Minimal torsion curves in geometric isogeny classes

TL;DR

This paper investigates minimal torsion points on modular curves within fixed geometric isogeny classes of elliptic curves, providing complete characterizations in specific cases and partial results more generally.

Contribution

It introduces the study of minimal torsion curves in isogeny classes and characterizes these points for rational and CM elliptic curves, especially for prime power levels.

Findings

Complete characterization for prime power levels in rational and CM cases.

Partial results for general isogeny classes and levels.

Determination of least degree points on modular curves within isogeny classes.

Abstract

In this paper, we introduce the study of minimal torsion curves within a fixed geometric isogeny class. For a $\overline{\mathbb{Q}}$-isogeny class $\mathcal{E}$ of elliptic curves and $N \in \mathbb{Z}^+$, we wish to determine the least degree of a point on the modular curve $X_1(N)$ associated to any $E \in \mathcal{E}$. In the present work, we consider the cases where $\mathcal{E}$ is rational, i.e., contains an elliptic curve with rational $j$-invariant, or where $\mathcal{E}$ consists of elliptic curves with complex multiplication (CM). If $N=\ell^k$ is a power of a single prime, we give a complete characterization upon restricting to points of odd degree, and also in the case where $\mathcal{E}$ is CM. We include various partial results in the more general setting.