# On towers of isogeny graphs with full level structures

Antonio Lei  Department of Mathematics and Statistics
University of Ottawa
150 Louis-Pasteur Pvt
Ottawa, ON
Canada K1N 6N5<EMAIL_ADDRESS>and  Katharina Müller  Département de
Mathématiques et de Statistique
Université Laval, Pavillion Alexandre-Vachon
1045 Avenue de la Médecine
Québec, QC
Canada G1V 0A6<EMAIL_ADDRESS>
###### Abstract.

Let $p,q,\ell$ be three distinct prime numbers and let $N$ be a positive
integer coprime to $pq\ell$. For an integer $n\geq 0$, we define the directed
graph $X_{\ell}^{q}(p^{n}N)$ whose vertices are given by isomorphism classes
of elliptic curves over a finite field of characteristic $q$ equipped with a
level $p^{n}N$ structure. The edges of $X_{\ell}^{q}(p^{n}N)$ are given by
$\ell$-isogenies. We are interested in when the connected components of
$X_{\ell}^{q}(p^{n}N)$ 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 Colò-Kohel, enhanced with a level structure.

###### Key words and phrases:

Isogeny graphs with level structure, Iwasawa theory for graphs, oriented
supersingular elliptic curves

###### 2020 Mathematics Subject Classification:

Primary: 05C25, 11G20 Secondary: 11R23, 14G17, 14K02

##  1\. Introduction

Let $p$ and $\ell$ be distinct prime numbers. Let $N$ be a positive integer
that is coprime to $p\ell$. Let $G_{N}^{n}$ be the directed graph whose
vertices are isomorphism classes $(E,P)$, where $E$ is an ordinary elliptic
curve defined over a fixed finite field of characteristic $p$ and $P$ is a
point of order $Np^{n}$ on $E$, and the edges of $G_{N}^{n}$ are given by
$\ell$-isogenies. In [LM23], we studied the tower of graph coverings

$G_{N}^{0}\leftarrow G_{N}^{1}\leftarrow G_{N}^{2}\leftarrow\cdots\leftarrow
G_{N}^{n}\leftarrow\cdots.$

Note that the degree of the covering $G_{N}^{n+1}/G_{N}^{n}$ is $p$ thanks to
the ordinarity condition. We studied the undirected graphs $\tilde{G}_{N}^{n}$
obtained from $G_{N}^{n}$ by forgetting the directions of the edges of
$G_{N}^{n}$. We showed that if $E$ is an elliptic curve representing a non-
isolated vertex of $G_{1}^{0}$ and $\tilde{{\mathcal{G}}}_{N}^{m}$ is a
connected component of ${\mathcal{G}}_{N}^{m}$ containing a vertex arising
from $E$, then there exists an integer $m_{0}$ such that

$\tilde{\mathcal{G}}_{N}^{m_{0}}\leftarrow\tilde{\mathcal{G}}_{N}^{m_{0}+1}\leftarrow\tilde{\mathcal{G}}_{N}^{m_{0}+2}\leftarrow\cdots\leftarrow\tilde{\mathcal{G}}_{N}^{m_{0}+n}\leftarrow\cdots$

is an abelian $p$-tower in the sense of Vallières and McGown-Vallières [Val21,
MV23a, MV23b], i.e. the covering
$\tilde{\mathcal{G}}_{N}^{m_{0}+r}/\tilde{\mathcal{G}}_{N}^{m_{0}}$ is Galois,
whose Galois group is isomorphic to the cyclic group
$\mathbb{Z}/p^{r}\mathbb{Z}$ for all $r\geq 0$. Such towers exhibit properties
that resemble $\mathbb{Z}_{p}$-towers of number fields studied in classical
Iwasawa theory [Iwa69, Iwa73]. For example, similar to the class numbers of
number fields inside a $\mathbb{Z}_{p}$-tower, the number of spanning trees of
$\tilde{\mathcal{G}}_{N}^{m_{0}+n}$ can be described explicitly for $n$
sufficiently large (see [MV23b, LV23]).

In this article, we study the following tower of isogeny graphs as $n$ varies,
resulting in non-commutative Galois covers under appropriate hypotheses.
Throughout, we fix a positive integer $k$ and a set of representatives of the
equivalence classes of elliptic curves over $\mathbb{F}_{q^{k}}$, which we
denote by $S$.

######  Definition 1.1.

Let $p,q,\ell$ be three distinct prime numbers and let $N$ be a positive
integer coprime to $pq\ell$. For an integer $n\geq 0$, we define the directed
graph $X_{\ell}^{q}(Np^{n})$ whose vertices are given by triples
$(E,Q_{1},Q_{2})$, where $E\in S$ and $\\{Q_{1},Q_{2}\\}$ is a basis of
$E[p^{n}N]\cong(\mathbb{Z}/p^{n}N\mathbb{Z})^{2}$. The edges of
$X_{\ell}^{q}(p^{n}N)$ are given by $\ell$-isogenies.

The basis $\\{Q_{1},Q_{2}\\}$ is called a $\Gamma(p^{n}N)$-level structure on
the elliptic curve $E$. As $pN$ is coprime to $q$, we can write
$Q_{i}=R_{i}+S_{i}$, where $i\in\\{1,2\\}$, $R_{i}\in E[N]$ and $S_{i}\in
E[p^{n}]$. We shall denote a vertex of $X_{\ell}^{q}(p^{n}N)$ by
$(E,R_{1},R_{2},P,Q)$ with $R_{1},R_{2}\in E[N]$ and $P,Q\in E[p^{n}]$ such
that $\\{R_{1}\oplus P,R_{2}\oplus Q\\}$ is a $\Gamma(p^{n}N)$-level structure
on $E$. When $n=0$, we may discard $P$ and $Q$, and simply write
$(E,R_{1},R_{2})$ for a vertex in $X_{\ell}^{q}(N)$.

The map $(E,R_{1},R_{2},P,Q)\mapsto(E,R_{1},R_{2},pP,pQ)$ induces a graph
covering $X_{\ell}^{q}(p^{n+1}N)/X_{\ell}^{q}(p^{n}N)$ (see Corollary 2.9). We
are interested in when the connected components of $X_{\ell}^{q}(p^{n}N)$ give
rise to a tower of Galois covers as $n$ varies. Let $X$ be a connected
component of $X_{\ell}^{q}(N)$. For $n\geq 1$, let $X_{n}$ be the pre-image of
$X$ under the covering $X_{\ell}^{q}(p^{n}N)/X_{\ell}^{q}(N)$. Note that if
$v_{1},v_{2}\in V(X_{n})$, the elliptic curves giving rise to $v_{1}$ and
$v_{2}$ are isogeneous. In particular, they have the same reduction type.
Therefore, it makes sense to refer to $X$ as ordinary or supersingular. It
turns out that the coverings $X_{n}/X$ exhibit different properties depending
on the reduction type. In particular, we prove:

######  Theorem A (Corollary 3.3).

If the connected component $X$ is ordinary, then $X_{n}/X$ is not a Galois
covering when $n$ is sufficiently large.

######  Theorem B (Corollaries 4.2 and 4.4).

Suppose that the connected component $X$ is supersingular, $p>2$ and that
$N\leq 2$. For a positive density set of primes $p$, the covering $X_{n}/X$ is
Galois, with Galois group isomorphic to
$\operatorname{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$.

In particular, in the setting of Theorem B,

$X_{1}\leftarrow X_{2}\leftarrow\cdots\leftarrow X_{n}\leftarrow\cdots$

gives an explicit example that fits in the framework of non-commutative
Iwasawa of graph coverings developed in [KM23].

In non-commutative Iwasawa theory of elliptic curves studied in [CFK+05], the
authors conjectured that the Pontryagin dual of the $p$-primary Selmer group
of a $p$-ordinary elliptic curve over a $p$-adic Lie extension
$\mathcal{K}_{\infty}$ of a number field $\mathcal{K}_{0}$ that contains the
cyclotomic $\mathbb{Z}_{p}$-extension should satisfy the so-called
$\mathfrak{M}_{H}(G)$-property. Here,
$G=\operatorname{Ga\ell}(\mathcal{K}_{\infty}/\mathcal{K}_{0})$ and $H$ is a
subgroup of $G$ such that $\mathcal{K}_{\infty}^{H}$ is the cyclotomic
$\mathbb{Z}_{p}$-extension of $\mathcal{K}_{0}$. The analogue of the
$\mathfrak{M}_{H}(G)$-property in the context of graphs has recently been
studied in [KM23]. It is therefore natural to seek an appropriate quotient
inside the tower given by Theorem B that would play the role of the cyclotomic
$\mathbb{Z}_{p}$-extension of a number field.

Indeed, the graph $X_{\ell}^{q}(p^{n}N)$ admits a natural quotient
$Y_{\ell}^{q}(p^{n}N)$ obtained by the map

$(E,R_{2},R_{2},P,Q)\mapsto(E,R_{1},R_{2},\langle P,Q\rangle),$

where $\langle-,-\rangle$ is the Weil pairing (see Definition 5.1). We show
that these graphs give rise to an abelian $\mathbb{Z}_{p}$-tower, similar to
the ordinary isogeny graphs studied in [LM23]. In particular, we prove:

######  Theorem C (Corollary 5.6).

For $N$ sufficiently large, there exists an integer $m_{0}$ such that if
$Y_{m_{0}}$ is a connected component of $Y_{\ell}^{q}(p^{m_{0}}N)$ and
$Y_{n+m_{0}}$ is the pre-image of $Y_{m_{0}}$ in $Y_{\ell}^{q}(p^{n+m_{0}}N)$,
then $Y_{n+m_{0}}/Y_{m_{0}}$ is Galois, with Galois group isomorphic to
$\mathbb{Z}/p^{n}\mathbb{Z}$ for all $n\geq 0$.

Note that unlike Theorems A and B, where a divergence between the ordinary and
supersingular cases emerges, Theorem C is independent of the reduction type of
the chosen connected component.

Finally, we study isogeny graphs arising from oriented supersingular elliptic
curves, which were first introduced in [CK20]. An orientation of a
supersingular elliptic curve $E$ is given by an embedding
$K\hookrightarrow\operatorname{End}(E)\otimes\mathbb{Q}$, where $K$ is an
imaginary quadratic field. The authors of [ACL+23] proved that these graphs
can be described as volcano graphs, similar to the ordinary isogeny graphs
studied in [Koh96].

Let $M\geq 1$ be an integer that is coprime to $q\ell$. We define the isogeny
graphs of oriented supersingular elliptic curves equipped with a
$\Gamma(M)$-level structure, which we denote by $\mathcal{X}_{\ell}^{q}(M)$
(see Definition 6.10). We describe the connected components of these graphs
explicitly for $M$ sufficiently large. In particular, we prove in Theorems
6.21, 6.22, 6.23 that depending on the splitting behaviour of $\ell$ in $K$, a
connected component of $\mathcal{X}_{\ell}^{q}(M)$ is a "double
intertwinement" of either a "volcano graph" or a "tectonic volcano" (see
Definitions 6.6, 6.16 and 6.20 where these concepts are introduced).

The final result of the present article is an analogue of Theorem B for
isogeny graphs of oriented supersingular elliptic curve with level structures:

######  Theorem D (Theorem 6.27).

Assume that $q\equiv 1\pmod{12}$, $N=1$ or $2$ is an integer coprime to
$pq\ell$ and that the subgraph $X_{\ell}^{q}(p^{n}N)^{\mathrm{ss}}$ of
$X_{\ell}^{q}(p^{n}N)$ generated by the vertices arising from supersingular
elliptic curves is connected for all $n$. Let $\mathcal{X}_{0}$ be a connected
component of $\mathcal{X}_{\ell}^{q}(N)$ and denote the pre-image of
$\mathcal{X}_{0}$ in $\mathcal{X}_{\ell}^{q}(p^{n}N)$ by $\mathcal{X}_{n}$ for
$n\geq 1$. The covering $\mathcal{X}_{n}/\mathcal{X}_{0}$ is a Galois covering
with Galois group isomorphic to $\textup{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$.

