Computing the Cassels-Tate pairing on the 2-Selmer group of a genus 2 Jacobian

2023/06/09 by Tom Fisher, Jiali Yan, Fisher, Tom +1 · 1 citation
Mathematics · Medicine · #11G10 #11G30 #Algebraic Geometry and Number Theory #Berberine and alkaloids research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2306.06011

openalex publication_date 2023/06/09 · openalex created_date 2023/06/13 · openalex updated_date 2026/07/28

Abstract

We describe a method for computing the Cassels-Tate pairing on the 2-Selmer group of the Jacobian of a genus 2 curve. This can be used to improve the upper bound coming from 2-descent for the rank of the group of rational points on the Jacobian. Our method remains practical regardless of the Galois action on the Weierstrass points of the genus 2 curve. It does however depend on being able to find a rational point on a certain twisted Kummer surface. The latter does not appear to be a severe restriction in practice. In particular, we have used our method to unconditionally determine the ranks of all genus 2 Jacobians in the L-functions and modular forms database (LMFDB).

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• Determining the rational points on a curve of genus 2 and Mordell-Weil rank 1

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