Ordinary isogeny graphs over $\mathbb{F}_p$: the inverse volcano problem

TL;DR

This paper investigates the structure of ordinary isogeny graphs over finite fields, proving that any abstract volcano graph can be realized as a component of such a graph for suitable primes.

Contribution

It establishes that every abstract volcano graph can be embedded as a component in an ordinary $ ext{ell}$-isogeny graph over some finite field, solving the inverse problem.

Findings

Any abstract volcano can be realized as a component of an ordinary isogeny graph.

Existence of primes p, ℓ for embedding any given volcano graph.

Provides a constructive approach to realize specific isogeny graph structures.

Abstract

We give a detailed presentation of $\ell$-isogeny graphs associated with ordinary elliptic curves defined over $\mathbb{F}_p$. We then focus on the following inverse problem: given an abstract volcano $V$, do there always exist primes $\ell, p \in \mathbb{N}$ such that the ordinary $\ell$-isogeny graph over $\mathbb{F}_p$ contains $V$ as a connected component? We provide an affirmative answer to this question.