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Experimental study of the rank of families of elliptic curves over \(\mathbb{Q}\). (Étude expérimentale du rang de familles de courbes elliptiques sur \(\mathbb{Q}\).) (French) Zbl 0873.11033

The author studies 93 elliptic curves over \({\mathbb{Q}}(t)\). These curves have Mordell-Weil groups of rank \(r\leq 4\). The author studies the ranks of the Mordell-Weil groups of the specializations of these curves for many \(t\in{\mathbb{Z}}\). His calculation involves computing the ranks of the groups of rational points of 66918 elliptic curves over \({\mathbb{Q}}\). Lower bounds for the ranks are established by searching for rational points of small height. Upper bounds are deduced from a parity criterion and certain explicit formulas [J.-F. Mestre, Compos. Math. 58, 209-232 (1986; Zbl 0607.14012)]. Here the author assumes the truth of the Shimura-Taniyama-Weil conjecture and of the Birch-Swinnerton-Dyer conjecture. The author makes various interesting observations concerning this extensive numerical experiment. One of which is the fact that the proportion of specialized elliptic curves \(E_t\) that have a Mordell-Weil group of rank \(r+s\) seems to be independent of the rank \(r\) of the elliptic curve \(E\) over \({\mathbb{Q}}(T)\).
Reviewer: R.Schoof (Roma)

MSC:

11G05 Elliptic curves over global fields
14H52 Elliptic curves
14Q05 Computational aspects of algebraic curves

Citations:

Zbl 0607.14012
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References:

[1] Atkin A. O. L., Math. Ann. 185 pp 134– (1970) · Zbl 0177.34901
[2] Billard Hervé, ”Sur la repartition des points rationnels de surfaces algébriques” (1995)
[3] Bull. Amer. Math. Soc. 23 pp 375– (1990) · Zbl 0741.14010
[4] Buhler J. P., Math. Comp. 44 pp 473– (1985)
[5] Deligne Pierre, Modular Functions of One Variable pp 501– (1973)
[6] Fermigier Stéfane, C. R. Acad. Sci. Paris 315 pp 719– (1992)
[7] Fouvry Etienne, Monatshefte für Mathematik 116 pp 111– (1993) · Zbl 0796.14022
[8] Gebel Josef, Elliptic curves and related topics pp 61– (1994)
[9] Geist Al, PVM: Parallel Virtual MachineA Users ’ Guide and Tutorial for Networked Parallel Computing (1994) · Zbl 0849.68032
[10] Mestre Jean-Frangois, Compositio Mathematica 58 pp 209– (1986)
[11] Mestre Jean-Frangois, C. R. Acad. Sci. Paris (série 1) 313 pp 139– (1991)
[12] Mestre Jean-Frangois, C. R. Acad. Sci. Paris (série 1) 313 pp 171– (1991)
[13] Mestre Jean-Frangois, C. R. Acad. Sci. Paris 314 pp 453– (1992)
[14] Michel Philippe, Monatshefte für Mathematik 120 pp 127– (1995) · Zbl 0869.11052
[15] Nagao Koh-ichi, Proc. Japan Acad. 68 pp 287– (1992) · Zbl 0791.14014
[16] Nagao Koh-ichi, Proc. Japan Acad. 69 pp 291– (1992) · Zbl 0794.14014
[17] Nagao Koh-ichi, ”An example of elliptic curve over Q with rank 21” (1993) · Zbl 0794.14014
[18] Nagao Koh-ichi, ”An example of elliptic curve over Q() with rank 13” (1993) · Zbl 0794.14014
[19] Rohrlich David E., Comm. Math. 87 pp 119– (1993)
[20] Shioda Tetsuji, Comm. Math. Univ. Sancti Pauli 39 (1990)
[21] Silverman Joseph H., The Arithmetic of Elliptic Curves (1986) · Zbl 0585.14026
[22] Tate, John. 1975.”Algorithm for Determining the Type of a Singular Fiber in an Elliptic Pencil”33–53. Berlin: Springer. [Tate 1975], dans Modular Functions of one Variable (Antwerp 1972), volume 4, Lect. Notes in Math 476
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