Chabauty–Coleman Experiments for Genus 3 Hyperelliptic Curves

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).

Rational Points on Genus 3 Real Hyperelliptic Curves

“…Our objective is to compute the rational points in the case of real hyperelliptic curves of genus 3 whose Jacobian have a Mordel-Weil rank equal to 0. This fits into the particular case where r g < has been considered by Chabauty and the techniques developed by Coleman in 1980 [1], allow us to use the p-adic integration to explicitly computation the set of rational points [2].…”

Geometric Quadratic Chabauty

“…For simplicity, we restrict ourselves in this article to the case K = Q. Chabauty's method (1941) for determining C(Q) is to intersect, for a prime number p, in the p-adic Lie group of p-adic points of the Jacobian, the closure of the Mordell-Weil group with the p-adic points of the curve. There is a fair amount of evidence (mainly hyperelliptic curves of small genus [3]) that Chabauty's method, in combination with other methods such as the Mordell-Weil sieve, does determine all rational points when r < g, with r the Mordell-Weil rank and g the genus of C.…”

Goormaghtigh's equation: small parameters

“…We prove the following. Equation (1) has been the subject of much study. A good survey of results up to 2001 or so can be found in the paper of Shorey [24].…”