×

The behavior of the Mordell-Weil group of elliptic curves. (English) Zbl 0741.14010

The authors have calculated via computer the (predicted) Mordell-Weil rank of elliptic curves which are defined over the rational numbers and are selected to be of prime discriminant. Then they found a lot of elliptic curves of Mordell-Weil rank 2 and many interesting facts. From these results they confirm the computational evidence of several conjetures about the arithmetic theory of elliptic curves.
Reviewer: H.Maeda (Tokyo)

MSC:

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14Q05 Computational aspects of algebraic curves
14H52 Elliptic curves
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G05 Elliptic curves over global fields
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] B. J. Birch and W. Kuyk , Modular functions of one variable. IV, Lecture Notes in Mathematics, Vol. 476, Springer-Verlag, Berlin-New York, 1975. · Zbl 0315.14014
[2] Armand Brumer and Kenneth Kramer, The rank of elliptic curves, Duke Math. J. 44 (1977), no. 4, 715 – 743. · Zbl 0376.14011
[3] Joe P. Buhler, Benedict H. Gross, and Don B. Zagier, On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3, Math. Comp. 44 (1985), no. 170, 473 – 481. · Zbl 0606.14021
[4] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II, Proc. Roy. Soc. London Ser. A 322 (1971), no. 1551, 405 – 420. · Zbl 0212.08101
[5] Dorian Goldfeld, Conjectures on elliptic curves over quadratic fields, Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979) Lecture Notes in Math., vol. 751, Springer, Berlin, 1979, pp. 108 – 118. · Zbl 0417.14031
[6] F. Gouvea and B. Mazur, The square-free sieve and the rank of Mordell-Weil, preprint, April 1989.
[7] Daniel R. Grayson, The arithogeometric mean, Arch. Math. (Basel) 52 (1989), no. 5, 507 – 512. · Zbl 0686.14040
[8] D. Zagier and G. Kramarz, Numerical investigations related to the \?-series of certain elliptic curves, J. Indian Math. Soc. (N.S.) 52 (1987), 51 – 69 (1988). · Zbl 0688.14016
[9] B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33 – 186 (1978). · Zbl 0394.14008
[10] Jean-François Mestre, Formules explicites et minorations de conducteurs de variétés algébriques, Compositio Math. 58 (1986), no. 2, 209 – 232 (French). · Zbl 0607.14012
[11] J.-F. Mestre, La méthode des graphes. Exemples et applications, Proceedings of the international conference on class numbers and fundamental units of algebraic number fields (Katata, 1986) Nagoya Univ., Nagoya, 1986, pp. 217 – 242 (French). · Zbl 0621.14021
[12] J.-F. Mestre and J. Oesterlé, Courbes de Weil semi-stables de discriminant une puissance \?-ième, J. Reine Angew. Math. 400 (1989), 173 – 184 (French). · Zbl 0693.14004
[13] Jordi Quer, Corps quadratiques de 3-rang 6 et courbes elliptiques de rang 12, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 6, 215 – 218 (French, with English summary). · Zbl 0622.14025
[14] Karl Rubin, The work of Kolyvagin on the arithmetic of elliptic curves, Arithmetic of complex manifolds (Erlangen, 1988) Lecture Notes in Math., vol. 1399, Springer, Berlin, 1989, pp. 128 – 136.
[15] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. · Zbl 0585.14026
[16] Joseph H. Silverman, Computing heights on elliptic curves, Math. Comp. 51 (1988), no. 183, 339 – 358. · Zbl 0656.14016
[17] John T. Tate, The arithmetic of elliptic curves, Invent. Math. 23 (1974), 179 – 206. · Zbl 0296.14018
[18] Lawrence C. Washington, Number fields and elliptic curves, Number theory and applications (Banff, AB, 1988) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 265, Kluwer Acad. Publ., Dordrecht, 1989, pp. 245 – 278. · Zbl 0693.14017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.