# An analogue of Kida’s formula in graph theory

Anwesh Ray  Anwesh Ray
Centre de recherches mathématiques, Université de Montréal, Pavillon André-
Aisenstadt, 2920 Chemin de la tour, Montréal (Québec) H3T 1J4, Canada  and
Daniel Vallières  Daniel Vallières
Mathematics and Statistics Department, California State University, Chico, CA
95929, USA<EMAIL_ADDRESS>
###### Abstract.

Let $\ell$ be a rational prime and let $p\mathrel{\mathop{\mathchar
58\relax}}Y\rightarrow X$ be a Galois cover of finite graphs whose Galois
group is a finite $\ell$-group. Consider a $\mathbb{Z}_{\ell}$-tower above $X$
and its pullback along $p$. Assuming that all the graphs in the pullback are
connected, one obtains a $\mathbb{Z}_{\ell}$-tower above $Y$. Under the
assumption that the Iwasawa $\mu$-invariant of the tower above $X$ vanishes,
we prove a formula relating the Iwasawa $\lambda$-invariant of the
$\mathbb{Z}_{\ell}$-tower above $X$ to the Iwasawa $\lambda$-invariant of the
pullback. This formula is analogous to Kida’s formula in classical Iwasawa
theory. We present an application to the study of structural properties of
certain noncommutative pro-$\ell$ towers of graphs, based on an analogy with
classical results of Cuoco on the growth of Iwasawa invariants in
$\mathbb{Z}_{\ell}^{2}$-extensions of number fields. Our investigations are
illustrated by explicit examples.

###### Key words and phrases:

Graph theory, Iwasawa invariants, Kida’s formula, the Riemann-Hurwitz formula,
noncommutative towers of graphs

###### 2020 Mathematics Subject Classification:

Primary: 05C25, 11R23; Secondary: 11Z05

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Graphs and covers of graphs
    2. 2.2 Artin-Ihara $L$-functions
    3. 2.3 Constructions of Galois covers via voltage assignments
      1. 2.3.1 Connectedness of graphs obtained from voltage assignments
    4. 2.4 The analogue of Iwasawa’s theorem
  3. 3 Pullbacks
    1. 3.1 Pullbacks and voltage assignments
  4. 4 The analogue of Kida’s formula
    1. 4.1 Main result on Kida’s formula
    2. 4.2 Examples
  5. 5 Growth of Iwasawa invariants in noncommutative pro-$\ell$ towers
    1. 5.1 Growth in uniform pro-$\ell$ towers
    2. 5.2 An Example

##  1\. Introduction

Let $f\mathrel{\mathop{\mathchar
58\relax}}\mathcal{S}_{1}\rightarrow\mathcal{S}_{2}$ be a branched cover of
compact Riemann surfaces of degree $d$. The Riemann-Hurwitz formula stipulates
that

(1.1)
$\chi(\mathcal{S}_{1})=d\chi(\mathcal{S}_{2})-\sum_{P\in\mathcal{S}_{1}}(e_{P}-1),$

where $\chi$ denotes the Euler characteristic of a compact Riemann surface,
and $e_{P}$ is the ramification index at the point $P$. Note that the above
sum is finite since $e_{P}=1$ for all but finitely many points $P$. If the
cover is unramified, then (1.1) simply becomes

(1.2) $\chi(\mathcal{S}_{1})=d\chi(\mathcal{S}_{2}).$

Using the relation $\chi(\mathcal{S})=2-2g(\mathcal{S})$, where
$g(\mathcal{S})$ denotes the genus of $\mathcal{S}$, (1.1) can also be written
as

(1.3)
$2g(\mathcal{S}_{1})-2=d(2g(\mathcal{S}_{2})-2)+\sum_{P\in\mathcal{S}_{1}}(e_{P}-1).$

Kida proved an analogous formula in the context of Iwasawa theory, which we
now explain. Throughout, $\ell$ will be a rational prime, and let
$\mu_{\ell^{\infty}}$ be the $\ell$-power roots of unity in
$\overline{\mathbb{Q}}\subseteq\mathbb{C}$. Let $K$ be a $CM$ number field.
The cyclotomic $\mathbb{Z}_{\ell}$-extension of $K$ is the unique
$\mathbb{Z}_{\ell}$-extension contained in $K(\mu_{\ell^{\infty}})$ and is
denoted by $K_{\infty}$. Associated to $K_{\infty}$ is a tower of $CM$-fields

$K=K_{0}\subseteq K_{1}\subseteq\ldots\subseteq K_{n}\subseteq\ldots$

for which $K_{n}/K$ is Galois with Galois group isomorphic to
$\mathbb{Z}/\ell^{n}\mathbb{Z}$. Let $K_{n}^{+}$ be the maximal totally real
subfield of $K_{n}$ and $h_{n}^{+}$ be the class number of $K_{n}^{+}$. Denote
by $h_{n}$, the class number of $K_{n}$ and let
$h_{n}^{-}\mathrel{\mathop{\mathchar 58\relax}}=h_{n}/h_{n}^{+}$ denote the
relative class number which is known to be an integer. Iwasawa showed (see
[Iwa59] and [Iwa73]) that there exist nonnegative integers
$\mu^{-}=\mu_{K}^{-},\lambda^{-}=\lambda_{K}^{-}$ and an integer
$\nu^{-}=\nu_{K}^{-}$ such that for $n$ large, one has

(1.4) ${\rm ord}_{\ell}(h_{n}^{-})=\mu^{-}\ell^{n}+\lambda^{-}n+\nu^{-},$

where ${\rm ord}_{\ell}$ is the usual $\ell$-adic valuation on $\mathbb{Q}$.
Let now $E/F$ be a Galois extension of $CM$-fields for which ${\rm Gal}(E/F)$
is an $\ell$-group, and assume for simplicity that $\ell>2$ and that $F$
contains the $\ell$-th roots of unity. Kida showed in [Kid80] that if
$\mu_{F}^{-}=0$, then $\mu_{E}^{-}=0$, and

(1.5) $2\lambda_{E}^{-}-2=[E_{\infty}\mathrel{\mathop{\mathchar
58\relax}}F_{\infty}](2\lambda_{F}^{-}-2)+\sum_{w}(e_{w}-1),$

where the sum is over all nonarchimedean places of $E_{\infty}$ that split
completely in $E_{\infty}/E_{\infty}^{+}$ and that do not lie above $\ell$.
Several authors pointed out the striking similarity between Kida’s formula
(1.5) and the Riemann-Hurwitz formula (1.3).

In this paper, we study analogies between Galois theoretic objects in
arithmetic and their topological counterparts in graph theory. The properties
of the Dedekind zeta function of a number field correlate to the arithmetic
distribution properties of its primes. On the other hand, the Ihara zeta
function in graph theory behaves much like the Dedekind zeta function, and has
many interesting combinatorial applications, cf. [Ter11]. There are also many
analogies between graphs and algebraic curves. For instance, [BN09]
establishes an analogue of the Abel-Jacobi map and the Riemann-Roch theorem in
graph theory. There is a natural notion of the class group in graph theory
(known in the literature under various names such as the critical group, the
sandpile group, the Jacobian group, the Picard group, etc), which exhibits
properties similar to its number theoretic and geometric counterparts. From an
Iwasawa theoretic viewpoint, it is of natural interest to study the behavior
of class numbers in various towers of graphs. Recently, a result similar to
Iwasawa’s theorem (1.4) above has been observed in graph theory. Let $X$ be a
connected graph (with loops and parallel edges allowed), and consider a
$\mathbb{Z}_{\ell}$-tower above $X$. This means that we are given a sequence
of covers of connected graphs

(1.6) $X=X_{0}\leftarrow X_{1}\leftarrow\ldots\leftarrow
X_{n}\leftarrow\ldots$

for which the cover $X_{n}/X$ obtained by composing the covers
$X_{n}\rightarrow\ldots\rightarrow X_{1}\rightarrow X$ is Galois with group of
covering transformations isomorphic to $\mathbb{Z}/\ell^{n}\mathbb{Z}$. Then,
[MV22b, Theorem 6.1] shows that there exist nonnegative integers $\mu,\lambda$
and an integer $\nu$ such that for $n$ large, one has

${\rm ord}_{\ell}(\kappa_{n})=\mu\ell^{n}+\lambda n+\nu,$

where $\kappa_{n}$ is the class number (i.e., number of spanning trees) of the
graph $X_{n}$. The approach is based on studying the properties of a suitable
analogue of the $\ell$-adic $L$-function in graph theory, obtained from the
$\ell$-adic interpolation of the special values of the Artin-Ihara
$L$-functions. For a more algebraic approach to this result, one is referred
to [Gon21, Gon22]. We shall refer to $\mu$ and $\lambda$ as the Iwasawa
invariants of the tower (1.6). Furthermore, it is known (see for instance
[Sun13, page 55]) that if $Y\rightarrow X$ is a cover of connected graphs of
degree $d$, then one has

(1.7) $\chi(Y)=d\chi(X),$

where $\chi$ here denotes the Euler characteristic of a graph. It becomes then
natural to study whether an analogue of Kida’s formula holds true in graph
theory as well. Our main result is the following theorem.

######  Theorem A (Theorem 4.1).

Let $X$ be a connected graph for which $\chi(X)\neq 0$ and consider a
$\mathbb{Z}_{\ell}$-tower

(1.8) $X=X_{0}\leftarrow X_{1}\leftarrow\ldots\leftarrow
X_{n}\leftarrow\ldots$

Let $p\mathrel{\mathop{\mathchar 58\relax}}Y\rightarrow X$ be a Galois cover
of graphs whose Galois group is an $\ell$-group and pull back the tower (1.8)
along $p$ to obtain another $\mathbb{Z}_{\ell}$-tower

(1.9) $Y\leftarrow Y\times_{X}X_{1}\leftarrow
Y\times_{X}X_{2}\leftarrow\ldots\leftarrow Y\times_{X}X_{n}\leftarrow\ldots$

Assume that all the graphs in both towers are connected. Further, let us
denote the Iwasawa invariants of the tower (1.8) by $\mu_{X}$ and
$\lambda_{X}$ and the Iwasawa invariants of the tower (1.9) by $\mu_{Y}$ and
$\lambda_{Y}$. Then $\mu_{X}=0$ if and only if $\mu_{Y}=0$ in which case

(1.10) $\lambda_{Y}+1=[Y\mathrel{\mathop{\mathchar
58\relax}}X](\lambda_{X}+1),$

where $[Y\mathrel{\mathop{\mathchar 58\relax}}X]$ is the degree of the cover
$Y/X$.

Theorem A can be viewed as an analogue of Kida’s formula (1.5), but in graph
theory. We are aware of three different proofs of Kida’s formula: the original
one by Kida in [Kid80] using genus theory, the one by Iwasawa in [Iwa81] where
he proves a result analogous to the Chevalley-Weil formula for a branched
Galois cover of compact Riemann surfaces, and the one by Sinnott in [Sin84]
using $\ell$-adic $L$-functions. Our approach here to proving Theorem A is
inspired by [Sin84].

In §5, we present an application of Theorem 4.1 to the study of certain growth
questions in noncommutative pro-$\ell$ towers of graphs. Our investigations
are inspired by a result of Cuoco [Cuo80], who studied the growth of Iwasawa
invariants in a $\mathbb{Z}_{\ell}^{2}$-tower over a number field $K$. This
provides us insight into the structural properties of certain non-commutative
pro-$\ell$ examples.

######  Remark 1.1.

In [BN09], Baker and Norine prove that if $\pi\mathrel{\mathop{\mathchar
58\relax}}Y\rightarrow X$ is a branched cover of graphs of degree $d$ in the
sense of [Sun13, Notes 5.6], then

(1.11) $\chi(Y)=d\chi(X)-\sum_{w\in V_{Y}}(e_{w}-1),$

where $e_{w}$ is the ramification index of the vertex $w$. In the present
paper (as well as in [Val21, MV22a, MV22b], [LV22], [DV22], [DRLV22], and
[Gon21, Gon22]), we restrict ourselves to unramified covers; therefore, the
Euler characteristic satisfies the simpler formula (1.7), and this might
explain the lack of ramification indices in (1.10).

