Rational Points on a Family of Genus 3 Hyperelliptic Curves Open Access
Hernandez, Roberto (Summer 2025)
Abstract
Let C/Q be a curve defined over the rational numbers of genus g ≥ 2. In 1922, Mordell conjectured that such a curve had only finitely many rational points. This question puzzled mathematicians for over 60 years until Faltings’ proved it in 1983. In fact, Faltings’ proved the more general version which said that the curve was allowed to be defined over any number field. This was a groundbreaking result and signified a huge advancement in arithmetic geometry. Unfortunately, Faltings’ theorem isn’t effective, meaning that the proof doesn’t actually tell us how to find the points, only that there’s finitely many. Despite new, simpler proofs of Faltings’ theorem by Voljta and Bombieri, we still do not have a grasp of an effective version of Faltings’ theorem. In practice, Chabauty–Coleman is a powerful tool for finding rational points on curves, but there are examples for which this method fails. We give a detailed exposition of the method developed by Dem’yanenko and Manin to explicitly find rational points on curves. To this end, we construct a family of genus 3 hyperelliptic curves for which we can compute the rational points on via the method of Dem’yanenko–Manin while avoiding the method of Chabauty–Coleman.
Table of Contents
Introduction Background Dem'yanenko-Manin Family of Genus 3 Curves Height Bounds
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