论文标题
无限的许多高ellip曲曲线，完全有两个合理点：第二部分
Infinitely many hyperelliptic curves with exactly two rational points: Part II
论文作者
论文摘要
在上一篇论文中，Hirakawa和作者确定了特定无限曲线曲线家族的合理点集合$ c^{（p; i，j）} $由Prime Number Number $ P $和Integers $ i $，$ J $进行了参数。在证明中，我们使用了标准的$ 2 $ deScent参数和由Grant证明的Lutz-Nagell定理。在本文中，我们扩展了上述工作。通过使用下降定理，$ j = 2 $的证明将减少到独立于$ p $的等级$ 0 $的椭圆曲线。另一方面，对于奇数$ j $，我们考虑了另一条过度曲线$ c'^{（p; i，j）} $，其雅各比的品种对$ c^{（p; i，j）} $是不相同的，并且证明了jacobian jacobian jacobian jacobian级别的$ c'^（p; i i是$ i是$ ies $ i是$ c; i，i，然后，我们通过使用lutz-nagell type定理来确定$ c^{（p; i，j）} $的有理点集。
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 $C'^{(p;i,j)}$ 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 $2$-descent. Then, we determine the set of rational points of $C^{(p;i,j)}$ by using the Lutz-Nagell type theorem.