_Organization:_ In §2, we gather together a few previous results that will be
used later on in the article. We remind the reader about graphs and covers of
graphs in §2.1, about Artin-Ihara $L$-functions in §2.2, about voltage
assignments in §2.3, and about the graph theoretical analogue of Iwasawa’s
theorem (1.4) in §2.4. In §3, we explain the notion of a pullback in the
category of graphs and some of its properties. We reinterpret pullbacks in
terms of voltage assignments in §3.1. We present our main result in §4, which
we illustrate through examples in §4.2. We present an application to the study
of the growth of Iwasawa invariants in noncommutative pro-$\ell$ towers in §5,
which we illustrate through an example in §5.2.

### Acknowledgement

AR acknowledges support from the CRM-Simons postdoctoral fellowship.

##  2\. Preliminaries

###  2.1. Graphs and covers of graphs

We think of a graph as presented in [Ser77] and [Sun13]. Thus, a graph $X$
consists of a set of vertices $V_{X}$, a set of directed edges
$\mathbf{E}_{X}$, an incidence map ${\rm inc}\mathrel{\mathop{\mathchar
58\relax}}\mathbf{E}_{X}\rightarrow V_{X}\times V_{X}$ given by $e\mapsto{\rm
inc}(e)=(o(e),t(e))$, and an inversion map
$\mathbf{E}_{X}\rightarrow\mathbf{E}_{X}$ given by $e\mapsto\bar{e}$
satisfying

  1. (1)

$\bar{e}\neq e$,

  2. (2)

$\bar{\bar{e}}=e$,

  3. (3)

$o(\bar{e})=t(e)$ and $t(\bar{e})=o(e)$,

for all $e\in\mathbf{E}_{X}$. For $v\in V_{X}$, we let

$\mathbf{E}_{X,v}=\\{e\in\mathbf{E}_{X}\,\mathrel{\mathop{\mathchar
58\relax}}\,o(e)=v\\}.$

The set of undirected edges is obtained by identifying $e$ with $\bar{e}$ and
will be denoted by $E_{X}$. Note that we allow loops and parallel edges. Such
graphs are sometimes called multigraphs, but we will refer to them as graphs
for simplicity. _We assume throughout this paper that all of our graphs are
finite unless otherwise stated. This means that both $V_{X}$ and
$\mathbf{E}_{X}$ are finite sets._ Setting $b_{i}(X)\mathrel{\mathop{\mathchar
58\relax}}={\rm rank}_{\mathbb{Z}}\,H_{i}(X,\mathbb{Z})$, the Euler
characteristic $\chi(X)$ is defined as follows

$\chi(X)=b_{0}(X)-b_{1}(X).$

We note that $b_{i}(X)=0$ for all $i\geq 2$. If $X$ is connected, then

$\chi(X)=|V_{X}|-|E_{X}|$

as explained for instance in [Sun13, Section 4.4]. In this case, we also note
that $\chi(X)=1-b_{1}(X)$. The valency of a vertex $v\in V_{X}$ is defined as
follows

${\rm val}_{X}(v)=|\mathbf{E}_{X,v}|.$

Henceforth, we choose a labeling of the vertices of $X$, say
$V_{X}=\\{v_{1},v_{2},\ldots,v_{g}\\}$. The degree (or valency) matrix
associated to $X$ is the diagonal matrix $D=(d_{ij})$ satisfying

(2.1) $d_{ii}={\rm val}_{X}(v_{i}),$

for all $i=1,\ldots,g$. The adjacency matrix $A=(a_{ij})$ of $X$ is the
symmetric matrix whose entries are specified as follows

(2.2) $a_{ij}=\begin{cases}\text{twice the number of undirected loops at
}v_{i},&\text{ if }i=j;\\\ \text{the number of undirected edges between
}v_{i}\text{ and }v_{j},&\text{ otherwise.}\end{cases}$

The matrix $Q\mathrel{\mathop{\mathchar 58\relax}}=D-A$ is called the
Laplacian matrix and is known to be singular. In fact, it is easy to see that
any vector whose entries are all the same is in the null-space of $Q$.

In order to introduce the Galois theory of graphs, we first recall the notion
of a morphism in the category of graphs. Let $X$ and $Y$ be two graphs. A
morphism of graphs $f\mathrel{\mathop{\mathchar 58\relax}}Y\rightarrow X$
consists of two functions $f_{V}\mathrel{\mathop{\mathchar
58\relax}}V_{Y}\rightarrow V_{X}$ and $f_{E}\mathrel{\mathop{\mathchar
58\relax}}\mathbf{E}_{Y}\rightarrow\mathbf{E}_{X}$ satisfying

  1. (1)

$f_{V}(o(e))=o(f_{E}(e))$,

  2. (2)

$f_{V}(t(e))=t(f_{E}(e))$,

  3. (3)

$\overline{f_{E}(e)}=f_{E}(\bar{e})$,

for all $e\in\mathbf{E}_{X}$. We will often write $f$ for both $f_{V}$ and
$f_{E}$.

######  Definition 2.1.

Let $X$ and $Y$ be two graphs and let $f\mathrel{\mathop{\mathchar
58\relax}}Y\rightarrow X$ be a morphism of graphs. If $f$ satisfies the
following conditions

  1. (1)

$f\mathrel{\mathop{\mathchar 58\relax}}V_{Y}\rightarrow V_{X}$ is surjective,

  2. (2)

for all $w\in V_{Y}$, the restriction $f|_{\mathbf{E}_{Y,w}}$ induces a
bijection

$f|_{\mathbf{E}_{Y,w}}\mathrel{\mathop{\mathchar
58\relax}}\mathbf{E}_{Y,w}\stackrel{{\scriptstyle\approx}}{{\rightarrow}}{\mathbf{E}_{X,f(w)}},$

then $f$ is said to be a cover. We shall often say that $Y/X$ is a cover when
the projection map $f$ is understood from the context.

######  Remark 2.2.

In [Sun13, Section 5.1], Sunada requires both $X$ and $Y$ to be connected in
the definition of a cover, but we do not make this requirement here. If $Y$ is
assumed to be connected, then so is $X$. Indeed, if $v_{1},v_{2}\in V_{X}$,
then let $w_{1},w_{2}\in V_{Y}$ be such that $f(w_{i})=v_{i}$ for $i=1,2$. If
$Y$ is connected, there exists a path $c$ going from $w_{i}$ to $w_{j}$. The
image $f(c)$ is a path in $X$ going from $v_{i}$ to $v_{j}$ showing that $X$
is connected as well. On the other hand, if $X$ is assumed to be connected,
then $Y$ is not necessarily connected.

If $Y/X$ is a cover and $Y$ is connected, then there exists a positive integer
$d$ such that $|f^{-1}(v)|=d$ for all $v\in V_{X}$. The integer $d$ is called
the degree of $Y/X$ and will be denoted by $[Y\mathrel{\mathop{\mathchar
58\relax}}X]$. If $f\mathrel{\mathop{\mathchar 58\relax}}Y\rightarrow X$ is a
cover, then we let

${\rm Aut}_{f}(Y/X)=\\{\sigma\in{\rm Aut}(Y)\,\mathrel{\mathop{\mathchar
58\relax}}\,f\circ\sigma=f\\}.$

######  Definition 2.3.

The cover $f\mathrel{\mathop{\mathchar 58\relax}}Y\rightarrow X$ is called
Galois if the following two conditions are satisfied.

  1. (1)

The graph $Y$ is connected (and hence also $X$ by Remark 2.2).

  2. (2)

The group ${\rm Aut}_{f}(Y/X)$ acts transitively on the fiber $f^{-1}(v)$ for
all $v\in V_{X}$.

If $Y/X$ is a Galois cover, then $[Y\mathrel{\mathop{\mathchar
58\relax}}X]=|{\rm Aut}_{f}(Y/X)|$, and we write ${\rm Gal}_{f}(Y/X)$, or
${\rm Gal}(Y/X)$ if $f$ is understood, instead of ${\rm Aut}_{f}(Y/X)$. One
has the usual Galois correspondence between subgroups of ${\rm Gal}(Y/X)$ and
equivalence classes of intermediate covers of $Y/X$.

######  Definition 2.4.

Let $\ell$ be a rational prime, and let $X$ be a connected graph. A
$\mathbb{Z}_{\ell}$-tower above $X$ consists of a sequence of covers of
connected graphs

$X=X_{0}\leftarrow X_{1}\leftarrow\ldots\leftarrow X_{n}\leftarrow\ldots$

such that for all positive integer $n$, the cover $X_{n}/X$ obtained from
composing the covers $X_{n}\rightarrow X_{n-1}\rightarrow\ldots\rightarrow
X_{1}\rightarrow X$ is Galois with Galois group ${\rm Gal}(X_{n}/X)$
isomorphic to $\mathbb{Z}/\ell^{n}\mathbb{Z}$.

###  2.2. Artin-Ihara $L$-functions

Let $Y/X$ be an abelian cover of graphs, meaning that $Y/X$ is Galois with
abelian group of covering transformations. The theory can be extended to a
non-abelian setting, but we restrict ourselves to the abelian setting for
simplicity, see [Ter11] for further details. Set $G\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(Y/X)$ and denote by $\widehat{G}={\rm
Hom}_{\mathbb{Z}}(G,\mathbb{C}^{\times})$ the group of complex valued
characters of $G$. If $\psi\in\widehat{G}$, then the corresponding Artin-Ihara
$L$-function is defined as follows

$L_{Y/X}(u,\psi)\mathrel{\mathop{\mathchar
58\relax}}=\prod_{\mathfrak{c}}\left(1-\psi\left(\frac{Y/X}{\mathfrak{c}}\right)u^{l(\mathfrak{c})}\right)^{-1}.$

The above product runs over all primes $\mathfrak{c}$ of $X$;
$l(\mathfrak{c})$ is the length of $\mathfrak{c}$, and
$\left(\frac{Y/X}{\mathfrak{c}}\right)\in G$ denotes the corresponding
Frobenius automorphism.

######  Remark 2.5.

For the definition of a prime in a graph, see [Ter11, Section 2.2], and for
the definition of the Frobenius automorphism, see [Ter11, Section 16].

Recall that we labeled the vertices of $X$ as follows
$V_{X}=\\{v_{1},v_{2},\ldots,v_{g}\\}$. For each $i=1,\ldots,g$, choose a
vertex $w_{i}$ of $Y$ in the fiber of $v_{i}$. For $\sigma\in G$, define the
matrix $A(\sigma)=(a_{ij}(\sigma))$ as follows

$a_{ij}(\sigma)\mathrel{\mathop{\mathchar 58\relax}}=\begin{cases}\text{twice
the number of undirected loops at the vertex }w_{i},&\text{if }i=j\text{ and
}\sigma=1;\\\ \text{the number of undirected edges connecting }w_{i}\text{ to
}w_{j}^{\sigma},&\text{otherwise}.\end{cases}$

If $\psi\in\widehat{G}$, one defines the _twisted adjacency matrix_ as follows

(2.3) $A_{\psi}\mathrel{\mathop{\mathchar 58\relax}}=\sum_{\sigma\in
G}\psi(\sigma)\cdot A(\sigma).$

Then the three-term determinant formula [Ter11, Theorem 18.15] asserts that

(2.4)
$L_{Y/X}(u,\psi)^{-1}=\left(1-u^{2}\right)^{-\chi(X)}\operatorname{det}\left(\operatorname{I}-A_{\psi}u+(D-\operatorname{I})u^{2}\right).$

In the above formula, we recall that $\chi(X)$ is the Euler characteristic of
$X$, $D$ is the degree matrix of $X$ (cf. (2.1)), $A_{\psi}$ the _twisted
adjacency matrix_ associated to $\psi$ and the cover $Y/X$ (cf. (2.3)), and
$\operatorname{I}$ is the identity matrix. For ease of notation, set

(2.5) $h_{Y/X}(u,\psi)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{det}\left(\operatorname{I}-A_{\psi}u+(D-\operatorname{I})u^{2}\right)\in\mathbb{Z}[\psi][u],$

where $\mathbb{Z}[\psi]$ denotes the ring of integers in the cyclotomic number
field

$\mathbb{Q}(\psi)\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{Q}(\\{\psi(\sigma)\,|\,\sigma\in G\\}).$

One can check that $h_{Y/X}(u,\psi)$ does not depend on the choice of the
vertices $w_{i}$. When $\psi=\psi_{0}$ is the trivial character, then we write
$h_{X}(u)$ instead of $h_{Y/X}(u,\psi_{0})$, and $Z_{X}(u)$ instead of
$L_{Y/X}(u,\psi_{0})$. The function $Z_{X}(u)$ is the Ihara zeta function of
the graph $X$ introduced in [Iha66] and reinterpreted in the context of graph
theory by Sunada in [Sun86] following a suggestion of Serre. Note that
$A_{\psi_{0}}=A$, the usual adjacency matrix of $X$ (cf. (2.2)). Note that
when the cover $Y/X$ is understood, it is suppressed in our notation, and we
shall simply write $h_{X}(u,\psi)$ instead of $h_{Y/X}(u,\psi)$.

