Computing the Cassels-Tate Pairing on the Selmer group of a Richelot Isogeny

In this paper, we study the Cassels-Tate pairing on Jacobians of genus two curves admitting a special type of isogenies called Richelot isogenies. Let ϕ:J→J ^ be a Richelot isogeny between two Jacobians of genus two curves. We give an explicit formula as well as a practical algorithm to compute the Cassels-Tate pairing on Sel ϕ ^ (J ^)×Sel ϕ ^ (J ^) where ϕ ^ is the dual isogeny of ϕ. The formula and algorithm are under the simplifying assumption that all two-torsion points on J are defined over K. We also include a worked example demonstrating we can turn the descent by Richelot isogeny into a 2-descent via computing the Cassels-Tate pairing.