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arXiv · 2407.14322

Minimal torsion curves in geometric isogeny classes

Abbey Bourdon·Nina Ryalls·Lori D. Watson
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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.

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Abbey Bourdon, Nina Ryalls, Lori D. Watson. 2025-11-24. Minimal torsion curves in geometric isogeny classes. https://doi.org/10.1017/s030500412610200x

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