2.1 Basic properties of multigraphs

Fix a prime number $\ell $ . We introduce basic notions in the theory of multigraphs and Iwasawa theory of $\mathbb {Z}_\ell $ -towers. For further details, the reader may refer to [Reference Lei and Vallières16, Sec. 2] and [Reference McGown and Vallières20, Sec. 4]. A finite undirected graph X consists of a finite set $V_X$ of vertices $\{v_1, \dots , v_n\}$ , and a finite number of edges $E_X$ joining the vertices. For such a graph, there is at most one edge which joins $v_i$ and $v_j$ . Thus, in this context, we may view $E_X$ as a subset of $V_X\times V_X$ . More specifically, if there is an edge joining $v_i$ to $v_j$ , then $(v_i, v_j)$ belongs to $E_X$ . Since the graph X in question is undirected, we have that $(v_i, v_j)\in E_X$ if and only if $(v_j, v_i)\in E_X$ . An edge from $v_i$ to $v_i$ is referred to as a loop. The graph X is fully described by the pair $(V_X, E_X)$ . On the other hand, a directed graph is a graph for which there is at most one directed edge from $v_i$ to $v_j$ . In this case, the set of edges is denoted $E_X^+$ , and is a subset of $V_X\times V_X$ . More specifically, there is a directed edge joining $v_i$ to $v_j$ precisely when $(v_i, v_j)\in E_X^+$ . A multigraph is a graph with possibly multiple directed edges joining any two vertices, subject to some further conditions. In this case, the set $E_X^+$ is not a subset of $V_X\times V_X$ , but simply admits a map to $V_X\times V_X$ . The notion of a multigraph is more general than that of a graph. We give the precise definition below.

Definition 2.1. A finite multigraph X is the data consisting of a finite set of vertices $V_X$ and a finite set of directed edges $E_{X}^+$ along with functions:

$$ \begin{align*}& i:E_{X}^{+} \to V_{X} \times V_{X},\\ &\iota:E_{X}^{+} \to E_{X}^{+}. \end{align*} $$

Here, i and $\iota $ are the incidence and inversion functions, respectively, and are subject to the following conditions:

1. 1. $\iota ^{2} = \mathrm {id}_{E_{X}^{+}}$ ,

2. 2. $\iota (e) \neq e$ for all $e \in E_{X}^{+}$ ,

3. 3. $i(\iota (e)) = \tau (i(e))$ for all $e \in E_{X}^{+}$ ,

where $\tau (x,y) := (y,x)$ .

Given a directed edge $e\in E_X^+$ , the incidence function maps e to $i(e)=(v,v')$ . The edge starts at v and ends at the vertex $v'$ . We define functions $o,t: E_X^+\rightarrow V_X$ , such that $i(e)=(o(e), t(e))$ . In other words, $o(e)$ (resp. $t(e)$ ) is the projection of $i(e)$ to the first (resp. second) coordinate. We refer to the function o (resp. t) as the source (resp. target) function. The inversion map $\iota $ satisfies the property that $i(\iota (e))=(t(e), o(e))$ . We shall occasionally denote the inverse $\iota (e)$ by $\bar {e}$ . These two conventions will be used interchangeably throughout.

The set of directed edges e starting at a vertex v and ending at a vertex $v'$ are the edges which satisfy the relation $i(e)=(v,v')$ . We note that there may be any number of directed edges joining two vertices v and $v'$ , thus the graph X is referred to as a multigraph. We allow for there to be edges to start and end at the same vertex v, and such edges are referred to as loops based at v. We note that for distinct vertices v and $v'$ , there are as many directed edges starting at v and ending at $v'$ as there are from $v'$ to v. Moreover, given a vertex v, there are an even number of directed loops starting and ending at v.

With respect to notation above, $i(a)=(v_1, v_1)$ , $i(b)=i(c)=(v_1, v_2)$ .

2.2 Iwasawa theory of multigraphs

In this section, we discuss the Iwasawa theory of multigraphs. The basic object of study is a $\mathbb {Z}_\ell $ -tower of multigraphs. Let X be a multigraph. Recall from the previous section that the data associated with X consists of a finite set of vertices $V_X=\{v_1, \dots , v_n\}$ and finitely many directed edges $E_X^+$ , along with an incidence map $i:E_X^+\rightarrow V_X\times V_X$ , and an inversion map $\iota : E_X^+\rightarrow E_X^+$ . These maps are subject to further conditions. Before introducing $\mathbb {Z}_\ell $ -towers of multigraphs, we first discuss the notion of a Galois cover of multigraphs. For $v\in V_X$ , let

$$\begin{align*}E_{X, v}:=\{e\in E_X^+\mid o(e)=v\}.\end{align*}$$

Let X and Y be two multigraphs. A morphism of graphs $f:Y \rightarrow X$ consists of two functions $f_{V}:V_{Y} \rightarrow V_{X}$ and $f_{E}:E_{Y}^+ \rightarrow E_{X}^+$ 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 $e \in E_{X}^+$ . We will often write f for both $f_{V}$ and $f_{E}$ .

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 $[Y:X]$ . If $f:Y \rightarrow X$ is a cover, then we let

$$ \begin{align*}\mathrm{Aut}_{f}(Y/X) = \{ \sigma \in \mathrm{Aut}(Y) \, : \, f \circ \sigma = f\}.\end{align*} $$

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

Galois covers of a multigraph can be constructed from certain combinatorial data associated with a multigraph, as we now explain. Define an equivalence relation on $E_X^+$ , by setting $e\sim e'$ if $e=e'$ or $e=\bar {e}'$ . The set of equivalence classes $E_X:=E_X^+/\iota $ is the set of undirected edges, and let $\pi :E_X^+\to E_{X}$ be the $2:1$ natural quotient map, mapping a directed edge to the associated undirected edge. Choose a section $\gamma :E_{X} \to E_{X}^+$ to $\pi $ , and set $S:=\gamma (E_{X})$ . Our main reference for voltage assignments and derived multigraphs is [Reference Gross and Tucker9], but we rephrase everything in the formalism used in [Reference Sunada24].

We extend $\alpha $ to all of $E_{X}^{+}$ by setting $\alpha (\bar {s}) = \alpha (s)^{-1}$ . To this data, one can associate a multigraph $X(G,S,\alpha )$ as follows. The set of vertices is $V = V_{X} \times G$ and the set of directed edges is $E^{+} = E_{X}^{+} \times G$ . The directed edge $(e,\sigma )$ connects $(o(e),\sigma )$ to $(t(e),\sigma \cdot \alpha (e))$ (where we recall that $o(e)$ and $t(e)$ denote the origin and the target of the edge e, respectively), and the inversion map is given by

$$ \begin{align*}\overline{(e,\sigma)} = (\bar{e},\sigma \cdot \alpha(e)).\end{align*} $$

If $G_{1}$ is another finite abelian group and $f:G \to G_{1}$ is a group morphism, then one obtains a morphism of multigraphs $f_{*}:X(G,S,\alpha ) \to X(G_{1},S,f \circ \alpha )$ that is defined via

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

If both $X(G,S,\alpha )$ and $X(G,S,f \circ \alpha )$ are connected and f is surjective, then $f_{*}$ is a cover in the sense of Definition 2.2 and in fact is a Galois cover with group of covering transformations isomorphic to $\mathrm {ker}(f)$ . In particular, if $f:G \to \{ 1\}$ is the group morphism into the trivial group and both X and $X(G,S,\alpha )$ are connected, then we get a Galois cover $f_{*}:X(G,S,\alpha ) \to X$ with group of covering transformation isomorphic to G.

We now recall the notion of a $\mathbb {Z}_\ell $ -tower of multigraphs.

Definition 2.5. 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

$$ \begin{align*}X = X_{0} \leftarrow X_{1} \leftarrow \cdots \leftarrow X_{n} \leftarrow \ldots,\end{align*} $$

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

We specialize the above construction to the case when $G=\mathbb {Z}_\ell $ . Set $\alpha _{/n}$ to denote the mod- $\ell ^n$ reduction of $\alpha $ . We obtain a $\mathbb {Z}_{\ell }$ -tower above X:

$$ \begin{align*}X \leftarrow X(\mathbb{Z}/\ell\mathbb{Z},S,\alpha_{/1}) \leftarrow X(\mathbb{Z}/\ell^{2}\mathbb{Z},S,\alpha_{/2}) \leftarrow \cdots \leftarrow X(\mathbb{Z}/\ell^{k}\mathbb{Z},S,\alpha_{/k}) \leftarrow \ldots.\end{align*} $$

Set $t:=|S|$ and enumerate $S=\{s_1,\dots , s_t\}$ , and let $\alpha _i$ denote $\alpha (s_i)$ . We may view $\alpha $ as a vector $\alpha =(\alpha _1, \alpha _2,\dots , \alpha _t)\in \mathbb {Z}_\ell ^t$ . The following assumption on the multigraphs $X(\mathbb {Z}/\ell ^n \mathbb {Z}, S, \alpha _{/n})$ is necessary to relate the growth patterns in Picard groups to Iwasawa invariants. In §2.3, we introduce conditions under which the following assumption is satisfied.