Theorem D illustrates an interesting relation between the two classes of
isogeny graphs studied in the present paper. We plan to study such relations
further in the future. Finally, we remark that a simple necessary and
sufficient condition of the connectivity of
$X_{\ell}^{q}(p^{n}N)^{\mathrm{ss}}$ is been proved by Roda [Rod19]. We shall
review this result in the main body of the article (see Theorem 4.1). This
result is also utilized to prove the non-triviality of the density of primes
satisfying Theorem B.

We conclude this introduction by highlighting the growing interest in isogeny
graphs of elliptic curves with level structures in recent years. One such
example is Goren–Kassaei's work [GK21] that delved into the dynamics of Hecke
operators on modular curves utilizing such graphs with $\Gamma_{1}(N)$-level
structure. On a different vein, Arpin [Arp23] investigated the implications of
isogeny graphs for equivalence classes of supersingular elliptic curves with
$\Gamma_{0}(N)$-level structures in the realm of isogeny-based cryptography.
During the preparation of the present article, we learnt about the recent work
of Codogni–Lido [CL23], where they considered more general level structures.
Analogous to Theorem 4.1, they studied the number of connected components in
these graphs. Moreover, they analyzed the eigenvalues of the associated
adjacency matrices, establishing connections between these graphs and modular
forms (the special case of $\Gamma_{0}(N)$-level structure was also studied in
[Sug17, LM23]).

### Acknowledgement

We thank Pete Clark and Daniel Vallières for helpful exchanges regarding the
content of the present article. We thank Sören Kleine for various helpful
remarks on the present article. The authors' research is supported by the
NSERC Discovery Grants Program RGPIN-2020-04259 and RGPAS-2020-00096.

##  2\. Voltage assignment and graph covering

In this section, we show that the isogeny graphs with level structures
introduced in Definition 1.1 can be naturally realized as voltage graphs. Our
discussion can be regarded as a natural extension of [LM23, Appendix A] to the
non-commutative setting.

###  2.1. Generalities of voltage graphs

Let us first recall the definition of a voltage assignment.

######  Definition 2.1.

Let $X$ be a directed graph.

The set of vertices and the set of edges of $X$ are denoted by $V(X)$ and
$\mathbb{E}(X)$, respectively.

Let $(G,\cdot)$ a group. A function $\alpha\mathrel{\mathop{\mathchar
58\relax}}\mathbb{E}(X)\rightarrow G$ is called a $G$-valued voltage
assignment on $X$.

To each voltage assignment $\alpha$ given as in ii), we define the derived
graph $X(G,\alpha)$ to be the graph whose vertices and edges are given by
$V(X)\times G$ and $\mathbb{E}(X)\times G$, respectively. If
$(e,\sigma)\in\mathbb{E}(X)\times G$, it links $(s,\sigma)$ to
$(t,\sigma\cdot\alpha(e))$, where $e$ is an edge in $X$ from $s$ to $t$.

A graph arises from a voltage assignment is called a voltage graph.

Let $G$, $X$ and $\alpha$ be given as in Definition 2.1. There is a natural
graph covering $X(G,\alpha)/X$ given by

$V(X(G,\alpha))\ni(v,\sigma)\mapsto
v,\quad\mathbb{E}(X(G,\alpha))\ni(e,\sigma)\mapsto e.$

If $G$ is finite, then the degree of this covering is $|G|$ (to which we may
also refer as a "$|G|$-sheeted covering"). Furthermore, there is a natural
left action by $G$ on $X(G,\alpha)$ given by
$g\cdot(v,\sigma)=(v,g\cdot\sigma)$ and $g\cdot(e,\sigma)=(e,g\cdot\sigma)$
for $g,\sigma\in G$, $v\in V(X)$ and $e\in\mathbb{E}(X)$.

######  Definition 2.2.

Let $X$ be a directed graph.

We write $X^{\prime}$ for the undirected graph obtained from $X$ by ignoring
the directions of the edges of $X$. The natural map from $X$ to $X^{\prime}$
is called the forgetful map.

We say that $X$ is connected if $X^{\prime}$ is connected.

A connected component of $X$ is the pre-image of a connected component of
$X^{\prime}$ in $X$ under the forgetful map.

We call a covering $Y/X$ of directed graphs Galois if the corresponding
covering of undirected graphs $Y^{\prime}/X^{\prime}$ is Galois. When $Y/X$ is
Galois, we call $\operatorname{Ga\ell}(Y^{\prime}/X^{\prime})$ the Galois
group of $Y/X$ and write
$\operatorname{Ga\ell}(Y/X)=\operatorname{Ga\ell}(Y^{\prime}/X^{\prime})$.

If $Y/X$ is a covering of directed graph, the group of deck transformations of
$Y/X$ is denoted by $\mathrm{Deck}(Y/X)$. Similarly, we denote the group of
deck transformations of $Y^{\prime}/X^{\prime}$ by
$\mathrm{Deck}(Y^{\prime}/X^{\prime})$.

The notion of connectedness here is sometimes referred to as "weakly
connected", as opposed to "strongly connected", where one requires the
existence of a path between any two given vertices. We shall see in Remark
2.11 that these two notions coincide for the graphs of interest in the present
paper.

######  Lemma 2.3.

Let $Y/X$ be a $d$-sheeted covering of connected directed graphs. Then
$Y^{\prime}/X^{\prime}$ is also a $d$-sheeted covering. Furthermore, there is
a natural injective group homomorphism

$\Xi\mathrel{\mathop{\mathchar
58\relax}}\textup{Deck}(Y/X)\hookrightarrow\textup{Deck}(Y^{\prime}/X^{\prime}).$

Furthermore, if $Y$ is the derived graph of a voltage assignment $\alpha$ on
$X$ and $Y^{\prime}/X^{\prime}$ is Galois, then $\Xi$ is a group isomorphism.

###### Proof.

As $V(X)=V(X^{\prime})$ and $V(Y)=V(Y^{\prime})$, it is clear that
$Y^{\prime}/X^{\prime}$ is a $d$-sheeted covering. It is also clear that every
deck transformation of $Y/X$ induces a deck transformation of
$Y^{\prime}/X^{\prime}$ since the edges of $Y^{\prime}$ and $X^{\prime}$ arise
from those in $Y$ and $X$, respectively, resulting in the injection $\Xi$.

Assume now that $Y$ is the derived graph of a voltage assignment $\alpha$ on
$X$ and that $Y^{\prime}/X^{\prime}$ (and so $Y/X$) is Galois. As $Y/X$ is a
$d$-sheeted cover, $\alpha$ takes values in a finite group $G$ of order $d$.
Thus, to show that $\Xi$ is surjective, it suffices to show that
$\textup{Deck}(Y/X)$ admits at least $d$ distinct elements. Since
$Y=X(G,\alpha)$, the elements of $V(Y)$ are of the form $(v,\sigma)$, where
$v\in V(X)$ and $\sigma\in G$. For any $g\in G$, the map
$(v,\sigma)\mapsto(v,g\sigma)$ induces a deck transformation of $Y/X$. This
results in $d$ distinct elements in $\textup{Deck}(Y/X)$, which proves the
surjectivity of $\Xi$. ∎

The following result allows us to determine if a graph covering arising from a
voltage assignment is Galois.

######  Proposition 2.4.

Let $X$ be a directed graph, $G$ a finite group equipped with a $G$-valued
voltage assignment $\alpha$ on $X$. Let $Y$ denote the derived graph
$X(G,\alpha)$.

If $X$ is connected, then the natural action of $G$ on $Y$ permutes
transitively the connected components of $Y$.

If $Y$ is connected, then $Y/X$ is a Galois cover. Furthermore, its Galois
group is isomorphic to $G$.

###### Proof.

This follows from [Gon22, Theorem 2.10] and [Gon21, §2.3], combined with the
identification of deck transformations of coverings of directed graphs and
undirected graphs given by Lemma 2.3. ∎

######  Corollary 2.5.

Let $X$, $Y$ and $G$ be as in Proposition 2.4i). Let $d$ be the number of
connected components of $Y$. If $d!>|G|$, then $Y/X$ is not Galois.

###### Proof.

Let $Y_{1}$ and $Y_{2}$ be two connected components of $Y$. Proposition 2.4i)
tells us that there exists $g\in G$ such that

$V(Y_{1})\ni(v,\sigma)\mapsto(v,g\cdot\sigma),\quad\mathbb{E}(Y_{1})\ni(e,\sigma)\mapsto(e,g\cdot\sigma)$

induces an isomorphism of graphs from $Y_{1}$ to $Y_{2}$. Thus, any
permutation of the connected components of $Y$ gives rise to an element of
$\mathrm{Deck}(Y/X)$. Since there are $d!$ such permutations and the degree of
the covering is $|G|$, the covering cannot be Galois if $d!>|G|$. ∎

######  Remark 2.6.

Assume that $G$ is finite. In the course of the paper we will frequently study
the number of connected components of $X(G,\alpha)$. To do so, it suffices to
fix a vertex $v\in V(X)$ and study the set

$\left\\{(v,g)\in V(X(G,\alpha))\mid(v,g)\textup{ lies in the same connected
component as $(v,1)$}\right\\}.$

Let $d_{v}$ be the cardinality of this set. As explained in the proof of
Corollary 2.5, all connected components of $X(G,\alpha)$ are isomorphic to
each other. Therefore, the number of connected components of $X(G,\alpha)$ is
given by $\frac{|G|}{d_{v}}$.

###  2.2. Realizing isogeny graphs as voltage graphs

The goal of this section is to realize the graph $X_{\ell}^{q}(p^{n}N)$ as
voltage graph arising from a
$\operatorname{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$-valued voltage assignment
on $X$, similar to the discussion in [LM23, Appendix A].

For each elliptic curve $E$ in $S$ defined over $\mathbb{F}_{q^{k}}$, we fix a
$\mathbb{Z}_{p}$-basis $\\{s_{E},t_{E}\\}$ of the Tate module $T_{p}(E)$. If
$e\in\mathbb{E}(X_{\ell}^{q}(N))$ with source $(E,R_{1},R_{2})$ and target
$(E^{\prime},R_{1}^{\prime},R_{2}^{\prime})$, it arises from an $\ell$-isogeny
$\phi\mathrel{\mathop{\mathchar 58\relax}}E\rightarrow E^{\prime}$. Since $p$
and $\ell$ are coprime, $\phi$ induces a $\mathbb{Z}_{p}$-isomorphism

$\phi^{*}\mathrel{\mathop{\mathchar 58\relax}}T_{p}(E)\rightarrow
T_{p}(E^{\prime}).$

This allows us to define the following voltage assignment:

######  Definition 2.7.

For any given $e\in\mathbb{E}(X_{\ell}^{q}(N))$ arising from an $\ell$-isogeny
$\phi\mathrel{\mathop{\mathchar 58\relax}}E\rightarrow E^{\prime}$, we define
$g_{e}\in\operatorname{GL}_{2}(\mathbb{Z}_{p})$ to be the transpose of the
matrix of $\phi^{*}$ with respect to our chosen bases $\\{s_{E},t_{E}\\}$ and
$\\{s_{E^{\prime}},t_{E^{\prime}}\\}$ so that the following equation holds

$\begin{pmatrix}\phi^{*}(s_{E})\\\
\phi^{*}(t_{E})\end{pmatrix}=g_{e}\cdot\begin{pmatrix}s_{E^{\prime}}\\\
t_{E^{\prime}}\end{pmatrix}.$

For an integer $n\geq 0$, we define $\alpha_{n}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{E}(X_{\ell}^{q}(N))\rightarrow\operatorname{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$
to be the voltage assignment sending $e$ to the image of $g_{e}$ under the
natural projection
$\operatorname{GL}_{2}(\mathbb{Z}_{p})\rightarrow\operatorname{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$.

