Abstract
The goal of this note is to give some complements to an article of Fouvery and Pomykala: by anad-hoc method, they bound on average the rank of elliptic curves over ℚ in polynomial families:y 2=x 3=a(t)x+b(t) whent varies in ℤ under some generic conditions on the polynomials (over ℤa(t),b(t). Here, by a more systematic treatment, we are able to relax most of these conditions, keeping only the natural one (the family is not geometricaly trivial). However, this result, specialized to the case treated by Fouvry and Pomykala, yields a better bound; our method depends on the distribution of the number of points in families of elliptic curves over finite fields (known as the “vertical” Sato-Tate law), which itself depends on the work of Deligne on the Weil conjectures.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Bibliographie
Birch, B. J.: How the number of points of an elliptic curve over a fixed prime field varies. J. London Math. Soc.43, 57–60 (1968).
Deligne, P.: La conjecture de Weil I. Publ. Math. IHES43, 273–308 (1974).
Deligne, P.: Courbes Elliptiques: Formulaire (d'après J. Tate). In Modular Functions of One Variable IV, Leol-Notes Math.476, 53–74. Berlin-Heidelberg New York: Springer. 1975.
Deligne, P.: La conjecture de Weil II. Publ. Math. IHES52, 313–428 (1981).
Fouvry, E., Pomykala, J.: Rang des courbes elliptiques et sommes d'exponentielles. Mh. Math.116, 111–125 (1993).
Katz, N. M.: Sommes d'exponentielles. Astérisque79. Soc. Math. de France 1978.
Katz, N. M.: Gauss Sums, Kloosterman Sums, and Monodromy Groups. Princeton: Univ. Press. 1987.
Katz, N. M.: Exponential sums and Differential equations. Princeton: Univ. Press. 1990.
Katz, N. M.: Exponential sums over finite fields and Differential equations over the complex numbers: some interactions. Bull. Am. Math. Soc.23, n2, p. 269.
Mestre, J.-F.: Formules explicites et minorations de conducteurs de variétés algébriques. Comp. Math.58, 209–232 (1986).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Michel, P. Rang moyen de familles de courbes elliptiques et lois de Sato-Tate. Monatshefte für Mathematik 120, 127–136 (1995). https://doi.org/10.1007/BF01585913
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01585913