It was shown by Stark and Terras (see [ST00, Proposition 3]) that the Artin-
Ihara $L$-functions satisfy the usual Artin formalism. It follows that

$Z_{Y}(u)=Z_{X}(u)\cdot\prod_{\psi\neq\psi_{0}}L_{Y/X}(u,\psi).$

Furthermore, (2.4) and (1.7) imply that

(2.6) $h_{Y}(u)=h_{X}(u)\prod_{\psi\neq\psi_{0}}h_{Y/X}(u,\psi).$

If $Z$ is an arbitrary connected graph, note that $h_{Z}(1)=0$ since the
Laplacian matrix $Q=D-A$ is singular, and it follows from (2.5) that
$h_{Z}(1)=\operatorname{det}(Q)$. In [Has90], Hashimoto showed that for a
connected graph $Z$, one has

(2.7) $h_{Z}^{\prime}(1)=-2\chi(Z)\kappa_{Z},$

where $\kappa_{Z}$ is the number of spanning trees of $Z$ (see also [Nor98]).

Coming back to our abelian cover $Y/X$, and assuming that $\chi(X)\neq 0$,
(1.7), (2.6), and (2.7) together imply that the following identity holds

(2.8)
$|G|\cdot\kappa_{Y}=\kappa_{X}\cdot\prod_{\psi\neq\psi_{0}}h_{Y/X}(1,\psi),$

(cf. [Val21, Equation (7)]). Very often, we shall make the following
assumption on our graphs.

######  Assumption 2.6.

Let $X$ be a connected graph. We assume that $\chi(X)\neq 0$.

######  Remark 2.7.

The special case when $\chi(X)=0$ can be treated separately. If $X$ is a
connected graph such that $\chi(X)=0$, note that removing all vertices of
degree one and their adjacent edge does not change the primes, the Ihara zeta
function, and the number of spanning trees. Thus we can assume that ${\rm
val}_{X}(v)\geq 2$ for all $v\in V_{X}$. In this case, $X$ is necessarily the
cycle graph on $g$ vertices, which we denote by $C_{g}$. Now,
$\pi_{1}(C_{g},v_{0})$ is a free group on one generator; hence, it is abelian,
and

$\pi_{1}(C_{g},v_{0})\simeq H_{1}(C_{g},\mathbb{Z})\simeq\mathbb{Z}.$

It follows that up to isomorphism, there is a unique Galois cover $Y/C_{g}$ of
degree $n$, it is abelian, with ${\rm
Gal}(Y/C_{g})\simeq\mathbb{Z}/n\mathbb{Z}$. By (1.7), we have $Y\simeq
C_{gn}$. Also note that the number of spanning trees of $C_{g}$ is $g$, and
thus, the number of spanning trees of $Y$ is $gn$. Thus, the growth patterns
in the number of spanning trees in any abelian tower is well understood, due
to this explicit description.

###  2.3. Constructions of Galois covers via voltage assignments

Our main reference for this section is [GT01]. If $X$ is a graph, then we say
that a subset $S$ of $\mathbf{E}_{X}$ is an orientation for $X$ if the
following two conditions are satisfied:

  1. (1)

$S\cap\overline{S}=\varnothing$,

  2. (2)

$\mathbf{E}_{X}=S\cup\overline{S}$.

In other words, an orientation $S$ is a complete set of representatives for
$E_{X}$.

######  Remark 2.8.

In [Val21, MV22a, MV22b], [LV22], [DV22], and [DRLV22], an orientation was
referred to instead as a _section_.

Given a finite group $G$ and a function $\alpha\mathrel{\mathop{\mathchar
58\relax}}S\rightarrow G$ (called a voltage assignment), one can associate a
graph $X(G,S,\alpha)$ constructed as follows. Extend $\alpha$ to
$\mathbf{E}_{X}$ by setting $\alpha(\bar{s})=\alpha(s)^{-1}$ for $s\in S$.
Then, the set of vertices of $X(G,S,\alpha)$ is $V_{X}\times G$ and the set of
directed edges is $\mathbf{E}_{X}\times G$. The edge $(e,\sigma)$ connects the
vertex $(o(e),\sigma)$ to the vertex $(t(e),\sigma\cdot\alpha(e))$ and the
inversion map is given by $\overline{(e,\sigma)}\mathrel{\mathop{\mathchar
58\relax}}=(\bar{e},\sigma\cdot\alpha(e))$. It is easy to check that
$X(G,S,\alpha)$ is a graph in the sense of §2.1, which we leave to the reader.
In what follows, $v$ (resp. $e$) shall denote a vertex (resp. edge) of $X$,
and $\sigma$ shall denote an element of $G$. The map
$p\mathrel{\mathop{\mathchar 58\relax}}X(G,S,\alpha)\rightarrow X$ defined via

(2.9) $p(v,\sigma)\mathrel{\mathop{\mathchar 58\relax}}=v\text{ and
}p(e,\sigma)\mathrel{\mathop{\mathchar 58\relax}}=e$

is a cover of graphs. Given $\tau\in G$, the map
$\phi_{\tau}\mathrel{\mathop{\mathchar 58\relax}}X(G,S,\alpha)\longrightarrow
X(G,S,\alpha)$ defined via

$\phi_{\tau}(v,\sigma)=(v,\tau\cdot\sigma)\text{ and
}\phi_{\tau}(e,\sigma)=(e,\tau\cdot\sigma)$

satisfies $\phi_{\tau}\in{\rm Aut}_{p}(X(G,S,\alpha)/X)$, and the function

(2.10) $G\hookrightarrow{\rm Aut}_{p}(X(G,S,\alpha)/X)$

given by $\tau\mapsto\phi_{\tau}$ is an injective group morphism. If one
assumes that $X(G,S,\alpha)$ is connected, then $X(G,S,\alpha)/X$ is a Galois
cover and it follows from the unique lifting theorem [Sun13, Theorem 5.1] that
(2.10) is an isomorphism.

If the base graph $X$ is a bouquet (i.e., has a single vertex), then the
graphs $X(G,S,\alpha)$ are precisely the Cayley-Serre graphs, a slight
generalization of the usual Cayley graphs. As explained in [DV22, Section 3],
given a fixed orientation $S$ of $X$, every abelian cover $Y/X$ with Galois
group $G$ is isomorphic as a cover to one of the form $X(G,S,\alpha)/X$ for
some function $\alpha\mathrel{\mathop{\mathchar 58\relax}}S\rightarrow G$.

If $G$ and $G_{1}$ are two finite groups and $f\mathrel{\mathop{\mathchar
58\relax}}G\rightarrow G_{1}$ is a group morphism, then one obtains a morphism
of graphs $f_{*}\mathrel{\mathop{\mathchar 58\relax}}X(G,S,\alpha)\rightarrow
X(G_{1},S,f\circ\alpha)$ by setting

$f_{*}(v,\sigma)=(v,f(\sigma))\text{ and }f_{*}(e,\sigma)=(e,f(\sigma)).$

If $f$ is surjective, then $f_{*}$ is a cover. For $\tau\in{\rm ker}(f)$, let
$\phi_{\tau}\mathrel{\mathop{\mathchar 58\relax}}X(G,S,\alpha)\longrightarrow
X(G,S,\alpha)$ be defined via

$\phi_{\tau}(v,\sigma)=(v,\tau\cdot\sigma)\text{ and
}\phi_{\tau}(e,\sigma)=(e,\tau\cdot\sigma).$

The map

(2.11) ${\rm ker}(f)\hookrightarrow{\rm
Aut}_{f_{*}}(X(G,S,\alpha)/X(G_{1},S,f\circ\alpha)),$

defined via $\tau\mapsto\phi_{\tau}$ is a well-defined injective group
morphism. If one assumes furthermore that $X(G,S,\alpha)$ is connected, then
$f_{*}\mathrel{\mathop{\mathchar 58\relax}}X(G,S,\alpha)\rightarrow
X(G_{1},S,f\circ\alpha)$ is a Galois cover and (2.11) is an isomorphism of
groups.

Starting with a function $\alpha\mathrel{\mathop{\mathchar
58\relax}}S\rightarrow\mathbb{Z}_{\ell}$, and letting $\alpha_{/n}$ be the
reduction of $\alpha$ modulo $\ell^{n}$, it follows from the above discussion
that we obtain a $\mathbb{Z}_{\ell}$-tower

(2.12) $X\leftarrow X(\mathbb{Z}/\ell\mathbb{Z},S,\alpha_{/1})\leftarrow
X(\mathbb{Z}/\ell^{2}\mathbb{Z},S,\alpha_{/2})\leftarrow\ldots\leftarrow
X(\mathbb{Z}/\ell^{n}\mathbb{Z},S,\alpha_{/n})\leftarrow\ldots$

in the sense of Definition 2.4, provided the graphs
$X(\mathbb{Z}/\ell^{n}\mathbb{Z},S,\alpha_{/n})$ are connected for all
nonnegative integers $n$. As explained in [DRLV22, Section 2.2], given an
orientation $S$ for $X$, every $\mathbb{Z}_{\ell}$-tower in the sense of
Definition 2.4 is isomorphic in a suitable sense to a
$\mathbb{Z}_{\ell}$-tower as in (2.12) for some function
$\alpha\mathrel{\mathop{\mathchar 58\relax}}S\rightarrow\mathbb{Z}_{\ell}$.

######  Assumption 2.9.

Let $\ell$ be a rational prime and let $\alpha\mathrel{\mathop{\mathchar
58\relax}}S\rightarrow\mathbb{Z}_{\ell}$ be a function. We assume that the
derived graphs $X(\mathbb{Z}/\ell^{n}\mathbb{Z},S,\alpha_{/n})$ are connected
for all $n\geq 0$.

######  Remark 2.10.

See [DRLV22, Theorem 2.11] for a condition that guarantees the validity of
Assumption 2.9. We explain the assumption in further detail in §2.3.1 below.

####  2.3.1. Connectedness of graphs obtained from voltage assignments

Let $X$ be a connected graph, $S$ an orientation for $X$, $G$ a finite group,
and let $\alpha\mathrel{\mathop{\mathchar 58\relax}}S\rightarrow G$ be a
function which we extend to all of $\mathbf{E}_{X}$ as usual by setting

(2.13) $\alpha(\bar{s})=\alpha(s)^{-1}.$

If $c=e_{1}e_{2}\ldots e_{n}$ is a path in $X$, then we set

$\alpha(c)=\alpha(e_{1})\cdot\ldots\cdot\alpha(e_{n})\in G.$

The condition (2.13) guarantees that if $c_{1}$ and $c_{2}$ are homotopically
equivalent, then $\alpha(c_{1})=\alpha(c_{2})$. Choosing an arbitrary vertex
$v_{0}\in V_{X}$, it follows that $\alpha$ induces a group morphism

(2.14) $\rho_{\alpha}\mathrel{\mathop{\mathchar
58\relax}}\pi_{1}(X,v_{0})\rightarrow G,$

defined via $\rho_{\alpha}([\gamma])=\alpha(\gamma)$.

######  Theorem 2.11.

Let $X$ be a connected graph and let $S,G,\alpha$ be as above. Then the graph
$X(G,S,\alpha)$ is connected if and only if the group morphism $\rho_{\alpha}$
defined above in (2.14) is surjective.