1. 1. Let X be a bouquet with two loops, and choose an orientation $S =\{s_{1}, s_{2} \}$ of $E_{X}^{+}$ . Let $\ell = 3$ , and consider the function $\alpha :E_{X}^{+} \rightarrow \mathbb {Z} \subseteq \mathbb {Z}_{3}$ given by

   $$ \begin{align*}s_{1} \mapsto \alpha(s_{1}) = 1 \text{ and } s_{2} \mapsto \alpha(s_{2}) = 11.\end{align*} $$
   By Corollary 2.16, all the graphs $X(\mathbb {Z}/3^{n}\mathbb {Z},S,\alpha _{/n})$ are connected. We obtain a $\mathbb {Z}_{3}$ -tower above the bouquet graph X which can be pictured as follows:
   

2. 2. Let $\ell =3$ again, and let now X be the dumbbell multigraph

   

   the section be

   

   and take the function $\alpha :S \rightarrow \mathbb {Z} \subseteq \mathbb {Z}_{3}$ given by

   $$ \begin{align*}s_{1} \mapsto \alpha(s_{1}) = 1, s_{2} \mapsto \alpha(s_{2}) =0, \text{ and } s_{3} \mapsto \alpha(s_{3}) = 11.\end{align*} $$
   By Theorem 2.15, all the multigraphs $X(\mathbb {Z}/3^{n}\mathbb {Z},S,\alpha _{/n})$ are connected. We obtain a $\mathbb {Z}_{3}$ -tower above the dumbbell multigraph X which can be pictured as follows:
   

   Note that the graphs $X(\mathbb {Z}/3^{n}\mathbb {Z},S,\alpha _{/n})$ are precisely the generalized Petersen graphs $G(3^{n},11)$ .

Let $u_X$ be the number of vertices of X and label $V_X=\{v_1,\dots , v_{u_X}\}$ . Let $o,t: E_X^+\rightarrow V_X$ be the origin and terminus functions taking an edge to its source and target vertex, respectively. Thus, we have that $i(e)=(o(e), t(e))$ . For $v\in V_X$ , set $E_v:=\{e\in E_X^+\mid o(e)=v\}$ , and let $\operatorname {val}_X(v):=\# E_v$ be the valency of v. The divisor group $\operatorname {Div}(X)$ is the free abelian group on the vertices $V_X$ of X. It consists of formal sums of the form $D=\sum _{v\in V_X} n_v v$ , where $n_v\in \mathbb {Z}$ for all v. The degree of D is the sum $\operatorname {deg}(D):=\sum _v n_v\in \mathbb {Z}$ , and this defines a homomorphism

$$\begin{align*}\operatorname{deg}:\operatorname{Div}(X)\rightarrow \mathbb{Z},\end{align*}$$

the kernel of which is denoted $\operatorname {Div}^0(X)$ . Let $\mathcal {M}(X)$ be the abelian group of $\mathbb {Z}$ -valued functions on $V_X$ . The discrete Laplacian map $\mathrm {div}:\mathcal {M}(X) \to \mathrm {Div}(X)$ is defined as follows. For $f \in \mathcal {M}(X)$ , one sets

$$ \begin{align*}\mathrm{div}(f) = -\sum_{v}m_{v}(f) \cdot v,\end{align*} $$

where

$$ \begin{align*}{m_{v}(f)} = \sum_{e \in E_{X,v}}\left(f(t(e)) - f(o(e))\right).\end{align*} $$

We let $\operatorname {Pr}(X)$ be the image of $\operatorname {div}$ . It is easy to see that $\operatorname {Pr}(X)$ is a subgroup of $\operatorname {Div}^0(X)$ , since the subjacent directed graph is Eulerian, meaning that every vertex has in-degree equals to its out-degree. The Picard group is taken to be the quotient $\operatorname {Pic}(X):=\operatorname {Div}(X)/\operatorname {Pr}(X)$ , and we set $\operatorname {Pic}^0(X):=\operatorname {Div}^0(X)/\operatorname {Pr}(X)$ . Set $\kappa _X$ to be the cardinality of $\operatorname {Pic}^0(X)$ , which is a finite group and is equal to the number of spanning trees in X.

The valency matrix is the $u_X\times u_X$ diagonal matrix $D=(d_{i,j})$ , with

$$\begin{align*}d_{i,j}=\begin{cases} \operatorname{val}_X(v_i), & \text{ for }i=j,\\ 0, & \text{ for }i\neq j. \end{cases}\end{align*}$$

The adjacency matrix $A=(a_{i,j})$ is given by

$$\begin{align*}a_{i,j}=\begin{cases} \text{Twice the number of undirected loops at }i,\text{ } & i=j,\\ \text{The number of undirected edges connecting the }i\text{th and }j\text{th vertices}, & i\neq j. \end{cases}\end{align*}$$

The matrix $Q:=D-A$ is called the Laplacian matrix.

We let $\mathbb {Z}_\ell [x]$ (resp. ) denote the polynomial ring (resp. formal power series ring) in the indeterminate x (resp. T) with coefficients in $\mathbb {Z}_\ell $ . Denote by $\mathbb {Z}_\ell [x;\mathbb {Z}_\ell ]$ the ring whose elements are expressions of the form $f(x)=\sum _{a} c_a x^a$ , where a runs over a finite set in $\mathbb {Z}_\ell $ and the coefficient $c_a$ is in $\mathbb {Z}_\ell $ . Here, multiplication is defined by setting $x^ax^b=x^{a+b}$ for $a,b\in \mathbb {Z}_\ell $ . Associate a matrix $M(x)$ to the pair $(X,\alpha )$ as follows:

(2.1) $$ \begin{align}M(x)=M_{X,\alpha}(x):=D - \left(\sum_{\substack{s \in S \\ {i}(s) = (v_{i},v_{j})}} x^{\alpha(s)} + \sum_{\substack{s \in S \\ { i}(s) = (v_{j},v_{i})}} x^{-\alpha(s)} \right) \in M_{u \times u}(\mathbb{Z}[x;\mathbb{Z}_{\ell}]).\end{align} $$

For $b\in \mathbb {Z}_\ell $ , let ${b\choose n}\in \mathbb {Z}_\ell $ be given by

$$\begin{align*}{b\choose n}:=\frac{b(b-1)\dots (b-n+1)}{n!}.\end{align*}$$

Let $(1+T)^b$ be the formal power series

$$\begin{align*}(1+T)^b:=\sum_{n=0}^\infty {b\choose n} T^n.\end{align*}$$

With this convention in place, we define the characteristic series to be

(2.2) $$ \begin{align} f(T)=f_{X,\alpha}(T)=\operatorname{det}M(1+T),\end{align} $$

which is a formal power series in . The convention used in this article differs from that used in [Reference McGown and Vallières20], where $-T$ is used in place of T. This convention simplifies our calculations.

The following result shows that $f_{X, \alpha }(T)\neq 0$ under the assumptions imposed on $\alpha $ . The proof is essentially contained in [Reference McGown and Vallières19], [Reference McGown and Vallières20], and [26], we recall these details. First, we introduce some further notation. Let $Y/X$ be an abelian cover of multigraphs and set $G:=\operatorname {Gal}(Y/X)$ . Let $\widehat {G}$ be the group of characters $\psi : G\rightarrow \mathbb {C}^\times $ . Given $\psi \in \widehat {G}$ , denote the Artin–Ihara L-function by

$$\begin{align*}L_X(z, \psi):=\prod_{\mathfrak{c}}\left(1-\psi\left(\left(\frac{Y/X}{\frak{c}}\right)\right)z^{l(\mathfrak{c})}\right)^{-1}.\end{align*}$$

Here, the product runs over all primes $\mathfrak {c}$ of X, $l(\mathfrak {c})$ is the length of $\mathfrak {c}$ and $\left (\frac {Y/X}{\frak {c}}\right )$ is the Frobenius automorphism at $\mathfrak {c}$ . (Recall that a prime in a multigraph is an equivalence class of closed paths that have no backtrack or tail, and that are not multiples of a strictly smaller closed path, where two such paths are identified if they are the same up to their starting vertex. For further details, the reader is referred to [Reference Terras25, Ch. 18].) Let $\chi (X)$ denote the Euler characteristic of X, which is defined as follows:

$$\begin{align*}\chi(X):=|V_X|-|E_X|.\end{align*}$$

We shall impose the following assumption without further mention.

That $\chi (X)\neq 0$ means that X does not consist of a single cycle, or a “cycle with hair,” compare [Reference Terras25, Fig. 2.1]. When $\chi (X)=0$ , X is referred to as a “bad graph,” compare [Reference Terras25] (and the references therein), since many arithmetic properties of zeta functions are not satisfied in this case. For instance, the analog of the prime number theorem for such bad graphs does not hold, compare [Reference Terras25, Th. 10.1].

