zbMATH — the first resource for mathematics

An infinite family of elliptic curves over \(\mathbb{Q}\) with large rank via Néron’s method. (English) Zbl 0766.14024
Here the author gives a simplified, complete and explicit (so explicit to have numerical examples) version of Néron’s construction [A. Néron, Proc. Internat. Congr. Math., Amsterdam 1954, Vol. III, 481-488 (1956; Zbl 0074.159)] of an infinite family of elliptic curves over \(\mathbb{Q}\) with rank 11: A very nice result and a very well written paper.
Reviewer: E.Ballico (Povo)

14H52 Elliptic curves
11G05 Elliptic curves over global fields
Full Text: DOI EuDML
[1] [DP] Demazure, M., Pinkham, H., Teissier, B.: Séminaire sur les Singularités des Surfaces. (Lect. Notes Math. vol. 777) Berlin Heidelberg New York: Springer 1980
[2] [F] Fried, M.: Construction arising from Néron’s high rank curves. Trans. Am. Math. Soc.281, 615-631 (1984) · Zbl 0609.14016
[3] [M] Manin, Ju.: Cubic Forms, 2nd ed., Amsterdam: North-Holland (1986) · Zbl 0582.14010
[4] [N1] Néron, A.: Propriétés arithmétiques de certaines familles de courbes algébriques. Proc. Int. Cong. Math., 1954, Amsterdam, vol. III, pp. 481-488
[5] [N2] Néron, A.: Modèles minimaux des variétés abéliennes sur les corps locaux et globaux. Publ. Math., Inst. Hautes Étud. Sci.21 (1964)
[6] [Se] Serre, J.-P.: Lectures on the Mordell-Weil Theorem. Braunschweig: Vieweg (1989)
[7] [S1] Shioda, T.: Mordell-Weil lattices and Galois representation, I, II, III. Proc. Japan Acad.65A, 268-271; 296-299; 300-303 (1989) · Zbl 0715.14015
[8] [S2] Shioda, T.: On the Mordell-Weil lattices. Comment. Math. Univ. St. Pauli39, 211-240 (1990) · Zbl 0725.14017
[9] [S3] Shioda, T.: Construction of elliptic curves with high rank via the invariants of the Weyl groups. Rikkyo University (Preprint 1990)
[10] [Si1] Silverman, J.: Heights and the specialization map for families of abelian varieties. J. Reine Angew Math.342, 197-211 (1983) · Zbl 0505.14035 · doi:10.1515/crll.1983.342.197
[11] [Si2] Silverman, J.: The Arithmetic of Elliptic Curves. Berlin Heidelberg New York: Springer 1986
[12] [T] Tate, J.: Variation of the canonical height of a point depending on a parameter. Am. J. Math.105, 287-294 (1983) · Zbl 0618.14019 · doi:10.2307/2374389
[13] [To] Top, J.: Néron’s proof of the existence of elliptic curves overQ with rank at least 11. University Utrecht (Preprint 1987)
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.