A local-global principle for isogenies of prime degree over number fields

TL;DR

This paper characterizes pairs of primes and elliptic curve invariants over number fields where local isogenies almost everywhere do not extend globally, providing bounds and finiteness results.

Contribution

It introduces a description and bounds for exceptional pairs of primes and elliptic curves over number fields where local-global isogeny behavior differs.

Findings

Upper bounds for prime e7 in exceptional pairs based on field degree and discriminant

Finiteness results for the number of such exceptional pairs

Characterization of local-global isogeny anomalies over number fields

Abstract

We give a description of the set of exceptional pairs for a number field $K$, that is the set of pairs $(\ell, j(E))$, where $\ell$ is a prime and $j(E)$ is the $j$-invariant of an elliptic curve $E$ over $K$ which admits an $\ell$-isogeny locally almost everywhere but not globally. We obtain an upper bound for $\ell$ in such pairs in terms of the degree and the discriminant of $K$. Moreover, we prove finiteness results about the number of exceptional pairs.