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

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 $\text{Ob}:H^1(G_K,J[2])\to \text{Br}(K)$ 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.