Infinitely many hyperelliptic curves with exactly two rational points. (arXiv:1904.00215v2 [math.NT] UPDATED)

In this paper, we construct some families of infinitely many hyperelliptic curves of genus $2$ with exactly two rational points. In the proof, we first show that the Mordell-Weil ranks of these hyperelliptic curves are $0$ and then determine the sets of rational points by using the Lutz-Nagell type theorem for hyperelliptic curves which was proven by Grant. 查看全文>>