Ordinary Isogeny Graphs with Level Structure

TL;DR

This paper investigates the structure of ordinary isogeny graphs of elliptic curves over finite fields with added level structures, revealing how parameters influence the crater structure of these graphs.

Contribution

It introduces a detailed analysis of isogeny graphs with level structures, extending the understanding of their crater structures and the action of ideal class groups.

Findings

Crater structure depends on parameter choices.

Level structures influence graph connectivity.

Ideal class group actions are characterized on these graphs.

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.