# Jacobians of Graphs via Edges and Iwasawa Theory

Jon Aycock

###### Abstract

The Jacobian is an algebraic invariant of a graph which is often seen in
analogy to the class group of a number field. In particular, there have been
multiple investigations into the Iwasawa theory of graphs with the Jacobian
playing the role of the class group, for example in [Gon22, Val21, MV23]. In
this paper, we construct an Iwasawa module related to the Jacobian of a
$\mathbb{Z}_{p}$-tower of connected graphs, and give examples where we use
this to compute asymptotic sizes of the Jacobians in this tower.

##  1 Introduction

The Jacobian is an algebraic invariant of a graph which is often seen in
analogy to the class group of a number field. Classically, the Jacobian is
defined in terms of the vertices of the graph; however, in this paper we give
another construction of the Jacobian in terms of the edges in Definition 2.11.
We prove that this gives an isomorphic group in Proposition 2.12.

Because of the analogy to the class group, there have been investigations into
a version of Iwasawa theory for graphs. In [Val21, MV23], the authors
$p$-adically interpolate the Ihara zeta function and use an analog of the
analytic class number formula to get asymptotics for the sizes of Jacobians in
a $\mathbb{Z}_{p}$-tower of graphs. In [Gon22], the author instead constructs
a $\Lambda$-module which is related to the Jacobian. This is more in line with
the methods of this paper; the main result here is Theorem 4.5, which is
roughly as follows.

######  Theorem 1.1.

Let $\Gamma$ be a group isomorphic to $\mathbb{Z}_{p}$, $\Gamma_{n}$ be its
quotient isomorphic to $\mathbb{Z}/p^{n}\mathbb{Z}$, and let $\Gamma^{(n)}$ be
the kernel of the projection $\Gamma\to\Gamma_{n}$. There is a module
$J(\infty)$ over the Iwasawa algebra $\Lambda=\mathbb{Z}_{p}[[\Gamma]]$ whose
coinvariants $J(n)=J(\infty)\otimes_{\Lambda}\mathbb{Z}_{p}[\Gamma_{n}]$ are
isomorphic to an extension of the $p$-power torsion subgroup of the Jacobian
$\mathcal{J}(X_{n})[p^{\infty}]$ by $\Gamma^{(n)}\cong\mathbb{Z}_{p}$.

One may contrast this with [Gon22, Proposition 4], which states that the
$\Lambda$-module constructed in that paper is an extension of $\mathbb{Z}_{p}$
by $\mathcal{J}(X_{n})[p^{\infty}]$. In addition to the extra copy of
$\mathbb{Z}_{p}$ being a submodule instead of a quotient, the way in which
these extra factors arise are different. In [Gon22], one must control for the
degree of the divisor. However, in Theorem 4.5, the extra factor of
$\mathbb{Z}_{p}$ comes from Galois theory; we first mention this defect in
Section 3.4, and we talk about it in more detail in Remark 4.6.

In Section 2, we explore the classical theory of the Jacobian of a graph. In
Section 3, we give some facts about the Galois theory of covering graphs,
which we use to construct the module in Section 4. Finally, we use this module
(with some small modifications that control for the fact that $J(n)$ is a
proper extension of the group we want to count) to actually get the
asymptotics for the size of $\mathcal{J}(X_{n})[p^{\infty}]$ for a specific
example in Section 5.

###  1.1 Acknowledgements

The author would like to thank Daniel Vallieres for his introduction to the
topic.In addition, the author would like to thank the person who produced the
figures, though that person would like to remain anonymous.

##  2 Jacobians

###  2.1 Graphs

Im this paper, all of our graphs will be finite, connected, and undirected.
Following conventions from e.g. [Gon22, Ter11], we define a graph as follows:

######  Definition 2.1.

A graph is a tuple $X=(V,\vec{E},o,t,\iota)$ of a set $V$ of vertices and a
set $\vec{E}$ of directed edges with incidence maps $o,t\colon\vec{E}\to V$.
The graph is finite if both $V$ and $\vec{E}$ are finite sets. A graph
undirected if it is equipped with a fixed-point-free involution
$\iota\colon\vec{E}\to\vec{E}$ with the properties that $o(\iota(e))=t(e)$ and
$t(\iota(e))=o(e)$.

We think of the edge $e$ as connecting the vertex $o(e)$ to the vertex $t(e)$,
so that $\iota(e)$ runs the other way, connecting $t(e)$ to $o(e)$. The pair
of directed edges $\\{e,\iota(e)\\}$ is thought of as a single undirected edge
connecting the two vertices. We will also fix an orientation of $X$, meaning a
partition $\vec{E}=E^{+}\sqcup E^{-}$ with $\iota(E^{+})=E^{-}$. We refer to
$E^{+}$ as being the set of positively oriented edges.

A path on $X$ is a sequence of edges $e_{1}\cdots e_{\ell}$ with
$o(e_{j+1})=t(e_{j})$ for each $i$. If, for any pair of vertices $v_{1}$ and
$v_{2}$, there is a path $e_{1}\cdots e_{\ell}$ with $o(e_{1})=v_{1}$ and
$t(e_{\ell})=v_{2}$, we say that $X$ is connected.

######  Remark 2.2.

What we call a graph is often called a multigraph, e.g. in [HMSV24]. We will
instead give the name “simple graph” to what they call a graph. For ease of
translation between papers, we give the definition of a simple graph here.

A loop on $X$ is an edge $e$ with $o(e)=t(e)$. Two distinct edges $e\neq
e^{\prime}$ are said to be parallel if $o(e)=o(e^{\prime})$ and
$t(e)=t(e^{\prime})$. A graph is simple if it has no loops and no two of its
edges are parallel.

Finally, we define what we mean by a morphism of graphs.

######  Definition 2.3.

Let $X=(V_{X},\vec{E}_{X},o_{X},t_{X},\iota_{X})$ and
$Y=(V_{Y},\vec{E}_{Y},o_{Y},t_{Y},\iota_{Y})$ be two graphs. A morphism
$\phi\colon X\to Y$ is a pair of functions $\phi=(\phi_{V},\phi_{E})$ with
$\phi_{V}\colon V_{X}\to V_{Y}$ and $\phi_{E}\colon\vec{E}_{X}\to\vec{E}_{Y}$
satisfying two compatibility relations. First,
$\phi_{E}(\iota_{X}(e))=\iota_{Y}(\phi_{E}(e))$. In addition, we should also
have $\phi_{V}(o_{X}(e))=o_{Y}(\phi_{E}(e))$ and
$\phi_{V}(t_{X}(e))=t_{Y}(\phi_{E}(e))$.

Note that the morphism $\phi$ is determined by the choice of $\phi_{E}$, and
in fact by its restriction to $E_{X}^{+}$. We may say that a morphism is
oriented if every positively oriented edge in $X$ is sent to a positively
oriented edge in $Y$; i.e., if $\phi_{E}({E}_{X}^{+})\subset{E}_{Y}^{+}$.

