## 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

Zbl 0607.14012
Full Text:

### 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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