Proof. The relationship between the power series $Q(T)$ of [Reference McGown and Vallières20, Th. 6.1] and $f_{X,\alpha }(T)$ is $Q(T) = f_{X,\alpha }(-T)$ . If $Y/X$ is an abelian cover of multigraphs and $\psi $ is a character of $G = \mathrm {Gal}(Y/X)$ , then the three-term determinant formula [Reference Terras25, Th. 18.15] for the Artin–Ihara L-function gives

$$ \begin{align*}L_{X}(z,\psi)^{-1} = (1-z^{2})^{-\chi(X)} \cdot \mathrm{det}(I - A_{\psi}z + (D-I)z^{2}),\end{align*} $$

where $A_{\psi }$ is the twisted adjacency matrix of Definition 2.9, D is the valency matrix of X, and $\chi (X)$ is the Euler characteristic of X. From now on, we let

$$ \begin{align*}h_{X}(z,\psi) = \mathrm{det}(I - A_{\psi}z + (D-I)z^{2}) \in \mathbb{Z}[\psi][z].\end{align*} $$

Coming back to the abelian $\ell $ -tower

$$ \begin{align*}X \leftarrow X(\mathbb{Z}/\ell\mathbb{Z},S,\alpha_{/1}) \leftarrow X(\mathbb{Z}/\ell^{2}\mathbb{Z},S,\alpha_{/2}) \leftarrow \cdots \leftarrow X(\mathbb{Z}/\ell^{n}\mathbb{Z},S,\alpha_{/n}) \leftarrow \ldots\end{align*} $$

associated with $\alpha $ , for all positive integer n, [Reference McGown and Vallières20, Cor. 5.6] implies

(2.3) $$ \begin{align} Q(1- \zeta_{\ell^{n}}) = h_{X}(1,\psi_{n}), \end{align} $$

where $\psi _{n}$ is the character of $\mathbb {Z}/\ell ^{n}\mathbb {Z} \simeq \mathrm {Gal}(X(\mathbb {Z}/\ell ^{n}\mathbb {Z},S,\alpha _{/n})/X)$ satisfying $\psi _{n}(\bar {1}) = \zeta _{\ell ^{n}}$ . Here, it is understood that $\zeta _{\ell ^{n}} = \exp (2 \pi i/\ell ^{n})$ and that an embedding of $\overline {\mathbb {Q}}$ into $\overline {\mathbb {Q}}_{\ell }$ has been fixed once and for all so that we view $\zeta _{\ell ^{n}}$ and the values of the characters $\psi _{n}$ inside $\overline {\mathbb {Q}}_{\ell }$ . From [Reference Vallières26, Sec. 3], one has

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

where the product is over all nontrivial characters of $\mathbb {Z}/\ell ^{n}\mathbb {Z}$ , and $\kappa _n$ denotes the number of spanning trees of $X(\mathbb {Z}/\ell ^{n}\mathbb {Z},S,\alpha _{/n})$ . The above formula crucially requires the Assumption 2.8. It follows that

$$ \begin{align*}h_{X}(1,\psi) \neq 0\end{align*} $$

for all nontrivial characters, and thus $Q(T) \neq 0$ by (2.3).

Associated with the characteristic series, are the Iwasawa $\mu $ and $\lambda $ invariants, which we now proceed to define. A polynomial is said to be a distinguished polynomial if it is monic and all non-leading coefficients are divisible by $\ell $ .

Definition 2.11. By the Weierstrass Preparation theorem, we may factor the characteristic polynomial as follows: $f(T)=\ell ^{\mu } g(T) u(T)$ , where $g(T)$ is a distinguished polynomial in , and $u(T)$ is a unit in . Since $u(T)$ is a unit in , the constant term $u(0)$ is a unit in $\mathbb {Z}_\ell $ . According to Lemma 2.7, the constant term of $f(T)$ is equal to $0$ , and hence, T divides $g(T)$ . In particular, we have that $\operatorname {deg} g\geq 1$ . The number $\mu \in \mathbb {Z}_{\geq 0}$ is the $\mu $ -invariant and depends on $(\ell ,\alpha )$ . Throughout, it is assumed that Assumption 2.6 is satisfied for $\alpha $ . Thus, we denote this invariant by $\mu _{\ell }(X,\alpha )$ , however, when $\ell $ , X and $\alpha :S\rightarrow \mathbb {Z}_\ell $ are understood, we simply denote it by $\mu $ . On the other hand, the $\lambda $ -invariant is defined to be the quantity

$$\begin{align*}\lambda=\lambda_{\ell}(X,\alpha):=\operatorname{deg} g(T)-1.\end{align*}$$

Let $\alpha :S\rightarrow \mathbb {Z}_\ell $ be a function such that Assumption 2.6 is satisfied. Given $n\geq 0$ , let $X_n=X(\mathbb {Z}/\ell ^n\mathbb {Z}, S, \alpha _{/n})$ and write $\kappa _n$ for the number of spanning trees of $X_n$ . Let $|\cdot |_\ell :\mathbb {Q}_\ell \rightarrow \mathbb {R}_{\geq 0}$ be the absolute value normalized by setting $|\ell |_\ell ^{-1}=\ell $ . The $\ell $ -part of $\kappa _n$ is defined as $\kappa _\ell (X_n):=|\kappa _n|_\ell ^{-1}$ . The following result is [Reference McGown and Vallières20, Th. 6.1].

Theorem 2.13 (McGown–Vallières).

Let $\ell $ be a prime and $\alpha $ be as above. Then, there exist $n_0>0$ and $\nu \in \mathbb {Z}$ , such that

(2.4) $$ \begin{align}\kappa_\ell(X_n)=\ell^{(\mu \ell^n+\lambda n+\nu)},\end{align} $$

for all $n\geq n_0$ . The invariants $\mu $ and $\lambda $ are given as in Definition 2.11.

2.3 Connectedness of multigraphs in abelian $\ell $ -towers

Let X be a connected multigraph, let $\ell $ be a prime number, and let $\alpha :S\rightarrow \mathbb {Z}_\ell $ . In this section, we introduce explicit conditions on X which are sufficient for Assumption 2.6 to hold. Let

$$ \begin{align*}X=X_0\leftarrow X_1\leftarrow X_2\leftarrow \dots \leftarrow X_n \leftarrow \dots\end{align*} $$

be an abelian $\ell $ -tower over X. Set also $G_{n} = \mathrm {Gal}(X_{n}/X)$ . We have natural group morphisms

$$ \begin{align*}\gamma_{n}:H_{1}(X,\mathbb{Z}) \to G_{n}\end{align*} $$

that are compatible so that we get a group morphism

$$ \begin{align*}\gamma:H_{1}(X,\mathbb{Z}) \to \varprojlim_n G_{n} \simeq \mathbb{Z}_{\ell}.\end{align*} $$

Fix now a spanning tree of X, say $\mathfrak {T}\subseteq E_X$ , a section $\gamma :E_{X} \to E_{X}^+$ , and set $S:=\gamma (E_{X})$ . We let $S(\mathfrak {T})$ be the collection of $s \in S$ such that $\pi (s) \in \mathfrak {T}$ . For each $s \in S$ , let $c_{s} = d_{s} \cdot s$ be the closed path in X, where $d_{s}$ is the unique geodesic path in $\mathfrak {T}$ going from $t(s)$ to $o(s)$ . Let $\langle c_{s} \rangle $ be the corresponding cycle in $H_{1}(X,\mathbb {Z})$ . Note that if $s \in S(\mathfrak {T})$ , then $\langle c_{s} \rangle = 0$ . The collection

$$ \begin{align*}\{ \langle c_{s} \rangle \, | \, s \in S \smallsetminus S(\mathfrak{T}) \}\end{align*} $$

forms a basis of $H_{1}(X,\mathbb {Z})$ (cf. [Reference Sunada24, Sec. 4.3]), so that, in particular,

$$ \begin{align*}\mathrm{rank}_{\mathbb{Z}}(H_{1}(X,\mathbb{Z})) = |S \smallsetminus S(\mathfrak{T})| = |E_{X}| - |V_{X}| + 1.\end{align*} $$

Define now $\alpha :S \to \mathbb {Z}_{\ell }$ via $s \mapsto \alpha (s) = \gamma (\langle c_{s} \rangle )$ . By successively lifting, the tree $\mathfrak {T}$ to $X_{1}$ , then to $X_{2}$ and so on, one can construct isomorphisms of multigraphs

$$ \begin{align*}\phi_{n}:X(\mathbb{Z}/\ell^{n},S,\alpha_{/n}) \xrightarrow{\sim} X_{n}\end{align*} $$

for which all the squares and the triangle in the diagram

commute. In other words, once a spanning tree $\mathfrak {T}$ for X has been fixed (which also specifies a $\mathbb {Z}$ -basis for $H_{1}(X,\mathbb {Z})$ ), every abelian $\ell $ -tower over X arises from a function $\alpha :S \to \mathbb {Z}_{\ell }$ satisfying $\alpha (s) = 0$ for all $s \in S(\mathfrak {T})$ .

