Rational Points on Genus 3 Real Hyperelliptic Curves

Basile G. R. Bossoto,Brice M. Miayoka,Regis F. Babindamana

doi:10.4236/ojdm.2021.114008

Abstract

We compute rational points on real hyperelliptic curves of genus 3 defined on whose Jacobian have Mordell-Weil rank r=0. We present an implementation in sagemath of an algorithm which describes the birational transformation of real hyperelliptic curves into imaginary hyperelliptic curves and the Chabauty-Coleman method to find C (). We run the algorithms in Sage on 47 real hyperelliptic curves of genus 3.

Highlights

IntroductionLet C be a hyperelliptic curve of the genus g ≥ 2 defined on. The Mordell conjecture proved by Fattings, gives that the set of rational points C ( ) is finite
Open AccessLet C be a hyperelliptic curve of the genus g ≥ 2 defined on
We present an implementation in sagemath of an algorithm which describes the birational transformation of real hyperelliptic curves into imaginary hyperelliptic curves and the Chabauty-Coleman method to find C ( )

Summary

Introduction

Let C be a hyperelliptic curve of the genus g ≥ 2 defined on. The Mordell conjecture proved by Fattings, gives that the set of rational points C ( ) is finite. 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. Algorithm 4 is this of Maria Inés de Frutos Fernaandez and Sachi Hashimoto in [5] and its implementation in [6] To the latter, we added a function allowing it to take the hyperelliptic curves of genus 3 and of rank 0 given by non-monic polynomials.

Background on Hyperelliptic Curves

Chabauty-Coleman Method

The Algorithm

Example

Conclusion