Rational Points on a Family of Genus 3 Hyperelliptic Curves

TL;DR

This paper determines rational points on a specific family of genus 3 hyperelliptic curves using classical methods, highlighting the efficiency of Dem'yanenko and Manin's approach over Chabauty--Coleman.

Contribution

It applies and adapts Dem'yanenko and Manin's methods to compute rational points, improving upon Chabauty--Coleman's approach for this family.

Findings

Successfully computed rational points for certain family members.

Dem'yanenko and Manin's methods outperform Chabauty--Coleman in this context.

Incorporated root numbers to refine rank estimates of Jacobian decompositions.

Abstract

We compute the rational points on certain members of the following family of hyperelliptic curves \[C_a \colon y^2 = x^8 + (4-4a^4) x^6 + (8a^4 + 6)x^4 + (4-4a^4)x^2 + 1\] via the method first developed by Dem'yanenko \cite{dem1966rational} and then further generalized by Manin \cite{manin1969p}. In particular, we show that the method of Chabauty--Coleman, while applicable to certain members of this family, is not the most efficient way of computing $C_a(\mathbb{Q})$. We adapt the approach of \cite{kulesz1999application}, incorporating root numbers to further restrict the possible ranks of the elliptic curves arising in the Jacobian decomposition.