######  Theorem 2.8.

For all $n\geq 0$, the derived graph
$X(\operatorname{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z}),\alpha_{n})$ is
isomorphic to $X_{\ell}^{q}(p^{n}N)$.

###### Proof.

To simplify notation, we write
$Y_{n}=X(\operatorname{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z}),\alpha_{n})$,
$Z_{n}=X_{\ell}^{q}(p^{n}N)$ and
$G_{n}=\operatorname{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$ in this proof.

We first identify the vertices of $Y_{n}$ and $Z_{n}$. Recall that
$V(Y_{n})=Z_{0}\times G_{n}$. Let $(E,R_{1},R_{2},P,Q)\in V(Z_{n})$. Since
$\\{P,Q\\}$ is a basis of $E[p^{n}]$, there exists a unique $\sigma\in G_{n}$
such that

(2.1) $\begin{pmatrix}P\\\
Q\end{pmatrix}=\sigma\cdot\begin{pmatrix}\overline{s}_{E}\\\
\overline{t}_{E}\end{pmatrix},$

where $\overline{s}_{E}$ and $\overline{t}_{E}$ denote the images of $s_{E}$
and $t_{E}$ in $E[p^{n}]$, respectively. Conversely, given any
$\left((E,R_{1},R_{2}),\sigma\right)\in V(Y_{n})$, we may define a basis of
$E[p^{n}]$ via (2.1). Hence, there is a natural bijection $\Phi$ from
$V(Z_{n})$ to $V(Y_{n})$.

It remains to show that the bijection $\Phi$ respects the edges of the two
graphs. That is, there is an edge in $Z_{n}$ from $v$ to $w$ induced by some
isogeny if and only if there is an edge in $Y_{n}$ from $\Phi(v)$ to $\Phi(w)$
induced by the same isogeny. Suppose that $e\in\mathbb{E}(Z_{0})$ with
$(E,R_{1},R_{2})$ and $(E^{\prime},R^{\prime}_{1},R^{\prime}_{2})$ as the
source and target, respectively. Let $\phi$ be the corresponding
$\ell$-isogeny. The same isogeny gives rise to an edge from
$(E,R_{1},R_{2},P,Q)$ to
$(E^{\prime},R^{\prime}_{1},R^{\prime}_{2},P^{\prime},Q^{\prime})$ in $Z_{n}$
if and only if

(2.2) $\phi(P)=P^{\prime}\quad\text{and}\quad\phi(Q)=Q^{\prime}.$

Let us write

$\displaystyle\Phi((E,R_{1},R_{2},P,Q))$
$\displaystyle=((E,R_{1},R_{2}),\sigma),$
$\displaystyle\Phi((E^{\prime},R^{\prime}_{1},R^{\prime}_{2},P^{\prime},Q^{\prime}))$
$\displaystyle=((E^{\prime},R^{\prime}_{1},R^{\prime}_{2}),\sigma^{\prime}).$

A direct calculation shows that (2.2) is equivalent to the equation

$\sigma\alpha_{n}(e)=\sigma^{\prime},$

which is precisely the condition for there to be an edge going from
$((E,R_{1},R_{2}),\sigma)$ to
$((E^{\prime},R^{\prime}_{1},R^{\prime}_{2}),\sigma^{\prime})$ in $Y_{n}$.
This shows that $\Phi$ respects the edges of $Y_{n}$ and $Z_{n}$, as desired.
∎

######  Corollary 2.9.

Let $n\geq 1$ be an integer. The map

$\displaystyle V(X_{\ell}^{q}(p^{n}N))$ $\displaystyle\rightarrow
V(X_{\ell}^{q}(p^{n-1}N)),$ $\displaystyle(E,R_{1},R_{2},P,Q)$
$\displaystyle\mapsto(E,R_{1},R_{2},pP,pQ)$

induces a graph covering $X_{n}\rightarrow X_{n-1}$.

###### Proof.

Let $X$, $G$ and $\alpha$ be as in Definition 2.1. Let $N$ be a normal
subgroup of $G$. We write $\tilde{\alpha}$ for the composition of $\alpha$
with the projection map $G\mapsto G/N$. Then $X(G/N,\tilde{\alpha})$ is an
intermediate covering of $X(G,\alpha)/X$; the covering map
$X(G,\alpha)\rightarrow X(G/N,\tilde{\alpha})$ is given by

$V(X(G,\alpha))\ni(v,\sigma)\mapsto(v,\sigma
N),\quad\mathbb{E}(X(G,\alpha))\ni(e,\sigma)\mapsto(e,\sigma N).$

Let $G_{n}=\operatorname{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$ and let
$\alpha_{n}$ be the voltage assignment on $X=X_{\ell}^{q}(N)$ given in
Definition 2.7. Let $\pi_{n}\mathrel{\mathop{\mathchar
58\relax}}G_{n}\rightarrow G_{n-1}$ be the natural projection map and
$N_{n}=\ker(\pi_{n})$. Then we have a natural graph covering
$X(G_{n},\alpha_{n})/X(G_{n}/N_{n},\tilde{\alpha}_{n})$ as discussed in the
previous paragraph.

Let $\Theta_{n}\mathrel{\mathop{\mathchar
58\relax}}V(X(G_{n},\alpha_{n}))\rightarrow V(X_{\ell}^{q}(p^{n}N))$ be the
inverse of the bijection $\Phi$ given in the proof of Theorem 2.8. Since
$G_{n-1}\cong G_{n}/N_{n}$ and $\tilde{\alpha}_{n}=\alpha_{n-1}$, we may
identify $X(G_{n}/N_{n},\tilde{\alpha}_{n})$ with $X(G_{n-1},\alpha_{n-1})$,
and hence with $X_{\ell}^{q}(p^{n-1}N)$. Finally, it follows from the
definition of the Tate module that

$\Theta_{n-1}\circ\pi_{n}=[p]\circ\Theta_{n},$

which concludes the proof. ∎

######  Corollary 2.10.

Let $X$ be a connected component of $X_{\ell}^{q}(N)$ (in the sense of
Definition 2.2). For an integer $n\geq 1$, let $X_{n}$ denote the pre-image of
$X$ in $X_{\ell}^{q}(p^{n}N)$ under the natural projection map. If $X_{n}$ is
connected, the covering $X_{n}/X$ is Galois with Galois group isomorphic to
$\textup{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$.

###### Proof.

This follows directly from Proposition 2.4ii). ∎

We conclude this section with a couple of remarks on the connected components
of $X_{\ell}^{q}(p^{n}N)$, which will be useful for subsequent sections.

######  Remark 2.11.

Let $X_{n}$ be a connected component of $X_{\ell}^{q}(p^{n}N)$. Let
$e\in\mathbb{E}(X_{n})$ with $v=(E,R_{1},R_{2},P,Q)$ and
$w=(E^{\prime},R^{\prime}_{1},R^{\prime}_{2},P^{\prime},Q^{\prime})$ as the
source and target, respectively. Then $e$ arises from an $\ell$-isogeny
$\phi\mathrel{\mathop{\mathchar 58\relax}}E\rightarrow E^{\prime}$ such that
$\phi(U)=U^{\prime}$ for $U\in\\{R_{1},R_{2},P,Q\\}$. The dual isogeny
$\hat{\phi}$ gives rise to an edge in $X_{n}$ from $w$ to $(E,\ell R_{1},\ell
R_{2},\ell P,\ell Q)$. Let $d$ be the order of $[\ell]$ as an automorphism on
$E[p^{n}N]$. On repeatedly applying $\phi$ and $\hat{\phi}$ alternatively $d$
times, we obtain a path from $(E,\ell R_{1},\ell R_{2},\ell P,\ell Q)$ to $v$.
Therefore, there exists a path starting at $w$ and terminating at $v$. In
other words, each directed edge in $X_{n}$ can be "reversed" via a path in the
opposite direction. Hence, $X_{n}$ is a strongly connected directed graph.

######  Remark 2.12.

Let $v=(E,R_{1},R_{2},P,Q)$ and
$w=(E^{\prime},R^{\prime}_{1},R^{\prime}_{2},P^{\prime},Q^{\prime})$ be two
vertices of $X_{\ell}^{q}(Np^{n})$. There is a path from $v$ to $w$ if and
only if there exists an isogeny $\phi\colon E\to E^{\prime}$ of $\ell$-power
degree such that $\phi(R_{1})=R^{\prime}_{1}$, $\phi(R_{2})=R^{\prime}_{2}$,
$\phi(P)=P^{\prime}$ and $\phi(Q)=Q^{\prime}$.

##  3\. The ordinary case

The goal of this section is to prove Theorem A stated in the introduction. We
fix a connected component $X$ of $X_{\ell}^{q}(N)$ and define $X_{n}$ to be
the pre-image of $X$ in $X_{\ell}^{q}(p^{n}N)$ under the natural projection
map. As explained in the introduction, all elliptic curves giving rise to a
vertex in $X_{n}$ have the same reduction type. We assume that these elliptic
curves are ordinary in this section. Furthermore, since these curves are
isogeneous to each other, their endomorphism rings are orders in the same
imaginary quadratic field, which we denote by $K$. We shall refer to $K$ as
the CM field of $X$.

We begin with the following lemma.

######  Lemma 3.1.

Let $n\geq 1$ be an integer. The order of the group
$\textup{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$ is
$p^{4(n-1)}(p^{2}-1)(p^{2}-p)$.

###### Proof.

Let $G_{n}=\operatorname{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$. Let
$\pi_{n}\mathrel{\mathop{\mathchar 58\relax}}G_{n}\rightarrow G_{1}$ be the
group homomorphism induced by the projection
$\mathbb{Z}/p^{n}\mathbb{Z}\rightarrow\mathbb{Z}/p\mathbb{Z}$. Then $\pi_{n}$
is surjective with

$\ker\pi_{n}=I_{4}+pM_{2\times 2}(\mathbb{Z}/p^{n}\mathbb{Z}).$

Thus, $|\ker(\pi_{n})|=p^{4(n-1)}$, and $|G_{n}|=p^{4(n-1)}|G_{1}|$. It is a
standard fact that $|G_{1}|=(p^{2}-1)(p^{2}-p)$. Hence the lemma follows. ∎

######  Theorem 3.2.

Assume that $X$ is an ordinary connected component of $X_{\ell}^{q}(N)$ and
let $K$ be the CM field of $X$. If $\ell$ splits in $K$, then there exists a
constant $c\geq(p+1)p$ such that the number of connected components of $X_{n}$
is equal to $cp^{2(n-1)}$ for $n$ sufficiently large. If $\ell$ is non-split,
then the same holds on replacing $cp^{2(n-1)}$ by $cp^{3(n-1)}$.

###### Proof.

The degree of the covering $X_{\ell}^{q}(p^{n}N)/X_{\ell}^{q}(N)$ equals
$|\operatorname{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})|$. As explained in Remark
2.6, the number of connected components of $X_{n}$ is given by
$|\operatorname{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})|$ divided the number of
pre-images of a fixed $(E,R_{1},R_{2})\in V(X)$ lying in the same connected
component of $X_{n}$.