###  2.2 Divisors and Principal Divisors

The divisor group of $X$ is group of formal linear combinations of vertices,
$\operatorname{Div}(X)=\mathbb{Z}V$. We have a degree homomorphism
$\operatorname{deg}\colon\operatorname{Div}(X)\to\mathbb{Z}$ with the formula

$\operatorname{deg}\left(\sum_{v\in V}a_{v}v\right)=\sum_{v\in V}a_{v}.$ (1)

The kernel of $\operatorname{deg}$ is the group of degree zero divisors,
$\operatorname{Div}^{0}(X)$.

For any pair of vertices $v\neq v^{\prime}\in V$, let
$\operatorname{deg}(v,v^{\prime})$ be the number of edges connecting $v$ to
$v^{\prime}$.111Note that, if $X$ is a simple graph,
$\operatorname{deg}(v,v^{\prime})\in\\{0,1\\}$ for any pair of vertices $v$
and $v^{\prime}$. We make no restriction to simple graphs, so it is possible
for multiple edges to connect the same pair of vertices. How we handle the
existence (or non-existence) of loops will be inconsequential, since we will
never ask for $\operatorname{deg}(v,v)$. Then let
$\operatorname{deg}(v)=\sum_{v^{\prime}\neq
v}\operatorname{deg}(v,v^{\prime})$ be the number of edges incident to $v$.
The Laplacian map $\Delta\colon\operatorname{Div}(X)\to\operatorname{Div}(X)$
is determined by the values, and extended $\mathbb{Z}$-linearly:

$\Delta(v)=\deg(v)v-\sum_{v^{\prime}\neq
v}\operatorname{deg}(v,v^{\prime})v^{\prime}.$ (2)

The image of $\Delta$ is known the group of principal divisors, and denoted
$\operatorname{Pr}(X)$. Note that $\operatorname{Pr}(X)$ is contained in
$\operatorname{Div}^{0}(X)$, since $\operatorname{deg}(\Delta v)$ is $0$ for
all $v$. Then we can define:

######  Definition 2.4.

The vertex Jacobian of the graph $X$ is the quotient
$\mathcal{J}_{v}(X)=\operatorname{Div}^{0}(X)/\operatorname{Pr}(X)$.

######  Remark 2.5.

This definition should be seen in analogy to the definition of the Picard
group for a curve. For intuition on why one might think that the image of the
Laplacian should give the principal divisors, see e.g. [BN07, Remark 1.4].

We can compute the size and the invariant factors of the Jacobian of a graph
by looking at the matrix for its Laplacian.

######  Example 2.6.

Consider the graph $X_{a,b}$ with three vertices $V=\\{v_{1},v_{2},v_{3}\\}$.
We connect $v_{1}$ to $v_{2}$ with $a$ undirected edges labeled
$e_{1},\dots,e_{a}$, and connect $v_{2}$ to $v_{3}$ with $b$ undirected edges
labeled $e_{1}^{\prime},\dots,e_{b}^{\prime}$. In the ordered basis
$v_{1},v_{2},v_{3}$, the Laplacian $\Delta_{X_{a,b}}$ has matrix

$\Delta=\begin{pmatrix}a&-a&0\\\ -a&a+b&-b\\\ 0&-b&b\end{pmatrix}$ (3)

This matrix is quickly seen to be singular. However, this is simply a
manifestation of the fact that its image is contained in the subgroup
$\operatorname{Div}^{0}(X)$ of $\operatorname{Div}(X)$, which has infinite
index. The standard trick to modify this matrix in order to obtain a
nonsingular matrix whose determinant is the size of $\mathcal{J}_{v}(X)$ is to
pick a vertex to be a “sink;” we will pick $v_{2}$ to be the sink, and this
gives rise to an isomorphism
$\mathbb{Z}v_{1}\oplus\mathbb{Z}v_{3}\xrightarrow{\sim}\operatorname{Div}^{0}(X)$
by the map $a_{1}v_{1}+a_{3}v_{3}\mapsto
a_{1}v_{1}+(-a_{1}-a_{3})v_{2}+a_{3}v_{3}$. This allows us to view $\Delta$ as
a map to $\mathbb{Z}v_{1}\oplus\mathbb{Z}v_{3}$; its matrix will be the same
as the original matrix for $\Delta$, but with the row corresponding to $v_{2}$
omitted. If we consider the minor that we get by also omitting the column
corresponding to $v_{2}$, we find that the cokernel of $\Delta$ will have
order $\det\begin{pmatrix}a&0\\\ 0&b\end{pmatrix}=ab$. Since
$\mathcal{J}_{v}(X_{ab})$ is the cokernel of $\Delta$ viewed as a map to
$\operatorname{Div}^{0}(X_{a,b})$ $\\#\mathcal{J}_{v}(X_{a,b})=ab$.

######  Remark 2.7.

The minor in Example 2.6 is independent of which column we omit. Omitting the
column corresponding to $v_{2}$ just gives the nicest remaining matrix, since
it is diagonal. In fact, the minors of $\Delta_{X_{a,b}}$ obtained from
omitting any single row and any single column are determined (up to unit)
independently of which row and which column you omit. This is some of the
content of Kirchhoff’s Matrix-Tree Theorem.

######  Example 2.8.

In this example, we compute the size of the Jacobian of the complete graph on
$4$ vertices, $K_{4}$. Its Laplacian is the following matrix:

$\Delta=\begin{pmatrix}3&-1&-1&-1\\\ -1&3&-1&-1\\\ -1&-1&3&-1\\\
-1&-1&-1&3\end{pmatrix}$ (4)

As in the previous example, we can omit any single row and any single column
and the determinant of the resulting matrix will give us the order of the
cokernel of $\Delta$ as a map to $\operatorname{Div}^{0}(X)$, which is then
the order of $\mathcal{J}_{v}(K_{4})$. A calculation of invariant factors in
Sage gives that $\mathcal{J}_{v}(K_{4})\cong(\mathbb{Z}/4\mathbb{Z})^{2}$, so
that $\\#\mathcal{J}_{v}(K_{4})=16$.

###  2.3 Stars and Homology

In Section 2.4, we will give another construction of the Jacobian of $X$.
Here, we define the pieces that go into that discussion. There are two
important maps between $\mathbb{Z}V$ and $\mathbb{Z}E^{+}$ that we want to
use.

######  Definition 2.9.

The boundary map $d\colon\mathbb{Z}E^{+}\to\mathbb{Z}V$ is determined by the
values $d(e)=t(e)-o(e)$ and extended linearly. The kernel $\ker d$ is the
first homology group of $X$, $H_{1}(X,\mathbb{Z})$. For any prime number $p$,
we can define
$H_{1}(X,\mathbb{Z}_{p})=H_{1}(X,\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{Z}_{p}$.

######  Definition 2.10.

The map $s\colon\mathbb{Z}V\to\mathbb{Z}E^{+}$ is determined by the following
values and extended linearly. The sums are over edges in $E^{+}$.

$s(v)=\sum_{t(e)=v}e-\sum_{o(e)=v}e.$ (5)

The image $\operatorname{im}s$ is denoted by $S(X,\mathbb{Z})$. For any prime
number $p$, we can define
$S(X,\mathbb{Z}_{p})=S(X,\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{Z}_{p}$.

The group $H_{1}(X,\mathbb{Z})$ is actually the first homology group of $X$
viewed as a topological space. However, $S(X,\mathbb{Z})$ is a more
combinatorial object that is specific to the situation of a graph. We will
refer to it as the group of stars, since it is generated by the elements
$s(v)$, and $s(v)$ is the star into the vertex $v$.

