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

TL;DR

This paper provides an explicit formula and practical algorithm for computing the Cassels-Tate pairing on Jacobians of genus two curves with Richelot isogenies, under specific torsion assumptions, enabling more effective descent calculations.

Contribution

It introduces a new explicit formula and algorithm for Cassels-Tate pairing computation in the Richelot isogeny case, simplifying descent procedures.

Findings

Derived an explicit formula for the pairing.

Developed a practical algorithm for computation.

Demonstrated application with a worked example.

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 $\phi: J \rightarrow \widehat{J}$ 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 $\text{Sel}^{\widehat{\phi}}(\widehat{J}) \times \text{Sel}^{\widehat{\phi}}(\widehat{J})$ where $\widehat{\phi}$ is the dual isogeny of $\phi$. 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.