve the upper bound coming from 2-descent for the rank of the group of rational points on the Jacobian. Our method remains practical regardless of the Galois action on the Weierstrass points of the genus 2 curve. It does however depend on being able to find a rational point on a certain twisted Kummer surface. The latter does not appear to be a severe restriction in practice. In particular, we have used our method to unconditionally determine the ranks of all genus 2 Jacobians in the L-functions and modular forms database (LMFDB).