Points rationnels et méthode de Chabauty elliptique
Journal de Théorie des Nombres de Bordeaux, Tome 15 (2003) no. 1, pp. 99-113.

La méthode de Chabauty elliptique permet de calculer les points rationnels sur une courbe définie sur un corps de nombres lorsque le théorème de Chabauty ne s’applique pas, c’est à dire lorsque le rang de la jacobienne est supérieur au genre de la courbe. Nous exposons cette méthode et nous la généralisons dans de nouveaux cas en écrivant une version explicite du théorème de préparation de Weierstrass en 2 variables. En particulier nous calculons tous les points rationnels d’une courbe de genre 4 dont le rang de la jacobienne vaut 4.

The elliptic curve Chabauty method allows to compute rational points on curves defined over a number field when the rank of the jacobian is greater than the genus of the curve. We explain this method and generalize it to some new cases. In particular, we are able to compute rational points on a curve of genus 4 and rank 4.

@article{JTNB_2003__15_1_99_0,
     author = {Duquesne, Sylvain},
     title = {Points rationnels et m\'ethode de {Chabauty} elliptique},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {99--113},
     publisher = {Universit\'e Bordeaux I},
     volume = {15},
     number = {1},
     year = {2003},
     doi = {10.5802/jtnb.389},
     zbl = {02058856},
     mrnumber = {2019003},
     language = {fr},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.389/}
}
Sylvain Duquesne. Points rationnels et méthode de Chabauty elliptique. Journal de Théorie des Nombres de Bordeaux, Tome 15 (2003) no. 1, pp. 99-113. doi : 10.5802/jtnb.389. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.389/

[1] N. Bruin, On Generalised Fermat Equations. PhD Dissertation, Leiden, 1999.

[2] J.W.S. Cassels, Local Fields. London Math. Soc. Student Text, Vol. 3, Cambridge University Press, 1986. | MR 861410 | Zbl 0595.12006

[3] J.W.S. Cassels, E.V. Flynn, Prolegomena to a middlebrow Arithmetic of Curves of Genus 2. LMS Lecture Note Series, Vol. 230, Cambridge University Press, 1996. | MR 1406090 | Zbl 0857.14018

[4] C. Chabauty, Sur les points rationnels des variétés algébriques dont l'irrégularité est supérieure à la dimension. C. R. Acad. Sci. Paris 212, (1941). 1022-1024 | JFM 67.0105.02 | MR 11005 | Zbl 0025.24903

[5] R.F. Coleman, Effective Chabauty. Duke Math. J. 52 (1985), 765-780. | MR 808103 | Zbl 0588.14015

[6] J.E. Cremona, mwrank, disponible sur .

[7] J.E. Cremona, P. Serf, Computing the rank of elliptic curves over real quadratic number fields of class number 1. Math. Comp. 68 (1999), 1187-1200. | MR 1627777 | Zbl 0927.11034

[8] Z. Djabri, E.F. Schaefer, N.P. Smart, Computing the p-Selmer group of an elliptic curve. Trans. Amer. Math. Soc. 352 (2000), 5583-5597. | MR 1694286 | Zbl 0954.11022

[9] S. Duquesne, Calculs effectifs des points entiers et rationnels sur les courbes. Thèse de l'université Bordeaux I, 2001, disponible sur .

[10] S. Duquesne, Rational Points on Hyperelliptic Curves and an Explicit Weierstrass Preparation Theorem. Manuscripta Math. 108 (2002), 191-204. | MR 1918586 | Zbl 01801610

[11] G. Faltings, Endlichtkeitssästze für abelsche Varietäten ûber Zahlenkörpen. Invent. Math. 73 (1983), 349-366. | MR 718935 | Zbl 0588.14026

[12] E.V. Flynn, A flexible method for applying Chabauty's Theorem. Compositio Math. 105 (1997), 79-94. | MR 1436746 | Zbl 0882.14009

[13] E.V. Flynn, On Q-Derived Polynomials. Proc. Edinb. Math. Soc. 44:1 (2001), 103-110. | MR 1879212 | Zbl 1058.11045

[14] E.V. Flynn, J.L. Wetherell, Finding rational points on bielliptic genus 2 curves. Manuscripta Math. 100 (1999), 519-533. | MR 1734798 | Zbl 1029.11024

[15] E.V. Flynn, J.L. Wetherell, Covering Collections and a Challenge Problem of Serre. Acta Arith. 98 (2001), 197-205. | MR 1831612 | Zbl 1049.11066

[16] S. Siksek, Infinite descent on elliptic curves. Rocky Mountain J. Math. 25 (1995), 1501-1538. | MR 1371352 | Zbl 0852.11028

[17] J.H. Silverman, The arithmetic of Elliptic Curves. Graduate Texts in Math., Vol. 106, Springer-Verlag, 1986. | MR 817210 | Zbl 0585.14026

[18] J.H. Silverman, Computing heights on elliptic curves. Math. Comp. 51 (1988), 339-358. | MR 942161 | Zbl 0656.14016

[19] D. Simon, Computing the rank of elliptic curves over number fields. LMS J. Comput. Math. 5 (2002), 7-17 (electronic). | MR 1916919 | Zbl 1067.11015

[20] M. Stoll, Implementing 2-descent for Jacobians of hyperelliptic curves. Acta Arith. 98 (2001), 245-277. | MR 1829626 | Zbl 0972.11058

[21] T. Sugatani, Rings of convergent power series and Weierstrass preparatian theorem. Nagoya Math. J. 81 (1981), 73-78. | MR 607075 | Zbl 0413.30039

[22] J.L. Wetherell, Bounding the Number of Rational Points on Certain Curves of High Rank. PhD dissertation, University of California at Berkeley, 1997.