2.1 Basic definitions in graph theory

A graph $X$ consists of a vertex set $V(X)$ and a set of directed edges $\mathbb {E}(X)$ along with an incidence function

$$ \begin{align*} \mathrm{inc} \colon \mathbb{E}(X) &\longrightarrow V(X) \times V(X)\\ e &\mapsto (o(e), t(e)) \end{align*} $$

and an inversion function

$$ \begin{align*} \mathrm{inv} \colon \mathbb{E}(X) &\longrightarrow \mathbb{E}(X)\\ e &\mapsto \overline{e} \end{align*} $$

satisfying the following conditions for all $e \in \mathbb {E}(X)$ :

1. (i) $\overline {e} \neq e$ ;

2. (ii) $\overline {\overline {e}} = e$ ;

3. (iii) $o(\overline {e}) = t(e)$ and $t(\overline {e}) = o(e)$ .

The vertex $o(e)$ is called the origin and the vertex $t(e)$ is called the terminus of the directed edge $e$ . Given $v \in V(X)$ , write

$$ \begin{align*} \mathbb{E}^o_v(X) & = \{e \in \mathbb{E}(X) : o(e) =v\},\\ \mathbb{E}^t_v(X) & = \{e \in \mathbb{E}(X) : t(e) =v\}. \end{align*} $$

The inversion map induces the bijection

$$ \begin{align*} \mathrm{inv} \colon \mathbb{E}^o_v(X) \longrightarrow \mathbb{E}^t_v(X)\quad \text{ for all } v \in V(X). \end{align*} $$

Note that graphs in the above sense are undirected due to the existence of the inversion map. In the following, a graph therefore always denotes an undirected graph. The set of undirected edges is obtained by identifying $e$ with $\overline {e}$ and is denoted by $E(X)$ . Finally, write $E_v(X)$ to denote the set of all undirected edges adjacent to $v$ . When it is clear from the context to which graph is being referred, we often drop it from the notation of $E$ , $E_v$ , $\mathbb {E}^o_v$ , $\mathbb {E}^t_v$ , and $V$ .

We write $f$ to denote both $f_V$ and $f_{\mathbb {E}}$ when there is no chance of confusion. Given a morphism of graphs $f \colon Y \to X$ and any vertex $v \in V(Y)$ , the restriction $f\vert _{\mathbb {E}^{o}_v(Y)}$ induces a function

$$ \begin{align*} f\vert_{\mathbb{E}^o_v(Y)} \colon \mathbb{E}^o_v(Y) \longrightarrow \mathbb{E}^o_{f(v)}(X). \end{align*} $$

2.2 Jacobian of graphs

Definition 2.2. A divisor on a (possibly infinite) graph $X$ is an element of the free abelian group on the vertices $V (X)$ defined as

$$ \begin{align*} \mathrm{Div}(X) = \bigg\{ \sum_{v\in V (X)} a_v v \,\bigg|\, a_v \in \mathbb{Z} \bigg\}, \end{align*} $$

where each $\sum _{v\in V (X)}a_v v$ is a formal linear combination of the vertices of $X$ with integer coefficients and only finitely many $a_v$ are nonzero (when $X$ is an infinite graph). When we consider infinite graphs, we require that the graph is locally finite.

The degree homomorphism is defined as

$$ \begin{align*} \deg \colon \mathrm{Div}(X) &\longrightarrow \mathbb{Z}\\ \sum_{v\in V (X)}a_v v &\mapsto \sum_{v\in V(X)} a_v. \end{align*} $$

The kernel of this degree map is the subgroup of divisors of degree 0 and is denoted as $\mathrm {Div}^0(X)$ .

Write $\mathcal {M}(X)$ to denote the set of $\mathbb {Z}$ -valued functions on $V(X)$ . Define the function $\psi _v\in \mathcal {M}(X)$ as follows:

$$ \begin{align*} \psi_v(v_0) = \begin{cases} 1 & \text{ if } v = v_0, \\ 0 & \text{ if } v \neq v_0. \end{cases} \end{align*} $$

