Computing the Cassels-Tate Pairing in the Case of a Richelot Isogeny

Abstract

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 $\varphi :J\to \hat{J}$ 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 ${\text{Sel}}^{\hat{\varphi }}(\hat{J})\times {\text{Sel}}^{\hat{\varphi }}(\hat{J})$ where $\hat{\varphi }$ is the dual isogeny of $\varphi$. 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.