Infinitely many hyperelliptic curves with exactly two rational points: Part II

In the previous paper, Hirakawa and the author determined the set of rational points of a certain infinite family of hyperelliptic curves $C^{(p;i,j)}$ parametrized by a prime number $p$ and integers $i$, $j$. In the proof, we used the standard $2$-descent argument and a Lutz-Nagell theorem that was proven by Grant. In this paper, we extend the above work. By using the descent theorem, the proof for $j=2$ is reduced to elliptic curves of rank $0$ that are independent of $p$. On the other hand, for odd $j$, we consider another hyperelliptic curve $C^{\mathrm{\prime }(p;i,j)}$ whose Jacobian variety is isogenous to that of $C^{(p;i,j)}$, and prove that the Mordell-Weil rank of the Jacobian variety of $C^{\mathrm{\prime }(p;i,j)}$ is $0$ by $2$-descent. Then, we determine the set of rational points of $C^{(p;i,j)}$ by using the Lutz-Nagell type theorem.