###### Proof.

Assume first that $\rho_{\alpha}$ is surjective. Since
$p\mathrel{\mathop{\mathchar 58\relax}}X(G,S,\alpha)\rightarrow X$ is a cover,
every directed edge $e$ of $X$ can be uniquely lifted to a directed edge of
$X(G,S,\alpha)$ once a vertex in the fiber of $o(e)$ has been chosen. The
connectedness of $X$ implies that given any vertices $v_{1},v_{2}\in V_{X}$
and a choice of $w_{1}\in p^{-1}(v_{1})$, there exists a path in
$X(G,S,\alpha)$ starting at $w_{1}$ and ending at a vertex in $p^{-1}(v_{2})$.
To show that $X(G,S,\alpha)$ is connected, it suffices then to show that every
vertex in $p^{-1}(v_{0})$ can be connected by a path in $X(G,S,\alpha)$. Let
thus $(v_{0},\sigma),(v_{0},\tau)\in p^{-1}(v_{0})$ be arbitrary. Since
$\rho_{\alpha}$ is assumed to be surjective, there exists a loop $\gamma$ in
$X$ based at $v_{0}$ satisfying $\rho_{\alpha}(v_{0})=\sigma^{-1}\cdot\tau$.
One has

$\gamma=e_{1}e_{2}\ldots e_{n},$

for some $e_{i}\in\mathbf{E}_{X}$. Then, the path

$(e_{1},\sigma)(e_{2},\sigma\cdot\alpha(e_{1}))\ldots(e_{n},\sigma\cdot\alpha(e_{1})\cdot\ldots\cdot\alpha(e_{n-1}))$

in $X(G,S,\alpha)$ connects $(v_{0},\sigma)$ to $(v_{0},\tau)$ showing the
claim.

Conversely, if $X(G,S,\alpha)$ is connected, then there is a path $c$ in
$X(G,S,\alpha)$ going from $(v_{0},1_{G})$ to $(v_{0},\sigma)$. The path $c$
is of the form

$(e_{1},1_{G})(e_{2},\alpha(e_{1}))\ldots(e_{n},\alpha(e_{1})\cdot\ldots\cdot\alpha(e_{n-1})),$

for some directed edges $e_{i}\in\mathbf{E}_{X}$ satisfying

  1. (1)

$o(e_{1})=v_{0}=t(e_{n})$,

  2. (2)

$t(e_{i})=o(e_{i+1})$ for $i=1,\ldots,n-1$,

  3. (3)

$\sigma=\alpha(e_{1})\cdot\ldots\cdot\alpha(e_{n})$.

Then, the loop $\gamma=p(c)$ satisfies $\rho_{\alpha}([\gamma])=\sigma$ and
since $\sigma\in G$ was chosen arbitrarily, this shows the surjectivity of
$\rho_{\alpha}$. ∎

Theorem 2.11 can be used to check if $X(G,S,\alpha)$ is connected provided a
basis for the free group $\pi_{1}(X,v_{0})$ is known. Indeed, say
$[\gamma_{1}],\ldots,[\gamma_{t}]$ is such a basis, then $X(G,S,\alpha)$ is
connected if and only if

$G=\langle\rho_{\alpha}([\gamma_{i}])\,\mathrel{\mathop{\mathchar
58\relax}}\,i=1,\ldots,t\rangle.$

Furthermore, we remind the reader that a basis for $\pi_{1}(X,v_{0})$ can be
constructed as follows. Let $\mathfrak{T}$ be a spanning tree for $X$ and let

$S(\mathfrak{T})=\\{s\in S\,|\,\text{The undirected edge corresponding to
}s\in\mathfrak{T}\\}.$

For $s\in S\smallsetminus S(\mathfrak{T})$, let $\gamma_{s}$ be the loop based
at $v_{0}$ obtained by going through the unique geodesic in $\mathfrak{T}$
going from $v_{0}$ to $o(s)$ followed by $s$ followed by the unique geodesic
in $\mathfrak{T}$ going from $t(s)$ to $v_{0}$. Then the collection

$\\{[\gamma_{s}]\,\mathrel{\mathop{\mathchar 58\relax}}\,s\in S\smallsetminus
S(\mathfrak{T})\\}$

is a basis for the free group $\pi_{1}(X,v_{0})$. (See [Sun13, page 68].)

######  Example 2.12.

Let $\ell=2$, $G=\mathbb{Z}/2\mathbb{Z}$ and consider the graph $X$ consisting
of four undirected edges connecting two distinct vertices $\\{v_{0},v_{1}\\}$.
Let $S=\\{s_{1},s_{2},s_{3},s_{4}\\}$, where $s_{1},s_{2}$ go from $v_{1}$ to
$v_{0}$ and $s_{3},s_{4}$ go from $v_{0}$ to $v_{1}$. Consider the function
$\alpha\mathrel{\mathop{\mathchar 58\relax}}S\rightarrow G$ defined via
$\alpha(s_{i})=\bar{1}$ for $i=1,2,3,4$. Choose the spanning tree
$\mathfrak{T}$ consisting of the undirected edge determined by $s_{4}$. Then,

$\gamma_{s_{1}}=s_{4}s_{1},\gamma_{s_{2}}=s_{4}s_{2},\text{ and
}\gamma_{s_{3}}=s_{3}\bar{s}_{4}.$

One has $\alpha(\gamma_{s_{i}})=\bar{0}$ for $i=1,2,3$, and thus
$X(\mathbb{Z}/2\mathbb{Z},S,\alpha)$ is not connected.

###  2.4. The analogue of Iwasawa’s theorem

Let $X$ be a connected graph satisfying Assumption 2.6, $S$ an orientation for
$X$, and $G$ a finite abelian group. Let also
$\alpha\mathrel{\mathop{\mathchar 58\relax}}S\rightarrow G$ be a function for
which $X(G,S,\alpha)$ is connected. If we label the vertices of $X$ as
$V_{X}=\\{v_{1},v_{2},\ldots,v_{g}\\}$ and we let $w_{i}=(v_{i},1_{G})\in
V_{X(G,S,\alpha)}$, then [MV22b, Corollary 5.3] shows that for
$\psi\in\widehat{G}$, the corresponding _twisted adjacency matrix_ is given by

(2.15) $A_{\psi}=\left(\sum_{\begin{subarray}{c}s\in S\\\ {\rm
inc}(s)=(v_{i},v_{j})\end{subarray}}\psi(\alpha(s))+\sum_{\begin{subarray}{c}s\in
S\\\ {\rm inc}(s)=(v_{j},v_{i})\end{subarray}}\psi(-\alpha(s))\right).$

Recall that $\mu_{\ell^{\infty}}$ is the group of $\ell$-th power roots of
unity in $\mathbb{C}^{\times}$. From now on, we fix an embedding
$\mathbb{Q}(\mu_{\ell^{\infty}})\hookrightarrow\overline{\mathbb{Q}}_{\ell}\subseteq\mathbb{C}_{\ell}$,
and we view all characters of a finite abelian $\ell$-group as taking values
in $\mathbb{C}_{\ell}^{\times}$ via this embedding. Let $|\cdot|_{\ell}$ be
the usual absolute value on $\mathbb{C}_{\ell}$ normalized so that
$|\ell|_{\ell}=\ell^{-1}$. Set $D\mathrel{\mathop{\mathchar
58\relax}}=\\{t\in\mathbb{C}_{\ell}\,\mathrel{\mathop{\mathchar
58\relax}}\,|t|_{\ell}<1\\}$, and for a continuous group morphism
$\psi\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}_{\ell}\rightarrow\mathbb{C}_{\ell}^{\times}$ of finite
order, let $t_{\psi}\in D$ be the element $t_{\psi}\mathrel{\mathop{\mathchar
58\relax}}=\psi(1)-1$. Consider now the continuous group morphism
$\rho\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}_{\ell}\rightarrow\mathbb{Z}_{\ell}\llbracket
T\rrbracket^{\times}$ given by $a\mapsto\rho(a)=(1+T)^{a}$ which satisfies

(2.16) $\rho(a)(t_{\psi})=\psi(a),$

for all continuous group morphisms $\psi\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}_{\ell}\rightarrow\mathbb{C}_{\ell}^{\times}$ of finite
order.

######  Remark 2.13.

In [MV22b] and [DV22], the authors work with the morphism $a\mapsto(1-T)^{a}$
rather than $\rho$ of the present paper, so our convention here is slightly
different.

Given a function $\alpha\mathrel{\mathop{\mathchar
58\relax}}S\rightarrow\mathbb{Z}_{\ell}$, we define

$A_{\rho}=\left(\sum_{\begin{subarray}{c}s\in S\\\ {\rm
inc}(s)=(v_{i},v_{j})\end{subarray}}\rho(\alpha(s))+\sum_{\begin{subarray}{c}s\in
S\\\ {\rm inc}(s)=(v_{j},v_{i})\end{subarray}}\rho(-\alpha(s))\right)\in
M_{g\times g}(\mathbb{Z}_{\ell}\llbracket T\rrbracket),$

and we let

(2.17)
$f_{X,\alpha}(T)=\operatorname{det}(D-A_{\rho})\in\mathbb{Z}_{\ell}\llbracket
T\rrbracket,$

where $D$ is the degree matrix of $X$. Using (2.5), (2.15), and (2.16), one
has

(2.18) $f_{X,\alpha}(t_{\psi})=h_{X}(1,\psi),$

for all continuous characters $\psi\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}_{\ell}\rightarrow\mathbb{C}_{\ell}^{\times}$ of finite
order. At the level $n$ of the tower (2.12), formula (2.8) becomes

(2.19)
$\ell^{n}\cdot\kappa_{n}=\kappa_{X}\prod_{\psi\neq\psi_{0}}h_{X}(1,\psi),$

where the product is over all non-trivial characters $\psi$ of
$\mathbb{Z}/\ell^{n}\mathbb{Z}\simeq{\rm
Gal}(X(\mathbb{Z}/\ell^{n}\mathbb{Z},S,\alpha_{/n})/X)$. Recall that to any
power series

$Q(T)=a_{0}+a_{1}T+a_{2}T^{2}+\ldots\in\mathbb{Z}_{\ell}\llbracket
T\rrbracket,$

one associates two invariants, namely

$\mu(Q(T))={\rm min}\\{{\rm ord}_{\ell}(a_{i})\,|\,i\geq 0\\},$

and

$\lambda(Q(T))={\rm min}\\{i\geq 0\,|\,{\rm ord}_{\ell}(a_{i})=\mu(Q(T))\\}.$

Putting (2.18), (2.19) together and using a classical argument about power
series in $\mathbb{Z}_{\ell}\llbracket T\rrbracket$ evaluated at points of the
form $t_{\psi}$ for continuous characters $\psi\mathrel{\mathop{\mathchar
58\relax}}\mathbb{Z}_{\ell}\rightarrow\mathbb{C}_{\ell}^{\times}$ of finite
order, one obtains the following result which is [MV22b, Theorem 6.1]:

######  Theorem 2.14.

Let $X$ be a connected graph satisfying Assumption 2.6, $\ell$ a rational
prime, $S$ an orientation for $X$, and $\alpha\mathrel{\mathop{\mathchar
58\relax}}S\rightarrow\mathbb{Z}_{\ell}$ a function for which Assumption 2.9
is satisfied. Consider the corresponding $\mathbb{Z}_{\ell}$-tower

(2.20) $X\leftarrow X(\mathbb{Z}/\ell\mathbb{Z},S,\alpha_{/1})\leftarrow
X(\mathbb{Z}/\ell^{2}\mathbb{Z},S,\alpha_{/2})\leftarrow\ldots\leftarrow
X(\mathbb{Z}/\ell^{n}\mathbb{Z},S,\alpha_{/n})\leftarrow\ldots,$

and let $\kappa_{n}$ denote the number of spanning trees of
$X(\mathbb{Z}/\ell^{n}\mathbb{Z},S,\alpha_{/n})$. Let also

$\mu=\mu(f(T))\text{ and }\lambda=\lambda(f(T))-1,$

where $f(T)=f_{X,\alpha}(T)$ was defined in (2.17) above. Then, there exist
$n_{0},\nu\in\mathbb{Z}$ for which

${\rm ord}_{\ell}(\kappa_{n})=\mu\ell^{n}+\lambda n+\nu,$

when $n\geq n_{0}$.