Note that the composition $d\circ s\colon\mathbb{Z}V\to\mathbb{Z}V$ is the
Laplacian $\Delta$.

###  2.4 The Jacobian via Edges

######  Definition 2.11.

The edge Jacobian of the graph $X$ is the quotient
$\mathcal{J}_{e}(X)=\mathbb{Z}E^{+}/[H_{1}(X,\mathbb{Z})+S(X,\mathbb{Z})]$.

######  Proposition 2.12.

The vertex Jacobian $\mathcal{J}_{v}(X)$ and the edge Jacobian
$\mathcal{J}_{e}(X)$ are isomorphic.

###### Proof.

The isomorphism will be induced by the boundary map $d$. Note that, since $X$
is connected, the image of $d$ is $\operatorname{Div}^{0}(X)$, so $d$ induces
an isomorphism
$\mathbb{Z}E^{+}/H_{1}(X,\mathbb{Z})\to\operatorname{Div}^{0}(X)$. In
addition, the Laplacian $\Delta\colon\mathbb{Z}V\to\mathbb{Z}V$ factors as the
composition $d\circ s\colon\mathbb{Z}V\to\mathbb{Z}E^{+}\to\mathbb{Z}V$; the
image $\operatorname{Pr}(X)$ of $\Delta$ is precisely the image of
$S(X,\mathbb{Z})$ under $d$. Thus we have that
$\mathcal{J}_{e}(X)=\mathbb{Z}E^{+}/[H_{1}(X,Z)+S(X,\mathbb{Z})]\cong\operatorname{Div}^{0}(X)/\operatorname{Pr}(X)=\mathcal{J}_{v}(X)$,
as desired. ∎

In light of this proposition, we will write $\mathcal{J}(X)$ for the edge
Jacobian. This will be better to set up the constructions in Section 4, and it
will not contradict any previous papers that use the vertex Jacobian.

The two methods to defining the Jacobian each have their advantages. The
vertex Jacobian gives an easier ability to compute the size of the group as in
Examples 2.6 and 2.8. However, the edge Jacobian’s prioritization of the edges
over the vertices aligns better with the theory of the Ihara zeta function; in
this theory the primes on $X$ (i.e., the points of the space corresponding to
$X$) are paths on $X$ rather than vertices. One can see this in e.g. [Ter11].

###  2.5 Spanning Trees

The Jacobian $\mathcal{J}(X)$ has a combinatorial meaning in counting specific
subgraphs of $X$.

######  Definition 2.13.

A graph $T$ is called a tree if it is connected and $H_{1}(T,\mathbb{Z})=0$. A
spanning tree of $X$ is a subgraph of $X$ which is a tree, and which includes
every vertex of $X$.

######  Example 2.14.

Consider the graphs $X_{a,b}$ from Example 2.6. Any spanning tree in $X_{a,b}$
needs to be spanning, meaning that it needs to contain all three vertices from
$X_{a,b}$; and it needs to be a tree, meaning that it needs to be connected
and contain no cycles. In order to specify such a subgraph, we just need to
specify the edges. This involves making two independent choices: first, choose
one edge out of the $a$ that connect $v_{1}$ to $v_{2}$, and then choose one
edge out of the $b$ that connect $v_{2}$ to $v_{3}$. This gives a total of
$ab$ spanning trees.

######  Example 2.15.

As in Example 2.8, we now consider the complete graph on $4$ vertices,
$K_{4}$. All $16$ spanning trees are shown in Figure 1.

Figure 1: The $16$ spanning trees on $K_{4}$, as in Example 2.15.

Recall that $\\#\mathcal{J}_{v}(X_{a,b})=ab$ and
$\\#\mathcal{J}_{v}(K_{4})=16$. This suggests the following theorem:

######  Theorem 2.16.

Let $\kappa(X)$ be the number of spanning trees of $X$. Then
$\kappa(X)=\\#\mathcal{J}(X)$.

In fact, we can prove this by realizing the set of spanning trees of $X$ as a
torsor for $\mathcal{J}(X)$. See [CCG15] for a description of this process,
which is known as rotor-routing.

######  Remark 2.17.

It is the point of view of this paper that viewing the Jacobian in terms of
edges is often profitable. However, this is one situation in which it is
highly unclear how this is helpful. The process of rotor-routing essentially
uses the vertices, and even relies on a choice of a base vertex. In [CCG15],
the action is described only on divisors of the form $(x-r)$ where $r$ is the
chosen base vertex and $x$ is another vertex, and extended linearly. It is not
described in a way that makes it clear what the action of a single edge $e$
should be, though we could describe it in terms of paths that end at the root.
Even more, changing the base vertex changes the action (for non-planar
graphs), which makes it even less likely that rotor-routing, as is, could be
modified directly in such a way that it gives an action of
$\mathcal{J}_{e}(X)$ without appealing to the isomorphism with
$\mathcal{J}_{v}(X)$.

It seems difficult in general to give an action of divisors on spaces of
subgraphs. We ask here whether or not there is a different process that gives
an action of $\mathcal{J}_{e}(X)$ intrinsically; later in Remark 4.6 we will
ask a similar question for a similar group.

##  3 Covers of Graphs

In this section, we discuss the Galois theory of graphs. We follow [Gon22] for
much of the exposition, though more detail is present in [Ter11].

###  3.1 Covering Graphs

######  Definition 3.1.

For an oriented graph $X$ and a chosen vertex $v\in V$, write
$S_{v}^{+}(X)=\\{e\in E^{+}\mid o(e)=v\text{ or }t(e)=v\\}$.

Let $\tilde{X}$ and $X$ be connected graphs. We say that a morphism
$\pi\colon\tilde{X}\to X$ is a covering map if the following two conditions
are true.

  * •

$\pi_{V}\colon V_{\tilde{X}}\to V_{X}$ is surjective, and

  * •

for each vertex ${v}\in V_{\tilde{X}}$, the restriction of $\pi_{E}$ to
$S_{{v}}^{+}(X)$ gives a bijection
$S_{{v}}^{+}(X)\xrightarrow{\sim}S_{\pi_{V}({v})}^{+}(B)$.

######  Remark 3.2.

Note that the second condition implicitly requires that $\pi$ be oriented. If
$X$ is oriented but $\tilde{X}$ is unoriented, one can weaken that condition,
and then induce an orientation on $\tilde{X}$ from the orientation on $X$ that
makes $\pi$ a covering map as follows.

Write $S_{v}^{\pm}(\tilde{X})=\\{e\in\vec{E}_{\tilde{X}}\mid o(e)=v\text{ or
}t(e)=v\\}$, and similarly for $X$. Then the second condition is that $\pi$
gives a bijection $S_{v}^{\pm}(\tilde{X})\to S_{\pi_{V}(v)}^{\pm}(X)$. Then we
pick $E_{\tilde{X}}^{+}$ to be the set of edges $e$ such that $\phi_{E}(e)\in
E_{X}^{+}$.

The theory of covering maps is a version of Galois theory. We refer the reader
to [Ter11] for a more in depth version of the discussion that follows,
including proofs of the main theorems. We record the following few facts and
notations for later use.