As all curves with complex multiplication by an order in $K$ are isogenous we
can assume that $\textup{End}(E)=\mathcal{O}$ is an order such that
$[\mathcal{O}_{K}\mathrel{\mathop{\mathchar 58\relax}}\mathcal{O}]$ is not
divisible by $\ell$. In particular, the number of prime ideals in
$\mathcal{O}$ that lie above $\ell$ is the same as the number of prime ideals
in $\mathcal{O}_{K}$ that lie above $\ell$.

Let $(E,R_{1},R_{2},P,Q)$ and $(E,R_{1},R_{2},P^{\prime},Q^{\prime})$ be two
pre-images of $(E,R_{1},R_{2})$ in $V(X_{n})$. By Remark 2.12, they lie in the
same connected component of $X_{n}$ if and only if there is an isogeny
$\phi\colon E\to E$ of $\ell$-power degree such that $\phi(R_{1})=R_{1}$,
$\phi(R_{2})=R_{2}$, $\phi(P)=P^{\prime}$ and $\phi(Q)=Q^{\prime}$. Recall
that $E[p^{n}]\cong\mathcal{O}/p^{n}$ and that $\mathcal{O}$ has a natural
action on $E[p^{n}]$. Isogenies of $\ell$-power degree are given by elements
of $\ell$-power norm in $\mathcal{O}$. Let $U$ be the (multiplicative) set of
elements of $\ell$-power norm that fix $E[N]$. If $\ell$ splits in $K$, there
exists a constant $c^{\prime}$ such that the image $U_{n}$ of $U$ in
$(\mathcal{O}/p^{n})^{\times}$ has $c^{\prime}p^{2(n-1)}$ elements for $n$
large enough, with $c^{\prime}\leq(p-1)^{2}$. In all other cases, $U_{n}$ has
$c^{\prime}p^{n-1}$ elements for $n$ large enough with $c^{\prime}\leq(p-1)$.
Hence, the theorem follows from Lemma 3.1 on setting
$c=(p^{2}-1)(p^{2}-p)/c^{\prime}\geq(p+1)p$. ∎

We are now ready to prove Theorem A.

######  Corollary 3.3.

The covering $X_{n}/X$ is not Galois for $n$ sufficiently large.

###### Proof.

We prove that $X_{n}/X_{n-1}$ is not Galois, which in turn implies that
$X_{n}/X$ is not Galois.

It follows from Theorem 3.2 that the number of connected components of $X_{n}$
lying above $X_{n-1}$ is at least $p^{2}$ when $n$ is sufficiently large.
Since the degree of the covering $X_{n}/X_{n-1}$ is $p^{4}$ by Lemma 3.1, it
suffices to show that $(p^{2})!>p^{4}$ in light of Corollary 2.5.

Indeed,

$(p^{2}-2)(p^{2}-3)>\frac{4}{3}\geq\frac{1}{1-\frac{1}{p^{2}}}.$

This in turn gives

$(p^{2})!\geq p^{2}(p^{2}-1)(p^{2}-2)(p^{2}-3)>p^{4},$

as required. ∎

##  4\. The supersingular case

We now turn our attention to the supersingular case and prove Theorem B. We
fix once again a connected component $X$ of $X_{\ell}^{q}(N)$ and define
$X_{n}$ accordingly. In contrast to the previous section, we assume that all
elliptic curves $E$ giving rise to a vertex in $X$ have supersingular
reduction. Let $E$ be any supersingular elliptic curve giving rise to a vertex
in $X$. Let $B=\textup{End}(E)\otimes\mathbb{Q}$. Then $B$ is a quaternion
algebra, which is only ramified at $q$ and $\infty$. The endomorphism ring
$\textup{End}(E)$ is a maximal order in $B$.

######  Theorem 4.1 (Roda).

The number of connected components of $X_{n}$ is given by the index
$(\mathbb{Z}/p^{n}N\mathbb{Z})^{\times}/\langle\ell\rangle$. In particular,
the number of connected components is uniformly bounded as $n$ varies.

This theorem is the main result of [Rod19, Section 3.3]. The proof follows
from the arguments given in [GK21]. We outline the proof for the convenience
of the reader.

###### Proof.

Let $(E,R_{1},R_{2},P,Q)$ and
$(E,R^{\prime}_{1},R^{\prime}_{2},P^{\prime},Q^{\prime})$ be two vertices in
$X_{n}$. Then there is a matrix $C$ in
$\textup{GL}_{2}(\mathbb{Z}/Np^{n}\mathbb{Z})$ such that

$C\begin{pmatrix}R_{1}+P\\\
R_{2}+Q\end{pmatrix}=\begin{pmatrix}R^{\prime}_{1}+P^{\prime}\\\
R^{\prime}_{2}+Q^{\prime}\end{pmatrix}.$

In light of Remark 2.12, we determine when $C$ is induced by an endomorphism
of $\ell$-power degree.

Let us first assume that $C\in\textup{SL}_{2}(\mathbb{Z}/p^{n}N\mathbb{Z})$.
Since $\textup{SL}_{2}(\mathbb{Z}/p^{n}N\mathbb{Z})$ has strong approximation,
there exists $x\in B$ of norm one that is integral at all places except
possibly at $\ell$ and such that $x$ acts as $C$ on $E[p^{n}N]$. Let $r$ be a
positive integer such that $\ell^{r}$ acts trivially on $E[p^{n}N]$ and such
that $\ell^{r}x$ is integral at all places. Then $y=\ell^{r}x$ is an
endomorphism of $E$ of $\ell$-power degree such that
$y(P+R_{1})=P^{\prime}+R^{\prime}_{1}$ and
$y(Q+R_{2})=Q^{\prime}+R^{\prime}_{2}$ (see [GK21, proof of Theorem 5.3.3] for
further details).

Assume now that $\det(C)$ is a power of $\ell$. By [GK21, proof of Theorem
5.33], $E$ admits an endomorphism $f$ of degree $\ell^{r}$ with $r$ odd. As
$[\ell]$ has degree $\ell^{2}$, it follows that there is a matrix
$D\in\textup{GL}_{2}(\mathbb{Z}/p^{n}N\mathbb{Z})$ induced by an $\ell$-power
degree endomorphism such that $\det(C)=\det(D)$. Thus, to determine whether
$C$ is induced by an $\ell$-power degree endomorphism, it suffices to consider
$C^{\prime}=CD^{-1}$. As $C^{\prime}$ has determinant one this is precisely
the first case.

Clearly, $C$ can only be induced by an $\ell$-power degree endomorphism if
$\det(C)=\ell^{r}$ for some $r$. Thus, we have shown that $C$ is induced by an
$\ell$-power degree endomorphism if and only if $\det(C)$ is an $\ell$-power,
which concludes the proof. ∎

######  Corollary 4.2.

Assume $N=1$ or $2$ and $p>2$. Let $T$ be the set of primes $\ell$ such that
$X_{n}$ is connected. Then $T$ has density $\frac{\varphi(p-1)}{p}$, where
$\varphi$ is the Euler totient function.

###### Proof.

Theorem 4.1 says that the graph $X_{n}$ is connected if and only if $\ell$
generates $(\mathbb{Z}/p^{n}N\mathbb{Z})^{\times}$ for all $n$. This is the
case if and only if $\ell$ generates the cyclic group
$(\mathbb{Z}/Np^{2}\mathbb{Z})^{\times}$, which is of order $p(p-1)$. The
density of primes generating $(\mathbb{Z}/Np^{2}\mathbb{Z})^{\times}$ is
therefore $\frac{\varphi(p(p-1))}{p(p-1)}=\frac{(p-1)\varphi(p-1)}{p(p-1)}$. ∎

######  Remark 4.3.

If $N>2$ or $p>2$, the group $(\mathbb{Z}/p^{n}N\mathbb{Z})^{\times}$ is not
cyclic. In particular, $X_{n}$ is not connected for any choices of $\ell$.

######  Corollary 4.4.

Assume that $X_{n}$ is connected for all $n$. Then $X_{n}/X$ is a Galois
covering with Galois group isomorphic to
$\textup{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$.

###### Proof.

This follows immediately from Proposition 2.4ii). ∎

Finally, we show that even if the graphs $X_{n}$ are not connected, we may
still obtain a tower of Galois covers.

######  Corollary 4.5.

There exists a minimal integer $m_{0}$ such that the number of connected
components of $X_{n}$ stabilizes and such that there are no multiple edges.
Let $n>m\geq m_{0}$ and let
$G_{n,m}\subset\textup{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$ be the subgroup of
matrices congruent to $\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}\pmod{p^{m}}$.
Let $Z_{n}$ and $Z_{m}$ be connected components of $X_{n}$ and $X_{m}$,
respectively such that $Z_{n}/Z_{m}$ is a graph covering. Then $Z_{n}/Z_{m}$
is Galois with $\operatorname{Ga\ell}(Z_{n}/Z_{m})\cong G_{n,m}$.

###### Proof.

We have already seen in Theorem 4.1 that the number of connected components in
$X_{n}$ stabilizes when $n$ is sufficiently large. The graph $X_{n}$ contains
multiple edges if and only if there are two distinct $\ell$-isogenies
$\phi,\psi\colon E\to E^{\prime}$, which coincide on $E[Np^{n}]$, which does
not happen when $n$ is large enough. This shows the existence of the integer
$m_{0}$.

Assume now that $n>m\geq m_{0}$. Clearly, $Z_{n}/Z_{m}$ is a
$|G_{n,m}|$-sheeted covering and there is a natural embedding

$\psi_{m}\colon G_{n,m}\hookrightarrow\textup{Deck}(Z_{n}/Z_{m}).$

It remains to show that this embedding is indeed an isomorphism.

Let $\sigma\in\textup{Deck}(Z_{n}/Z_{m})$ and let $v$ be a vertex of $Z_{n}$.
Let $\pi\colon Z_{n}\to Z_{m}$ be the projection map. By definition

$\pi(\sigma(v))=\pi(v).$

The graph $Z_{n}$ is a subgraph of $X_{n}$. Therefore, there exists a vertex
$v_{0}$ of $X_{0}$ and elements
$h,h^{\prime}\in\textup{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$ such that
$v=(v_{0},h)$ and $\sigma(v)=(v_{0},h^{\prime})$. As $\pi(\sigma(v))=\pi(v)$,
the elements $h$ and $h^{\prime}$ have the same image in
$\textup{GL}_{2}(\mathbb{Z}/p^{m}\mathbb{Z})$. In particular, there exists an
element $g\in G_{n,m}$ such that $h^{\prime}=gh$, which is equivalent to
$g(v)=\sigma(v)$. We shall show that the action of $g$ on $Z_{n}$ coincides
with that of $\sigma$.

Let $w$ be a vertex of $Z_{n}$ admitting an edge whose source and target are
$v$ and $w$, respectively. We need to show that $\sigma(w)=g(w)$. As $Z_{m}$
does not have any multiple edges by assumption, it admits a unique edge from
$\pi(v)$ to $\pi(w)$. It follows that there is a unique edge in $Z_{n}$ from
$\sigma(v)$ to some vertex $w^{\prime}$ such that $\pi(w)=\pi(w^{\prime})$. As
$\sigma$ and $g$ are both deck transformations of $Z_{n}/Z_{m}$, we have
$\sigma(w)=g(w)=w^{\prime}$. Since $Z_{n}$ is strongly connected by Remark
2.11, this property propagates to all vertices of $Z_{n}$. Thus, the actions
of $\sigma$ and $g$ agree as desired, proving that $\psi_{m}$ is indeed an
isomorphism. ∎

##  5\. An abelian $p$-covering arising from quotients of
$X_{\ell}^{q}(p^{n}N)$

Let $\alpha_{n}$ be the voltage assignment given in Definition 2.7. Let
$\beta_{n}=\det\circ\alpha_{n}$. Then $\beta_{n}$ is a voltage assignment with
values in $\mathbb{Z}_{p}^{\times}$. As explained in the first paragraph of
the proof of Corollary 2.9, the derived graph $X(X_{\ell}^{q}(N),\beta_{n})$
is a subcovering of $X(X_{\ell}^{q}(N),\alpha_{n})/X_{\ell}^{q}(N)$. We study
this subcovering in detail in this section.

