# zbMATH — the first resource for mathematics

The average rank of elliptic curves. I. (With an appendix by Oisín McGuinness: The explicit formula for elliptic curves over function fields). (English) Zbl 0783.14019
There has been much interest in the question of what to “expect” the rank of an elliptic curve over $$\mathbb{Q}$$ to be. The traditional philosophy is that most elliptic curves would have as small of a rank as allowed by the sign of their functional equations, namely either 0 or 1. More recent work has uncovered families of elliptic curves with somewhat higher rank than this philosophy predicts. – This paper studies the average analytic rank of elliptic curves over $$\mathbb{Q}$$ (as given by their Hasse-Weil $$L$$- functions), modulo several standard conjectures. The elliptic curves are ordered, essentially by their Faltings’ height. The paper shows that the average rank is then at most 2.3.
The techniques of this paper are analytic making use of the explicit formulae. It assumes the Taniyama-Weil conjecture and a form of generalized Riemann hypothesis. For the results to apply to the ranks of the Mordell-Weil groups of the elliptic curves, one needs to assume the weak Birch and Swinnerton-Dyer conjecture as well. While these conjectures are substantial, they are generally expected to be true.
Reviewer: J.Jones (Tempe)

##### MSC:
 14H52 Elliptic curves 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14H05 Algebraic functions and function fields in algebraic geometry 11G05 Elliptic curves over global fields
Full Text:
##### References:
 [1] [BrZ] Brumer, A.: The Average Behaviour of the Zeroes ofL-functions of Elliptic Curves. (In preparation) [2] [BMcG] Brumer, A., McGuinness, O.: The Behaviour of the Mordelll-Weil Group of Elliptic Curves, Bull. Am. Math. Soc.23, 375-381 (1990) · Zbl 0741.14010 · doi:10.1090/S0273-0979-1990-15937-3 [3] [Dav] Davenport, H.: Multiplicative Number Theory, 2nd edition. Berlin Heidelberg New York: Springer 1980 [4] [De1] Deligne, P.: Les constantes des ?quations fonctionelles des fonctions L. In: Deligne, P., Kuyk, W. (eds.) Modular Functions of One Variable II. (Lect. Notes Math., vol. 349) Berlin Heidelberg New York: Springer 1973 [5] [De2] Deligne, P.: La Conjecture de Weil I, II. Publ. Math., Inst. Hautes ?tud. Sci.43,52 (1974, 1981) [6] [FNT] Fouvry, E., Nair, M., Tenenbaum, G.: L’ensemble Exceptionnel dans la Conjecture de Szpiro. (Preprint, received February 1991) · Zbl 0770.11030 [7] [Go1] Goldfeld, D.: Sur les produits partiels eul?riens attach?s aux courbes elliptiques. C.R. Acad. Sci., Paris294, 471-474 (1982) · Zbl 0494.14007 [8] [Go2] Goldfeld, D.: Conjectures on Elliptic Curves over Quadratic Fields. In: Nathanson, M.B. (ed.) Number Theory Carbondale, pp. 108-118 (Lect. Notes Math., vol. 751) Berlin Heidelberg New York: Springer 1979 [9] [GoH] Goldfeld, D., Hoffstein, J.: On the Number of Fourier Coefficients that Determine a Modular Form. (to appear) · Zbl 0805.11040 [10] [La] Lang, S.: Algebraic Number Theory. Reading: Addison-Wesley 1970 · Zbl 0211.38404 [11] [HSi] Hindry, M., Silverman, J.H.: The canonical height and integral points on elliptic curves. Invent. Math.93, 419-450 (1988) · Zbl 0657.14018 · doi:10.1007/BF01394340 [12] [KZ] Kramarz, G., Zagier, D.: Numerical Investigations Related to theL-series of Certain Elliptic Curves. J. Indian Math. Soc., New. Ser.52, 51-60 (1987) · Zbl 0688.14016 [13] [Mo] Moreno, C.: Explicit Formulas in the Theory of Automorphic Forms. (Lect. Notes Math., vol. 626) Berlin Heidelberg New York: Springer 1977 · Zbl 0367.10023 [14] [Me] Mestre, J.-F.: Formules Explicites et Minorations de Conducteurs de vari?t?s Alg?briques. Compos. Math.58 (1982) [15] [Mu] Murty, R.: On Simple Zeroes of CertainL-Series. In: Mollin, R.A. (ed.) Proc. First Conf. Can. Number Theory Assn, pp 427-439. Alberta: Banff 1988 [16] [Oes] Oesterl?, J.: Emplilements de sph?res. S?minaire Bourbaki, Expos? 727, June 1990 [17] [Pa] Patterson, S.J.: An Introduction to the theory of the Riemann zeta-function, Cambridge: Cambridge University Press 1988 · Zbl 0641.10029 [18] [Se] Serre, J.-P.: Sur le nombre des points rationnels d’une courbe alg?brique sur un corps fini. C.R. Acad. Sci., Paris296, 397-402 (1983) · Zbl 0538.14015 [19] [Si] Silverman, J.H.: Heights and Elliptic Curves. In: Cornell, G., Silverman, J. (eds.) Arithmetic Geometry, pp. 253-265. Berlin Heidelberg New York: Springer 1986 [20] [Ta] Tate, J.: Number-theoretic Background. In: Borel, A., Casselman, W. (eds.) Automorphic Forms, Representations andL-Functions. (Proc. Symp. Pure Math., vol. 33, Part 2) Providence, RI: Am. Math. Soc. 1979 [21] [TaBS] Tate, J.: On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. S?minaire Bourbaki, Expos? 306, 1966 [22] [We1] Weil, A.: On the Analog of the modular group in characteristic p. In: Functional Analysis, In: Browder, F.E. (ed.) Proceedings of a conference in honor of M. Stone, pp. 211-223. Berlin Heidelberg New York: Springer, Collected Papers, vol. III, pp. 201-213 (1980) · Zbl 0226.10031 [23] [We2] Weil, A.: Basic Number Theory, 3rd edition. Berlin Heidelberg New York: Springer 1974 [24] [we3] Weil, A.. Dirichlet Series and Automorphic Forms. (Lect. Notes Math., vol. 189) Berlin Heidelberg New York: Springer 1971 · Zbl 0218.10046 [25] [We4] weil, A.: Sur les formules explicites de la th?orie des nombres Izv. Akad. Nauk SSSR, Ser. Mat.36, 3-18 (1972), Collected Papers, vol. III, pp. 249-264 (1980) · Zbl 0245.12010
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.