# Torsion points and isogenies on CM elliptic curves

## TL;DR

This paper investigates the minimal degrees of CM points on certain modular curves and introduces new theorems on rational cyclic isogenies of CM elliptic curves, extending previous results.

## Contribution

It provides explicit degrees of CM points on modular curves and extends known results on rational cyclic isogenies of CM elliptic curves.

## Key findings

- Determined least degrees of CM points on X(M,N) over specific fields.
- Established new theorems on rational cyclic isogenies of CM elliptic curves.
- Extended Kwon's classification of N for rational N-isogenies.

## Abstract

Let $\mathcal{O}$ be an order in the imaginary quadratic field $K$. For positive integers $M \mid N$, we determine the least degree of an $\mathcal{O}$-CM point on the modular curve $X(M,N)_{/K(\zeta_M)}$ and also on the modular curve $X(M,N)_{/\mathbb{Q}(\zeta_M)}$: that is, we treat both the case in which the complex multiplication is rationally defined and the case in which we do not assume that the complex multiplication is rationally defined. To prove these results we establish several new theorems on rational cyclic isogenies of CM elliptic curves. In particular, we extend a result of Kwon that determines the set of positive integers $N$ for which there is an $\mathcal{O}$-CM elliptic curve $E$ admitting a cyclic, $\mathbb{Q}(j(E))$-rational $N$-isogeny.