Abstract
Lety 2=x 3+a(t)x+b(t) be the equation of an elliptic curve over ℚ, wherea(t) andb(t) are polynomials overℤ. We give an upper bound on average of the rank of such curves whent varies overℤ under the assumption that theL-function attached to these curves satisfies classical assumptions. The proof is based on estimates of exponential sums, over varieties, which are treated by Deligne's theorem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Bibliographie
[A-S]Adolphson, A., Sperber, S.: Exponential sums and Newton polyhedra: Cohomology and estimates. Ann. Math.130, 367–406 (1989).
[B]Brumer, A.: The average rank of elliptic curves I. (Appendix byO. Mc Guinness). Invent. Math.109, 445–472 (1992).
[DH]Davenport, H., Heilbronn, H.: On the density of discriminants of cubic fields, II. Proc. Roy. Soc. London Ser. A322, 405–420 (1971).
[F]Fouvry, E.: Sur le comportement en moyenne du rang des courbesy 2=x 3+k. In: Séminaire de Théorie des Nombres de Paris, 1990–91, pp. 61–84. Basel: Birkhäuser. 1993.
[Go]Goldfeld, D.: Conjectures on elliptic curves over quadratic fields. In: Number Theory Carbondale 1979, pp 108–118. New York: Springer. 1979 (Lecture Notes in Mathematics 751).
[Ho]Hooley, C.: On exponential sums and certain of their applications. In: Journées Arithmétiques, pp 92–122. London: Math. Soc. 1980 (Lecture Notes Series 56).
[Ma]Mai, L.: Thèse de Doctorat. University of Toronto. 1991.
[M1]Mestre, J.-F.: Courbes elliptiques et formules explicites. In: Séminaire de Théorie des Nombres de Paris, 1981–82, pp. 179–187. Basel: Birkhaüser. 1983.
[M2]Mestre, J.-F.: Formules explicites et minorations de conducteurs de variétés algébriques. Compositio Math.58, 209–232 (1986).
[Sa]Satgé, P.: Groupes de Selmer et corps cubiques. J. Number Theory23, 294–317 (1986).
[Sh]Shioda, T.: On elliptic modular surfaces. J. Math. Soc. Japan24, 20–59 (1972).
[Si]Silverman, J.: The Arithmetic of Elliptic Curves. New York: Springer. 1986.
[W]Weil, A.: Sur les formules explicites de la théorie des nombres premiers. Comm. Sem. Math. Lund (vol. déd. à M. Riesz), Lund, 252–265 (1952).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Fouvry, E., Pomykala, J. Rang des courbes elliptiques et sommes d'exponentielles. Monatshefte für Mathematik 116, 111–125 (1993). https://doi.org/10.1007/BF01404006
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01404006