Comptes Rendus
no. 11
Number Theory/Algebraic Geometry
Almost all reductions modulo p of an elliptic curve have a large exponent
[Presque toutes les réductions mod p d'une courbe elliptique sur ℚ ont un groupe de points qui est presque cyclique]

Présenté par : Jean-Pierre Serre

William Duke1
1 UCLA Mathematics Department, Box 951555, Los Angeles, CA 90095-1555, USA
Comptes Rendus. Mathématique, Volume 337 (2003) no. 11, pp. 689-692.

Soit E une courbe elliptique sur ℚ. Soit f(x) une fonction réelle positive tendant vers l'infini. Nous montrons (sous GRH) que, pour presque tout p, le groupe des 𝔽 p -points de la réduction de E mod p contient un groupe cyclique d'ordre au moins p/f(p).

Let E be an elliptic curve defined over ℚ. Suppose that f(x) is any positive function tending to infinity with x. It is shown (under GRH) that for almost all p, the group of 𝔽 p -points of the reduction of E mod p contains a cyclic group of order at least p/f(p).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2003.10.006
William Duke 1

1 UCLA Mathematics Department, Box 951555, Los Angeles, CA 90095-1555, USA 
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William Duke. Almost all reductions modulo p of an elliptic curve have a large exponent. Comptes Rendus. Mathématique, Volume 337 (2003) no. 11, pp. 689-692. doi : 10.1016/j.crma.2003.10.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.10.006/

[1] W. Duke; Á Tóth The splitting of primes in division fields of elliptic curves, Experiment. Math., Volume 11 (2003), pp. 555-565

[2] J. Hinz; M. Lodemann On Siegel zeros of Hecke–Landau zeta-functions, Monatsh. Math., Volume 118 (1994), pp. 231-248

[3] R. Schoof The exponents of the groups of points on the reductions of an elliptic curve, Arithmetic Algebraic Geometry (Texel, 1989), Progr. Math., 89, Birkhäuser, Boston, MA, 1991, pp. 325-335

[4] J.-P. Serre Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. (Collected Papers), Volume 15 (1972), pp. 259-331 (also in, III, 1985)

[5] J.-P. Serre Quelques applications du théorème de densité de Chebotarev, Publ. Math. I. H. E. S. (Collected Papers), Volume 54 (1981), pp. 123-201 (also in, III, 1985)

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