Comptes Rendus
Théorie des nombres
Rational points on a certain genus 2 curve
Xuan Tho Nguyen1
1 Hanoi University of Science and Technology
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1071-1073.

We give a correct proof to the fact that all rational points on the curve

y 2 =(x 2 +1)(x 2 +3)(x 2 +7)

are ±∞ and (±1,±8). This corrects previous works of Cohen [3] and Duquesne [4, 5].

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.471
Classification : 14G05, 14H99
Xuan Tho Nguyen 1

1 Hanoi University of Science and Technology
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Rational points on a certain genus 2 curve},
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Xuan Tho Nguyen. Rational points on a certain genus 2 curve. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1071-1073. doi : 10.5802/crmath.471. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.471/

[1] Wieb Bosma; John Cannon; Catherine Playoust The MAGMA algebra system. I. The user language, J. Symb. Comput., Volume 24 (1997) no. 3-4, pp. 235-265 | DOI | MR | Zbl

[2] Nils Bruin; Michael Stoll Two cover descent on hyperelliptic curves, Math. Comput., Volume 78 (2009) no. 268, pp. 2347-2370 | DOI | MR | Zbl

[3] Henri Cohen Number Theory. Volume II: Analytic and Mordern Tools, Graduate Texts in Mathematics, 240, Springer, 2007

[4] Sylvain Duquesne Calculs Effectifs des Points Entiers et Rationnels sur les Courbes, Ph. D. Thesis, Université Bordeaux I (2001)

[5] Sylvain Duquesne Points rationnels et méthode de Chabauty elliptique, J. Théor. Nombres Bordeaux, Volume 15 (2003) no. 1, pp. 99-113 | DOI | Numdam | MR | Zbl

[6] Eugene Victor Flynn; Joseph L. Wetherell Finding Rational Points on Bielliptic Genus 2 Curves, Manuscr. Math., Volume 100 (1999) no. 4, pp. 519-533 | DOI | MR | Zbl

[7] Michael Stoll Slides of the talk Rational Diophantine quintuples and diagonal genus 5 curves (https://www.mathe2.uni-bayreuth.de/stoll/talks/Manchester-2017-print.pdf)

[8] Michael Stoll Rational Diophantine quintuples and diagonal genus 5 curves, Acta Arith., Volume 190 (2019) no. 3, pp. 239-261 | DOI | MR | Zbl

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