Ordinary Isogeny Graphs with Level Structure

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 $\hat{I}{“}_0(N),$ $\hat{I}{“}_1(N),$ and $\hat{I}“(N)$-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.