We shall refer to $\mu,\lambda,\nu$ as the Iwasawa invariants associated to
the $\mathbb{Z}_{\ell}$-tower (2.20), and they will be denoted by

$\mu_{\ell}(X,\alpha),\lambda_{\ell}(X,\alpha),\text{ and
}\nu_{\ell}(X,\alpha).$

We shall often drop the $\ell$ from the notation if it is understood from the
context. Note that

$\mu_{\ell}(X,\alpha)=\mu(f_{X,\alpha}(T)),\text{ but
}\lambda_{\ell}(X,\alpha)=\lambda(f_{X,\alpha}(T))-1.$

######  Remark 2.15.

If $\ell$ is odd, then [DRLV22, Theorem 2.14] shows that
$\lambda_{\ell}(X,\alpha)$ is odd.

##  3\. Pullbacks

In the category of graphs, there is a well-defined notion of pullback as was
pointed out for instance in [Sta83]. If one starts with a diagram

${Y_{1}}$${X}$${Y_{2},}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$

where $p_{1}$ and $p_{2}$ are morphisms of graphs, one defines the pullback
$Y_{1}\times_{X}Y_{2}$ by defining the set of vertices to be

$V=\\{(w_{1},w_{2})\in V_{Y_{1}}\times V_{Y_{2}}\,\mathrel{\mathop{\mathchar
58\relax}}\,p_{1}(w_{1})=p_{2}(w_{2})\\},$

and the set of directed edges to be

$\mathbf{E}=\\{(e_{1},e_{2})\in\mathbf{E}_{Y_{1}}\times\mathbf{E}_{Y_{2}}\,\mathrel{\mathop{\mathchar
58\relax}}\,p_{1}(e_{1})=p_{2}(e_{2})\\}.$

Further, define

$o(e_{1},e_{2})=(o(e_{1}),o(e_{2})),t(e_{1},e_{2})=(t(e_{1}),t(e_{2}))\text{
and }\overline{(e_{1},e_{2})}=(\bar{e}_{1},\bar{e}_{2}).$

We leave it to the reader to check that $Y_{1}\times_{X}Y_{2}$ is a graph. For
$i=1,2$, the maps $\pi_{i}\mathrel{\mathop{\mathchar
58\relax}}Y_{1}\times_{X}Y_{2}\rightarrow Y_{i}$ given by

$\pi_{i}(w_{1},w_{2})=w_{i}\text{ and }\pi_{i}(e_{1},e_{2})=e_{i}$

are morphisms of graphs, and the triple
$(Y_{1}\times_{X}Y_{2},\pi_{1},\pi_{2})$ satisfies the usual universal
property satisfied by a pullback. We leave the details to the reader.

######  Remark 3.1.

Even if $X$, $Y_{1}$ and $Y_{2}$ are connected graphs, then the pullback
$Y_{1}\times_{X}Y_{2}$ is not necessarily connected. For example, take
$X=B_{1}$, the bouquet with one undirected loop, and $Y$ the graph consisting
of two vertices $\\{v_{1},v_{2}\\}$ and two parallel undirected edges between
the two vertices. Say $\mathbf{E}_{X}=\\{s,\bar{s}\\}$ and
$\mathbf{E}_{Y}=\\{e_{1},e_{2},\bar{e}_{1},\bar{e}_{2}\\}$, where $e_{1}$ goes
from $v_{1}$ to $v_{2}$ and $e_{2}$ from $v_{2}$ to $v_{1}$. Consider also the
morphism of graphs $p\mathrel{\mathop{\mathchar 58\relax}}Y\rightarrow X$
given by $e_{i}\mapsto s$ and by mapping the two vertices to the unique one of
$X$. If we take $Y_{1}=Y_{2}=Y$ and $p_{1}=p_{2}=p$, then the graph
$Y\times_{X}Y$ is disconnected; it is two disjoint copies of $Y$.

######  Proposition 3.2.

Consider the diagram

${Y_{1}}$${Y_{1}\times_{X}Y_{2}}$${X}$${Y_{2},}$$\scriptstyle{p_{1}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$$\scriptstyle{p_{2}}$

in the category of graphs. If $p_{2}$ is a cover, then so is
$\pi_{1}\mathrel{\mathop{\mathchar 58\relax}}Y_{1}\times_{X}Y_{2}\rightarrow
Y_{1}$.

###### Proof.

For the first property, let $w_{1}\in V_{Y_{1}}$. Since
$p_{2}\mathrel{\mathop{\mathchar 58\relax}}V_{Y_{2}}\rightarrow V_{X}$ is
assumed to be surjective, there exists $w_{2}\in V_{Y_{2}}$ such that
$p_{2}(w_{2})=p_{1}(w_{1})$. Then $(w_{1},w_{2})\in V_{Y_{1}\times_{X}Y_{2}}$,
and $\pi_{1}(w_{1},w_{2})=w_{1}$.

For the second property, let $w=(w_{1},w_{2})\in V_{Y_{1}\times_{X}Y_{2}}$. We
show first that

(3.1)
$\pi_{1}|_{\mathbf{E}_{Y_{1}\times_{X}Y_{2}},w}\mathrel{\mathop{\mathchar
58\relax}}\mathbf{E}_{Y_{1}\times_{X}Y_{2},w}\rightarrow\mathbf{E}_{Y_{1},w_{1}}$

is injective. Let
$(e_{1},e_{2}),(\varepsilon_{1},\varepsilon_{2})\in\mathbf{E}_{Y_{1}\times_{X}Y_{2},w}$,
then in particular $o(e_{2})=o(\varepsilon_{2})=w_{2}$. If
$\pi_{1}(e_{1},e_{2})=\pi_{1}(\varepsilon_{1},\varepsilon_{2})$, then
$e_{1}=\varepsilon_{1}$. It follows that
$p_{2}(e_{2})=p_{1}(e_{1})=p_{1}(\varepsilon_{1})=p_{2}(\varepsilon_{2})$ and
thus $p_{2}(e_{2})=p_{2}(\varepsilon_{2})$. Since
$e_{2},\varepsilon_{2}\in\mathbf{E}_{Y_{2},w_{2}}$, the assumption on $p_{2}$
implies that $e_{2}=\varepsilon_{2}$ and this shows that (3.1) is injective.
In order to show that (3.1) is surjective, let
$e_{1}\in\mathbf{E}_{Y_{1},w_{1}}$. Then
$p_{1}(e_{1})\in\mathbf{E}_{X,p_{1}(w_{1})}=\mathbf{E}_{X,p_{2}(w_{2})}$. The
assumption on $p_{2}$ implies that there exists
$e_{2}\in\mathbf{E}_{Y_{2},w_{2}}$ such that $p_{2}(e_{2})=p_{1}(e_{1})$. But
then $(e_{1},e_{2})\in\mathbf{E}_{Y_{1}\times_{X}Y_{2},w}$ and
$\pi_{1}(e_{1},e_{2})=e_{1}$. This ends the proof. ∎

######  Proposition 3.3.

Consider the diagram

${Y}$${Y\times_{X}X_{1}}$${X}$${X_{1}}$${X_{2},}$$\scriptstyle{f}$$\scriptstyle{\pi_{1}}$$\scriptstyle{f_{1}}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$

and let $p=p_{1}\circ p_{2}$. Consider also the pullback
$(Y\times_{X}X_{2},\pi,f_{2})$ of the diagram

${Y}$${X}$${X_{2}.}$$\scriptstyle{f}$$\scriptstyle{p}$

By the universal property satisfied by $(Y\times_{X}X_{1},\pi_{1},f_{1})$, we
have a unique morphism of graphs $\pi_{2}\mathrel{\mathop{\mathchar
58\relax}}Y\times_{X}X_{2}\rightarrow Y\times_{X}X_{1}$ which fits into the
following commutative diagram

${Y}$${Y\times_{X}X_{1}}$${Y\times_{X}X_{2}}$${X}$${X_{1}}$${X_{2},}$$\scriptstyle{f}$$\scriptstyle{\pi_{1}}$$\scriptstyle{f_{1}}$$\scriptstyle{\pi_{2}}$$\scriptstyle{f_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$

and for which $\pi=\pi_{1}\circ\pi_{2}$. If $p_{1}$ and $p_{2}$ are covers,
then so is $\pi_{2}$.

###### Proof.

Note that $p$ is a cover, since a composition of covers is a cover. By
Proposition 3.2, we know that $\pi$ is a cover as well. The graph morphism
$\pi_{2}$ is given by $\pi_{2}(w,v_{2})=(w,p_{2}(v_{2}))$. Since $p_{2}$ is
surjective on the vertices, it follows directly that $\pi_{2}$ is surjective
on the vertices as well. If $(w,v_{2})\in V_{Y\times_{X}X_{2}}$, and since
$\pi_{1}$, $\pi_{2}$ are morphisms of graphs, they induce functions

$\mathbf{E}_{Y\times_{X}X_{2},(w,v_{2})}\stackrel{{\scriptstyle\pi_{2}}}{{\longrightarrow}}\mathbf{E}_{Y\times_{X}X_{1},(w,p_{2}(v_{2}))}\stackrel{{\scriptstyle\pi_{1}}}{{\longrightarrow}}\mathbf{E}_{Y,w},$

where $\pi_{1}$ is a bijection by Proposition 3.2. Since
$\pi=\pi_{1}\circ\pi_{2}$ (more precisely the restriction of $\pi$ to the set
of directed edges on the left of the displayed line above with target space
the set of directed edges one the right of the displayed line above) and $\pi$
is also a bijection, we deduce that $\pi_{2}\mathrel{\mathop{\mathchar
58\relax}}\mathbf{E}_{Y\times_{X}X_{2},(w,v_{2})}\rightarrow\mathbf{E}_{Y\times_{X}X_{1},(w,p_{2}(v_{2}))}$
is also a bijection, and the result follows.

∎

######  Proposition 3.4.

Consider the diagram

${Y_{1}}$${Y_{1}\times_{X}Y_{2}}$${X}$${Y_{2},}$$\scriptstyle{p_{1}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$$\scriptstyle{p_{2}}$

in the category of graphs. Assume furthermore that $Y_{1}\times_{X}Y_{2}$ is
connected. If $p_{2}\mathrel{\mathop{\mathchar 58\relax}}Y_{2}\rightarrow X$
is a Galois cover, then so is $\pi_{1}\mathrel{\mathop{\mathchar
58\relax}}Y_{1}\times_{X}Y_{2}\rightarrow Y_{1}$. Furthermore, the map ${\rm
Gal}(Y_{2}/X)\rightarrow{\rm Gal}(Y_{1}\times_{X}Y_{2}/Y_{1})$ given by
$\sigma\mapsto\tilde{\sigma}$, where $\tilde{\sigma}$ is defined via

$\tilde{\sigma}(w_{1},w_{2})=(w_{1},\sigma(w_{2}))\text{ and
}\tilde{\sigma}(e_{1},e_{2})=(e_{1},\sigma(e_{2}))$

is a group isomorphism.