Proof. Let us assume first that there exists $s \in S \smallsetminus S(\mathfrak {T})$ such that $(\alpha (s),\ell )=1$ . Let also $X_{n} = X(\mathbb {Z}/\ell ^{n}\mathbb {Z},S,\alpha _{/n})$ . Let $v\in V_X$ , $e\in E_X^+$ and $\bar {a}\in \mathbb {Z}/\ell ^n\mathbb {Z}$ . The map $p_{n}:X_{n} \to X$ defined via

$$ \begin{align*}p_{n}\left((v,\bar{a})\right) = v \text{ and } p_{n}\left((e,\bar{a})\right) = e\end{align*} $$

is a morphism of multigraphs (recall that the vertices and (directed) edges of $X_n$ consist of elements in $V_X\times \mathbb {Z}/\ell ^n\mathbb {Z}$ and $E_X^+\times \mathbb {Z}/\ell ^n\mathbb {Z}$ , respectively). Furthermore, for all $w \in V_{X_{n}}$ , the restriction $p_{n}|_{E_{X_{n},w}}$ induces a bijection

$$ \begin{align*}p_{n}|_{E_{X_{n}},w}: E_{X_{n},w} \xrightarrow{\sim} E_{X,p_{n}(w)}.\end{align*} $$

It follows that every path c in X has a unique lift $\tilde {c}$ to $X_{n}$ once the starting point in the fiber of $o(c)$ is fixed. Furthermore, since $p_{n}$ is a morphism of multigraphs one has

(2.5) $$ \begin{align} p_{n}(o(\tilde{c})) = o(c) \text{ and } p_{n}(t(\tilde{c})) = t(c). \end{align} $$

Now, $\mathbb {Z}/\ell ^{n}\mathbb {Z}$ acts on $X_{n}$ via

$$ \begin{align*}\bar{a} \cdot (v,\bar{b}) = (v, \bar{a} + \bar{b}) \text{ and } \bar{a} \cdot (e,\bar{b}) = (e,\bar{a} + \bar{b}),\end{align*} $$

which gives an injective group morphism

$$ \begin{align*}\mathbb{Z}/\ell^{n}\mathbb{Z} \hookrightarrow \mathrm{Aut}_{p_{n}}(X_{n}/X).\end{align*} $$

Let $v_{1},v_{2} \in V_{X}$ be arbitrary. Since X is connected, there is a path c in X satisfying $o(c)=v_{1}$ and $t(c) = v_{2}$ . Thus, by lifting c and using (2.5), for any vertex $w \in p_{n}^{-1}(v_{1})$ , there exists a path $\tilde {c}$ in $X_{n}$ going from w to another vertex in $p_{n}^{-1}(v_{2})$ . It follows that in order to show that $X_{n}$ is connected, it suffices to show that every vertex in a single fiber can be joined via a path in $X_{n}$ . Under our assumption, there exists $s_{0} \in S \smallsetminus S(\mathfrak {T})$ such that $(\alpha (s_{0}),\ell )=1$ , and thus, $\alpha _{/n}(s_{0})$ is a generator for $\mathbb {Z}/\ell ^{n}\mathbb {Z}$ . Let $v \in V_{X}$ be such that $v = t(s_{0}),$ and let $\sigma _{0} = \alpha _{/n}(s_{0}) \in \mathbb {Z}/\ell ^{n}\mathbb {Z}$ . The closed path $c_{s_{0}} = d_{s_{0}}\cdot s_{0}$ can be lifted to a path $\tilde {c}_{s_{0}}$ in $X_{n}$ starting at $(v,\bar {0})$ . Furthermore, since $\alpha (s) = 0$ for all $s \in S(\mathfrak {T})$ , we have $t(\tilde {c}_{s_{0}}) = (v,\sigma _{0})$ . The path $\tilde {c}_{s_{0}} \cdot \sigma _{0}(\tilde {c}_{s_{0}})$ will start at $(v,\bar {0})$ and end at $(v,2\cdot \sigma _{0})$ . Keeping going like this, we can find a path in $X_{n}$ going from $(v,\bar {0})$ to $(v,m\cdot \sigma _{0})$ for all positive integer m. Since $\sigma _{0}$ generates $\mathbb {Z}/\ell ^{n}\mathbb {Z}$ , this shows that every vertex in the fiber of v can be joined via a path in $X_{n}$ . It follows that $X_{n}$ is connected.

Conversely, assume that all multigraphs $X_{n}$ are connected. In particular, $X_{1}$ is connected. To simplify the notation, write $S \smallsetminus S(\mathfrak {T}) = \{s_{1},\ldots ,s_{t} \}$ . The function $\alpha $ induces a function on $C_{1}(X,\mathbb {Z})$ and thus also on $H_{1}(X,\mathbb {Z})$ by restriction. Let also $v \in V_{X}$ be a fixed vertex. Since we are assuming that $X_{1}$ is connected, there exists a path $\tilde {c}$ in $X_{1}$ going from $(v,\bar {0})$ to $(v,\bar {1})$ . The path $c = p_{1}(\tilde {c})$ is then a closed path in X, and $\alpha (\langle c \rangle ) = \bar {1}$ . Since the $\langle c_{s_{i}} \rangle $ for $i=1,\ldots ,t$ form a $\mathbb {Z}$ -basis of $H_{1}(X,\mathbb {Z})$ , we have

$$ \begin{align*}\langle c \rangle = n_{1} \langle c_{s_{1}} \rangle + \cdots + n_{t} \langle c_{s_{t}}\rangle,\end{align*} $$

for some $n_{i} \in \mathbb {Z}$ . Applying $\alpha _{/1}$ to this last equation gives

$$ \begin{align*}\bar{1} = n_{1} \alpha_{/1}(s_{1}) + \cdots + n_{t}\alpha_{/1}(s_{t}).\end{align*} $$

Now, if all the $\alpha (s_{i})$ were divisible by $\ell $ , then we would get a contradiction.

The following corollary is most likely well known, we list it here for convenience.

2.4 Basic properties of Iwasawa invariants

In this section, we prove a number of basic results about the Iwasawa $\mu $ and $\lambda $ invariants associated with multigraphs. The graph X is called a bouquet if there is one vertex, in which case $M(x)$ is a $1\times 1$ -matrix. Suppose that there are t loops in total, labeled $e_1,\dots , e_t$ and $\alpha =(\alpha _1,\dots , \alpha _t)$ sends $e_i\mapsto \alpha _i$ . We denote the corresponding bouquet by $X_t$ . We shall set $\mu :=\mu _\ell (X_t;\alpha )$ and $\lambda :=\lambda _\ell (X_t;\alpha )$ . Given $a\in \mathbb {Z}_\ell $ , let

$$\begin{align*}f_a(T)=2-(1+T)^a-(1+T)^{-a}=-\sum_{i\geq 2}\left({a\choose i}+{-a\choose i}\right) T^i.\end{align*}$$

We have that $f(T)=\sum _{j=1}^t f_{\alpha _j}(T)$ , and therefore, we may write

$$\begin{align*}f(T)=\sum_{n\geq 2} \beta_{n} T^n\end{align*}$$

with $\beta _2=-\left (\sum _{j} \alpha _j^2\right )$ . In particular, we find that $\operatorname {ord}_{T=0} f(T)\geq 2$ , and thus $\lambda \geq 1$ . It is easy to see that $\mu =0$ and $\lambda =1$ if and only if $\beta _2$ is a unit in $\mathbb {Z}_\ell $ . These are the minimal values of the Iwasawa invariants attainable by a bouquet. The following result shows that the Iwasawa invariants $\mu $ and $\lambda $ can be arbitrarily large.

Lemma 2.17. Let $n_1\in \mathbb {Z}_{\geq 0}$ , $n_2\in \mathbb {Z}_{\geq 1}$ , and let $\ell $ be a prime number. There is a bouquet $X_t$ consisting of $t = \ell ^{n_{1}+1} + \ell ^{n_{1}}$ loops and a choice of vector $\alpha \in \left (\mathbb {Z}_{\ell }^t\backslash (\ell \mathbb {Z}_{\ell })^t\right )$ (which depends on both $n_1$ and $n_2$ ) such that the associated Iwasawa invariants satisfy

$$\begin{align*}\mu_{\ell}(X_t, \alpha)= n_1, \quad\lambda_{\ell}(X_t, \alpha)= 2\ell^{n_2}-1.\end{align*}$$

Proof. Set $t=\ell ^{n_1+1}+\ell ^{n_1}$ and let $\alpha =(\alpha _1, \dots , \alpha _t)$ with

$$\begin{align*}\alpha_i=\begin{cases} 1,&\text{ if }i\leq \ell^{n_1+1},\\ \ell^{n_2}, &\text{ if }i> \ell^{n_1+1}.\\ \end{cases}\end{align*}$$

We have that

$$\begin{align*}f_1(T)=2-(1+T)-(1+T)^{-1}=-(1+T)^{-1}T^2,\end{align*}$$

and that

$$\begin{align*}f_{\ell^{n_2}}(T)=2-(1+T)^{\ell^{n_2}}-(1+T)^{-\ell^{n_2}}=-(1+T)^{-\ell^{n_2}}\left(1-(1+T)^{\ell^{n_2}}\right)^2.\end{align*}$$

As a result, we find that

$$ \begin{align*}f(T)=&\ \ell^{n_1}\left(\ell f_1(T)+f_{\ell ^{n_2}}(T)\right),\\ = & -\ell^{n_1} (1+T)^{-\ell ^{n_2}}\left(\ell (1+T)^{\ell ^{n_2-1}}T^2 +\left(1-(1+T)^{\ell ^{n_2}}\right)^2\right).\end{align*} $$

The degree of $\left (1-(1+T)^{\ell ^{n_2}}\right )^2$ is $2 \ell ^{n_2}$ , which is larger than the degree of $(1+T)^{\ell ^{n_2-1}}T^2$ . Furthermore, all non-leading coefficients of $\left (1-(1+T)^{\ell ^{n_2}}\right )^2$ are divisible by $\ell $ . We have shown that $P(T):=\ell (1+T)^{\ell ^{n_2-1}}T^2 +\left (1-(1+T)^{\ell ^{n_2}}\right )^2$ is a distinguished polynomial. Thus, $f(T)=\ell ^{n_1}u(T)P(T)$ , where $u(T)=-(1+T)^{-\ell ^{n_2}}$ is a unit in the Iwasawa algebra and $P(T)$ is a distinguished polynomial of degree $2\ell ^{n_2}$ . Thus, we have shown that $\mu =n_1$ and $\lambda =2\ell ^{n_2}-1$ .

