Soit E une courbe elliptique sur Soit f(x) une fonction réelle positive tendant vers l'infini. Nous montrons (sous GRH) que, pour presque tout p, le groupe des -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 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 -points of the reduction of E mod p contains a cyclic group of order at least p/f(p).
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@article{CRMATH_2003__337_11_689_0, author = {Duke, William}, title = {Almost all reductions modulo \protect\emph{p} of an elliptic curve have a large exponent}, journal = {Comptes Rendus. Math\'ematique}, pages = {689--692}, publisher = {Elsevier}, volume = {337}, number = {11}, year = {2003}, doi = {10.1016/j.crma.2003.10.006}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2003.10.006/} }
TY - JOUR AU - Duke, William TI - Almost all reductions modulo p of an elliptic curve have a large exponent JO - Comptes Rendus. Mathématique PY - 2003 SP - 689 EP - 692 VL - 337 IS - 11 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2003.10.006/ DO - 10.1016/j.crma.2003.10.006 LA - en ID - CRMATH_2003__337_11_689_0 ER -
%0 Journal Article %A Duke, William %T Almost all reductions modulo p of an elliptic curve have a large exponent %J Comptes Rendus. Mathématique %D 2003 %P 689-692 %V 337 %N 11 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2003.10.006/ %R 10.1016/j.crma.2003.10.006 %G en %F CRMATH_2003__337_11_689_0
Duke, William. Almost all reductions modulo p of an elliptic curve have a large exponent. Comptes Rendus. Mathématique, Tome 337 (2003) no. 11, pp. 689-692. doi : 10.1016/j.crma.2003.10.006. http://archive.numdam.org/articles/10.1016/j.crma.2003.10.006/
[1] The splitting of primes in division fields of elliptic curves, Experiment. Math., Volume 11 (2003), pp. 555-565
[2] On Siegel zeros of Hecke–Landau zeta-functions, Monatsh. Math., Volume 118 (1994), pp. 231-248
[3] 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] 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] 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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