Note that $\psi _v$ forms a $\mathbb {Z}$ -basis of $\mathcal {M}(X)$ as $v$ runs over all vertices $v\in V(X)$ . We can then define a group morphism

$$ \begin{align*} \mathrm{div}: \mathcal{M}(X) & \longrightarrow \mathrm{Div}(X)\\ \psi_{v_0} & \mapsto \sum_{v\in V(X)} \rho_{v}(v_0) \cdot v, \end{align*} $$

where

$$ \begin{align*} \rho_v(v_0) = \begin{cases} \vert{\mathbb{E}_{v_0}^o(X)}\vert - 2\times \#(\text{undirected loops at }v_0) & \text{ if } v=v_0,\\ -\#(\text{undirected edges from }v \text{ to }v_0) & \text{ if } v\neq v_0. \end{cases} \end{align*} $$

Definition 2.3. With notation as introduced above, define the principal divisors on X as

$$ \begin{align*} \Pr(X) = \mathrm{div}(\mathcal{M}(X)). \end{align*} $$

The Picard group of X is then defined as

$$ \begin{align*} \mathrm{Pic}(X) = \mathrm{Div}(X)/\Pr(X). \end{align*} $$

The Jacobian group of X is defined as

$$ \begin{align*} \mathrm{Jac}(X) = \mathrm{Div}^0(X)/\Pr(X). \end{align*} $$

In the literature, this is also referred to as a Picard group of degree zero and denoted by $\mathrm {Pic}^0(X)$ .

2.3 Group actions on graphs, Galois theory, and coverings of graphs

Write $\mathrm {Aut}(X)$ to denote the group of automorphisms of a graph $X$ .

Definition 2.6. Let $f\colon Y\to X$ be morphism of graphs that is surjective on the set of edges and on the set of vertices. The morphism $f$ is a branched cover if for each vertex $v\in V(Y)$ , the cardinality $m_v= (f\vert _{\mathbb {E}^o_v}^{-1}(e))$ does not depend on the choice of $e\in \mathbb {E}^o_{f(v)}(X)$ . The cardinality $m_v$ is called the ramification index at $v$ . If $m_v=1$ , the morphism $f$ is said to be an unramified cover.

The function $f\vert _{\mathbb {E}^o_v} \colon \mathbb {E}^o_v(Y) \to \mathbb {E}^o_{f(v)}(X)$ is $m_v$ -to- $1$ . The graph $Y$ is called a d-sheeted covering of a connected graph $X$ if for all $w\in V(X)$ ,

$$ \begin{align*} d = [Y:X] = \sum_{v \in f^{-1}(w)} m_{v}. \end{align*} $$

When $Y$ is a $d$ -sheeted unramified cover of a connected graph $X$ , for all ${w\in V(X)}$ ,

$$ \begin{align*} d = [Y:X] = \sum_{v\in f^{-1}(w)} 1 = \# \{v \in V(Y) \mid f(v) = w\}. \end{align*} $$

2.4 Iwasawa theory of pro- $p$ tower of graphs obtained from a voltage assignment

We describe the construction of uniform pro- $p\, p$ -adic Lie extensions of connected graphs. Let $\mathcal {G}$ be a (possibly non-Abelian) uniform pro- $p\, p$ -adic Lie group of dimension $d$ . This means $\mathcal {G}$ is topologically generated by $d$ elements and has a filtration

$$ \begin{align*} \mathcal{G} \supseteq \mathcal{G}^p \supseteq \mathcal{G}^{p^2} \supseteq \cdots \supseteq \mathcal{G}^{p^n} \supseteq \cdots \end{align*} $$

such that $\mathcal {G}^{p^{n+1}} \trianglelefteq \mathcal {G}^{p^n}$ and each subsequent quotient is isomorphic to $(\mathbb {Z}/p\mathbb {Z})^d$ . When $\mathcal {G}$ is abelian, we write $\mathcal {G} = \Gamma ^{\oplus d} \simeq \mathbb {Z}^{\oplus d}_p$ , where $\Gamma $ is a multiplicative group topologically isomorphic to $\mathbb {Z}_p$ .

Let $X$ be a finite connected graph with vertex set $V(X)$ and set of edges $\mathbb {E}(X)$ . We define a voltage assignment function as follows:

$$ \begin{align*} \alpha \colon \mathbb{E}(X) \longrightarrow \mathcal{G} \text{ satisfying } \alpha(\overline{e}) = \alpha(e)^{-1}. \end{align*} $$

