Crit\`eres d'irr\'eductibilit\'e pour les repr\'esentations des courbes elliptiques

TL;DR

This paper investigates the primes for which an elliptic curve over a number field admits a $p$-isogeny, providing explicit criteria and an algorithm to determine the finite exceptional set when the curve lacks complex multiplication.

Contribution

It establishes explicit bounds and criteria for the exceptional primes of elliptic curves over number fields, improving computational methods for identifying these primes.

Findings

The exceptional set of primes is contained within divisors of an explicit list of integers.

An efficient algorithm is provided for computing the exceptional set in the non-CM case.

Numerical examples illustrate the practical application of the criteria.

Abstract

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.