Recall that we fixed for any elliptic curve $E$ in $S$ defined over
$\mathbb{F}_{q^{k}}$ a basis $\\{s_{E},t_{E}\\}$ of the $p$-adic Tate-module
$T_{p}(E)$. Let

$\langle\cdot,\cdot\rangle\colon T_{p}(E)\times
T_{p}(E)\longrightarrow\mathbb{Z}_{p}(1)$

be the Weil pairing. In this section, we make the additional assumption that
for all $E$ the pairing $\langle s_{E},t_{E}\rangle$ is a topological
generator $\xi$ of $\mathbb{Z}_{p}(1)$. Let $\xi_{n}$ denote the primitive
$p^{n}$-th root of unity given by the image of $\xi$ under the natural map
$\mathbb{Z}_{p}(1)\rightarrow\mu_{p^{n}}$.

######  Definition 5.1.

We define a directed graph $Y_{\ell}^{q}(p^{n}N)$ as follows. The set of
vertices of $Y_{\ell}^{q}(p^{n}N)$ is given by tuples $(E,R_{1},R_{2},\zeta)$,
where $E,R_{1},R_{2}$ are as before and $\zeta$ is a primitive $p^{n}$-th root
of unity. Let $\Psi_{n}$ be the following surjection

$\Psi_{n}\colon V(X_{\ell}^{q}(N))\to
V(Y_{\ell}^{q}(p^{n}N)),\quad(E,R_{1},R_{2},P,Q)\mapsto(E,R_{1},R_{2},\langle
P,Q\rangle).$

For every vertex $v\in V(Y_{\ell}^{q}(Np^{n}))$, we fix a pre-image
$v^{\prime}\in V(X_{\ell}^{q}(p^{n}N))$ under $\Psi_{n}$. The edges of
$Y_{\ell}^{q}(Np^{n})$ are given as follows: there is an edge from $v$ to $w$
if and only if there is an edge from $v^{\prime}$ to some element
$w^{\prime\prime}\in\Psi_{n}^{-1}(w)$ in $X_{\ell}^{q}(Np^{n})$.

Given $(E,R_{1},R_{2},\zeta)\in V(Y_{\ell}^{q}(p^{n}N))$, we may write
$\zeta=\xi_{n}^{a}$ for a unique $a\in(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}$.
We will frequently denote $(E,R_{1},R_{2},\zeta)$ by $(E,R_{1},R_{2},a)$.

######  Remark 5.2.

For every elliptic curve $E$ defined over $\mathbb{F}_{q^{k}}$, the order of
the group $\textup{Aut}(E)$ is bounded by a constant $C_{q}$ that depends only
on $q$. Thus, if $N>C_{q}$, the graph $Y_{\ell}^{q}(Np^{n})$ does not admit
any multiple edges for all $n\geq 0$.

######  Proposition 5.3.

Let $N>C_{q}$. Then the graph $Y_{\ell}^{q}(Np^{n})$ is isomorphic to
$X(X_{\ell}^{q}(N),\beta_{n})$.

###### Proof.

By definition, there is a natural bijective map

$\Theta_{n}\colon V(Y_{\ell}^{q}(Np^{n}))\longrightarrow
V(X(X_{\ell}^{q}(N),\beta_{n})).$

Thus, it remains to show that $\Theta_{n}$ respects the edges of the two
graphs, as in the proof of Theorem 2.8.

Let $v=(E,R_{1},R_{2},a)$ and
$w=(E^{\prime},R^{\prime}_{1},R^{\prime}_{2},a^{\prime})$ be two vertices of
$Y_{\ell}^{q}(Np^{n})$. Let $v^{\prime}$ be the fixed pre-image in
$X_{\ell}^{q}(Np^{n})$ under $\Psi_{n}$ as in Definition 5.1. There is an edge
from $v$ to $w$ if and only if there is an edge from $v^{\prime}$ to
$w^{\prime\prime}$ for some $w^{\prime\prime}\in\Psi_{n}^{-1}(w)$.

Theorem 2.8 allows us to identify $V(X_{\ell}^{q}(p^{n}N))$ with
$V(X_{\ell}^{q}(N))\times\operatorname{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$.
Two bases of $E[p^{n}]$ have the same image in $\mu_{p^{n}}$ under the Weil
pairing if and only if the corresponding matrices in
$\operatorname{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$ have the same determinant.
This implies the existence of
$\sigma,\sigma^{\prime}\in\textup{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$ such
that $a=\det(\sigma)$, $a^{\prime}=\det(\sigma^{\prime})$ and that

$\displaystyle\Psi_{n}^{-1}(v)$
$\displaystyle=\left\\{(E,R_{1},R_{2},\sigma\tau)\mid\tau\in\textup{SL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})\right\\},$
$\displaystyle\Psi_{n}^{-1}(w)$
$\displaystyle=\left\\{(E^{\prime},R^{\prime}_{1},R^{\prime}_{2},\sigma^{\prime}\tau)\mid\tau\in\textup{SL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})\right\\}.$

Without loss of generality, we may assume that the chosen pre-image of $v$ is
given by $v^{\prime}=(E,R_{1},R_{2},\sigma)$. By the definition of
$X(X_{\ell}^{q}(N),\alpha_{n})$, there exists an edge from $v^{\prime}$ to an
element $w^{\prime\prime}\in\Psi_{n}^{-1}(w)$ if and only if there exists an
edge $e$ from $(E,R_{1},R_{2})$ to
$(E^{\prime},R^{\prime}_{1},R^{\prime}_{2})$ in $X_{\ell}^{q}(N)$ and an
element $\tau\in\textup{SL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$ such that

$\sigma^{\prime}\tau=\sigma\alpha_{n}(e).$

The existence of $\tau$ is equivalent to

$a^{\prime}=\det(\sigma^{\prime})=\det(\sigma)\det(\alpha_{n}(e))=a\beta_{n}(e).$

This condition is equivalent to the existence of an edge from
$(E,R_{1},R_{2},a)$ to $(E^{\prime},R^{\prime}_{1},R^{\prime}_{2},a^{\prime})$
in $X(X_{\ell}^{q}(N),\beta_{n})$. Therefore, $\Theta_{n}$ indeed respects the
edges of $Y_{\ell}^{q}(Np^{n})$ and $X(X_{\ell}^{q}(N),\beta_{n})$, as
desired. ∎

######  Remark 5.4.

Proposition 5.3 tells us that the graph $Y_{\ell}^{q}(p^{n}N)$ is independent
of the choice of pre-image $v^{\prime}\in V(X_{\ell}^{q}(p^{n}N))$ of $v\in
V(Y_{\ell}^{q}(p^{n}N))$ when $N$ is sufficiently large.

######  Theorem 5.5.

The number of connected components of $Y_{\ell}^{q}(Np^{n})$ is uniformly
bounded in $n$.

###### Proof.

Let $u$ be the minimal non-negative integer such that $\ell^{u}\equiv
1\pmod{N}$. Since the multiplication by $\ell^{u}$ is an endomorphism of
$\ell$-power degree, it follows from Remark 2.12 that there is a path from
$(E,R_{1},R_{2},a)$ to $(E,R_{1},R_{2},\ell^{2u}a)$ in $Y_{\ell}^{q}(Np^{n})$
for all choices of $(E,R_{1},R_{2},a)$. Furthermore, Remark 2.6 tells us that
the number of connected components of $Y_{\ell}^{q}(p^{n}N)$ is bounded by
$|(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}/\langle\ell^{2u}\rangle|$. This index
is uniformly bounded in $n$, which concludes the proof of the theorem. ∎

We are now ready to prove Theorem C.

######  Corollary 5.6.

Assume that $N>C_{q}$. Let $m_{0}$ be the minimal integer such that the number
of connected components of $Y_{\ell}^{q}(Np^{n})$ stabilizes. Let $n>m\geq
m_{0}$. Let $Y_{m}$ be a connected component of $Y_{\ell}^{q}(p^{m}N)$ and let
$Y_{n}$ be a connected component of $Y_{\ell}^{q}(p^{n}N)$ above it. Then
$Y_{n}/Y_{m}$ is Galois and $\operatorname{Ga\ell}(Y_{n}/Y_{m})$ is the
subgroup ${\mathcal{G}}_{n,m}$ of $(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}$ of
elements $x\equiv 1\pmod{p^{m}}$.

###### Proof.

Clearly, $Y_{n}/Y_{m}$ is a $|{\mathcal{G}}_{n,m}|$-sheeted cover. Via the
voltage assignment $\beta_{n}$, we see that there is an injective group
homomorphism

${\mathcal{G}}_{n,m}\hookrightarrow\textup{Deck}(Y_{n}/Y_{m}).$

It suffices to show that this map is surjective.

Let $\omega\in\mathrm{Deck}(Y_{n}/Y_{m})$ and $v=(E,R_{1},R_{2},a)\in
V(Y_{n})$. Let $v^{\prime}=(E,R_{1},R_{2},\sigma)\in X_{\ell}^{q}(Np^{n})$ be
the fixed pre-image under the map $\Psi_{n}$ given in Definition 5.1. Then
there exists $g\in{\mathcal{G}}_{n,m}$ such that $gv=\omega(v)$. Let
$g^{\prime}$ be a lift to $\textup{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$, i.e.
$g^{\prime}$ is a matrix with $\det(g^{\prime})=g$. In particular,

$\Psi_{n}(E,R_{1},R_{2},g^{\prime}\sigma)=gv.$

Let $\phi$ be a $\ell$-isogeny on $E$ and let $w=\phi(v)$. Let

$w^{\prime}=(\phi(E),\phi(R_{1}),\phi(R_{2}),\sigma\alpha_{n}(e_{\phi})),$

where $e_{\phi}$ is the edge induced by $\phi$. By definition,

$w=\Psi_{n}(w^{\prime})=(\phi(E),\phi(R_{1}),\phi(R_{2}),a\beta_{n}(e_{\phi})).$

As $Y_{\ell}^{q}(N)$ does not have multiple edges we see that
$\omega(w)=\phi(\omega(v))$. Furthermore,

$\displaystyle\omega(w)$
$\displaystyle=\phi(\omega(v))=\phi(gv)=\Psi_{n}(\phi(g^{\prime}v^{\prime}))=\Psi_{n}(\phi(E,R_{1},R_{2},g^{\prime}\sigma))$
$\displaystyle=\Psi_{n}(\phi(E),\phi(R_{1}),\phi(R_{2}),g^{\prime}\sigma\alpha_{n}(e))=(\phi(E),\phi(R_{1}),\phi(R_{2}),ga\beta_{n}(e_{\phi}))$
$\displaystyle=gw.$

Thus, $\omega$ is uniquely determined by $\omega(v)$. Hence,
${\mathcal{G}}_{n,m}\cong\textup{Deck}(Y_{n}/Y_{m})$, which concludes the
proof. ∎

In particular, we see that

$Y_{m_{0}}\leftarrow Y_{m_{0}+1}\leftarrow
Y_{m_{0}+2}\leftarrow\cdots\leftarrow Y_{m_{0}+n}\leftarrow\cdots$

form an abelian $p$-tower in the sense of [Val21, Definition 4.1].

