A Local-global principle for isogenies of composite degree

TL;DR

This paper investigates whether local conditions on elliptic curves over number fields imply the existence of global cyclic isogenies of the same degree, extending previous prime degree results to arbitrary degrees.

Contribution

It generalizes the local-global principle for cyclic isogenies from prime degrees to arbitrary degrees over number fields.

Findings

Established conditions under which local isogeny data implies global isogeny existence.

Extended previous prime degree results to composite degrees.

Provided new insights into the structure of elliptic curves with local isogenies.

Abstract

Let $E$ be an elliptic curve over a number field $K$. If for almost all primes of $K$, the reduction of $E$ modulo that prime has rational cyclic isogeny of fixed degree, we can ask if this forces $E$ to have a cyclic isogeny of that degree over $K$. Building upon the work of Sutherland, Anni, and Banwait-Cremona in the case of prime degree, we consider this question for cyclic isogenies of arbitrary degree.