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

2021/09/17 by Yan, Jiali
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2109.08257

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 ϕ: J → \widehatJ 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^\widehatϕ(\widehatJ) × Sel^\widehatϕ(\widehatJ) where \widehatϕ 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.

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