On towers of Isogeny graphs with full level structure

TL;DR

This paper investigates the structure of isogeny graphs with full level structure over finite fields, identifying conditions under which they form towers of Galois covers, particularly in the supersingular case.

Contribution

It characterizes when towers of isogeny graphs with level structures form Galois covers, focusing on supersingular elliptic curves and extending to oriented supersingular cases.

Findings

Only supersingular cases produce towers of Galois covers.

Analysis of isogeny graphs with level structures over finite fields.

Extension to oriented supersingular curves with level structure.

Abstract

Let $p,q,l$ be three distinct prime numbers and let $N$ be a positive integer coprime to $pql$. For an integer $n\ge 0$, we define the directed graph $X_l^q(p^nN)$ whose vertices are given by isomorphism classes of elliptic curves over a finite field of characteristic $q$ equipped with a level $p^nN$ structure. The edges of $X_l^q(p^nN)$ are given by $l$-isogenies. We are interested in when the connected components of $X_l^q(p^nN)$ give rise to a tower of Galois covers as $n$ varies. We show that only in the supersingular case we do get a tower of Galois covers. We also study similar towers of isogeny graphs given by oriented supersingular curves, as introduced by Col\`o-Kohel, enhanced with a level structure.