###### Proof.

First, let $\sigma\in{\rm Gal}(Y_{2}/X)$ and define $\tilde{\sigma}$ as above.
We leave it to the reader to check that $\tilde{\sigma}\in{\rm
Aut}_{\pi_{1}}(Y_{1}\times_{X}Y_{2}/X)$. Let
$(w_{1},w_{2}),(w_{1},w_{2}^{\prime})\in\pi_{1}^{-1}(w_{1})$. We have
$p_{2}(w_{2})=p_{2}(w_{2}^{\prime})$ and thus $w_{2},w_{2}^{\prime}$ are in
the same fiber of $p_{2}$. By assumption, $p_{2}$ is a Galois cover and it
follows that there exists $\sigma\in{\rm Gal}(Y_{2}/X)$ such that
$\sigma(w_{2})=w_{2}^{\prime}$. But then
$\tilde{\sigma}(w_{1},w_{2})=(w_{1},w_{2}^{\prime})$ and this shows that ${\rm
Aut}_{\pi_{1}}(Y_{1}\times_{X}Y_{2}/Y_{1})$ acts transitively on the fibers of
$\pi_{1}$. Combining with Proposition 3.2, we deduce that $\pi_{1}$ is also a
Galois cover. We now have a well-defined function

(3.2) ${\rm Gal}(Y_{2}/X)\rightarrow{\rm Gal}(Y_{1}\times_{X}Y_{2}/Y_{1}),$

given by $\sigma\mapsto\tilde{\sigma}$. Let us show first that (3.2) is
injective. If $\tilde{\sigma}_{1}=\tilde{\sigma}_{2}$ and if $(w_{1},w_{2})\in
V_{Y_{1}\times_{X}Y_{2}}$, then we have $\sigma_{1}(w_{2})=\sigma_{2}(w_{2})$.
Since $Y_{2}/X$ is assumed to be Galois, $Y_{2}$ is connected by definition.
The unique lifting theorem ([Sun13, Theorem 5.1]) implies that
$\sigma_{1}=\sigma_{2}$, and this shows the injectivity of (3.2). For the
surjectivity of (3.2), let $\tau\in{\rm Gal}(Y_{1}\times_{X}Y_{2}/Y_{1})$ and
let $(w_{1},w_{2})\in V_{Y_{1}\times_{X}Y_{2}}$. Then
$\tau(w_{1},w_{2})=(w_{1}^{\prime},w_{2}^{\prime})$ for some
$(w_{1}^{\prime},w_{2}^{\prime})\in V_{Y_{1}\times_{X}Y_{2}}$. Since $\tau$
commutes with $\pi_{1}$, we have $w_{1}=w_{1}^{\prime}$. Furthermore, $w_{2}$
and $w_{2}^{\prime}$ are in the same fiber of $p_{2}$, and since $Y_{2}/X$ is
a Galois cover, there exists $\sigma\in{\rm Gal}(Y_{2}/X)$ such that
$\sigma(w_{2})=w_{2}^{\prime}$. Therefore,
$\tau(w_{1},w_{2})=\tilde{\sigma}(w_{1},w_{2})$ and the unique lifting theorem
implies that $\tau=\tilde{\sigma}$. The fact that (3.2) is a group morphism is
clear and this ends the proof. ∎

######  Proposition 3.5.

Consider the diagram

${Y_{1}}$${Y_{1}\times_{X}Y_{2}}$${X}$${Y_{2},}$$\scriptstyle{p_{1}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$$\scriptstyle{p_{2}}$

in the category of graphs. Assume furthermore that $Y_{1}\times_{X}Y_{2}$ is
connected and let $p=p_{i}\circ\pi_{i},(i=1,2)$. If both $p_{1}$ and $p_{2}$
are Galois cover, then so is

$p\mathrel{\mathop{\mathchar 58\relax}}Y_{1}\times_{X}Y_{2}\rightarrow X.$

Furthermore, the map ${\rm Gal}(Y_{1}/X)\times{\rm
Gal}(Y_{2}/X)\rightarrow{\rm Gal}(Y_{1}\times_{X}Y_{2}/X)$ given by
$\sigma=(\sigma_{1},\sigma_{2})\mapsto\tilde{\sigma}$, where $\tilde{\sigma}$
is defined via

$\tilde{\sigma}(w_{1},w_{2})=(\sigma_{1}(w_{1}),\sigma_{2}(w_{2}))\text{ and
}\tilde{\sigma}(e_{1},e_{2})=(\sigma_{1}(e_{1}),\sigma_{2}(e_{2}))$

is a group isomorphism.

###### Proof.

First, note that $\pi$ is a cover by using Proposition 3.2 and the fact that
compositions of covers is a cover. We leave it to the reader to check that
$\tilde{\sigma}$ is an isomorphism of graphs and that $\tilde{\sigma}\in{\rm
Aut}(Y_{1}\times_{X}Y_{2}/X)$. To show that $Y_{1}\times_{X}Y_{2}/X$ is
Galois, we proceed as follows. Let $w=(w_{1},w_{2})$ and
$w^{\prime}=(w_{1}^{\prime},w_{2}^{\prime})$ be two vertices in the same fiber
of $p$. Then, we have $p_{i}(w_{i})=w_{i}^{\prime}$ for $i=1,2$. Since both
$Y_{1}/X$ and $Y_{2}/X$ are assumed to be Galois, there exists
$\sigma_{i}\in{\rm Gal}(Y_{i}/X)$ satisfying
$\sigma_{i}(w_{i})=w_{i}^{\prime}$ for $i=1,2$. Letting
$\sigma=(\sigma_{1},\sigma_{2})$, we have $\tilde{\sigma}(w)=w^{\prime}$ and
this shows that $Y_{1}\times_{X}Y_{2}/X$ is Galois. Using the unique lifting
theorem [Sun13, Theorem 5.1], one shows using a similar argument as in the
proof of Proposition 3.4 that the map ${\rm Gal}(Y_{1}/X)\times{\rm
Gal}(Y_{2}/X)\rightarrow{\rm Gal}(Y_{1}\times_{X}Y_{2}/X)$ given by
$\sigma=(\sigma_{1},\sigma_{2})\mapsto\tilde{\sigma}$ is a group isomorphism.
The details are left to the reader. ∎

Let now $X$ and $Y$ be connected graphs, and $f\mathrel{\mathop{\mathchar
58\relax}}Y\rightarrow X$ a morphism of graphs. If we start with a
$\mathbb{Z}_{\ell}$-tower

(3.3) $X=X_{0}\leftarrow X_{1}\leftarrow\ldots\leftarrow
X_{n}\leftarrow\ldots,$

and if we assume that the graphs $Y\times_{X}X_{n}$ are connected for all
nonnegative integers $n$, it follows from Propositions 3.3, 3.4, and the
Galois correspondence that we can pull back the tower (3.3) along $f$ to
obtain a $\mathbb{Z}_{\ell}$-tower

$Y\leftarrow Y\times_{X}X_{1}\leftarrow
Y\times_{X}X_{2}\leftarrow\ldots\leftarrow Y\times_{X}X_{n}\leftarrow\ldots$

which is above $Y$ now.

###  3.1. Pullbacks and voltage assignments

If $f\mathrel{\mathop{\mathchar 58\relax}}Y\rightarrow X$ is a morphism of
graphs for which the induced map $f\mathrel{\mathop{\mathchar
58\relax}}\mathbf{E}_{Y}\rightarrow\mathbf{E}_{X}$ is surjective, then any
orientation $S$ for $X$ induces a unique orientation $S_{Y}$ for $Y$ by
setting

$S_{Y}=f^{-1}(S).$

Let now $G$ be a finite abelian group, $\alpha\mathrel{\mathop{\mathchar
58\relax}}S\rightarrow G$ a function, and consider the diagram

${Y}$${X}$${X(G,S,\alpha),}$$\scriptstyle{f}$$\scriptstyle{p}$

where the morphism $p$ is given by (2.9) above. We have an induced function
$\alpha\circ f\mathrel{\mathop{\mathchar 58\relax}}S_{Y}\rightarrow G$ and a
natural graph morphism

$\pi_{1}\mathrel{\mathop{\mathchar 58\relax}}Y(G,S_{Y},\alpha\circ
f)\rightarrow Y,$

as defined above in (2.9) again. Furthermore, define
$\pi_{2}\mathrel{\mathop{\mathchar 58\relax}}Y(G,S_{Y},\alpha\circ
f)\rightarrow X(G,S,\alpha)$ via

$\pi_{2}(w,\sigma)=(f(w),\sigma)\text{ and }\pi_{2}(e,\sigma)=(f(e),\sigma).$

We leave it to the reader to check that $\pi_{2}$ is a morphism of graphs and
that the diagram

${Y}$${Y(G,S_{Y},\alpha\circ
f)}$${X}$${X(G,S,\alpha).}$$\scriptstyle{f}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$$\scriptstyle{p}$

commutes.

######  Proposition 3.6.

With the notation as above, the triple $(Y(G,S_{Y},\alpha\circ
f),\pi_{1},\pi_{2})$ is a pullback for the diagram

${Y}$${X}$${X(G,S,\alpha).}$$\scriptstyle{f}$$\scriptstyle{p}$

###### Proof.

Let $(Z,\alpha_{1},\alpha_{2})$, where $Z$ is a graph, and let
$\alpha_{1}\mathrel{\mathop{\mathchar 58\relax}}Z\rightarrow Y$,
$\alpha_{2}\mathrel{\mathop{\mathchar 58\relax}}Z\rightarrow X(G,S,\alpha)$ be
morphisms of graphs satisfying $f\circ\alpha_{1}=p\circ\alpha_{2}$. Then, for
$w\in V_{Z}$ and $e\in\mathbf{E}_{Z}$, one has

$\alpha_{2}(w)=(V_{2}(w),G_{2}(w))\text{ and
}\alpha_{2}(e)=(E_{2}(e),G_{2}(e)),$

for some $V_{2}(w)\in V_{X}$, $E_{2}(e)\in\mathbf{E}_{X}$, and some
$G_{2}(w),G_{2}(e)\in G$. Define

$\phi\mathrel{\mathop{\mathchar 58\relax}}Z\rightarrow Y(G,S_{Y},\alpha\circ
f)$

via

$w\mapsto(\alpha_{1}(w),G_{2}(w))\text{ and
}e\mapsto(\alpha_{1}(e),G_{2}(e)).$

We leave it to the reader to check that $\phi$ is the unique morphism of
graphs for which $\alpha_{i}=\pi_{i}\circ\phi$ for $i=1,2$, and this concludes
the proof. ∎

Starting now with a function $\alpha\mathrel{\mathop{\mathchar
58\relax}}S\rightarrow\mathbb{Z}_{\ell}$ for which Assumption 2.9 is
satisfied, let us assume that Assumption 2.9 is satisfied as well for
$\alpha\circ f\mathrel{\mathop{\mathchar
58\relax}}S_{Y}\rightarrow\mathbb{Z}_{\ell}$. Then, we obtain two
$\mathbb{Z}_{\ell}$-towers that fit into the following diagram

(3.4) ${Y}$${Y(\mathbb{Z}/\ell\mathbb{Z},S_{Y},\alpha\circ
f_{/1})}$${\ldots}$${Y(\mathbb{Z}/\ell^{n}\mathbb{Z},S_{Y},\alpha\circ
f_{/n})}$${\ldots}$${X}$${X(\mathbb{Z}/\ell\mathbb{Z},S,\alpha_{/1})}$${\ldots}$${X(\mathbb{Z}/\ell^{n}\mathbb{Z},S,\alpha_{/n})}$${\ldots}$$\scriptstyle{f}$

We would like to understand how the Iwasawa invariants for the
$\mathbb{Z}_{\ell}$-tower above $X$ are related to the Iwasawa invariants for
the $\mathbb{Z}_{\ell}$-tower above $Y$.

######  Remark 3.7.

If the map $f\mathrel{\mathop{\mathchar 58\relax}}Y\rightarrow X$ in the
diagram (3.4) is a cover, and if we assume that $\alpha\circ
f\mathrel{\mathop{\mathchar 58\relax}}S_{Y}\rightarrow\mathbb{Z}_{\ell}$
satisfies Assumption 2.9, then Assumption 2.9 for
$\alpha\mathrel{\mathop{\mathchar 58\relax}}S\rightarrow\mathbb{Z}_{\ell}$ is
also satisfied by Propositions 3.2, 3.6, and Remark 2.2.

Before stating and proving our main theorem, namely Theorem 4.1 below, we end
this section with the following proposition which will be used in the proof of
Theorem 4.1.

######  Proposition 3.8.

Let $X$ be a connected graph, and let $G_{1}$, $G_{2}$ be two finite abelian
groups. Let also $S$ be an orientation for $X$. Suppose we are given two
functions $\beta\mathrel{\mathop{\mathchar 58\relax}}S\rightarrow G_{2}$,
$\alpha\mathrel{\mathop{\mathchar 58\relax}}S\rightarrow G_{1}$ and assume
that both graphs $X(G_{2},S,\beta)$ and $X(G_{1},S,\alpha)$ are connected. Let
$Y=X(G_{2},S,\beta)$, and consider the diagram

${Y}$${Y(G_{1},S_{Y},\alpha\circ
p_{2})}$${X}$${X(G_{1},S,\alpha),}$$\scriptstyle{p_{2}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$$\scriptstyle{p_{1}}$

coming from Proposition 3.6. We set $p=p_{1}\circ\pi_{2}=p_{2}\circ\pi_{1}$.
Consider now the function $\gamma\mathrel{\mathop{\mathchar
58\relax}}S\rightarrow G_{2}\times G_{1}$ given by

$s\mapsto\gamma(s)=(\beta(s),\alpha(s)),$

and the natural morphism of graphs $\pi\mathrel{\mathop{\mathchar
58\relax}}X(G_{2}\times G_{1},S,\gamma)\rightarrow X$. Then, there is an
isomorphism of graphs

$\phi\mathrel{\mathop{\mathchar 58\relax}}X(G_{2}\times
G_{1},S,\gamma)\rightarrow Y(G_{1},S_{Y},\alpha\circ p_{2})$

satisfying $p\circ\phi=\pi$.

