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

Yan, Jiali

Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 659-674

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 $\varphi :J\to \widehat{J}$ 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 ${\mathrm{Sel}}^{\widehat{\varphi }}\left(\widehat{J}\right)\times {\mathrm{Sel}}^{\widehat{\varphi }}\left(\widehat{J}\right)$ where $\widehat{\varphi }$ 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.

Dans cet article, nous étudions l’accouplement de Cassels-Tate sur les jacobiennes des courbes de genre $2$ possédant une isogénie dite de Richelot. Soit $\varphi :J\to \widehat{J}$ une isogénie de Richelot entre les jacobiennes de deux courbes de genre $2$. Nous donnons une formule explicite et un algorithme pratique pour calculer l’accouplement de Cassels-Tate sur ${\mathrm{Sel}}^{\widehat{\varphi }}\left(\widehat{J}\right)\times {\mathrm{Sel}}^{\widehat{\varphi }}\left(\widehat{J}\right)$ où $\widehat{\varphi }$ est l’isogénie duale de $\varphi$. Ces résultats sont obtenus sous l’hypothèse simplificatrice que tous les points de $2$-torsion sur $J$ sont définis sur $K$. Nous donnons un exemple explicite qui montre que nous pouvons transformer la descente par l’isogénie de Richelot en $2$-descente en calculant l’accouplement de Cassels-Tate.

Classification : 11N56, 14G42
Keywords: Genus 2 Curve, Cassels-Tate Pairing, Richelot isogeny

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