Proof. Set $\dot {T}:=(1+T)^{-1}-1$ and note that

(2.6) $$ \begin{align}(1+\dot{T})^{a}+(1+\dot{T})^{-a}=(1+T)^{-a}+(1+T)^a\end{align} $$

for all $a\in \mathbb {Z}_\ell $ . Note that $M(x)^t=M(x^{-1})$ and it follows from (2.6) that

$$\begin{align*}f(\dot{T})=\operatorname{det}M(1+\dot{T})=\operatorname{det}M\left((1+T)^{-1}\right)=\operatorname{det}M(1+T)=f(T).\end{align*}$$

Write $f(T)=\ell ^\mu P(T) u(T)$ , where $P(T)=\sum _{j=0}^{\lambda } \xi _j T^j+ T^{\lambda +1}$ and $\xi _j$ are all divisible by $\ell $ . Note that

$$\begin{align*}(1+T)^{\lambda+1} P(\dot{T})=\sum_{j=0}^{\lambda} (-1)^{j}\xi_j (1+T)^{\lambda+1-j}T^j+ (-1)^{\lambda+1} T^{\lambda+1}.\end{align*}$$

We see that $P(-1)=\sum _{j=0}^{\lambda } \xi _j (-1)^j+(-1)^{\lambda +1}\equiv (-1)^{\lambda +1} \bmod {\ell }$ , and hence, $P(-1)$ is a unit in $\mathbb {Z}_\ell $ . The above relation shows that $P(-1)^{-1}(1+T)^{\lambda +1}P(\dot {T})$ is the distinguished polynomial dividing $f(\dot {T})=f(T)$ , thus

$$\begin{align*}P(T)=P(-1)^{-1}(1+T)^{\lambda+1}P(\dot{T})\end{align*}$$

and

$$\begin{align*}u(T)=P(-1)(1+T)^{-\lambda-1} u(\dot{T}).\end{align*}$$

The latter relation specialized to $T=0$ gives $P(-1)=1$ . Thus, we have shown that $1\equiv (-1)^{\lambda +1} \bmod {\ell }$ , and since $\ell $ is assumed to be odd, it follows that $\lambda $ is odd.

3.1 Systems of equations

We begin with a generalization of the Chevalley–Warning theorem to a system of equations in many variables. Associate to $I=(i_1, \dots , i_n)\in \mathbb {Z}_{\geq 0}^n$ , the monomial $x_I:=x_1^{i_1}x_2^{i_2}\dots x_n^{i_n}$ , then define the degree to be the sum

$$\begin{align*}\operatorname{deg}\left(x_I\right):=i_1+i_2+\dots +i_n.\end{align*}$$

Given $f(x_1, \dots , x_n)=\sum _I c_I x_I$ , the degree of f is defined as follows:

$$\begin{align*}\operatorname{deg} f:=\operatorname{max}\left\{\operatorname{deg}(x_I)\mid c_I\neq 0\right\}.\end{align*}$$

The polynomial is homogenous of degree d if $\operatorname {deg}(x_I)=d$ for all I such that $c_I\neq 0$ . The following is often referred to as Warning’s second theorem, compare [Reference Warning27].

Theorem 3.1. Let $m, n\in \mathbb {Z}_{\geq 1}$ and $f_1,\dots , f_m\in \mathbb {F}_q[x_1, \dots , x_n]$ . Assume that

$$\begin{align*}d:=\sum_{i=1}^m \operatorname{deg}(f_i)<n,\end{align*}$$

and that for all i, we have that $f_i(0,0,\dots , 0)=0$ . Let N be the number of solutions of $f_1=f_2=\dots =f_m=0$ . Then, N is divisible by $\ell $ and $N\geq q^{n-d}$ .

3.2 Quadratic forms and their solutions

A quadratic form is a homogenous polynomial of degree $2$ . Assume throughout that q is odd, let $\eta :\mathbb {F}_q^\times \rightarrow \{\pm 1\}$ be the quadratic character of $\mathbb {F}_q$ , and extend $\eta $ to a function on $\mathbb {F}_q$ by setting $\eta (0)=0$ . Given a quadratic form $Q(x_1,\dots , x_n)$ over $\mathbb {F}_q$ , express Q as

$$\begin{align*}Q(x_1,\dots, x_n)=\sum_{i,j} a_{i,j} x_i x_j,\end{align*}$$

with $a_{i,j}=a_{j,i}$ . The rank of Q is the rank of the associated matrix $A=(a_{i,j})$ over $\mathbb {F}_q$ . We let $\operatorname {det} Q$ denote the determinant of A. The form Q is nondegenerate if $\operatorname {rank}A=n$ , that is, A is non-singular. On the other hand, Q is diagonal if $a_{i,j}=0$ unless $i=j$ .

Let $v:\mathbb {F}_q\rightarrow \mathbb {Z}$ be the function defined by

$$\begin{align*}v(b):=\begin{cases} -1, &\text{ if }b\neq 0, \\ (q-1),&\text{ if }b=0. \end{cases}\end{align*}$$

Theorem 3.3. Let $Q(x_1, \dots , x_n)$ be a nonzero quadratic form over $\mathbb {F}_q$ , and assume that q is odd and n is even. For $b\in \mathbb {F}_q$ , let $N\left (Q(x_1,\dots , x_n)=b\right )$ be the number of solutions to $Q(x_1,\dots , x_n)=b$ . Then, we have that

$$\begin{align*}N\left(Q(x_1,\dots, x_n)=b\right)= q^{n-1}+ v(b)q^{\frac{n-2}{2}}\eta\left((-1)^{n/2} \Delta\right),\end{align*}$$

where $\eta $ is the quadratic character of $\mathbb {F}_q$ and $\Delta =\operatorname {det} Q$ .

Given a degenerate quadratic form $Q(x_1, \dots , x_n)$ , we let $\mathfrak {r}$ denote the rank of Q. According to Proposition 3.2, Q is equivalent to the form $\sum _{i=1}^{\mathfrak {r}} a_i x_i^2$ , where $a_i$ are all nonzero. We set $Q_0(x_1,\dots , x_{\mathfrak {r}})\in \mathbb {F}_q[x_1, \dots , x_{\mathfrak {r}}]$ to be the nondegenerate form $Q_0(x_1, \dots , x_{\mathfrak {r}})=\sum _{i=1}^{\mathfrak {r}} a_i x_i^2$ , with discriminant $\Delta (Q_0)=\prod _{i=1}^{\mathfrak {r}} a_i\neq 0$ .

Proposition 3.4. Let $\ell $ be odd, and let $Q(x_1,\dots , x_n)\in \mathbb {F}_q[x_1,\dots , x_n]$ be a quadratic form. Setting $\mathfrak {r}:=\operatorname {rank} Q$ , we denote by $N(Q=0)$ the number of solutions of $Q=0$ . Then, we have that

$$\begin{align*}N(Q=0)=\begin{cases} q^{n-1}+(q-1)q^{n-\mathfrak{r}/2-1}\eta\left((-1)^{\mathfrak{r}/2}\Delta(Q_0)\right),& \text{ if }\mathfrak{r}\text{ is even,}\\ q^{n-1}, & \text{ if }\mathfrak{r}\text{ is odd.}\end{cases}\end{align*}$$

Proof. Let $\mathfrak {r}$ denote the rank of Q, and first consider the case when $\mathfrak {r}$ is even. According to Proposition 3.2, Q is equivalent to a diagonal form. Thus, without loss of generality, assume that there are nonzero constants $a_1, \dots , a_{\mathfrak {r}}$ such that

$$\begin{align*}Q(x_1, \dots, x_{\mathfrak{r}}, \dots, x_n)=\sum_{i=1}^{\mathfrak{r}} a_i x_i^2.\end{align*}$$

Thus, for every solution to

$$\begin{align*}Q_0(x_1, \dots, x_{\mathfrak{r}})=\sum_{i=1}^{\mathfrak{r}} a_i x_i^2=0,\end{align*}$$

there are $q^{n-\frak {r}}$ solutions to $Q(x_1, \dots , x_{\mathfrak {r}}, \dots , x_n)=0$ . As a consequence, we find that $N(Q=0)=q^{n-\mathfrak {r}} N(Q_0=0)$ . Since $Q_0$ is nondegenerate, the result follows from Theorem 3.3, according to which

$$\begin{align*}N(Q_0=0)=q^{\mathfrak{r}-1}+(q-1)q^{(\mathfrak{r}-2)/2}\eta\left((-1)^{\mathfrak{r}/2}\Delta(Q_0)\right).\end{align*}$$

On the other hand, if $\mathfrak {r}$ is odd, we have that $N(Q_0=0)=q^{\mathfrak {r}-1}$ , according to [Reference Lidl and Niederreiter17, Th. 6.27].