For each vertex $v\in V(X)$ , choose a closed subgroup $I_v$ of $\mathcal {G}$ and define the set

$$ \begin{align*} \mathcal{I} = \{(v,I_v) : v \in V(X)\}. \end{align*} $$

This gives a graph $X_\infty = X(\mathcal {G},\mathcal {I},\alpha )$ with vertex set equal to the disjoint union

$$ \begin{align*} V(X_\infty) = \bigsqcup_{v\in V(X)} \{v\} \times \mathcal{G}/I_v, \end{align*} $$

and the collection of directed edges is given by

$$ \begin{align*} \mathbb{E}(X_\infty)=\mathbb{E}(X)\times \mathcal{G}. \end{align*} $$

Let $g\in \mathcal {G}$ . The directed edge $(e,g)$ connects the vertex $(o(e),gI_{o(e)})$ to the vertex $(t(e), g \alpha (e)I_{t(e)})$ . Finally, let $\overline {(e, g)} = (\overline {e},g \alpha (e))$ . This is a branched cover $X_\infty \to X$ . Moreover, the graph $X_\infty $ is infinite. Note that $X_\infty $ is not locally finite.

To get finite graphs, for each positive integer $n$ , consider the natural surjective group morphism $\pi _n \colon \mathcal {G} \twoheadrightarrow \mathcal {G}_n$ , where $\mathcal {G}_n = \mathcal {G}/\mathcal {G}^{p^n}$ . Set $\alpha _n = \pi _n \circ \alpha \colon \mathbb {E}(X) \to \mathcal {G}_n$ and define subgroups

$$ \begin{align*} \mathcal{I}_n = \{(v,\pi_n(I_v)) \ : \ v \in V(X)\} \end{align*} $$

of $\mathcal {G}_n$ indexed by the vertices of $X$ . This gives a family of graphs $X_n =X(\mathcal {G}_n,I_n,\alpha _n)$ that are finite. The natural surjective group morphisms $\mathcal {G}_{n+1} \twoheadrightarrow \mathcal {G}_n$ induce branched covers $X_{n+1} \to X_n$ for each nonnegative integer $n$ . Therefore, we obtain a (Galois) tower of graphs

$$ \begin{align*} X = X_0 \longleftarrow X_1 \longleftarrow X_2 \longleftarrow \cdots \longleftarrow X_n \longleftarrow \cdots \longleftarrow X_\infty, \end{align*} $$

where each map $X_{n+1} \to X_n$ is a branched cover satisfying $[X_{n+1} : X_n] = p^d$ . Such a tower is referred to as a branched $\mathcal {G}$ -tower of finite graphs provided that all $X_n$ are connected.

When the closed subgroups $I_v$ are all trivial, the graph $X_\infty = X(\mathcal {G},\alpha )$ is the unramified $\mathcal {G}$ -tower.

We call a vertex $v\in V(X_n)$ ramified in $X_\infty /X_n$ if $I_v\cap \mathcal {G}_n$ is nontrivial. Otherwise, we call $v$ unramified.

Lemma 2.9. Let $(X_n)_{n\in \mathbb {N}}$ be a $\mathcal {G}$ -tower of graphs constructed as above and assume that all $X_n$ are connected. Suppose that there exists an index $n_0$ such that the number of ramified vertices in $X_\infty /X_{n_0}$ stabilizes. Let $\mathsf {v}_r$ be the number of ramified vertices and let $\mathsf {v}_u$ be the number of unramified vertices of $X_\infty /X_{n_0}$ . Then, the number of vertices of $X_n$ for $n\ge n_0 $ is given by

$$ \begin{align*} \mathsf{v}_r+\frac{\vert \mathcal{G}_n\vert }{\vert \mathcal{G}_{n_0}\vert } \mathsf{v}_u. \end{align*} $$

2.5 Matrices associated to graphs

We now remind the reader of some matrices associated to graphs that are important throughout the discussion. We often prefer to work with a multiplicative notation, so we write $\Gamma $ in place of $\mathbb {Z}_p$ .

2.6 Growth patterns in $\mathbb {Z}_p^d$ -towers of graphs

A strong analogy between number theory and graph theory has been observed by mathematicians. This suggests that it might be possible to study the variation of the $p$ -part of the number of spanning trees as one goes up a $\mathbb {Z}^d_p$ -tower of graphs. In fact, there is a perfect analog of Iwasawa’s asymptotic class number formula in the setting of graph theory; see [Reference GonetGon22, Reference VallièresVal21]. For example, when $d=1$ , there exist nonnegative integers $\mu , \lambda , n_0$ , and $\nu \in \mathbb {Z}$ such that for $n\geq n_0$ ,

$$ \begin{align*} \mathrm{ord}_p(\kappa(X_n)) = \mu p^n + \lambda n + \nu, \end{align*} $$

where $\kappa (X_n)$ is the number of spanning trees of $X_n$ .

