Cyclic isogenies of elliptic curves over fixed quadratic fields

TL;DR

This work tackles the problem of classifying cyclic isogenies of elliptic curves over fixed quadratic fields $K=\mathbb{Q}(\sqrt{d})$, extending Mazur’s classical rational-isogeny results. It develops a generalised framework based on minimally finite levels MF$(K)$ and MF$^f(K)$, and couples it with a multi‑tool computational pipeline (Quotient Method, quadratic-point catalogues, Özman sieve, Trbović filter, No growth in plus-part, and Symmetric Chabauty with Mordell–Weil sieve) to study $X_0(N)$ over $K$. Conditional on GRH, the authors determine all cyclic isogenies over $19$ quadratic fields (including $\mathbb{Q}(\sqrt{213})$ and $\mathbb{Q}(\sqrt{-2289})$) and compute all quadratic points on $X_0(125)$ and $X_0(169)$, including a non‑CM point over $\mathbb{Q}(\sqrt{509})$ with an explicit $j$-invariant. The results showcase how a carefully curated, automated toolkit can extend isogeny classifications to quadratic fields and provide a public codebase for broader exploration in arithmetic geometry.

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.

Theorems & Definitions (26)

• Theorem 1.1: Mazur, Kenku
• Theorem 1.2
• Remark 1.3
• Theorem 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Definition 3.1
• Lemma 3.2