##  6\. Isogeny graphs of oriented supersingular elliptic curves and Volcano
graphs

Let $M$ be a positive integer coprime to $q\ell$. While the structure of an
$\ell$-isogeny graph of ordinary elliptic curves equipped with a
$\Gamma_{1}(M)$-level structure is related to the so-called (tectonic) volcano
graphs (see [LM23, §4]), analogous results for the supersingular counterpart
are not available. In [CK20], Colò–Kohel introduced the concept of
orientations of supersingular elliptic curves. The authors in [ACL+23] showed
that the isogeny graph of supersingular elliptic curves equipped with an
orientation (without any level structures) is a volcano graph, analogous to
the ordinary counterpart studied in [Koh96]. In this section, we review the
notion of orientations and describe the structure of oriented supersingular
isogeny graphs enhanced with a $\Gamma(M)$-level structure for $M>2$.
Furthermore, we will show that the voltage assignment $\alpha_{n}$ defined in
§2.2 gives rise to a "$\textup{GL}_{2}(\mathbb{Z}_{p})$-tower" of
$\ell$-isogeny graphs of oriented elliptic curves with $\Gamma(Np^{n})$-level
structure as $n$ varies.

Throughout this section, we assume that $q\equiv 1\pmod{12}$ and that $B$ is
the quaternion algebra only ramified at $q$ and at $\infty$. Furthermore, we
assume that our fixed integer $k$ is even so that
$\mathbb{F}_{q^{2}}\subset\mathbb{F}_{q^{k}}$. All elliptic curves
$E/\mathbb{F}_{q^{k}}$ are assumed to be supersingular. As $q\equiv
1\pmod{12}$, we have $\textup{Aut}(E)=\\{\pm 1\\}$.

###  6.1. The $\ell$-isogeny graph of oriented supersingular elliptic curves

We recall the notion of oriented supersingular elliptic curves and their
isogeny graphs, as given in [CK20]. Furthermore, we review a result of
Arpin–Chen–Lauter–Scheidler–Stange–Tran [ACL+23], which describes these graphs
in terms of "volcanoes".

We fix once and for all an imaginary quadratic field $K$ in which $q$ does not
split. Then there is a natural embedding $\iota\colon K\to B$, where $B$ is a
quaternion algebra only ramified at $q$ and $\infty$ as in §4. Let $E$ be a
supersingular elliptic curve over ${\mathbb{F}_{q^{k}}}$. Then
$\textup{End}(E)\otimes\mathbb{Q}\cong B$. Thus, we can regard $\iota$ as an
embedding of $K$ into $\textup{End}(E)\otimes\mathbb{Q}$.

######  Definition 6.1.

Let $E/\mathbb{F}_{q^{k}}$ be a supersingular elliptic curve and $\iota$ be an
embedding of $K$ into $\operatorname{End}(E)\otimes B$.

We call the pair $(E,\iota)$ an oriented elliptic curve.

We call an order $\mathcal{O}$ of $K$ primitive for $(E,\iota)$, if
$\iota(\mathcal{O})=\iota(K)\bigcap\textup{End}(E)$.

We say that $\mathcal{O}$ is $\ell$-primitive if $\ell$ does not divide
$[\iota(K)\bigcap\textup{End}(E)\mathrel{\mathop{\mathchar
58\relax}}\iota(\mathcal{O})]$.

If $\varphi\colon E\to E^{\prime}$ is an isogeny of elliptic curves, we define
the orientation $\varphi_{*}\iota$ on $E^{\prime}$ by

$\displaystyle\varphi_{*}\iota\mathrel{\mathop{\mathchar 58\relax}}K$
$\displaystyle\rightarrow\operatorname{End}(E^{\prime})\otimes\mathbb{Q},$
$\displaystyle\alpha$
$\displaystyle\mapsto\frac{1}{\deg(\varphi)}\widehat{\varphi}\iota(\alpha)\varphi.$

An isogeny of oriented elliptic curves
$\varphi\colon(E,\iota)\to(E^{\prime},\iota^{\prime})$ is an isogeny of
elliptic curves $\varphi\colon E^{\prime}\to E$ such that
$\varphi_{*}\iota=\iota^{\prime}$.

We say that $\ell$-isogenies $\phi,\psi\colon E\to E^{\prime}$ are equivalent
isogenies if they have the same kernel.

We say that $(E,\iota)$ and $(E,\iota^{\prime})$ are equivalent oriented
elliptic curves if there exists an isomorphism of elliptic curves $\phi\colon
E\to E^{\prime}$ such that $\iota^{\prime}=\phi_{*}\iota$.

######  Remark 6.2.

Let $\phi,\psi\colon E\to E^{\prime}$ be equivalent isogenies. Then
$(\phi(E),\phi_{*}\iota)$ and $(\psi(E),\psi_{*}\iota)$ are equivalent.

We are now ready to define the $\ell$-isogeny graph of oriented elliptic
curves.

######  Definition 6.3.

We define $\mathcal{X}_{\ell}^{q}$ to be the directed graph whose vertices are
equivalence classes of oriented supersingular elliptic curves $(E,\iota)$ and
whose edges are given by $\ell$-isogenies up to equivalence.

The following notions will allow us to describe the structure of
$\mathcal{X}_{\ell}^{q}$.

######  Definition 6.4.

Let $\varphi\mathrel{\mathop{\mathchar
58\relax}}(E,\iota)\rightarrow(E^{\prime},\iota^{\prime})$ be an isogeny of
oriented supersingular elliptic curves. Assume that $\mathcal{O}$ and
$\mathcal{O}^{\prime}$ are orders of $K$ that are primitive for $(E,\iota)$
and $(E^{\prime},\iota^{\prime})$, respectively. We say that $\varphi$ is

  * •

horizontal if $\mathcal{O}=\mathcal{O}^{\prime}$,

  * •

descending if $\mathcal{O}^{\prime}\subset\mathcal{O}$,

  * •

ascending if $\mathcal{O}^{\prime}\supset\mathcal{O}$.

The notion of horizontal, ascending and descending isogenies allows us to
define the depth of an oriented elliptic curve.

######  Definition 6.5.

Let $(E,\iota)$ be an oriented elliptic curve. Assume that $\mathcal{O}$ is
primitive with respect to $(E,\iota)$. We say that $(E,\iota)$ is of depth
zero if the conductor of $\mathcal{O}$ is not divisible by $\ell$. We say that
$(E,\iota)$ is of depth $k$ if there is a path of $k$ descending isogenies of
degree $\ell$ starting at a depth zero oriented elliptic curve
$(E^{\prime},\iota^{\prime})$ and ending at $(E,\iota)$ .

Finally, we introduce the following notions of graphs, which are the last
ingredients for our description of $\mathcal{X}_{\ell}^{q}$.

######  Definition 6.6.

An abstract crater is either a connected and directed cycle graph or a totally
disconnected finite graph.

A volcano graph (or simply a volcano) $G$ is a directed graph whose vertices
may be decomposed as $V(G)=\bigcup_{i=0}^{\infty}V_{i}$ such that

  * •

The out-degree of every vertex is $\ell+1$;

  * •

The subgraph generated by $V_{0}$ is an abstract crater;

  * •

If there is an edge between $v_{i}$ and $v_{j}$ then $i-j\in\\{\pm 1\\}$ or
$i=j=0$;

  * •

Each $v\in V_{i}$ with $i\geq 1$ has exactly one edge starting at $v$ and
ending at a vertex in $V_{i-1}$ and $\ell$ edges starting at $v$ and ending at
vertices in $V_{i+1}$.

An undirected volcano $G$ is an undirected $(\ell+1)$-regular graph whose
vertices may be decomposed as $V(G)=\bigcup_{i=0}^{\infty}V_{i}$ such that

  * •

The subgraph generated by $V_{0}$, called the crater of $G$, is an undirected
cycle graph;

  * •

If there is an edge between $v_{i}$ and $v_{j}$ then $i-j\in\\{\pm 1\\}$ or
$i=j=0$;

  * •

Each $v\in V_{i}$ with $i\geq 1$ has exactly one neighbour in $V_{i-1}$ and
$\ell$ neighbours in $V_{i+1}$.

######  Remark 6.7.

Note that (undirected) volcanoes are always infinite graphs.

######  Theorem 6.8.

[ACL+23, Section 3.4] The graph $\mathcal{X}_{\ell}^{q}$ is an infinite graph.
Its connected components are volcano graphs whose craters are generated by the
depth zero vertices.

###  6.2. The $\ell$-isogeny graph of oriented supersingular elliptic curve
with level structure

We would like to consider oriented elliptic curves enhanced with a
$\Gamma(M)$-level structure. In other words, instead of pairs $(E,\iota)$ we
consider tuples $(E,\iota,R_{1},R_{2})$, where $\\{R_{1},R_{2}\\}$ is a basis
for $E[M]$.

######  Definition 6.9.

An isogeny of oriented elliptic curves with a $\Gamma(M)$-level structure
$\varphi\colon(E,\iota,R_{1},R_{2})\to(E^{\prime},\iota^{\prime},R^{\prime}_{1},R^{\prime}_{2})$
is an isogeny of oriented elliptic curves
$(E,\iota)\to(E^{\prime},\iota^{\prime})$ such that
$\varphi(R_{1})=R^{\prime}_{1}$ and $\varphi(R_{2})=R^{\prime}_{2}$.

######  Definition 6.10.

We fix a set of representatives $S^{\prime}$ for the equivalence classes of
oriented elliptic curves over $\mathbb{F}_{q^{k}}$.

Let $\mathcal{X}_{\ell}^{q}(M)$ be the directed graph whose vertices are
quadruples $(E,\iota,R_{1},R_{2})$, where $(E,\iota)\in S^{\prime}$ and
$\\{R_{1},R_{2}\\}$ is a basis for $E[M]$, and such that the edges are given
by $\ell$-isogenies defined as in Definition 6.9.

Let $e\in\mathbb{E}(\mathcal{X}_{\ell}^{q}(M))$. We say that $e$ is
horizontal/descending/ascending if the isogeny giving rise to $e$ is
horizontal/descending/ascending.

######  Remark 6.11.

If $\phi\mathrel{\mathop{\mathchar 58\relax}}E\rightarrow E^{\prime}$ is an
$\ell$-isogeny of elliptic curves, the composition
$\phi\circ[-1]=[-1]\circ\phi$ is also an $\ell$-isogeny. Note that $[-1]$ acts
non-trivially on a basis of $E[M]$ for $M>2$. Thus, equivalent isogenies
induce edges with different targets if $M>2$. For this reason, we have decided
to distinguish edges arising from $\phi$ and $\phi\circ[-1]$ in our definition
of $\mathcal{X}_{\ell}^{q}(M)$ for all positive integers $M$, even though
these two isogenies are equivalent. In particular, the graphs
$\mathcal{X}_{\ell}^{q}(1)$ and $\mathcal{X}_{\ell}^{q}$ are not isomorphic to
each other. They share the same vertex set, but each edge of
$\mathcal{X}_{\ell}^{q}$ corresponds to two distinct edges in
$\mathcal{X}_{\ell}^{q}(1)$.

In order to describe the structure of $\mathcal{X}_{\ell}^{q}(M)$ in more
detail, we recall the following notions from [ACL+23].

######  Definition 6.12.

Let $E/_{\mathbb{F}_{q^{k}}}$ be a supersingular elliptic curve and
$\theta\in\textup{End}(E)$. Let $f_{\theta}$ be the minimal polynomial of
$\theta$, i.e. the unique polynomial $X^{2}+aX+b\in\mathbb{Z}[x]$ such that
$f_{\theta}(\theta)=0$. Let $\alpha$ be the (unique up to conjugation) element
generating a quadratic extension of $\mathbb{Q}$ with minimal polynomial
$f_{\theta}$. Let $K=\mathbb{Q}(\alpha)$ and let

$\omega_{K}=\begin{cases}\frac{1+\sqrt{\Delta}}{2}\quad&\Delta\equiv
1\pmod{4}\\\ \frac{\sqrt{\Delta}}{2}\quad&\Delta\equiv 0\pmod{4},\end{cases}$

where $\Delta$ is the fundamental discriminant of $K$.

