Rational Points on a Family of Genus 3 Hyperelliptic Curves
Roberto Hernandez
TL;DR
This work addresses the problem of determining $C_a(\mathbb{Q})$ for a family of genus $3$ hyperelliptic curves $C_a$ by exploiting a Dem'yanenko–Manin framework rather than the Chabauty–Coleman method. The authors show that $\mathrm{Jac}(C_a)$ decomposes as $E_a \times E_a \times E'$, with two independent maps $\varphi_1, \varphi_2: C_a \to E_a$, enabling an effective finite-search approach even when $\mathrm{Rank}\,\mathrm{Jac}(C_a)$ is large. They obtain conditions under which $\mathrm{Rank}(E_a)=1$ and $\mathrm{Rank}(E')$ is nonzero (via parity/root-number arguments and descent), and they derive explicit height-difference bounds $|\hat h_{E_a}(\varphi_1(P)) - \hat h_{E_a}(\varphi_2(P))| \le 51.18 + 2\log(a)$ to restrict possible images of rational points. The paper provides a concrete computation for $C_{237}$, determining $C_{237}(\mathbb{Q})$, and reports that a large proportion of curves in the family (e.g., $a \le 1000$ with certain primality conditions) admit the Dem'yanenko–Manin method, illustrating a scalable strategy for higher-genus rational-point computation via Jacobian decomposition and elliptic-factor control.
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.