Chabauty-Coleman experiments for genus 3 hyperelliptic curves

2018/05/09 by Jennifer S. Balakrishnan, Francesca Bianchi, Balakrishnan, Jennifer S. +7 · 1 citation
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1805.03361

Abstract

We describe a computation of rational points on genus 3 hyperelliptic curves C defined over ℚ whose Jacobians have Mordell-Weil rank 1. Using the method of Chabauty and Coleman, we present and implement an algorithm in Sage to compute the zero locus of two Coleman integrals and analyze the finite set of points cut out by the vanishing of these integrals. We run the algorithm on approximately 17,000 curves from a forthcoming database of genus 3 hyperelliptic curves and discuss some interesting examples where the zero set includes global points not found in C(ℚ).

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• Rational and isolated quadratic points on hyperelliptic curves of rank 0 and small genus

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