Computing the Cassels-Tate Pairing for Genus Two Jacobians with Rational Two Torsion Points

TL;DR

This paper presents explicit formulas and algorithms for computing the Cassels-Tate pairing and Obstruction map on genus two Jacobians with all 2-torsion points rational, improving rank bounds via practical computations.

Contribution

It provides the first explicit formulas and algorithms for the Cassels-Tate pairing on genus two Jacobians with rational 2-torsion points, along with a worked example.

Findings

Explicit formulas for Cassels-Tate pairing and Obstruction map.

Practical algorithms for computing these pairings.

Improved rank bounds demonstrated through example.

Abstract

In this paper, we give an explicit formula as well as a practical algorithm for computing the Cassels-Tate pairing on $\text{Sel}^{2}(J) \times \text{Sel}^{2}(J)$ where $J$ is the Jacobian variety of a genus two curve under the assumption that all points in $J[2]$ are $K$-rational. We also give an explicit formula for the Obstruction map $\text{Ob}: H^1(G_K, J[2]) \rightarrow \text{Br}(K)$ under the same assumption. Finally, we include a worked example demonstrating we can indeed improve the rank bound given by a 2-descent via computing the Cassels-Tate pairing.