Infinitely many hyperelliptic curves with exactly two rational points $(2)$

Hideki Matsumura

Published 2020-05-05Version 1

In the previous paper, Hirakawa and the author considered a certain infinite family of hyperelliptic curves $C^{(p;i,j)}$ parametrized by a prime number $p$ and integers $i$, $j$, and proved that some of them have exactly two obvious rational points. In this paper, we extend the above work. In the proof, we consider another hyperelliptic curve $C'^{(p;i,j)}$ whose Jacobian variety is isogenous to that of $C^{(p;i,j)}$, and prove that the Mordell-Weil rank of the Jacobian variety of $C'^{(p;i,j)}$ is $0$ by the standard $2$-descent argument. Then, we determine the set of rational points of $C^{(p;i,j)}$ by using the Lutz-Nagell type theorem for hyperelliptic curves that was proven by Grant.