A local-global principle for rational isogenies of prime degree

Sutherland, Andrew V.

Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 2, pp. 475-485.

Soit $K$ un corps de nombres. Nous étudions un principe local-global pour les courbes elliptiques $E/K$ admettant ou non une isogénie rationnelle de degré premier $\ell$. Pour des corps $K$ convenables (dont $K=\mathbb{Q}$), nous démontrons ce principe pour tout $\ell \equiv 1mod4$ et tout $\ell <7$ mais exhibons une courbe elliptique d’invariant modulaire $2268945/128$ comme contre-exemple pour $\ell =7$. Nous montrons alors qu’il s’agit du seul contre-exemple à isomorphisme près lorsque $K=\mathbb{Q}$.

Let $K$ be a number field. We consider a local-global principle for elliptic curves $E/K$ that admit (or do not admit) a rational isogeny of prime degree $\ell$. For suitable $K$ (including $K=\mathbb{Q}$), we prove that this principle holds for all $\ell \equiv 1mod4$, and for $\ell <7$, but find a counterexample when $\ell =7$ for an elliptic curve with $j$-invariant $2268945/128$. For $K=\mathbb{Q}$ we show that, up to isomorphism, this is the only counterexample.

Classification : 11G05
Mots clés : elliptic curve, isogeny, local-global principle

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