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arXiv · 2411.02732

Ordinary Isogeny Graphs with Level Structure

Derek Perrin·José Felipe Voloch
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Abstract

We study $\ell$-isogeny graphs of ordinary elliptic curves defined over $\mathbb{F}_q$ with an added level structure. Given an integer $N$ coprime to $p$ and $\ell,$ we look at the graphs obtained by adding $\Gamma_0(N),$ $\Gamma_1(N),$ and $\Gamma(N)$-level structures to volcanoes. Given an order $\mathcal{O}$ in an imaginary quadratic field $K,$ we look at the action of generalised ideal class groups of $\mathcal{O}$ on the set of elliptic curves whose endomorphism rings are $\mathcal{O}$ along with a given level structure. We show how the structure of the craters of these graphs is determined by the choice of parameters.

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Derek Perrin, José Felipe Voloch. 2024-11-05. Ordinary Isogeny Graphs with Level Structure. https://doi.org/10.4153/s0008439525000372

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