Let $\pi\colon\tilde{X}\to X$ be a covering map. The automorphism group of
$\pi$ is the group
$\operatorname{Aut}(\tilde{X}/X)=\\{\sigma\in\operatorname{Aut}(\tilde{X})\mid\pi\circ\sigma=\pi\\}$.
If $\tilde{X}$ and $X$ are both finite graphs, then $\pi$ is $n$-to-one for
some $n$. We call this the degree, and denote it $[\tilde{X}:X]$. When
$\tilde{X}$ is connected, we have the bound
$\\#\operatorname{Aut}(\tilde{X}/X)\leq[\tilde{X}:X]$; if
$\\#\operatorname{Aut}(\tilde{X}/X)=[\tilde{X}:X]$, then we say $\pi$ is
Galois.

######  Definition 3.3.

Let $\pi\colon Y\to X$ and $\rho\colon Z\to X$ be two covering maps. A
morphism of covering maps is a graph morphism $\phi\colon Y\to Z$ such that
$\pi=\rho\circ\phi$. If $\phi$ is invertible, we say that $\pi$ and $\rho$ are
isomorphic. If $\phi$ is surjective, we say that $\rho$ is a subcover of
$\pi$.

Note that a morphism of covering graphs is necessarily oriented. Compare this
definition with Definition 7 from [Gon22].

###  3.2 Constructing Covers: Voltage Graphs

We construct covering graphs by using the following tool.

######  Definition 3.4.

A $G$-valued voltage assignment for a graph $X$ is a function
$\alpha\colon\vec{E}_{X}\to G$ satisfying $\alpha(\iota(e))=\alpha(e)^{-1}$.

If the graph $X$ is oriented, the voltage assignment function $\alpha$ is
determined by its values on $E^{+}$. Given a $G$-valued voltage assignment
$\alpha$ for a graph $X$, we may refer to the triple $(X,G,\alpha)$ as a
voltage graph.

The purpose of a voltage assignment is to build the derived graph; if it is
connected, the derived graph will be a Galois cover of the base graph $X$ with
Galois group $G$. Let $X=(V,\vec{E},o,t,\iota)$ be a graph, and fix a
$G$-valued voltage assignment $\alpha$ for $X$. The derived graph is the graph
$\tilde{X}=(\tilde{V},\tilde{E},\tilde{o},\tilde{t},\tilde{\iota})$ where
$\tilde{V}=V\times G$, $\tilde{E}=\vec{E}\times G$, and the incidence
functions are defined by the following formulas:

$\tilde{o}(e,g)=(o(e),g),\qquad\tilde{t}(e,g)=(t(e),g\alpha(e)),\qquad\tilde{\iota}(e,g)=(\iota(e),g\alpha(e)).$
(6)

######  Remark 3.5.

We get a graph morphism $\pi\colon\tilde{X}\to X$ with $\pi_{V}(v,g)=v$ and
$\pi_{E}(e,g)=e$. If $G$ is finite, this has degree $[\tilde{X}:X]=\\#G$.
There is an action of $G$ on $\tilde{X}$ given by the formulas
$h\cdot(v,g)=(v,hg)$ and $h\cdot(e,g)=(e,hg)$; this embeds $G$ into
$\operatorname{Aut}(\tilde{X}/X)$. If $\tilde{X}$ is connected, then this
shows that $\tilde{X}$ is a Galois cover of $X$.

######  Remark 3.6.

For the derived graph $\tilde{X}$ of a voltage graph $(X,G,\alpha)$ to be
connected, it is neccessary but not sufficient that the image of $\alpha$
generate $G$. For example, if we drop the condition that $G$ be finite, there
is a universal abelian voltage assignment
$\alpha\colon\vec{E}\to\mathbb{Z}E^{+}$. Certainly the image of $\alpha$
generates $\mathbb{Z}E^{+}$, but in this case, there will be a path from the
vertex $(v,0)$ to the vertex $(v,g)$ only if $g\in
H_{1}(X,\mathbb{Z})\subset\mathbb{Z}E^{+}$.

We conclude this section with the statement of [Gon22, Theorem 4], which
explains the focus on voltage assignments for the construction of Galois
covers.

######  Theorem 3.7.

Let $(X,G,\alpha)$ be a voltage graph with $\tilde{X}$ the derived graph. If
$\tilde{X}$ is connected, then $\tilde{X}/X$ is a normal cover with
$\operatorname{Gal}(\tilde{X}/X)\cong G$. Conversely, given a Galois cover
$\pi\colon\tilde{X}\to X$, then $\tilde{X}$ is the derived graph for some
$G$-valued voltage assignment $\alpha$ on $X$.

###  3.3 Intermediate Voltage Covers

Given a group homomorphism $f\colon G\to H$ and a voltage graph
$(X,G,\alpha)$, we can build a new voltage graph $(X,H,f\circ\alpha)$. There
is a morphism on the level of derived graphs given by applying $f$ to the
second component. Write $\tilde{X}_{G}$ for the derived graph of the voltage
assignment $\alpha$ and $\tilde{X}_{H}$ for the derived graph of the voltage
assignment $f\circ\alpha$; if $\tilde{X}_{G}$ is connected and $f$ is
surjective, then $\tilde{X}_{H}$ will be connected as well, and it will be an
intermediate cover via the morphism of covering graphs described above.

######  Example 3.8.

Let $X=B_{2}=(V,\vec{E},o,t,\iota)$ be the bouquet of two loops; here
$V=\\{*\\}$ and $\vec{E}=\\{e_{1},\overline{e}_{1},e_{2},\overline{e}_{2}\\}$,
so that there are two undirected edges, each of which is a loop on the unique
vertex. Let $G=\mathbb{Z}/4\mathbb{Z}$, and fix the voltage assignment
$\alpha(e_{1})=\alpha(e_{2})=1$. The derived graph $\tilde{X}$ has vertices
$\tilde{V}=\\{v_{0},v_{1},v_{2},v_{3}\\}$ indexed by $G$; the edges are as in
the table below, with only the forward oriented edges listed.

$\begin{array}[]{c|c c c c c c c
c}e&e_{1,0}&e_{1,1}&e_{1,2}&e_{1,3}&e_{2,0}&e_{2,1}&e_{2,2}&e_{2,3}\\\
\hline\cr o(e)&v_{0}&v_{1}&v_{2}&v_{3}&v_{0}&v_{1}&v_{2}&v_{3}\\\
t(e)&v_{1}&v_{2}&v_{3}&v_{0}&v_{1}&v_{2}&v_{3}&v_{0}\end{array}$ (7)

There is one intermediate cover $X_{I}$ of degree $2$ over $X$, given by the
projection from $G$ to $\mathbb{Z}/2\mathbb{Z}$. We get two vertices
$V_{I}=\\{w_{0},w_{1}\\}$ indexed by the elements of $\mathbb{Z}/2\mathbb{Z}$,
and four undirected edges; they are attached as in the table below, with only
the forward oriented edges listed.

