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≡1mod4, 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.