4.1 The vanishing of the $\mu $ -invariant

In this section, we study when the $\mu $ -invariant of a $\mathbb {Z}_\ell $ -tower of a bouquet $X_t$ vanishes. We are interested in the power series

(4.1) $$ \begin{align} f(T)=f_{X_t,\alpha}(T)=\sum_{k=1}^tf_{\alpha_k}(T)\in\mathbb{Z}_\ell[\![ T]\!], \end{align} $$

where we recall that

$$\begin{align*}f_{a}(T)=2-(1+T)^{a}-(1+T)^{-a}=-\sum_{n\ge2}\left(\binom{a}{n}+\binom{-a}{n}\right)T^n. \end{align*}$$

Upon expanding the binomial coefficients, we deduce that the coefficient of $T^n$ in $f_a(T)$ is of the form

$$\begin{align*}\frac{2}{n!}\sum_{i=1}^{\lfloor n/2\rfloor}c_{n,i}a^{2i}, \end{align*}$$

where $c_{n,i}\in \mathbb {Z}$ is independent of a. More precisely, $c_{n,i}$ is the coefficient of $x^{2i}$ in $-x(x-1) \dots (x-n+1)$ . Furthermore, if $n=2m$ is even, then the highest term in a is $a^{2m}$ with coefficient $c_{2m, m}=- 1$ , whereas if $n=2m+1$ is odd, then the highest term is $a^{2m}$ with coefficient $c_{2m+1,m}=m(2m+1)$ . We write $f(T)=\sum _{n\geq 2} \beta _{n} T^{n}$ , in particular, we have

$$\begin{align*}\beta_n=\sum_{k=1}^t\frac{2}{n!}\sum_{i=1}^{\lfloor n/2\rfloor}c_{n,i}\alpha_k^{2i}. \end{align*}$$

Lemma 4.1. Let $\ell $ be an odd prime number, and let $f(T)$ be the power series associated with the $\mathbb {Z}_{\ell }$ -tower of the bouquet $X_t$ given by the admissible vector $\alpha =(\alpha _1, \dots , \alpha _t)$ . Suppose that the associated $\mu $ -invariant $\mu _\ell (X_t, \alpha )$ is positive. Then, for all $m\geq 1$ , we have that

$$\begin{align*}\sum_{k} \alpha_k^{2m}\equiv 0 \bmod{\ell^{\mu}}.\end{align*}$$

Proof. If $\mu>0$ , then, all coefficients $\beta _n$ are divisible by $\ell ^{\mu }$ . Setting $p_i:=\sum _{k} \alpha _k^{2i}$ , we find that

$$\begin{align*}\beta_n=\frac{2}{n!}\sum_{i=1}^{\lfloor n/2\rfloor} c_{n, i} p_i\equiv 0 \bmod{\ell^{\mu}}.\end{align*}$$

Then, we find that $\sum _{i=1}^{\lfloor n/2\rfloor } c_{n, i} p_i\equiv 0\bmod {\ell ^{\mu }}$ . Set $n=2m$ , and obtain from the above that

$$\begin{align*}p_m=-c_{2m, m}p_m\equiv \sum_{i=1}^{m-1} c_{2m, i} p_i\bmod{\ell^{\mu}}.\end{align*}$$

Therefore, by induction on m, we find that $p_m\equiv 0\bmod {\ell ^\mu }$ for all m.

Next, we prove a result on the $\mu $ -invariant. Note that since $\alpha $ is assumed to be admissible, we have that $\#\{k:\alpha _k\not \equiv 0\bmod \ell \}>0$ .

Proof. Let us write $p_i:=\sum _{k=1}^t\alpha _k^{2i}$ . By Lemma 4.1, we find that $p_i\equiv 0\bmod \ell ^{\mu }$ for all $i\ge 1$ .

Let us write $N=\#\{k:\alpha _k\not \equiv 0\bmod \ell \}$ . For $1\le i\le N$ , let $p_i'$ denote the power sum $\displaystyle \sum _{\ell \nmid \alpha _k}\alpha _k^{2i}$ and let $e_i$ denote the symmetric sum $\displaystyle \sum _{k_1<\cdots <k_i} \alpha _{k_1}^2\ldots \alpha _{k_i}^2$ of the same elements. In particular, $p_i'\equiv 0\bmod \ell $ for all i. Newton’s identities (see [Reference Mead21]) say that

$$\begin{align*}ie_i=\sum_{j=1}^i(-1)^{j-1}e_{i-j}p'_j,\quad 1\le i\le N, \end{align*}$$

where $e_0$ is defined to be $1$ . In particular, $Ne_N\equiv 0\bmod \ell $ . By definition,

$$ \begin{align*}e_N=\prod_{\ell\nmid \alpha_k}\alpha_k^2\not\equiv 0\bmod \ell.\end{align*} $$

Thus, we deduce that $\ell |N$ .

The result below improves upon (see [Reference Vallières26, Cor. 5.8]).

Proof. Note that since there exists some value $\alpha _k$ such that $\ell \nmid \alpha _k$ , we find that $\#\{k:\alpha _k\not \equiv 0\bmod \ell \}>0$ . According to Proposition 4.2,

$$\begin{align*}t\geq \#\{k:\alpha_k\not\equiv 0\bmod \ell\}\geq \ell,\end{align*}$$

and the result follows.

The following result is a refinement of Proposition 4.2.

Proof. Recall from the proof Proposition 4.2 that we have

$$\begin{align*}\sum_{k=1}^t \alpha_k^{2i}\equiv0\bmod \ell^{\mu},\quad i\ge 1. \end{align*}$$

This implies that

$$\begin{align*}\sum_{x\in Q_\ell} r_x \cdot x^{i}\equiv 0\bmod\ell^{\mu}, \quad i=1,\ldots, (\ell-1)/2. \end{align*}$$

The determinant of the Vandermonde matrix $(x^i)_{x\in Q_\ell ,1\le i\le (\ell -1)/2}$ is nonzero mod $\ell $ . Therefore, we deduce that $r_x\equiv 0\bmod \ell ^{\mu }$ for all x.

Remark 4.5. The converse to the statement in Proposition 4.4 is not true. For instance, let $\ell =t=3$ and $\alpha =(1,1,2)$ , we find that

$$ \begin{align*}f(T)=& 2\left(2-(1+T)-(1+T)^{-1}\right)+\left(2-(1+T)^2-(1+T)^{-2}\right)\\ \equiv & \frac{2T^4}{(T + 1)^2}\bmod{3}.\end{align*} $$

In particular, the $\mu $ -invariant is zero. This also gives a counterexample for the converse of both Lemma 4.1 and Proposition 4.2.

Theorem 4.6. Suppose that $\ell $ is odd, and let $m=(\ell -1)/2$ . We have

$$\begin{align*}\operatorname{Prob}_{\ell}(X_t;\mu>0)\le \sum_{i=1}^{\lfloor t/\ell\rfloor }2^{i\ell}\binom{t}{i\ell}\sum_{a_1+\cdots+a_m=i}\frac{(i\ell)!}{\prod_{i=1}^m(a_j\ell)!(\ell^t-1)}, \end{align*}$$

where the second sum runs over nonnegative integers $a_j\ge 0$ .

Proof. If $\mu>0$ , then Proposition 4.2 tells us that $\#\{k:\alpha _k\not \equiv 0\bmod \ell \}$ is a multiple of $\ell $ . Since each residue class modulo $\ell $ in $\alpha \in \mathbb {Z}_\ell ^t\setminus (\ell \mathbb {Z}_\ell )^t$ has the same measure, it is enough to count the number of elements in $(r_1,\ldots , r_t)\in \mathbb {F}_\ell ^t\setminus \{0\}$ such that $\#\{k:r_k\neq 0\}$ is a multiple of $\ell $ and $\sum _{k=1}^t r_k^{2i}= 0$ , for all $i\ge 1$ .

For each $1\le i\le \lfloor t/\ell \rfloor $ , we may choose $i\ell $ indices $k_1,\dots , k_{i\ell }$ so that $r_j\neq 0$ if and only if $j\in \{k_b\mid b=1,\dots , i\ell \}$ . By Proposition 4.4, amongst these $i\ell $ elements, we have to distribute $r_j^2$ over the m classes in $Q_\ell $ so that the number of times each quadratic residue occurs is a multiple of $\ell $ .

