Rational Points on a Family of Genus 3 Hyperelliptic Curves

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.

17 pages, 1 table