Minimal torsion curves in geometric 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.

This version has improved exposition in many places, including a significantly revised introduction. 25 pages