Cyclic isogenies of elliptic curves over fixed quadratic fields

TL;DR

This paper develops a method to classify cyclic isogenies of elliptic curves over quadratic fields, successfully applying it to 19 such fields under the GRH, and determines all quadratic points on specific modular curves.

Contribution

It introduces a new procedure for determining cyclic isogenies over quadratic fields and applies it to a large class of fields, expanding the known classifications.

Findings

Classified cyclic isogenies over 19 quadratic fields under GRH.

Determined all quadratic points on X_0(125) and X_0(169).

Extended the understanding of elliptic curve isogenies beyond rational fields.

Abstract

Building on Mazur's 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over $\mathbb{Q}$. Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised. In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields $\mathbb{Q}(\sqrt{d})$ with $|d| < 10^4$ we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over $19$ quadratic fields, including $\mathbb{Q}(\sqrt{213})$ and $\mathbb{Q}(\sqrt{-2289})$. To make this procedure work, we determine all of the finitely many quadratic points on the modular curves $X_0(125)$ and $X_0(169)$, which may be of independent interest.