$\begin{array}[]{c|c c c c}e&e_{1,0}&e_{1,1}&e_{2,0}&e_{2,1}\\\ \hline\cr
o(e)&w_{0}&w_{1}&w_{0}&w_{1}\\\ t(e)&w_{1}&w_{0}&w_{1}&w_{0}\end{array}$ (8)

The covering morphism sends both vertices to the unique vertex $*$ of $X$, and
sends the edges $e_{i,g}$ to the loop $e_{i}$. See Figure 2 for a visual
representation of the three graphs $\tilde{X}$, $X_{I}$, and $X$.

Figure 2: From left to right, the graphs $\tilde{X}$, $X_{I}$, and $X$ from
Example 3.8.

###  3.4 Relationships Between Homology Groups for Abelian Covers

The absolute Galois group of a graph $X$ is isomorphic to the profinite
completion of its fundamental group $\pi_{1}(X,v_{0})$. If we restrict to
abelian covers $\pi\colon\tilde{X}\to X$, we will always have that
$\operatorname{Gal}(\tilde{X}/X)$ is a quotient of the abelianization
$\pi_{1}(X,v_{0})^{ab}\cong H_{1}(X,\mathbb{Z})$. In fact, we can describe
exactly which quotient: pushforward along $\pi$ gives a morphism
$\pi_{*}\colon H_{1}(\tilde{X},\mathbb{Z})\to H_{1}(X,\mathbb{Z})$, whose
cokernel is naturally isomorphic to $\operatorname{Gal}(\tilde{X}/X)$.

######  Example 3.9.

Let $X=B_{2}$, and $\tilde{X}$ be the Galois cover of $X$ with Galois group
$\mathbb{Z}/4\mathbb{Z}$ as in Example 3.8 above.

We have that $H_{1}(\tilde{X},\mathbb{Z})\cong\mathbb{Z}^{5}$, and it is
generated by the following cycles:

$c_{0}=e_{1,0}-e_{2,0},\quad c_{1}=e_{1,1}-e_{2,1},\quad
c_{2}=e_{1,2}-e_{2,2},\quad c_{3}=e_{1,3}-e_{2,3},\quad$ (9)

along with the cycle $c^{\prime}=e_{1,0}+e_{1,1}+e_{1,2}+e_{1,3}$. Pushed
forward, these become the following elements of
$H_{1}(X,\mathbb{Z})=\mathbb{Z}e_{1}\oplus\mathbb{Z}e_{2}$:

$\begin{array}[]{c|ccccc}c&c_{0}&c_{1}&c_{2}&c_{3}&c^{\prime}\\\
\hline\cr\pi_{*}(c)&e_{1}-e_{2}&e_{1}-e_{2}&e_{1}-e_{2}&e_{1}-e_{2}&4e_{1}\end{array}$
(10)

Their images generate the subgroup
$\pi_{*}(H_{1}(\tilde{X},\mathbb{Z}))=\langle e_{1}-e_{2},4e_{1}\rangle$; the
quotient of $H_{1}(X,\mathbb{Z})=\mathbb{Z}\\{e_{1},e_{2}\\}$ by this subgroup
identifies $e_{1}$ with $e_{2}$ and then kills $4e_{1}$, so

$H_{1}(X,\mathbb{Z})/\pi_{*}(H_{1}(\tilde{X},\mathbb{Z}))\cong\mathbb{Z}/4\mathbb{Z}.$
(11)

This was $\operatorname{Gal}(\tilde{X}/X)$ by construction.

##  4 The Iwasawa Module

###  4.1 $\mathbb{Z}_{p}$-covers

Classically, Iwasawa theory concerns itself with infinite field extensions,
specifically those with Galois group isomorphic to $\mathbb{Z}_{p}$. In the
present work, all graphs are assumed to be finite, which means that we will
never have a covering map with infinite Galois group. However, we will still
obtain some infinitary results by considering infinite towers of finite
graphs.

######  Definition 4.1.

A $\mathbb{Z}_{p}$-tower of graphs is a collection of graphs
$\\{X_{i}\\}_{i=0}^{\infty}$ and Galois covering morphisms $X_{0}\leftarrow
X_{1}\leftarrow X_{2}\leftarrow\dots$ with the property that
$\operatorname{Gal}(X_{j}/X_{i})\cong\mathbb{Z}/p^{j-i}\mathbb{Z}$ for each
$j\geq i$.

######  Remark 4.2.

We will create $\mathbb{Z}_{p}$-towers of graphs by assigning voltages in the
infinite group $\mathbb{Z}_{p}$ and applying the method in Section 3.3 to
produce the infinite tower. Careful application of Theorem 3.7 shows that all
$\mathbb{Z}_{p}$-towers of graphs can be obtained in this way.

Write $\Gamma_{n}=\operatorname{Gal}(X_{n}/X_{0})$, and let
$\Gamma=\varprojlim\Gamma_{n}$, so that $\Gamma\cong\mathbb{Z}_{p}$. Fix a
topological generator $\gamma$ of $\Gamma$, so that the class of $\gamma$ in
the quotient $\Gamma_{n}$ will generate $\Gamma_{n}$. In addition, let
$\Gamma^{(n)}=\overline{\langle\gamma^{p^{n}}\rangle}\subset\Gamma$ be the
kernel of the projection to $\Gamma_{n}$. We let
$\Lambda=\varprojlim\mathbb{Z}_{p}[\Gamma_{n}]$ be the completed group ring,
which we also denote as $\Lambda=\mathbb{Z}_{p}[[\Gamma]]$. $\Lambda$ is
isomorphic to the standard power series ring $\mathbb{Z}_{p}[[T]]$ by the
continuous homomorphism $\gamma\mapsto T+1$.

######  Remark 4.3.

Since we focus on finite graphs here, there is no graph
$X_{\infty}=\varprojlim X_{n}$. However, there is a theory of profinite graphs
into which this graph would fit. For intuition, one may think of an infinite
graph $X_{\infty}$ living at the top of the $\mathbb{Z}_{p}$ tower, though
this is not necessary for what we do here. With this, we have the added
intuition that $\Gamma^{(n)}=\operatorname{Gal}(X_{\infty}/X_{n})$.

###  4.2 Cycles, Stars, and Homology

As foreshadowed in the Remark 4.2, we fix a voltage graph
$(X_{0},\Gamma,\alpha)$. Let $\alpha_{n}$ be the composition of $\alpha$ with
the projection $\Gamma\to\Gamma_{n}$, and let $X_{n}$ be the derived graph of
the voltage assignment $(X_{0},\Gamma_{n},\alpha_{n})$; we assume that we have
chosen the voltage assignment such that each $X_{n}$ is connected, in which
case $X_{n}$ is a Galois cover of $X_{0}$ with Galois group
$\Gamma_{n}\cong\mathbb{Z}/p^{n}\mathbb{Z}$. Thus we have a
$\mathbb{Z}_{p}$-tower of graphs in the sense of Definition 4.1.

For each $n$, let $V_{n}$ be the set of vertices of $X_{n}$ and $\vec{E}_{n}$
its set of edges. Fix an orientation $E_{0}^{+}$ for $X_{0}$, and note that
this determines an orientation $E_{n}^{+}$ for each $X_{n}$. For each $n$,
write $C(n)=\mathbb{Z}_{p}E_{n}^{+}$ and $D(n)=\mathbb{Z}_{p}V_{n}$.

