Infinitely many hyperelliptic curves with exactly two rational points

Abstract

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.Contents2), p ≡ −3 (mod 8) 18 3. Application of the Lutz-Nagell type theorem for hyperelliptic curves 21 4. Concluding remarks 23 Appendix A. 23 References 24

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

“…By taking Theorem 2.1 into account, Theorem 1.1 is an immediate consequence of the following proposition that we proved in [7].…”

“…> Points(C: Bound:=10ˆ5); > end if; > end for; (OUTPUTS) {@ (1 : 0 : 0), (0 : 0 : 1), (8 : -252 : 1), (8 : 252 : 1) @} for p = 17, {@ (1 : 0 : 0), (0 : 0 : 1) @} for p = 17. 7 The isomorphisms are given by the following:…”

Two-descent on some genus two curves

“…4.3.4, this already suffices to conclude for the curves discussed in the present paper that C p (Q) consists of the 6 Weierstrass points only, whenever p ≡ 7 mod 24. The recent preprint [9] studies some similar families of genus 2 curves, but with only 2 rational Weierstrass points. Again, computing the 2-Selmer group over Q allow the authors to identify congruence conditions on the prime p such that the corresponding Mordell-Weil group is finite.…”