Infinitely many hyperelliptic curves with exactly two rational points

Abstract

We construct some infinite families of 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 a Lutz–Nagell type theorem for hyperelliptic curves which was proven by Grant.