Now fix $n$, and for each vertex and each edge of $X_{0}$ pick a lift to
$X_{n}$. Note that, for a fixed vertex $v_{0}\in V_{0}$, there are exactly
$p^{n}$ choices for the lift $v_{n}$, and for a fixed choice of the lift
$v_{n}$, the set of choices is $\\{gv_{n}\mid g\in\Gamma_{n}\\}$. Similarly,
for a fixed (positively oriented) edge $e_{0}\in E_{0}^{+}$, there are exactly
$p^{n}$ choices for the lift $e_{n}$, and for a fixed choice of the lift
$e_{n}$, the set of choices is $\\{ge_{n}\mid g\in\Gamma_{n}\\}$. From this,
we see that a choice of lifts gives an isomorphism
$C(n)\cong\mathbb{Z}_{p}[\Gamma_{n}]E_{0}^{+}$ and
$D(n)\cong\mathbb{Z}_{p}[\Gamma_{n}]V_{0}$.

Now fix a compatible system of lifts $v_{n}\in V_{n}$ for each $n$ such that,
for every $m\leq n$, $v_{n}$ maps to $v_{m}$ under the projection from $X_{n}$
to $X_{m}$. Then do the same with each (positively oriented) edge $e_{0}\in
E_{0}^{+}$. The above line of reasoning, along with the compatibility of the
lifts, gives a compatible system of isomorphisms
$C(n)\cong\mathbb{Z}_{p}[\Gamma_{n}]E_{0}^{+}$ and
$D(n)\cong\mathbb{Z}_{p}[\Gamma_{n}]V_{0}$.

Recall the maps $d\colon\mathbb{Z}E^{+}\to\mathbb{Z}V$ and
$s\colon\mathbb{Z}V\to\mathbb{Z}E^{+}$ from Section 2.4; let $d$ and $s$ also
denote the maps on the extensions of scalars to $\mathbb{Z}_{p}$ as in this
section. Note that these functions are Galois equivariant — in fact, the
identifications
$\mathbb{Z}_{p}E_{n}^{+}\cong\mathbb{Z}_{p}[\Gamma_{n}]E_{0}^{+}$ and
$\mathbb{Z}_{p}V_{n}\cong\mathbb{Z}_{p}[\Gamma_{n}]V_{0}$ give commuting
diagrams

${\mathbb{Z}_{p}E_{n}^{+}}$${\mathbb{Z}_{p}V_{n}}$${\mathbb{Z}_{p}V_{n}}$${\mathbb{Z}_{p}E_{n}^{+}}$${\mathbb{Z}_{p}[\Gamma_{n}]E_{0}^{+}}$${\mathbb{Z}_{p}[\Gamma_{n}]V_{0}}$${\mathbb{Z}_{p}[\Gamma_{n}]V_{0}}$${\mathbb{Z}_{p}[\Gamma_{n}]E_{0}^{+}}$$d$$d$$s$$s$

We deduce from this that $H_{1}(X_{n},\mathbb{Z}_{p})$ can be computed as the
kernel of
$d\colon\mathbb{Z}_{p}[\Gamma_{n}]E_{0}^{+}\to\mathbb{Z}_{p}[\Gamma_{n}]V_{0}$,
and that $S(X_{n},\mathbb{Z}_{p})$ can be computed as the image of
$s\colon\mathbb{Z}_{p}[\Gamma_{n}]V_{0}\to\mathbb{Z}_{p}[\Gamma_{n}]E_{0}^{+}$.
Note that
$S(X_{n},\mathbb{Z}_{p})=\mathbb{Z}_{p}[\Gamma_{n}]S(X_{0},\mathbb{Z}_{p})$,
but $H_{1}(X_{n},\mathbb{Z}_{p})\subset H_{1}(X_{0},\mathbb{Z}_{p})$ will be a
submodule with quotient
$H_{1}(X_{0},\mathbb{Z}_{p})/H_{1}(X_{n},\mathbb{Z}_{p})\cong\operatorname{Gal}(X_{n}/X_{0})=\Gamma_{n}$.

###  4.3 The Module

Now we patch together the finitary data from the previous section into a
$\Lambda$-module. Write $C(\infty)=\Lambda E_{0}^{+}$ and $D(\infty)=\Lambda
V_{0}$, noting that we still have the maps $d$ and $s$ on this level. Write
$H(\infty)=\ker\left(d\colon C(\infty)\to D(\infty)\right)$ and
$S(\infty)=\operatorname{im}\left(s\colon D(\infty)\to C(\infty)\right)$.
Since these are $\Lambda$-submodules of $C(\infty)$, we can define
$J(\infty)=C(\infty)/\left(S(\infty)+H(\infty)\right)$.

######  Remark 4.4.

For the sake of intuition, let $X_{\infty}=\varprojlim X_{n}$ be the profinite
graph at the top of the tower. We can look at $C(\infty)=\Lambda E_{0}^{+}$ as
some completion of $\mathbb{Z}_{p}E_{\infty}^{+}$, with $S(\infty)$ some
completed group of stars and $H(\infty)$ some completed version of
$H_{1}(X_{\infty},\mathbb{Z}_{p})$. In this lens, we see that $J(\infty)$
should be (a completed version of the $p$-part of) the Jacobian of
$X_{\infty}$; however, it is not quite so simple on the level of the finite
graphs. In fact, while this could be thought of as
$\mathcal{J}_{e}(X_{\infty})[p^{\infty}]$, the corresponding
$\mathbb{Z}_{p}[\Gamma_{n}]$-modules
$J(\infty)\otimes_{\Lambda}\mathbb{Z}_{p}[\Gamma_{n}]$ at each finite step of
the tower will be an extension of $\mathcal{J}(X_{n})[p^{\infty}]$ by
$\Gamma^{(n)}$. The discrepancy comes from the fact that
$H_{1}(X_{n},\mathbb{Z}_{p})/H(\infty)\cong\operatorname{Gal}(X_{\infty}/X_{n})=\Gamma^{(n)}\cong\mathbb{Z}_{p}$.

Write $J(n)=J(\infty)\otimes_{\Lambda}\mathbb{Z}_{p}[\Gamma_{n}]$, and
similarly define $C(n)$, $D(n)$, $H(n)$, and $S(n)$. Since tensor products
preserve cokernels, we have that $J(n)\cong C(n)/[H(n)+S(n)]$. Note that
$C(n)\cong\mathbb{Z}_{p}E_{n}^{+}$, $D(n)=\mathbb{Z}_{p}V_{n}$, and
$S(n)=S(X_{n},\mathbb{Z}_{p})$ by the above argument, but $H(n)\subset
H_{1}(X_{n},\mathbb{Z}_{p})$ is just a submodule. We get an exact sequence

$0\to H(n)\to H_{1}(X_{n},\mathbb{Z}_{p})\to
J(n)\to\mathcal{J}(X_{n})\otimes\mathbb{Z}_{p}\to 0.$ (12)

Since $\mathcal{J}(X_{n})$ is finite, we have that
$\mathcal{J}(X_{n})\otimes\mathbb{Z}_{p}\cong\mathcal{J}(X_{n})[p^{\infty}]$
is the $p$-power torsion subgroup of $\mathcal{J}(X_{n})$. We can also replace
$H_{1}(X_{n},\mathbb{Z}_{p})$ by the quotient
$H_{1}(X_{n},\mathbb{Z}_{p})/H(n)\cong\Gamma^{(n)}$ to obtain the following
short exact sequence:

$0\to\Gamma^{(n)}\to J(n)\to\mathcal{J}(X_{n})[p^{\infty}]\to 0.$ (13)

We have proven the following.