We say that $\theta$ is $\ell$-primitive if $\mathbb{Z}[\alpha]$ is
$\ell$-primitive with respect to the orientation $K\to\textup{End}(E)$ given
by $\alpha\mapsto\theta$.

We say that $\theta$ is $\ell$-suitable if $\alpha$ is of the form
$f\omega_{K}+m\ell$, where $m$ is some integer and $f$ is the conductor of
$\mathbb{Z}[\alpha]$.

######  Remark 6.13.

Given a fixed orientation $\iota\colon K\to B$ and a supersingular elliptic
curve $E$, there always exists $\theta\in\textup{End}(E)$ that is
$\ell$-primitive and $\ell$-suitable. Indeed, let $\mathcal{O}$ be an order of
$K$, which is primitive with respect to $(E,\iota)$. Let $f$ be the conductor
of $\mathcal{O}$ and let $\alpha=f\omega_{K}$. Then
$\mathcal{O}=\mathbb{Z}[f\omega_{K}]=\mathbb{Z}[\alpha]$. If we set
$\theta=\iota(\alpha)$, the endomorphism $\theta$ is $\ell$-primitive and
$\ell$-suitable.

######  Proposition 6.14.

[ACL+23, Proposition 4.8] Let $\theta$ be an element that is $\ell$-primitive
and $\ell$-suitable for $(E,\iota)$. Assume that $(E,\iota)$ is of depth zero
and let $\lambda_{1},\lambda_{2}$ be the eigenvalues of the action of $\theta$
on $E[\ell]$.

  * i)

If $\ell$ splits in $K$, then
$\lambda_{1}\neq\lambda_{2}\in\mathbb{F}_{\ell}$. Let $P_{\lambda_{i}}$ be
generators of the $\lambda_{i}$-eigenspaces, then $E\to E/\langle
P_{\lambda_{i}}\rangle$ are horizontal isogenies for $i=1,2$.

  * ii)

If $\ell$ ramifies in $K$, then
$0\neq\lambda_{1}=\lambda_{2}\in\mathbb{F}_{\ell}$. The
$\lambda_{i}$-eigenspace is one-dimensional, where $i=1,2$. Let $P$ be a
generator of this eigenspace. Then $E\to E/\langle P\rangle$ is a horizontal
isogeny.

  * iii)

If $\ell$ is inert in $K$, then
$\lambda_{1},\lambda_{2}\in\mathbb{F}_{\ell^{2}}\setminus\mathbb{F}_{\ell}$
and there are no horizontal isogenies on $E$.

All other $\ell$-isogenies are descending in all three cases.

######  Lemma 6.15.

Assume that $\ell$ splits in $K$. Let $(E,\iota)$ be a depth zero vertex. Let
$\theta\in\operatorname{End}(E)$ be $\ell$-primitive and $\ell$-suitable for
$(E,\iota)$ and let $\alpha$ be the element attached to $\theta$ given in
Definition 6.12. Let $\lambda_{1}$ and $\lambda_{2}$ be given as in
Proposition 6.14. For $i\in\\{1,2\\}$, let $P^{E}_{i}$ be a generator of the
$\lambda_{i}$-eigenspace and $\phi^{E}_{i}\colon E\to E/\langle
P_{i}^{E}\rangle$ be the natural projection.

  * i)

Let $\phi\colon(E,\iota)\to(E^{\prime},\iota^{\prime})$ be a horizontal
isogeny. Then $\theta^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=\iota^{\prime}(\alpha)$ is $\ell$-primitive and $\ell$-suitable for
$(E^{\prime},\iota^{\prime})$. In particular, $\lambda_{1}$ and $\lambda_{2}$
are the eigenvalues of the action of $\theta^{\prime}$ on $E^{\prime}[\ell]$.

  * ii)

Assume that $\lambda_{1}\neq\lambda_{2}$ and let $P_{i}^{E^{\prime}}$ be a
generator of the $\lambda_{i}$-eigenspace of $\phi^{E}_{i}(E)$. Let $i\neq j$.
Then $\phi_{i}^{E}(P_{j})=cP_{j}^{E^{\prime}}$, for some $c$ coprime to
$\ell$.

  * iii)

Fix $i\in\\{1,2\\}$. Let
$v_{0}=(E_{0},\iota_{0},R_{0,1},R_{0,2})=(E,\iota,R_{1},R_{2})\in
V(\mathcal{X}_{\ell}^{q}(M))$ and define recursively

$v_{j}=(E_{j},\iota_{j},R_{j,1},R_{j,2})=\phi_{i}^{E_{j-1}}(v_{j-1}),\ j\geq
1.$

Then there exists a non-negative integer $r$ such that $v_{r}=v_{0}$.

###### Proof.

i) and ii) follow directly from definitions. It remains to prove iii). By
Theorem 6.8, there are only finitely many depth zero vertices in
$\mathcal{X}_{\ell}^{q}$ that lie in the same connected component as $v$.
Thus, there is a non-negative integer $r_{0}$ such that $E_{r_{0}}=E$ and
$\iota_{r_{0}}=\iota$. Hence, there is an isogeny
$\gamma\mathrel{\mathop{\mathchar 58\relax}}(E,\iota)\rightarrow(E,\iota)$
whose degree is an $\ell$-power such that $\gamma(R_{i})=R_{r_{0},i}$ for
$i=1,2$. Let $s$ be the minimal non-negative integer such that
$\gamma^{s}(R_{i})=R_{i}$ for both $i$. Then part iii) follows on setting
$r=r_{0}s$. ∎

We now introduce several notions of graphs that will be utilized in our
description of $\mathcal{X}_{\ell}^{q}(M)$.

######  Definition 6.16.

Let $Z$ be a directed graph. We define the double intertwinement $Z^{+-}$ of
$Z$ to be the graph such that

$V(Z^{+-})=\\{+v,-v\mathrel{\mathop{\mathchar 58\relax}}v\in V(Z)\\}$

(each vertex of $V(Z)$ gives rise to two vertices in $Z^{+-}$) and that

$\mathbb{E}(Z^{+-})=\\{e^{++},e^{+-},e^{-+},e^{--}\mathrel{\mathop{\mathchar
58\relax}}e\in\mathbb{E}(Z)\\},$

where $e^{\bullet\circ}$ denotes an edge whose source and target are $\bullet
v$ and $\circ w$, respectively if $e$ is an edge from $v$ to $w$ in the
original graph $Z$.

######  Example 6.17.

Let us illustrate the double intertwinement of the following graph:

$v_{1}$$v_{2}$Z

Then $Z^{+-}$ is given by

$+v_{1}$$+v_{2}$$-v_{1}$$-v_{2}$$Z^{+-}$

The double intertwinement of the directed cycle graph with $4$ edges and $4$
vertices is given by:

$+v_{1}$$+v_{2}$$+v_{3}$$+v_{4}$$-v_{2}$$-v_{3}$$-v_{4}$$-v_{1}$

The following definition was introduced in [LM23, §5].

######  Definition 6.18.

Let $\mathfrak{r},\mathfrak{s},\mathfrak{t},\mathfrak{c}$ be nonnegative
integers. We say that a directed graph is an abstract tectonic crater of
parameters $(\mathfrak{r},\mathfrak{s},\mathfrak{t},\mathfrak{c})$ if it
satisfies

  * a)

There are $\mathfrak{r}\mathfrak{s}\mathfrak{t}$ vertices;

  * b)

Each edge is assigned a color – blue or green;

  * c)

At each vertex $v$, there is exactly one blue edge with $v$ as the source, and
exactly one blue edge with $v$ as the target, and similarly for green edges;

  * d)

Starting at each vertex, there is exactly one closed blue (resp. green) path
without backtracks of length $\mathfrak{r}\mathfrak{s}$ (resp.
$\mathfrak{r}\mathfrak{t}$);

  * e)

After every $\mathfrak{s}$ (resp. $\mathfrak{c}\mathfrak{t}$) steps in the
closed blue (resp. green) paths given in d), the two paths meet at a common
vertex.

######  Example 6.19.

An abstract tectonic crater with $\mathfrak{t}=\mathfrak{s}=1$,
$\mathfrak{c}=2$ and $\mathfrak{r}=5$.

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$

For further examples, the reader is referred to [LM23, §4.2].

We may enhance the definition of an abstract tectonic crater, giving the
"tectonic" version of a volcano graph:

######  Definition 6.20.

A tectonic volcano $G$ is a directed graph whose vertices may be decomposed
into $V(G)=\bigcup_{i=0}^{\infty}V_{i}$ such that

  * •

The out-degree of every vertex is $\ell+1$;

  * •

The subgraph generated by $V_{0}$ is an abstract tectonic crater (we shall
refer to $V_{0}$ as the tectonic crater of $G$);

  * •

If there is an edge between $v_{i}$ and $v_{j}$ then $i-j\in\\{\pm 1\\}$ or
$i=j=0$;

  * •

Every $v\in V_{i}$ with $i\geq 1$ has exactly one edge starting at $v$ and
ending at a vertex in $V_{i-1}$ and $\ell$ edges starting at $v$ and ending at
a vertex in $V_{i+1}$.

We are now ready to prove our first theorem on the structure of
$\mathcal{X}_{\ell}^{q}(M)$.

######  Theorem 6.21.

Assume that $\ell$ splits in $K$ and that $M>2$. Let $\mathcal{Y}_{M}$ be a
connected component of $\mathcal{X}_{\ell}^{q}(M)$. Then $\mathcal{Y}_{M}$ is
the double intertwinement of a tectonic volcano graph.

###### Proof.

There is a natural map $\pi\colon\mathcal{Y}_{M}\to\mathcal{X}_{\ell}^{q}$
given by

$(E,\iota,R_{1},R_{2})\mapsto(E,\iota).$

Note that this is not a covering of graphs, as the vertices of
$\mathcal{Y}_{M}$ have out-degree $2(\ell+1)$, while those in
$\mathcal{X}_{\ell}^{q}$ have out-degree $(\ell+1)$. It follows from Theorem
6.8 that $\mathcal{X}_{\ell}^{q}$ is the union of volcanoes.

Let $(E,\iota)$ be a depth zero vertex of $\mathcal{X}_{\ell}^{q}$ and let $v$
be a lift of $(E,\iota)$ in $\mathcal{Y}_{M}$. For each pair $(E_{1},E_{2})$
of supersingular elliptic curves, we fix a set of representatives
$S(E_{1},E_{2})$ of $\\{\textup{$\ell$-isogenies from $E_{1}$ to
$E_{2}$}\\}/\\{\pm 1\\}$.

Let $\theta\in\textup{End}(E)$ be $\ell$-primitive and $\ell$-suitable for
$(E,\iota)$ (such $\theta$ always exists as explained in Remark 6.13). Let
$\alpha\in K$ be the element given as in Definition 6.12. It follows from
Proposition 6.14i) that the eigenvalues $\lambda_{1}$ and $\lambda_{2}$ of the
action of $\theta$ on $E[\ell]$ are distinct elements of $\mathbb{F}_{\ell}$.
Let $S_{0}(E_{1},E_{2})\subset S(E_{1},E_{2})$ be the subset of horizontal
isogenies. Starting at $v$ and only propagating along the edges given by
$S_{0}(E_{1},E_{2})$, we obtain a finite connected subgraph $Z_{0}$ of
$\mathcal{Y}_{M}$. We will show that $Z_{0}$ is an abstract tectonic crater.

