A local-global principle for rational isogenies of prime degree

Abstract

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 ℓ. For suitable K (including K = Q), we prove that this principle holds for all ℓ ≡ 1 mod 4, and for ℓ < 7, but find a counterexample when ℓ = 7 for an elliptic curve with j-invariant 2268945/128. For K = Q we show that, up to isomorphism, this is the only counterexample. © Société Arithmétique de Bordeaux, 2012.