Let $a_1\ell ,\ldots ,a_{m}\ell $ be the number of times the residue classes in $Q_\ell $ occur. We have

$$\begin{align*}a_1+\cdots+a_{m}=i. \end{align*}$$

There are $(i\ell )!/\prod _{i=1}^m(a_j\ell )!$ ways to distribute $i\ell $ distinguishable elements in m distinguishable sets so that the number of elements in these sets are given by $a_1\ell ,\ldots ,a_m\ell $ . Once we have distributed the $i\ell $ elements amongst the m classes, each element has two possible values modulo $\ell $ , giving us $2^{i\ell }$ choices. Summing up all possible choices of $a_j$ ’s gives the expression on the right-hand side of the inequality.

Remark 4.7. Our numerical calculations suggest that for a given $\ell $ , the right-hand side of the inequality given by Theorem 4.6 tends to $\frac {1}{\ell ^{(\ell -1)/2}}$ as $t\rightarrow \infty $ .

In what follows, we prove a necessary and sufficient condition for $\mu>0$ in the special case where $\alpha _k\in \mathbb {Z}$ for all k. Given an integer $a\in \mathbb {Z}$ , let us write $d_k(a)$ for the coefficient of $X^k$ in the shifted Chebyshev polynomial $P_a(X)$ used in [Reference McGown and Vallières19, Sec. 2]. More explicitly,

$$\begin{align*}P_a(X)=2-2 T_a(1-X/2)=d_1(a)X+d_2(a)X^2+\cdots+d_a(a)X^a, \end{align*}$$

where the Chebyshev polynomials $T_a(X)\in \mathbb {Z}[X]$ of the first kind are defined by the relation

$$\begin{align*}T_a(\cos(\theta))=\cos(a\theta) \end{align*}$$

for all $\theta \in \mathbb {R}$ (see [Reference McGown and Vallières19, Sec. 2]).

Lemma 4.8. Let $m\ge 1$ be an integer. Let $b_{1},\ldots ,b_{m}$ be distinct positive integers, then

$$ \begin{align*}(d_{b_{i}}(b_{j})) \in \mathrm{GL}(m,\mathbb{Z}).\end{align*} $$

Proof. We have $P_{a}(X) = P_{-a}(X)$ for all $a \in \mathbb {Z}$ . Thus,

(4.2) $$ \begin{align} \sum_{s\in S}P_{\alpha(s)}(X) = \sum_{j=1}^{m}t_{j}P_{b_{j}}(X). \end{align} $$

It is clear that $\ell \, | \, t_{j}$ for all $j=1,\ldots ,m$ implies that the polynomial above belongs to $ \ell \mathbb {Z}[X]$ .

Conversely, if $\sum _{s\in S}P_{\alpha (s)}(X) \in \ell \mathbb {Z}[X]$ , then (4.2) implies that

(4.3) $$ \begin{align} \sum_{j=1}^{m}d_{k}(b_{j})t_{j} \equiv 0\quad \pmod{\ell} \end{align} $$

for all $k \ge 1$ . If we let b be the maximal value of the $b_{j}$ , then the matrix $(d_{k}(b_{j}) + \ell \mathbb {Z}) \in M_{b\times m}(\mathbb {F}_{\ell })$ has rank m over $\mathbb {F}_{\ell }$ by Lemma 4.8. Therefore, the system of linear equations (4.3) has a unique solution modulo $\ell $ . This tells us that $\ell \, | \, t_{j}$ for all $j=1,\ldots ,m$ and concludes the proof.

4.2 The distribution of Iwasawa invariants of a fixed $X_t$ varying $\alpha $

In this section, we fix an odd prime number $\ell $ and fix an integer $t\geq 2$ . We let $X_t$ be the bouquet with t undirected loops and we study question (1). Note that Assumption 2.8 requires that $\chi (X_{t})\neq 0$ , and hence, we avoid the case when $t=1$ (see §4.3 for a separate discussion on this case).

We begin by considering $\mathcal {S}(\mu _0, \lambda _0, \ell , t)$ . Throughout, it shall be assumed that $\alpha :S\rightarrow \mathbb {Z}_\ell $ is an admissible function. We compute the probability $\operatorname {Prob}(X_t;\mu =\mu _0,\lambda = \lambda _0)$ for various values of $(\mu _0, \lambda _0)\in \mathbb {Z}_{\geq 0}\times \mathbb {Z}_{\geq 0}$ . Note that according to Theorem 2.18, $\lambda $ is odd, and thus, the minimal values of $\mu $ and $\lambda $ are $0$ and $1$ , respectively.

Proof. First, assume that $\mu =0$ and $\lambda =2k-1$ . Setting $p_i:=\sum _{j} \alpha _j^{2i}$ , we find that

$$\begin{align*}\beta_n=\frac{2}{n!}\sum_{i=1}^{\lfloor n/2\rfloor} c_{n, i} p_i.\end{align*}$$

According to Lemma 4.11, $\beta _n$ is divisible by $\ell $ for all values of $n<2k$ , and $\beta _{2k}$ is not divisible by $\ell $ . The assertion of part (2) follows immediately from this.

We now prove (1). In particular, we assume that $2k<\ell $ . Then, $\frac {2}{n!}$ is an $\ell $ -adic unit for all integers n such that $0\le n\leq 2k$ .

Suppose that (b) holds. Then, $p_i$ is divisible by $\ell $ for all $i<k$ , whereas $p_k$ is not divisible by $\ell $ . It follows that $\ell $ divides $\beta _n$ for all $n<2k$ , but $\beta _{2k}$ is not divisible by $\ell $ . Thus, (a) holds by Lemma 4.11.

Conversely, suppose that (a) holds. Then

$$\begin{align*}\ell\ | \sum_{i=1}^{\lfloor n/2\rfloor}c_{n,i}p_i,\ n<2k,\quad \ell \nmid \sum_{i=1}^{k}c_{2k,i}p_i.\end{align*}$$

Recall that $c_{2m,m}=-1$ for all $m\ge 1$ . An inductive argument shows that $\ell |p_i$ for $i<k$ , and $\ell \nmid p_k$ . In particular, (b) holds.

Proof. By Corollary 4.3, $\ell>t$ implies that the $\mu $ -invariant of $f(T)$ is zero. Assume by way of contradiction that $\lambda \geq 2t$ . Then, $\beta _2,\ldots , \beta _{2t}\equiv 0\bmod \ell $ . Under the same notation as in the proof of Proposition 4.2, we have

$$\begin{align*}p_i\equiv 0\bmod \ell, \quad i=1,2,\ldots, t \end{align*}$$

forcing $Ne_N\equiv 0\bmod \ell $ once again by Newton’s identities (see [Reference Mead21]). But this is impossible if $\ell>t$ , proving the affirmation of the theorem.

Remark 4.14. Without the condition $\ell>t$ , it is possible that $\mu =0$ and $\lambda> 2t$ . In fact, applying Lemma 2.17 with $n_{1}=0$ shows that it is possible to find examples with $\mu =0$ and $\lambda $ arbitrarily large when $t = \ell +1$ . When $t=\ell =2$ , we have found examples with $\mu =0$ and

$$ \begin{align*}\lambda = 5,9,17,33,65,129,257.\end{align*} $$

Whereas when $t=3$ , for $\ell =2,$ we have found examples with $\mu =0$ and

$$ \begin{align*}\lambda=3,5,7,9,15,17,31,33,63,65,127,129,255,257,\end{align*} $$

and for $\ell = 3,$ we have found examples with $\mu =0$ and

$$ \begin{align*}\lambda = 3,5,9,17,27,53,81,161,243.\end{align*} $$

For instance, if we take $t=3$ , $\ell = 3$ , with the voltage assignment on the three edges given by 1, 8, and 10, respectively, then we get:

The power series $f(T)$ starts as follows:

$$ \begin{align*}f(T) = -165T^2 +165T^{3}-1,326T^{4} + \cdots +3,470,655T^{17} - 5,167,526T^{18}+ \ldots,\end{align*} $$

where $3$ divides all the coefficients up to and including $3,470,655$ , but it does not divide $5,167,526$ . Thus, $\mu = 0$ and $\lambda = 17$ . Furthermore, since

$$ \begin{align*}\log_{3}\left(\frac{3}{2} \cdot 18 \right) = 3,\end{align*} $$

[Reference McGown and Vallières19, Th. 4.1] implies that if $n \ge 3$ , we have

$$ \begin{align*}\mathrm{ord}_{3}(\kappa_{n}) = 17n + \nu\end{align*} $$

for some $\nu \in \mathbb {Z}$ . Using SageMath [Reference Stein23], we calculate

$$ \begin{align*}\kappa_{0} = 1, \kappa_{1} = 3^{3} , \kappa_{2} = 3^{10}, \kappa_{3} = 2^{18} \cdot 3^{27},\end{align*} $$

and

$$ \begin{align*}\kappa_{4} = 2^{18} \cdot 3^{44} \cdot 163^{2} \cdot 487^{2} \cdot 37,907^{2} \cdot 799,471^{2}.\end{align*} $$

Thus, we have

$$ \begin{align*}\mathrm{ord}_{3}(\kappa_{n}) = 17n - 24,\end{align*} $$

for all $n \ge 2$ .

