On an analogue of the Lutz–Nagell Theorem for hyperelliptic curves

Grant, David

doi:10.1016/j.jnt.2012.02.023

Cited by 9 publications

(8 citation statements)

References 17 publications

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“…This proposition follows from the following Lutz-Nagell type theorem. Note that Grant [5] proved the above Lutz-Nagell type theorem in more general settings.…”

Section: Application Of the Lutz-nagell Type Theorem For Hyperellipti...mentioning

confidence: 92%

“…Since the F 2 -dimension of the 2-torsion part J (p;i,j) (Q) [2] is also 2, we obtain that rank(J (p;i,j) (Q)) = 0. Next, we determined the set of rational points on C (p;i,j) which map to torsion points on J (p;i,j) via the Abel-Jacobi map by the Lutz-Nagell type theorem [5,Theorem 3].…”

Section: Main Theoremmentioning

confidence: 99%

“…In §2, we carry out this task by the 2-descent argument [12], and in §3, we determine the set of rational points on C (p;i,j) which map to torsion points on J (p;i,j) via the Abel-Jacobi map by the Lutz-Nagell type theorem [5,Theorem 3]. Finally, we state some conjectures in §4.…”

Section: Main Theoremmentioning

confidence: 99%

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

Matsumura

2020

Preprint

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In the previous paper, Hirakawa and the author considered a certain infinite family of hyperelliptic curves C (p;i,j) parametrized by a prime number p and integers i, j, and proved that some of them have exactly two obvious rational points. In this paper, we extend the above work. In the proof, 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 the standard 2-descent argument. Then, we determine the set of rational points of C (p;i,j) by using the Lutz-Nagell type theorem for hyperelliptic curves that was proven by Grant.

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“…This proposition follows from the following Lutz-Nagell type theorem. Note that Grant [5] proved the above Lutz-Nagell type theorem in more general settings.…”

Section: Application Of the Lutz-nagell Type Theorem For Hyperellipti...mentioning

confidence: 92%

Section: Main Theoremmentioning

confidence: 99%

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

Matsumura

2020

Preprint

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“…Our proof of Theorem 1.1 is based on natural generalizations of the 2-descent argument and the Lutz-Nagell theorem (cf. [11], [5,Theorem 3]). Recall that, in the case of elliptic curves, the 2-descent argument makes it possible to bound the Mordell-Weil rank of an elliptic curve by means of the 2-Selmer group, and the Lutz-Nagell theorem makes it possible to determine the torsion points of an elliptic curve by means of the discriminant.…”

Section: Main Theoremmentioning

confidence: 99%

“…In §3, we carry out this task by applying the Lutz-Nagell type theorem for hyperelliptic curves which was proven by Grant [5]. 2 Remark 1.2.…”

Section: Main Theoremmentioning

confidence: 99%

Infinitely many hyperelliptic curves with exactly two rational points

Hirakawa¹,

Matsumura²

2019

Preprint

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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

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Chabauty–Coleman Experiments for Genus 3 Hyperelliptic Curves

Balakrishnan

Bianchi

Cantoral-Farfán

et al. 2019

Association for Women in Mathematics Series

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We describe a computation of rational points on genus 3 hyperelliptic curves C defined over Q whose Jacobians have Mordell-Weil rank 1. Using the method of Chabauty and Coleman, we present and implement an algorithm in Sage to compute the zero locus of two Coleman integrals and analyze the finite set of points cut out by the vanishing of these integrals. We run the algorithm on approximately 17,000 curves from a forthcoming database of genus 3 hyperelliptic curves and discuss some interesting examples where the zero set includes global points not found in C(Q).

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On an analogue of the Lutz–Nagell Theorem for hyperelliptic curves

Cited by 9 publications

References 17 publications

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

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

Infinitely many hyperelliptic curves with exactly two rational points

Chabauty–Coleman Experiments for Genus 3 Hyperelliptic Curves

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