Let E be an elliptic curve defined over a number field K and let S be a density-one set of primes of K of good reduction for E. Faltings proved in 1983 that the K-isogeny class of E is characterized by the function , which maps a prime to the order of the group of points of E over the corresponding field . We show that, in this statement, the integer can be replaced by its radical.
Soit E une courbe elliptique définie sur un corps de nombres K, et soit S un ensemble de densité 1 de places de K en lesquelles E a bonne réduction. Faltings a montré en 1983 que la classe de K-isogénie de E est caracterisée par la fonction , qui envoie chaque place sur lʼordre du groupe des points de E sur le corps résiduel correspondant. On montre quʼil suffit de considérer les nombres premiers divisant cet ordre.
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Chris Hall 1; Antonella Perucca 2
@article{CRMATH_2013__351_1-2_1_0, author = {Chris Hall and Antonella Perucca}, title = {On the prime divisors of the number of points on an elliptic curve}, journal = {Comptes Rendus. Math\'ematique}, pages = {1--3}, publisher = {Elsevier}, volume = {351}, number = {1-2}, year = {2013}, doi = {10.1016/j.crma.2013.01.003}, language = {en}, }
Chris Hall; Antonella Perucca. On the prime divisors of the number of points on an elliptic curve. Comptes Rendus. Mathématique, Volume 351 (2013) no. 1-2, pp. 1-3. doi : 10.1016/j.crma.2013.01.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.01.003/
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