######  Theorem 4.5.

There is a module $J(\infty)$ over the Iwasawa algebra $\Lambda$ whose
coinvariants $J(n)=J(\infty)\otimes_{\Lambda}\mathbb{Z}_{p}[\Gamma_{n}]$ are
isomorphic to an extension of the $p$-power torsion subgroup of the Jacobian
$\mathcal{J}(X_{n})[p^{\infty}]$ by the group $\Gamma^{(n)}$.

######  Remark 4.6.

For any cover $\pi\colon\tilde{X}\to X$ with abelian Galois group $G$ we can
build a natural $G$-action on $\mathcal{J}(\tilde{X})$; by the same arguments
above, taking coinvariants will give us an extension of $\mathcal{J}(X)$ by
$G$. Call this $\mathcal{J}(\pi)$.

In Section 2.5, we referenced the fact that the set of spanning trees on a
graph $X$ is a torsor for the Jacobian $\mathcal{J}({X})$, which means that we
can count spanning trees on ${X}$ by finding $\\#\mathcal{J}({X})$. One may
ask what $\mathcal{J}(\pi)$ counts.

It is not too difficult to see the fact that $\\#\mathcal{J}(\pi)$ is the
number of lifts of spanning trees on $X$ to (non-spanning) subtrees of
$\tilde{X}$: $\\#\mathcal{J}(B)$ is the number of spanning trees on $B$, and
$\\#G$ is how many different choices we have for a lift of any chosen
basepoint, so there are $(\\#\mathcal{J}(X))(\\#G)=\\#\mathcal{J}(\pi)$ many
lifts. However, it seems more difficult to define such an action intrinsically
on $\tilde{X}$, due to the fact that rotor-routing essentially needs the trees
to be spanning in order for the process to be well-defined.

We ask a similar question here to the question in Remark 2.17: Is there a
natural action of $\mathcal{J}(\pi)$ on the set of lifts of spanning trees
from $X$ to $\tilde{X}$?

##  5 Asymptotics

###  5.1 Difficulties with the Asymptotics

The asymptotics for $\\#\mathcal{J}(X_{n})$ are complicated by the fact that
$\\#J(n)=\infty$ for all $n$. We need to leverage the connection between
different homology groups in the tower to extract the finite groups
$\mathcal{J}(X_{n})[p^{\infty}]$ from the infinite groups $J(n)$. It is
difficult to do in generality, which is why we dedicate the next section to an
example. In this section, we give an idea of how one might deal with this
discrepancy in a uniform way.

Let $(X_{0},\Gamma,\alpha)$ be a voltage assignment as in the beginning of
Section 4.2, with the $\mathbb{Z}_{p}$-tower $\\{X_{n}\\}_{n=0}^{\infty}$
constructed as described there. Pick a base vertex $v_{0}$ on $X_{0}$, and let
$v_{1}$ be a chosen lift of $v_{0}$ to $X_{1}$. Since $X_{1}$ is connected,
there is a path $\delta_{1}=e_{1}\cdots e_{\ell}$ that connects $v_{1}$ to
$\gamma v_{1}$. This projects to a path $\delta_{0}$ on $X_{0}$, and
$\delta_{0}\in H_{1}(X_{0},\mathbb{Z}_{p})$.

This $\delta_{0}$ will generate $H_{1}(X_{0},\mathbb{Z}_{p})/H(0)$ as a
$\mathbb{Z}_{p}$-module. In fact, if we let $\delta_{n}$ be any lift of
$\delta_{0}$ to a path on $X_{n}$, then
$\sum_{i=0}^{p^{n}-1}\gamma^{i}\delta_{n}$ will be a loop on $X_{n}$, and it
will generate $H_{1}(X_{n},\mathbb{Z}_{p})/H(n)$ as a $\mathbb{Z}_{p}$-module.
Writing $N_{n}(\gamma)=\sum_{i=0}^{p^{n}-1}\gamma^{i}$, we have argued that
$\mathcal{J}(X_{n})[p^{\infty}]\cong
J(n)/\mathbb{Z}_{p}N_{n}(\gamma)\delta_{0}$. In the next section, we describe
how to use this isomorphism to get asymptotics for
$\\#\mathcal{J}(X_{n})[p^{\infty}]$ in an example.

###  5.2 The Example

Let $X_{0}=B_{k}$ be the bouquet of $k$ loops, with $V_{0}=\\{*\\}$ and
$\vec{E}_{0}=\\{e_{0},\overline{e}_{0},\dots,e_{k-1},\overline{e}_{k-1}\\}$.
Choose the orientation $E_{0}^{+}=\\{e_{0},\dots,e_{k-1}\\}$. Then fix a prime
$p$. Let $\Gamma$ be a group isomorphic to $\mathbb{Z}_{p}$, let $\gamma$ be a
chosen topological generator, and consider the $\mathbb{Z}_{p}$ tower of
graphs determined by the voltage assignment $\alpha\colon E_{0}^{+}\to\Gamma$
with $\alpha(e_{i})=\gamma$ for each $i$. Consider the projection
$f_{n}\colon\Gamma\to\Gamma_{n}=\Gamma/\langle\gamma^{p^{n}}\rangle$.
Replacing $\alpha$ by $\alpha_{n}=f_{n}\circ\alpha$ gives rise to a voltage
graph $(X_{0},\Gamma_{n},\alpha_{n})$ whose derived graph we will call
$X_{n}$.

In fact, $X_{n}$ is fairly easy to describe, as is the projection map
$\pi_{n}\colon X_{n}\to X_{0}$. $X_{n}$ has $p^{n}$ vertices, indexed by the
elements of $\mathbb{Z}/p^{n}\mathbb{Z}$, and each vertex $v_{i}$ is connected
to the vertex $v_{i+1}$ by $k$ edges $e_{i,j}$ for $j=0,\dots,k-1$. The
projection map $\pi_{n}$ sends every vertex of $X_{n}$ to the unique vertex of
$X_{0}$, and sends the edge $e_{i,j}$ of $X_{n}$ to the edge $e_{j}$ of
$X_{0}$.

We have $C(\infty)=\Lambda\\{e_{0},\dots,e_{k-1}\\}$ is the free $\Lambda$
module on the basis $E_{0}^{+}$. $S(\infty)$ is spanned over $\Lambda$ by the
star into $v_{1}$, which is $e_{0}+\dots+e_{k-1}-\gamma e_{0}-\dots-\gamma
e_{k-1}$. Finally, $H(\infty)$ is spanned over $\Lambda$ by the elements
$e_{i}-e_{0}$, where $i=1,\dots,k-1$. There is a map
$C(\infty)\to\Lambda/(k-k\gamma)$ which sends each edge $e_{i}$ to $1$; the
kernel certainly contains $H(\infty)$ since it kills the differences
$e_{i}-e_{0}$. Once those differences are killed, the generator of $S(\infty)$
becomes $k-k\gamma$, so the map factors through $J(\infty)$; it is not
difficult to show that the induced map is an isomorphism
$J(\infty)\xrightarrow{\sim}\Lambda/(k-k\gamma)$.