###### Proof.

The vertex and directed edge sets of $X(G_{2}\times G_{1},S,\gamma)$ are

$V_{X}\times(G_{2}\times G_{1})\text{ and }\mathbf{E}_{X}\times(G_{2}\times
G_{1}),$

whereas the vertex and directed edge sets of $Y(G_{1},S_{Y},\alpha\circ
p_{2})$ are

$(V_{X}\times G_{2})\times G_{1}\text{ and }(\mathbf{E}_{X}\times G_{2})\times
G_{1}.$

It is then obvious what the isomorphism of graphs

$\phi\mathrel{\mathop{\mathchar 58\relax}}X(G_{2}\times
G_{1},S,\gamma)\rightarrow Y(G_{1},S_{Y},\alpha\circ p_{2})$

is, and we leave the details to the reader. ∎

##  4\. The analogue of Kida’s formula

###  4.1. Main result on Kida’s formula

In this section, we state and prove the main result.

######  Theorem 4.1.

Let $\ell$ be a rational prime, $X$ a connected graph satisfying Assumption
2.6, $S$ an orientation for $X$, and $\alpha\mathrel{\mathop{\mathchar
58\relax}}S\rightarrow\mathbb{Z}_{\ell}$ a function. Let
$p\mathrel{\mathop{\mathchar 58\relax}}Y\rightarrow X$ be a Galois cover for
which $G={\rm Gal}(Y/X)$ is an $\ell$-group. Consider the orientation $S_{Y}$
of $Y$, the induced function $\alpha\circ p\mathrel{\mathop{\mathchar
58\relax}}S_{Y}\rightarrow\mathbb{Z}_{\ell}$, and assume that $\alpha\circ p$
satisfies Assumption 2.9. By Remark 3.7, $\alpha\mathrel{\mathop{\mathchar
58\relax}}S\rightarrow\mathbb{Z}_{\ell}$ also satisfies Assumption 2.9, and we
have two $\mathbb{Z}_{\ell}$-towers that fit into the diagram

${Y}$${Y(\mathbb{Z}/\ell\mathbb{Z},S_{Y},\alpha\circ
p_{/1})}$${\ldots}$${Y(\mathbb{Z}/\ell^{n}\mathbb{Z},S_{Y},\alpha\circ
p_{/n})}$${\ldots}$${X}$${X(\mathbb{Z}/\ell\mathbb{Z},S,\alpha_{/1})}$${\ldots}$${X(\mathbb{Z}/\ell^{n}\mathbb{Z},S,\alpha_{/n})}$${\ldots}$$\scriptstyle{p}$

Then, $\mu(X,\alpha)=0$ if and only if $\mu(Y,\alpha\circ p)=0$, in which case

(4.1) $\lambda(Y,\alpha\circ p)+1=[Y\mathrel{\mathop{\mathchar
58\relax}}X]\cdot(\lambda(X,\alpha)+1).$

######  Remark 4.2.

Note that in the above formula, the degree "$[Y\mathrel{\mathop{\mathchar
58\relax}}X]$" plays a role, while in (1.5), the analogue is the degree
"$[E_{\infty}\mathrel{\mathop{\mathchar 58\relax}}F_{\infty}]$". If one
specializes (1.5) to the case when $E$ and $F_{\infty}$ are linearly disjoint
over $F$, then of course, $[E_{\infty}\mathrel{\mathop{\mathchar
58\relax}}F_{\infty}]=[E\mathrel{\mathop{\mathchar 58\relax}}F]$. The
Assumption 2.9 on the pullback tower over $Y$, may be viewed as a sort of
"disjointedness" assumption. For instance, if $p\mathrel{\mathop{\mathchar
58\relax}}Y\rightarrow X$ is the map $X_{n}\rightarrow X$, then, the tower
over $X_{n}$ obtained by pullback fails to be connected.

###### Proof.

Every $\ell$-group $G$ has a non-trivial center $Z(G)$, and by Cauchy’s
theorem, there exists a subgroup $H\leq Z(G)$ such that $|H|=\ell$. Since
$H\subseteq Z(G)$, $H$ is a normal subgroup of $G$. By the Galois
correspondence, there exists an intermediate cover $(Z,q_{1},q_{2})$
corresponding to the subgroup $H$ fitting into a commutative diagram

${Y}$${Z}$${X}$$\scriptstyle{p}$$\scriptstyle{q_{1}}$$\scriptstyle{q_{2}}$

and the normality of $H$ implies that both $q_{1}$ and $q_{2}$ are Galois
covers. The cover $Y/Z$ has Galois group $H$, and $Z/X$ is again a Galois
cover whose Galois group $G/H$ is an $\ell$-group. Note that $\alpha\circ
q_{2}\mathrel{\mathop{\mathchar 58\relax}}S_{Z}\rightarrow\mathbb{Z}_{\ell}$
satisfies Assumption 2.9 by Remark 3.7. Repeating the argument above, we can
construct a finite tower of covers of connected graphs

$X=Z_{0}\leftarrow Z_{1}\leftarrow\ldots\leftarrow Z_{t}=Y,$

for which $Z_{i}/Z_{i-1}$ is Galois with Galois group isomorphic to
$\mathbb{Z}/\ell\mathbb{Z}$ ($i=1,\ldots,t$). Furthermore,
$[Y\mathrel{\mathop{\mathchar 58\relax}}X]=\ell^{t}=|{\rm Gal}(Y/X)|$.
Therefore, it suffices to show the theorem in the situation where $Y/X$ is a
cyclic cover of order $\ell$, and we assume so from now on.

As explained in [DV22, Section 3], there exists a function
$\beta\mathrel{\mathop{\mathchar 58\relax}}S\rightarrow G$ for which the cover
$X(G,S,\beta)/X$ is isomorphic to the cover $Y/X$, so we can replace $Y$ by
$X(G,S,\beta)$. From now on, let $\Gamma_{n}={\rm
Gal}(Y(\mathbb{Z}/\ell^{n}\mathbb{Z},S_{Y},\alpha\circ
p_{/n})/Y)\simeq\mathbb{Z}/\ell^{n}\mathbb{Z}$. Note that by Proposition 3.5,
the cover $Y(\mathbb{Z}/\ell^{n}\mathbb{Z},S_{Y},\alpha\circ p_{/n})/X$ is an
abelian cover with Galois group isomorphic to the cartesian product
$G\times\Gamma_{n}$, and by Proposition 3.8, we can replace the graphs
$Y(\mathbb{Z}/\ell^{n}\mathbb{Z},S_{Y},\alpha\circ p_{/n})$ by the graphs
$X(G\times\Gamma_{n},S,\gamma_{/n})$, where $\gamma\mathrel{\mathop{\mathchar
58\relax}}S\rightarrow G\times\mathbb{Z}_{\ell}$ is given by
$\gamma(s)=(\beta(s),\alpha(s))$, and $\gamma_{/n}$ represents the reduction
modulo $\ell^{n}$ of $\gamma$ in the second component.

Let $\mathfrak{o}$ be the ring of integers of the local field
$K=\mathbb{Q}_{\ell}(\mu_{\ell})$. If $\psi\in\widehat{G}$, then we define

$A_{\psi,\rho}=\left(\sum_{\begin{subarray}{c}s\in S\\\ {\rm
inc}(s)=(v_{i},v_{j})\end{subarray}}\psi(\beta(s))\rho(\alpha(s))+\sum_{\begin{subarray}{c}s\in
S\\\ {\rm
inc}(s)=(v_{j},v_{i})\end{subarray}}\psi(-\beta(s))\rho(-\alpha(s))\right)\in
M_{g\times g}(\mathfrak{o}\llbracket T\rrbracket),$

and set

$f_{\psi}(T)=\operatorname{det}(D-A_{\psi,\rho})\in\mathfrak{o}\llbracket
T\rrbracket,$

where $D$ is as usual the degree matrix of $X$. Note that

$f_{\psi_{0}}(T)=f_{X,\alpha}(T)\in\mathbb{Z}_{\ell}\llbracket T\rrbracket.$

The characters of $G\times\Gamma_{n}$ are of the form $\psi\otimes\phi$, where
$\psi\in\widehat{G}$ and $\phi\in\widehat{\Gamma}_{n}$. Now, view $\phi$ as a
character on $1\times\Gamma_{n}\leq G\times\Gamma_{n}$. We claim that

(4.2) ${\rm
Ind}_{1\times\Gamma_{n}}^{G\times\Gamma_{n}}(\phi)\simeq\bigoplus_{\psi\in\widehat{G}}\psi\otimes\phi.$

To prove the claim, let $W$ be the $\mathbb{C}[\Gamma_{n}]$-module
corresponding to the character $\phi$. Then

$\displaystyle\mathbb{C}[G\times\Gamma_{n}]\otimes_{\mathbb{C}[\Gamma_{n}]}W$
$\displaystyle\simeq\mathbb{C}[G]\otimes_{\mathbb{C}}\mathbb{C}[\Gamma_{n}]\otimes_{\mathbb{C}[\Gamma_{n}]}W$
$\displaystyle\simeq\mathbb{C}[G]\otimes_{\mathbb{C}}W$
$\displaystyle\simeq\Big{(}\bigoplus_{\psi\in\widehat{G}}\mathbb{C}e_{\psi}\Big{)}\otimes_{\mathbb{C}}W$
$\displaystyle\simeq\bigoplus_{\psi\in\widehat{G}}\mathbb{C}e_{\psi}\otimes_{\mathbb{C}}W,$

and this shows the claim (4.2) above. Using the Artin formalism satisfied by
the Artin-Ihara $L$-functions [ST00, Proposition 3] and (1.7), we get

(4.3) $h_{Y}(u,\phi)=\prod_{\psi\in\widehat{G}}h_{Y/X}(u,\psi\otimes\phi),$

for all characters $\phi\in\widehat{\Gamma}_{n}$ and for all $n\geq 0$.

Combining (2.5), (2.15), and (2.16) give

(4.4) $f_{\psi}(t_{\phi})=h_{X}(1,\psi\otimes\phi),$

for all characters $\phi\in\widehat{\Gamma}_{n}$, for all
$\psi\in\widehat{G}$, and for all $n\geq 0$. Combining further with (4.3), we
obtain

$f_{Y,\alpha\circ p}(t_{\phi})=\prod_{\psi\in\widehat{G}}f_{\psi}(t_{\phi})$

for all characters $\phi\in\widehat{\Gamma}_{n}$ and for all $n\geq 0$. Since
a power series in $\mathfrak{o}\llbracket T\rrbracket$ has finitely many zeros
in the unit disk $D=\\{t\in\mathbb{C}_{\ell}\,\mathrel{\mathop{\mathchar
58\relax}}\,|t|_{\ell}<1\\}$, the equalities (4.4) above imply

(4.5) $f_{Y,\alpha\circ p}(T)=\prod_{\psi\in\widehat{G}}f_{\psi}(T).$

Let ${\rm ord}_{\ell}$ be the valuation on $\mathbb{C}_{\ell}$ normalized so
that ${\rm ord}_{\ell}(\ell)=1$. One defines the Iwasawa invariant of a non-
zero power series

$Q(T)=a_{0}+a_{1}T+a_{2}T^{2}+\ldots\in\mathfrak{o}\llbracket T\rrbracket$

similarly as before, namely

$\mu(Q(T))={\rm min}\\{{\rm ord}_{\ell}(a_{i})\,|\,i\geq 0\\},$

and

$\lambda(Q(T))={\rm min}\\{i\geq 0\,|\,{\rm ord}_{\ell}(a_{i})=\mu(Q(T))\\}.$

Note that $\mu(Q(T))$ is a nonnegative rational number and $\lambda(Q(T))$ is
an integer. If $Q_{1}(T),Q_{2}(T)\in\mathfrak{o}\llbracket T\rrbracket$, then

(4.6) $\mu(Q_{1}\cdot Q_{2})=\mu(Q_{1})+\mu(Q_{2})\text{ and
}\lambda(Q_{1}\cdot Q_{2})=\lambda(Q_{1})+\lambda(Q_{2}).$

If $\psi$ is a non-trivial character of $G$, and $\mathcal{L}$ is the unique
maximal ideal of $\mathfrak{o}$, then

(4.7) $A_{\psi,\rho}\equiv A_{\rho}\pmod{M_{g\times g}(\mathcal{L}\llbracket
T\rrbracket)}.$

Let us denote the reduction map $\mathfrak{o}\llbracket
T\rrbracket\longrightarrow\mathbb{F}_{\ell}\llbracket T\rrbracket$ by
$f(T)\mapsto\bar{f}(T)$. Equation (4.7) implies

$\bar{A}_{\psi,\rho}=\bar{A}_{\rho}\text{ in }M_{g\times
g}(\mathbb{F}_{\ell}\llbracket T\rrbracket),$

and it follows that

(4.8) $\bar{f}_{\psi}(T)=\bar{f}_{X,\alpha}(T)\text{ in
}\mathbb{F}_{\ell}\llbracket T\rrbracket,$

for all non-trivial characters $\psi$ of $G$. We deduce that if
$\mu(f_{X,\alpha}(T))=0$, then $\mu(f_{\psi}(T))=0$ for all non-trivial
characters $\psi$ of $G$. Combined with (4.5) and (4.6) above, one gets that
if $\mu(f_{X,\alpha}(T))=0$, then $\mu(f_{Y,\alpha\circ p}(T))=0$. Since the
$\mu$ invariant is nonnegative, the converse is clear. Under the assumption
that $\mu(X,\alpha)=0$, it follows from (4.8) that

$\lambda(f_{\psi}(T))=\lambda(f_{X,\alpha}(T)),$

for all non-trivial characters $\psi$ of $G$. Therefore, (4.5) and (4.6) again
imply the desired result.