We wish to estimate the probability $\operatorname {Prob}_\ell (X_t; \mu =0, \lambda =\lambda _0)$ . Note that by Theorem 2.18, $\operatorname {Prob}_\ell (X_t; \lambda =\lambda _0)=0$ whenever $\lambda _0$ is even. Therefore, we assume that $\lambda _0=2k-1$ is odd.

Theorem 4.16. Let $X_t$ be a bouquet and $k>1$ . Assume that $\ell $ is odd, $2k-1<\ell $ and $k(k-1)<t$ . Then, we have that

$$\begin{align*}\operatorname{Prob}_\ell(X_t; \mu=0, \lambda< 2k-1)\leq (1-\ell^{-t})^{-1}(1-\ell^{-k(k-1)}).\end{align*}$$

Proof. Note that since $\ell $ is assumed to be odd and $2k-1<\ell $ , it is automatically true that $2k-1<\ell -1$ . Let $\alpha :S\rightarrow \mathbb {Z}_\ell $ be an admissible function for which $\mu =0$ and $\lambda <2k-1$ . We set $\lambda _0:=2k-1$ , note that by assumption, $\lambda _0<\ell -1$ . Let $\bar {\alpha }=(\bar {\alpha }_1, \dots , \bar {\alpha }_t)$ be the mod- $\ell $ reduction of $\alpha $ . According to Lemma 4.12, $\bar {\alpha }$ is not a solution to all of the equations

$$\begin{align*}Q_i(x_1,\dots, x_t)=\sum_{j=1}^{t} x_j^{2i}= 0\end{align*}$$

for $i<k$ . Then, since $\lambda _0<\ell -1$ , it follows that $\beta _n$ is divisible by $\ell $ for $n\leq \lambda _0$ . Let N be the number of nonzero solutions $\bar {\alpha }$ such that $Q_i(\bar {\alpha })=0$ for all $i<k$ . We find that

$$\begin{align*}\operatorname{Prob}_\ell(X_t; \mu=0, \lambda< 2k-1)\leq 1-\frac{N}{\ell^{t}-1}.\end{align*}$$

The sum of degrees $\sum _{i=1}^{k-1} \operatorname {deg} (Q_i)=\sum _{i=1}^{k-1} 2i=k(k-1)$ . Since $k(k-1)<t$ by assumption, it follows from Theorem 3.1 that $N\geq \ell ^{t-k(k-1)}-1$ , and therefore,

$$\begin{align*}\operatorname{Prob}_\ell(X_t; \mu=0, \lambda< 2k-1)\leq 1-\frac{\ell^{t-k(k-1)}-1}{\ell^{t}-1}=(1-\ell^{-t})^{-1}(1-\ell^{-k(k-1)}).\end{align*}$$

Example 4.17. Let us now illustrate the above result through an example. Let $t=10$ and $k=3$ . We study the proportion of admissible vectors for which $\mu =0$ and $\lambda <5$ . Note that $k(k-1)=6<t=10$ . Thus, for all primes $\ell>2k-1=7$ , we have that

$$\begin{align*}\operatorname{Prob}_\ell(X_t; \mu=0, \lambda< 5)\leq (1-\ell^{-10})^{-1}(1-\ell^{-6}).\end{align*}$$

Next, we study the proportion of admissible vectors $\alpha =(\alpha _1,\dots , \alpha _t)$ such that $\mu =0$ and $\lambda =1$ . This case is of particular interest, since it is indeed the case when the Iwasawa invariants take on their minimal values.

Theorem 4.18. Let $\ell $ be an odd prime number and $t \in \mathbb {Z}_{\ge 2}$ . Then, we have that

$$\begin{align*}\operatorname{Prob}_{\ell}(X_t;\mu=0, \lambda=1)=\begin{cases} 1-\frac{\ell^{t-1}+(-1)^{\frac{t(\ell-1)}{4}}(\ell-1)\ell^{\frac{t}{2}-1}-1}{\ell^{t}-1},& \text{ if }t \text{ is even}, \\ 1-\frac{\ell^{t-1}-1}{\ell^t-1}, & \text{ if }t\text{ is odd.}\end{cases}\end{align*}$$

Proof. Consider the quadratic form $Q(x_1,\dots , x_t)=\sum _{i=1}^t x_i^2$ defined over $\mathbb {F}_\ell $ . Then $\mu =0$ and $\lambda =1$ if and only if $Q(\bar a_1,\ldots ,\bar a_t)\ne 0$ . Therefore,

$$\begin{align*}\operatorname{Prob}_{\ell}(X_t;\mu=0, \lambda=1)=1-\frac{N(Q=0)-1}{\ell^t-1}. \end{align*}$$

Therefore, the theorem is now a direct consequence of Proposition 3.4.

4.4 Vary t and $\alpha $

In this section, we study question (2) mentioned at the start of the section. In other words, we prove results about the variation of $\mu _{t,\alpha }=\mu _\ell (X_t,\alpha )$ and $\lambda _{t,\alpha }=\lambda _\ell (X_t,\alpha )$ when both t and $\alpha $ vary. Recall that the prime $\ell $ is fixed. We shall assume that $\ell $ is odd. Given $t\in \mathbb {Z}_{\geq 2}$ , define $\mathcal {T}(\mu ,\lambda ,\ell ,t)$ to be the set of integral vectors $\alpha = (\alpha _1,\ldots ,\alpha _t)\in \mathbb {Z}^t$ such that:

• • for some coordinate $\alpha _i$ , we have $\ell \nmid \alpha _i$ ,

• • $\mu _{t,\alpha }=\mu $ and $\lambda _{t,\alpha }=\lambda $ .

Set $\mathcal {T}_{\leq x}(\mu ,\lambda ,\ell ,t)$ to be the set of $\alpha \in \mathcal {T}(\mu ,\lambda ,\ell ,t)$ such that $\max \{|\alpha _1|,\ldots , |\alpha _t|\} \leq x$ . Let $\mathcal {A}_{\leq x}(\ell , t)$ be the set of all tuples $\alpha $ such that:

• • $\max \{|\alpha _1|,\ldots , |\alpha _t|\}\leq x$ ,

• • $\ell \nmid \alpha _i$ for some $1\leq i\leq t$ .

On the other hand, fix $\delta \in (0,1)$ and set

$$\begin{align*}\mathcal{T}_{\leq x}(\mu, \lambda, \ell):=\bigcup_{t\leq x^\delta} \mathcal{T}_{\leq x}(\mu, \lambda, \ell,t)\text{ and } \mathcal{A}_{\leq x}(\ell):=\bigcup_{t\leq x^{\delta}} \mathcal{A}_{\leq x}(\ell,t).\end{align*}$$

We now state the main result of this section.

Theorem 4.21. With respect to notation above,

$$\begin{align*}\lim_{x\rightarrow \infty}\frac{\# \mathcal{T}_{\leq x}(0, 1, \ell)}{\# \mathcal{A}_{\leq x}(\ell)}=1-\ell^{-1}. \end{align*}$$

Lemma 4.23. With respect to notation above,

$$\begin{align*}\left|\#\mathcal{A}_{\leq x}(\ell, t)-2^t\left(1-\ell^{-t}\right)x^t\right|< ctx^{t-1}\end{align*}$$

for some absolute constant $c>0$ which is independent of t and x.

Proof. The number of $\alpha $ such that $\operatorname {max}\left \{|\alpha _1|, |\alpha _2|, \dots , |\alpha _t|\right \}\leq x$ is $\left (2\lfloor x\rfloor +1\right )^t$ . The number of tuples such that such that $\operatorname {max}\left \{|\alpha _1|, |\alpha _2|, \dots , |\alpha _t|\right \}\leq x$ and $\ell $ divides all $\alpha _i$ is $\left (2\lfloor x/\ell \rfloor +1\right )^t$ . Therefore, we find that

$$\begin{align*}\#\mathcal{A}_{\leq x}(\ell, t)=\left(2\lfloor x\rfloor+1\right)^t-\left(2\lfloor x/\ell\rfloor+1\right)^t.\end{align*}$$

We have that

$$ \begin{align*} & \left|\#\mathcal{A}_{\leq x}(\ell, t)-2^t\left(1-\ell^{-t}\right)x^t\right|\\ = & \left| \left(2\lfloor x\rfloor+1\right)^t-\left(2\lfloor x/\ell\rfloor+1\right)^t-2^t\left(1-\ell^{-t}\right)x^t\right|\\ \leq & \left|(2\lfloor x\rfloor+1)^t-(2x)^t\right|+\left|(2\lfloor x/\ell\rfloor+1)^t-(2x/\ell)^t\right|\\ \leq & t\left|(2\lfloor x\rfloor+1)\right|{}^{t-1}+t\left|(2\lfloor x/\ell\rfloor+1)\right|{}^{t-1}<ct x^{t-1}\\ \end{align*} $$

for an absolute constant $c>0$ . In the above inequalities, we have used the mean value theorem to obtain the bound

$$\begin{align*}\left|(2\lfloor x\rfloor+1)^t-(2x)^t\right|\leq t\left|(2\lfloor x\rfloor+1)\right|{}^{t-1}\end{align*}$$

and similarly with x replaced by $x/\ell $ .