Computing the Cassels-Tate pairing for genus two jacobians with rational two-torsion points

In this paper, we give an explicit formula as well as a practical algorithm for computing the Cassels-Tate pairing on Sel(2)(J) x 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 Ob:H-1(G(K), J[2]) -> Br(K) under the same assumption. Finally, we include a worked example demonstrating that we can improve the rank bound given by a 2-descent via computing the Cassels-Tate pairing.