Critères d'irréductibilité pour les représentations des courbes elliptiques

Details Critères d'irréductibilité pour les représentations des courbes elliptiques Critères d'irréductibilité pour les représentations des courbes elliptiques

2009-08-07

arxiv.org/abs/0908.1084v3

Mathematics

Number Theory

33 pages; french; v3: shortened and improved version. Paper slightly reorganized; new result (thm. 2) added.

Scientific paper

Let $E$ be an elliptic curve defined over a number field $K$. We say that a prime number $p$ is exceptional for $(E,K)$ if $E$ admits a $p$-isogeny defined over $K$. The so-called exceptional set of all such prime numbers is finite if and only if $E$ does not have complex multiplication over K. In this paper, we prove that the exceptional set is included in the set of prime divisors of an explicit list of integers (depending on $E$ and $K$), whose infinitely many of them are non-zero. It provides an efficient algorithm for computing it in the finite case. Other less general but rather useful criteria are given, as well as several numerical examples.

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