Let $(E^{\prime},\iota^{\prime},R^{\prime}_{1},R^{\prime}_{2})$ be an
arbitrary vertex in $Z_{0}$. We define
$\theta^{\prime}=\iota^{\prime}(\alpha)$. By Lemma 6.15i) the eigenvalues of
the action of $\theta^{\prime}$ on $E^{\prime}[\ell]$ are $\lambda_{1}$ and
$\lambda_{2}$. For $1\leq i\leq 2$, let $P^{E^{\prime}}_{i}$ be a
$\lambda_{i}$-eigenvector of the action of $\theta^{\prime}$ on $E[\ell]$. Let

$\phi^{E^{\prime}}_{i}\colon E^{\prime}\to E^{\prime}/\langle
P^{E^{\prime}}_{i}\rangle$

be the natural projection. We say that an edge in $Z_{0}$ is blue (resp.
green) if it is induced by $\phi^{E^{\prime}}_{1}$ or by
$\phi^{E^{\prime}}_{1}\circ[-1]$ (resp. by $\phi^{E^{\prime}}_{2}$ or
$\phi^{E^{\prime}}_{2}\circ[-1]$). By the definition of $S_{0}(E_{1},E_{2})$,
exactly one of the two elements in
$\\{\phi^{E^{\prime}}_{i},\phi^{E^{\prime}}_{i}\circ[-1]\\}$ induces an edge
in $Z_{0}$. It follows from Lemma 6.15ii) that blue and green edges commute
with each other, i.e. starting at a vertex $v^{\prime}$ any path consisting of
$k_{1}$ blue and $k_{2}$ green edges will end at the same vertex $w$ – no
matter in which order one follows green and blue edges.

For every vertex $v^{\prime}$ in $Z_{0}$ we denote by $\phi_{i,j}(v^{\prime})$
the vertex at the end of a path consisting of $i$ blue and $j$ green edges
that starts at $v^{\prime}$. By a similar proof as the one given in [LM23,
Lemma 4.4], we have:

(6.1) If $\phi_{i,j}(v)=\phi_{i^{\prime},j^{\prime}}(v)$, then
$\phi_{i,j}(v^{\prime})=\phi_{i^{\prime},j^{\prime}}(v^{\prime})$ for all
$v^{\prime}\in V(Z_{0})$.

By Lemma 6.15iii) there exists a non negative integer $h_{1}$ (resp. $h_{2}$)
such that a blue (resp. green) path of length $h_{1}$ (resp. $h_{2}$) sends
$v$ to itself. Note that $\phi_{h_{1},0}(v)=\phi_{0,h_{2}}(v)=v$. Choose now
$0<s<h_{1}$ minimal with the property that there exists an integer $j$ such
that $\phi_{s,0}(v)=\phi_{0,j}(v)$. Let $t=\gcd(h_{2},j)$ and let $ct=j$. Let
now $(s^{\prime},t^{\prime},c^{\prime})$ be a triple of integers with the
following properties

  * •

$s^{\prime}\mid h_{1}$.

  * •

$t^{\prime}\mid h_{2}$.

  * •

$c^{\prime}$ is coprime to $h_{2}/t^{\prime}$.

  * •

$\phi_{s^{\prime},0}(v)=\phi_{0,c^{\prime}t^{\prime}}(v)$.

Using (6.1) instead of [LM23, Proposition 4.4] in the proof of [LM23,
proposition 4.6] we can deduce the following claim:

(6.2) There exists an integer $d$ such that $s^{\prime}=ds$ and
$t^{\prime}=dt$.

Substituting Proposition 4.4 in loc. cit by (6.1) and Proposition 4.6 by
(6.2), it now follows from the same line of argument as given in [LM23,
Section 4.2] that $Z_{0}$ is indeed a tectonic volcano.

Let $Z$ be the subgraph obtained by starting at $v$ and only propagating along
the edges in $S(E_{1},E_{2})$. Then $Z_{0}$ is a subgraph of $Z$ and all
horizontal edges of $Z$ lie in $Z_{0}$. By Proposition 6.14, every vertex in
$Z_{0}$ is the source of exactly $\ell-1$ descending edges in $Z$. By [ACL+23,
Proposition 2.15], every vertex in $V(Z)\setminus V(Z_{0})$ has one ascending
edge and $\ell-1$ descending ones. Therefore, $Z$ is indeed a tectonic volcano
with $Z_{0}$ as its tectonic crater. The last step of our proof constitutes of
showing that $\mathcal{Y}_{M}$ is the double intertwinement of $Z$.

For every vertex $v=(E,\iota,R_{1},R_{2})$ in $Z$, the definition of
$S(E_{1},E_{2})$ implies that the "opposite" vertex
$v^{\prime}\mathrel{\mathop{\mathchar 58\relax}}=(E,\iota,-R_{1},-R_{2})$ does
not lie in $Z$. We may find a path from $v$ to $v^{\prime}$ by composing an
appropriate power of $[\ell]$ with $[-1]$. Thus, $v^{\prime}\in
V(\mathcal{Y}_{M})\setminus V(Z)$. In particular, we may identify $v$ and
$v^{\prime}$ with $+v$ and $-v$ in $V(Z^{+-})$, respectively.

If there is an edge $e$ from $v$ to $w$ in $Z$, there is one from $v$ to $-w$
in $\mathcal{Y}_{M}$ simply by composing the path from $v$ to $w$ with $[-1]$.
Similarly, there are edges from $-v$ to $-w$ and from $-v$ to $w$. Therefore,
$e\in\mathbb{E}(Z)$ gives rise to four edges in $\mathcal{Y}_{M}$, which we
may identify with the edges $e^{\bullet\circ}$, $\bullet,\circ\in\\{+,-\\}$ in
$Z^{+-}$.

As all edges in $\mathcal{Y}_{M}$ are given by compositions of the edges in
the set $S(E_{1},E_{2})$ and $[-1]$, we conclude that
$\mathcal{Y}_{M}=Z^{+-}$. ∎

In a similar manner, one can prove:

######  Theorem 6.22.

Assume that $\ell$ ramifies in $K$ and $M>2$. Then $\mathcal{Y}_{M}$ is the
double intertwinement of a volcano with a connected crater.

######  Theorem 6.23.

Assume that $\ell$ is inert in $K$ and $M>2$. Then $\mathcal{Y}_{M}$ is the
double intertwinement of a volcano whose crater is disconnected.

######  Remark 6.24.

If $M=2$, the vertices $(E,\iota,R_{1},R_{2})$ and $(E,\iota,-R_{1},-R_{2})$
coincide. Furthermore $\phi$ and $\phi\circ[-1]$ define the same edge. So
instead of the double intertwinement of a (tectonic) volcano, the graph
$\mathcal{Y}_{2}$ is derived from a (tectonic) volcano by drawing every edge
with multiplicity $2$.

######  Definition 6.25.

We define $X_{\ell}^{q}(M)^{\mathrm{ss}}$ to be the subgraph of
$X_{\ell}^{q}(M)$ generated by the vertices arising from supersingular
elliptic curves.

######  Corollary 6.26.

Suppose that $M>2$. Then each connected component of
$X_{\ell}^{q}(M)^{\mathrm{ss}}$ is the quotient of the double intertwinement
of either a volcano or a tectonic volcano.

###### Proof.

This follows from the fact that each connected component of
$X_{\ell}^{q}(M)^{\mathrm{ss}}$ is the quotient of certain $\mathcal{Y}_{M}$
given in Theorems 6.21, 6.22 and 6.23, where the covering map is given by
sending the vertex $(E,\iota,R_{1},R_{2})$ to $(E,R_{1},R_{2})$. ∎

###  6.3. Towers of isogeny graphs of oriented supersingular elliptic curves

Let $N$ be a positive integer that is coprime to $pq\ell$. We now turn our
attention to "$\textup{GL}_{2}(\mathbb{Z}_{p})$-towers" that arise from
$\mathcal{X}_{\ell}^{q}(p^{n}N)$ as $n$ varies, proving Theorem D.

######  Theorem 6.27.

Assume that $q\equiv 1\pmod{12}$, $N=1$ or $2$ and that
$X_{\ell}^{q}(p^{n}N)^{\mathrm{ss}}$ is connected for all $n$. Let
$\mathcal{X}_{0}$ be a connected component of $\mathcal{X}_{\ell}^{q}(N)$ and
denote the pre-image of $\mathcal{X}_{0}$ in $\mathcal{X}_{\ell}^{q}(p^{n}N)$
by $\mathcal{X}_{n}$ for $n\geq 1$. The covering
$\mathcal{X}_{n}/\mathcal{X}_{0}$ is Galois with Galois group isomorphic to
$\textup{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$.

######  Remark 6.28.

The assumption that $X_{\ell}^{q}(p^{n}N)^{\mathrm{ss}}$ is connected for all
$n$ forces $N\in\\{1,2\\}$ (see Remark 4.3).

###### Proof.

By an abuse of notation, we denote the restriction of the voltage assignment
of $\alpha_{n}$ to the supersingular vertices by the same notation. We have as
before
$X(\textup{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z}),\alpha_{n})=X_{\ell}^{q}(p^{n}N)^{\mathrm{ss}}$.

By Corollary 6.26 there is a natural projection

$\pi_{n}\colon\mathcal{X}_{n}\to X_{\ell}^{q}(p^{n}N)^{\mathrm{ss}}.$

Let $\alpha^{\prime}_{n}=\alpha_{n}\circ\pi_{0}$. By an argument similar to
the one employed in the proof of Theorem 2.8, we can identify the graph
$\mathcal{X}_{n}$ with
$X(\textup{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z}),\alpha^{\prime}_{n})$. Thus,
there is an embedding

$\textup{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})\hookrightarrow\textup{Deck}(\mathcal{X}_{n}/\mathcal{X}_{0}).$

Hence, the theorem would follow from the surjectivity of this embedding, which
we show below.

The projection $\pi$ induces a natural projection

$\pi^{\prime}_{n}\colon\textup{Deck}(\mathcal{X}_{n}/\mathcal{X}_{0})\to\textup{Deck}\left(X_{\ell}^{q}(p^{n}N)^{\mathrm{ss}}/X_{\ell}^{q}(N)^{\mathrm{ss}}\right)\cong\textup{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z}),$

where the last isomorphism follows from Corollary 4.4. Therefore, it suffices
to show that $\pi^{\prime}_{n}$ is injective.

Let $\sigma\in\ker(\pi^{\prime}_{n})$. Let $v=(E,\iota,R_{1},R_{2},P,Q)\in
V(\mathcal{X}_{n})$. As $\sigma$ is a deck transformation,
$\sigma(v)=(E,\iota,R_{1},R_{2},P,^{\prime}Q^{\prime})$ for some basis
$\\{P^{\prime},Q^{\prime}\\}$ of $E[p^{n}]$. Furthermore,
$\sigma\in\ker(\pi^{\prime}_{n})$ means that

$(E,R_{1},R_{2},P^{\prime},Q^{\prime})=\pi_{n}(\sigma(v))=\pi^{\prime}_{n}(\sigma)\pi_{n}(v)=(E,R_{1},R_{2},P,Q).$

It follows that $P=P^{\prime}$ and $Q=Q^{\prime}$. Thus, $\sigma(v)=v$. Since
this holds for any choice of $v$, the deck transformation $\sigma$ is trivial.
Thus, $\pi^{\prime}_{n}$ is injective and
$\textup{Deck}(\mathcal{X}_{n}/\mathcal{X}_{0})\cong\textup{GL}_{2}(\mathbb{Z}/p^{n}\mathbb{Z})$
as desired. ∎

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