∎

###  4.2. Examples

######  Example 4.3.

Consider the bouquet with three loops $X=B_{3}$, and pick an orientation
$S=\\{s_{1},s_{2},s_{3}\\}$. Let $\ell=3$, and consider
$\alpha\mathrel{\mathop{\mathchar
58\relax}}S\rightarrow\mathbb{Z}\subseteq\mathbb{Z}_{3}$ given by

$s_{1}\mapsto 1,s_{2}\mapsto 4,\text{ and }s_{3}\mapsto 20.$

This is Example $3$ on page $451$ of [Val21]. It was shown in _loc. cit._ that

${\rm ord}_{3}(\kappa_{n})=5n-2,$

for all $n\geq 1$. Thus,

$\mu(B_{3},\alpha)=0\text{ and }\lambda(B_{3},\alpha)=5.$

The $\mathbb{Z}_{3}$-tower above $X=B_{3}$ is the following:

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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{}{}{}}
{}{}{}\pgfsys@moveto{-26.31056pt}{11.34909pt}\pgfsys@lineto{12.86015pt}{25.6062pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{}{}{}}
{}{}{}\pgfsys@moveto{-28.21907pt}{4.97563pt}\pgfsys@lineto{6.60825pt}{27.88153pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{}{}{}}
{}{}{}\pgfsys@moveto{-28.60558pt}{-1.66583pt}\pgfsys@lineto{0.0pt}{28.65411pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{}{}{}}
{}{}{}\pgfsys@moveto{-15.74585pt}{-23.94035pt}\pgfsys@lineto{-22.98415pt}{17.11087pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{}{}{}}
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{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{-3.32642pt}{-28.46042pt}\pgfsys@lineto{-28.21907pt}{4.97563pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{15.74585pt}{-23.94035pt}\pgfsys@lineto{-24.81525pt}{-14.32706pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{}{}{}}
{}{}{}\pgfsys@moveto{20.84216pt}{-19.6634pt}\pgfsys@lineto{-20.84216pt}{-19.6634pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{}{}{}}
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{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{27.45042pt}{-8.21811pt}\pgfsys@lineto{-9.80043pt}{-26.92618pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{}{}{}}
{}{}{}\pgfsys@moveto{28.60558pt}{-1.66626pt}\pgfsys@lineto{-3.32642pt}{-28.46042pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{}{}{}}
{}{}{}\pgfsys@moveto{28.21907pt}{4.97563pt}\pgfsys@lineto{3.32642pt}{-28.46042pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{}{}{}}
{}{}{}\pgfsys@moveto{26.31056pt}{11.34909pt}\pgfsys@lineto{9.80043pt}{-26.92618pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{}{}{}}
{}{}{}\pgfsys@moveto{22.98415pt}{17.11087pt}\pgfsys@lineto{15.74585pt}{-23.94035pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{}{}{}}
{}{}{}\pgfsys@moveto{18.41862pt}{21.9501pt}\pgfsys@lineto{20.84216pt}{-19.6634pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{}{}{}}
{}{}{}\pgfsys@moveto{12.86015pt}{25.6062pt}\pgfsys@lineto{24.81482pt}{-14.32706pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}{}{}}{}{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{}{}{}}
{}{}{}\pgfsys@moveto{6.60825pt}{27.88153pt}\pgfsys@lineto{27.45042pt}{-8.21811pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \par\par
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\,\longleftarrow\ldots$

Now let $\beta\mathrel{\mathop{\mathchar
58\relax}}S\rightarrow\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$ be
defined via

$s_{1},s_{3}\mapsto(\bar{1},\bar{0})\text{ and
}s_{2}\mapsto(\bar{0},\bar{1}).$

Since $\beta(S)$ generates
$\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$, the graph
$Y=X(\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z},S,\beta)$ is connected
and thus the cover $p\mathrel{\mathop{\mathchar 58\relax}}Y\rightarrow X$ is
Galois with Galois group isomorphic to
$\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$:

$\leavevmode\hbox to36.98pt{\vbox
to22.7pt{\pgfpicture\makeatletter\hbox{\hskip 18.48918pt\lower 41.77914pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{}{}{}{{}}{}{{}{}}{
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}{}{}}{}{{}}{}{{{}}
{}{}{}{}{}{}{}{}
}\pgfsys@moveto{0.0pt}{42.47914pt}\pgfsys@moveto{0.7pt}{42.47914pt}\pgfsys@curveto{0.7pt}{42.86574pt}{0.3866pt}{43.17914pt}{0.0pt}{43.17914pt}\pgfsys@curveto{-0.3866pt}{43.17914pt}{-0.7pt}{42.86574pt}{-0.7pt}{42.47914pt}\pgfsys@curveto{-0.7pt}{42.09254pt}{-0.3866pt}{41.77914pt}{0.0pt}{41.77914pt}\pgfsys@curveto{0.3866pt}{41.77914pt}{0.7pt}{42.09254pt}{0.7pt}{42.47914pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{42.47914pt}\pgfsys@fill\pgfsys@invoke{
} {{}{}{}}{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{42.47914pt}\pgfsys@curveto{-18.28918pt}{64.275pt}{18.28918pt}{64.275pt}{0.0pt}{42.47914pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}{}}{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{42.47914pt}\pgfsys@curveto{-14.63135pt}{59.91583pt}{14.63135pt}{59.91583pt}{0.0pt}{42.47914pt}\pgfsys@stroke\pgfsys@invoke{
} {{}{}{}}{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{42.47914pt}\pgfsys@curveto{-9.14459pt}{53.37706pt}{9.14459pt}{53.37706pt}{0.0pt}{42.47914pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\stackrel{{\scriptstyle
p}}{{\longleftarrow}}\,\,\,\leavevmode\hbox to57.86pt{\vbox
to58.09pt{\pgfpicture\makeatletter\hbox{\hskip
28.93036pt\lower-28.72684pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{}{}{}{{}}{}{{}{}}{{}{}}{}{{}{}}{}{{}{}}{}{{}{}}{}{{}{}}{}{{}{}}{}{{}{}}{}{{}{}}{}{
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\par{{}{}{}}{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}\pgfsys@moveto{0.0pt}{28.66559pt}\pgfsys@moveto{0.7pt}{28.66559pt}\pgfsys@curveto{0.7pt}{29.05219pt}{0.3866pt}{29.36559pt}{0.0pt}{29.36559pt}\pgfsys@curveto{-0.3866pt}{29.36559pt}{-0.7pt}{29.05219pt}{-0.7pt}{28.66559pt}\pgfsys@curveto{-0.7pt}{28.27899pt}{-0.3866pt}{27.96559pt}{0.0pt}{27.96559pt}\pgfsys@curveto{0.3866pt}{27.96559pt}{0.7pt}{28.27899pt}{0.7pt}{28.66559pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.66559pt}\pgfsys@fill\pgfsys@invoke{
} {{}{}{}}{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}\pgfsys@moveto{-18.426pt}{21.9589pt}\pgfsys@moveto{-17.726pt}{21.9589pt}\pgfsys@curveto{-17.726pt}{22.34549pt}{-18.0394pt}{22.65889pt}{-18.426pt}{22.65889pt}\pgfsys@curveto{-18.81259pt}{22.65889pt}{-19.12599pt}{22.34549pt}{-19.12599pt}{21.9589pt}\pgfsys@curveto{-19.12599pt}{21.5723pt}{-18.81259pt}{21.2589pt}{-18.426pt}{21.2589pt}\pgfsys@curveto{-18.0394pt}{21.2589pt}{-17.726pt}{21.5723pt}{-17.726pt}{21.9589pt}\pgfsys@closepath\pgfsys@moveto{-18.426pt}{21.9589pt}\pgfsys@fill\pgfsys@invoke{
} {{}{}{}}{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}\pgfsys@moveto{-28.23036pt}{4.97763pt}\pgfsys@moveto{-27.53036pt}{4.97763pt}\pgfsys@curveto{-27.53036pt}{5.36423pt}{-27.84377pt}{5.67763pt}{-28.23036pt}{5.67763pt}\pgfsys@curveto{-28.61696pt}{5.67763pt}{-28.93036pt}{5.36423pt}{-28.93036pt}{4.97763pt}\pgfsys@curveto{-28.93036pt}{4.59103pt}{-28.61696pt}{4.27763pt}{-28.23036pt}{4.27763pt}\pgfsys@curveto{-27.84377pt}{4.27763pt}{-27.53036pt}{4.59103pt}{-27.53036pt}{4.97763pt}\pgfsys@closepath\pgfsys@moveto{-28.23036pt}{4.97763pt}\pgfsys@fill\pgfsys@invoke{
} {{}{}{}}{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}\pgfsys@moveto{-24.8252pt}{-14.3328pt}\pgfsys@moveto{-24.1252pt}{-14.3328pt}\pgfsys@curveto{-24.1252pt}{-13.9462pt}{-24.4386pt}{-13.6328pt}{-24.8252pt}{-13.6328pt}\pgfsys@curveto{-25.21179pt}{-13.6328pt}{-25.52519pt}{-13.9462pt}{-25.52519pt}{-14.3328pt}\pgfsys@curveto{-25.52519pt}{-14.71939pt}{-25.21179pt}{-15.03279pt}{-24.8252pt}{-15.03279pt}\pgfsys@curveto{-24.4386pt}{-15.03279pt}{-24.1252pt}{-14.71939pt}{-24.1252pt}{-14.3328pt}\pgfsys@closepath\pgfsys@moveto{-24.8252pt}{-14.3328pt}\pgfsys@fill\pgfsys@invoke{
} {{}{}{}}{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}\pgfsys@moveto{-9.80437pt}{-26.93697pt}\pgfsys@moveto{-9.10437pt}{-26.93697pt}\pgfsys@curveto{-9.10437pt}{-26.55037pt}{-9.41777pt}{-26.23697pt}{-9.80437pt}{-26.23697pt}\pgfsys@curveto{-10.19096pt}{-26.23697pt}{-10.50436pt}{-26.55037pt}{-10.50436pt}{-26.93697pt}\pgfsys@curveto{-10.50436pt}{-27.32356pt}{-10.19096pt}{-27.63696pt}{-9.80437pt}{-27.63696pt}\pgfsys@curveto{-9.41777pt}{-27.63696pt}{-9.10437pt}{-27.32356pt}{-9.10437pt}{-26.93697pt}\pgfsys@closepath\pgfsys@moveto{-9.80437pt}{-26.93697pt}\pgfsys@fill\pgfsys@invoke{
} {{}{}{}}{}{{}}{}{{{}} {}{}{}{}{}{}{}{}