A local-global principle for rational isogenies of prime degree

TL;DR

This paper investigates a local-global principle for elliptic curves over number fields regarding rational isogenies of prime degree, establishing conditions under which the principle holds and identifying specific counterexamples.

Contribution

It proves the local-global principle for certain prime degrees and characterizes all counterexamples over Q, including a unique case for n=7.

Findings

The principle holds for n ≡ 1 mod 4 and n < 7.

Counterexamples exist for n=7, with a unique case over Q.

Identifies the only counterexample over Q up to isomorphism.

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