As we noted before,
$H(n)=H(\infty)\otimes_{\Lambda}\mathbb{Z}_{p}[\Gamma_{n}]$ is not all of
$H_{1}(X_{n},\mathbb{Z}_{p})$, and in fact the quotient
$H_{1}(X_{n},\mathbb{Z}_{p})/H(n)$ will be isomorphic to $\mathbb{Z}_{p}$.
Following the ideas in Section 5.1, we notice that it is spanned over
$\mathbb{Z}_{p}$ by the element $N_{n}(\gamma)e_{0}$, where
$N_{n}(\gamma)=1+\gamma+\dots+\gamma^{p^{n}-1}$. Writing
$J(n)=J(\infty)\otimes_{\Lambda}\mathbb{Z}_{p}[\Gamma_{n}]$, we have

$\mathcal{J}(X_{n})[p^{\infty}]\cong
J(n)/H_{1}(X_{n},\mathbb{Z}_{p})\cong\mathbb{Z}_{p}[\Gamma_{n}]/(k-k\gamma,1+\gamma+\dots+\gamma^{p^{n}-1}).$
(14)

If $k$ is prime to $p$, the calculation is easier. $k$ is a unit in
$\mathbb{Z}_{p}$, so

$\begin{array}[]{rcl}\mathcal{J}(X_{n})[p^{\infty}]&\cong&\mathbb{Z}_{p}[\Gamma_{n}]/(k-k\gamma,1+\dots+\gamma^{p^{n}-1})\\\\[6.0pt]
&\cong&\mathbb{Z}_{p}[\Gamma_{n}]/(1-\gamma,1+\dots+\gamma^{p^{n}-1})\\\\[6.0pt]
&\cong&\mathbb{Z}_{p}/(p^{n}).\end{array}$ (15)

Thus, when $p\nmid k$, we have
$\mathcal{J}(X_{n})[p^{\infty}]\cong\mathbb{Z}/p^{n}\mathbb{Z}$, so
$\\#\mathcal{J}(X_{n})[p^{\infty}]=p^{n}$.

If $p\mid k$, the relationship is more complicated. Let $u=v_{p}(k)$, so that
$p^{u}$ is the largest power of $p$ that divides $k$. Then we instead write

$\begin{array}[]{rcl}\mathcal{J}(X_{n})[p^{\infty}]&\cong&\mathbb{Z}_{p}[\Gamma_{n}]/(k-k\gamma,1+\dots+\gamma^{p^{n}-1})\\\\[6.0pt]
&\cong&\mathbb{Z}_{p}[\Gamma_{n}]/(p^{u}(1-\gamma),1+\dots+\gamma^{p^{n}-1}).\end{array}$
(16)

We have a map
$f=(f_{1},f_{2})\colon\mathbb{Z}_{p}[\Gamma_{n}]\to\mathbb{Z}_{p}\times(\mathbb{Z}/p^{u}\mathbb{Z})[\Gamma_{n}]$,

$\sum_{i=0}^{p^{n}-1}a_{i}\gamma^{i}\mapsto\left(\sum_{i=0}^{p^{n}-1}a_{i},\sum_{i=0}^{p^{n}-1}[a_{i}]\gamma^{i}\right).$
(17)

Here the $[a_{i}]$ in brackets is the class of $a_{i}$ in
$\mathbb{Z}/p^{u}\mathbb{Z}$. The kernel of the first component $f_{1}$ is the
ideal generated by $\gamma-1$, and the kernel of $f_{2}$ is the ideal
generated by $p^{u}$, so the kernel of $f=(f_{1},f_{2})$ is
$(\gamma-1)\cap(p^{u})=(p^{u}(\gamma-1))$. Then the image is isomorphic to
$J(n)$. This is the set of tuples

$\operatorname{im}(f)=\left\\{\left(m,\sum[a_{i}]\gamma^{i}\right)\mid
m\equiv\sum[a_{i}]\pmod{p^{u}}\right\\}$ (18)

The group $\mathcal{J}(X_{n})[p^{\infty}]$ is a quotient of
$\operatorname{im}(f)$ by the $\mathbb{Z}_{p}[\Gamma_{n}]$-submodule generated
by $f(N_{n}(\gamma))=(p^{n},N_{n}(\gamma))$. Each coset of this submodule has
a unique representative $\left(m,\sum\alpha_{i}\gamma^{i}\right)$ with $m$ an
integer, $0\leq m<p^{n}$. Thus, at least as a set, we have that
$\mathcal{J}(X_{n})[p^{\infty}]$ is isomorphic to

$\left\\{\left(m,\sum_{i}\alpha_{i}\gamma^{i}\right)\in\mathbb{Z}/p^{n}\mathbb{Z}\times(\mathbb{Z}/p^{u}\mathbb{Z})[\Gamma_{n}]\mid
m\equiv\sum_{i}\alpha_{i}\pmod{p^{u}}\right\\}.$ (19)

Let $n\geq u$. For each of the $(p^{u})^{p^{n}}$ elements
$\sum_{i=0}^{p^{n}-1}\alpha_{i}\gamma^{i}$ of
$(\mathbb{Z}/p^{u}\mathbb{Z})[\Gamma_{n}]$, there will be $p^{n-u}$ choices
for $m$. Thus
$\\#\mathcal{J}(X_{n})[p^{\infty}]=p^{n-u}(p^{u})^{p^{n}}=p^{n}(p^{u})^{p^{n}-1}$.

Note that in the case $u=0$, i.e., when $p\nmid k$, we recover the simpler
result that $\\#\mathcal{J}(X_{n})[p^{\infty}]=p^{n}$.

######  Remark 5.1.

In fact, in this example, we can actually count the spanning trees of $X_{n}$.
Recall that $X_{n}$ is a graph with $p^{n}$ vertices labeled by the elements
of $\mathbb{Z}/p^{n}\mathbb{Z}$, where each vertex $v_{i}$ is connected to the
vertex $v_{i+1}$ by $k$ distinct edges. We can specify a spanning tree with
the following choices: first, a vertex $v_{i_{0}}$, and then, for all $i\neq
i_{0}$, a choice of one of the $k$ edges that connects $v_{i}$ to $v_{i+1}$.
This gives us a total of $p^{n}k^{p^{n}-1}$ choices. The $p$-part of this
number is obtained by replacing $k$ with $p^{u}$, which gives us
$p^{n}(p^{u})^{p^{n}-1}$ as the target number. This is exactly what we
calculated for $\\#\mathcal{J}(X_{n})[p^{\infty}]$.

######  Remark 5.2.

Though we have to make the adjustment with $H_{1}(X_{0},\mathbb{Z}_{p})$, we
do still end up getting an asymptotic of the standard type for
$\\#\mathcal{J}(X_{n})[p^{\infty}]$. For $n\geq u$, we have
$\\#\mathcal{J}(X_{n})[p^{\infty}]=p^{up^{n}+n-u}$, which is of the form
$p^{\lambda p^{n}+\mu n+\nu}$ for $\lambda=u$, $\mu=1$, and $\nu=-u$.

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