In what follows, we summarize some relevant notions from [Reference Kleine and MüllerKM24]. Henceforth, let $\mathcal {G} = \Gamma ^{\oplus d} \simeq \mathbb {Z}_p^{\oplus d}$ for some $d\geq 1$ . As discussed previously, to every finite connected graph $X$ , we can attach a finite abelian group called the Jacobian of $X$ . Moreover, any cover of finite connected graphs $f \colon Y \to X$ induces a surjective group morphism of divisor groups,

$$ \begin{align*} f_*\colon \mathrm{Div}(Y)\to \mathrm{Div}(X), \quad \sum_{v\in V(Y)}a_vv\to \sum_{v\in V(Y)}a_vf(v). \end{align*} $$

Furthermore,

$$ \begin{align*} f_*(\text{div}(v))=\text{div}(f_*(v)),\end{align*} $$

which shows that $f_*$ maps principal divisors onto principal divisors. Therefore, $f_*$ factors through the Jacobian and induces a surjection

$$ \begin{align*} f_* \colon \mathrm{Jac}(Y) \longrightarrow \mathrm{Jac}(X). \end{align*} $$

Starting with a $\mathbb {Z}^d_p$ -cover of a finite connected graph $X$ , we obtain a compatible system of maps between the Sylow- $p$ subgroups of the Jacobians, that is,

$$ \begin{align*} \mathrm{Jac}(X_{n+1})[p^\infty] \longrightarrow \mathrm{Jac}(X_n)[p^\infty]. \end{align*} $$

Recall that $\mathrm {Jac}(X_n)[p^\infty ]$ is a $\mathbb {Z}_p[(\mathbb {Z}/p^n\mathbb {Z})^d]$ -module. Since $X$ is assumed to be finite and connected,

$$ \begin{align*} \mathrm{Jac}(X)[p^\infty] \simeq \mathbb{Z}_p \otimes_{\mathbb{Z}} \mathrm{Jac}(X); \end{align*} $$

the same holds for each subsequent layer $X_n$ . We then define the inverse limit

$$ \begin{align*} \mathrm{Jac}_{\Lambda}(X_\infty) := \varprojlim_{n} \mathrm{Jac}(X_n)[p^\infty] = \varprojlim_{n} \Big( \mathbb{Z}_p \otimes_{\mathbb{Z}} \mathrm{Jac}(X_n) \Big); \end{align*} $$

this is a finitely generated torsion module over the Iwasawa algebra $\Lambda =\mathbb {Z}_p[\![ \mathcal {G}]\!] $ . Indeed, by [Reference Kleine and MüllerKM24, Lemma 5.2], $\mathrm {Jac}_\Lambda (X_\infty )$ is isomorphic to $\mathrm {Jac}(X_\infty )\otimes _{\mathbb {Z}_p[\mathcal {G}]} \Lambda $ . By [Reference Kleine and MüllerKM24, Lemma 3.2], we conclude that $\mathrm {Pic}(X_\infty )\otimes _{\mathbb {Z}_p[\mathcal {G}]} \Lambda $ is the quotient of two free $\Lambda $ -modules of the same rank. Thus, $\mathrm {Pic}(X_\infty )\otimes _{\mathbb {Z}_p[\mathcal {G}]} \Lambda $ is a finitely generated $\Lambda $ -torsion module and the same is true for $\mathrm {Jac}_\Lambda (X_\infty )$ .

It is well known [Reference GreenbergGre73] that there is a noncanonical isomorphism

$$ \begin{align*} \Lambda = \mathbb{Z}_p[\![ \mathcal{G} ]\!] &\simeq \mathbb{Z}_p [\![ T_1,\ldots, T_d ]\!]\\ \gamma_i & \mapsto 1+T_i, \end{align*} $$

where $\gamma _1,\ldots ,\gamma _d$ are topological generators of $\mathcal {G} = \Gamma ^{\oplus d}$ . Henceforth, we go between the Iwasawa algebra and its identification with the formal power series ring in $d$ -variables as is convenient.

A finitely generated, torsion $\Lambda $ -module $M$ is said to be pseudo-null if its annihilator $\mathrm {Ann}_{\Lambda }(M)$ has height at least 2. Equivalently, for a pseudo-null module $M$ , the annihilator $\mathrm {Ann}_{\Lambda }(M)$ is generated by two co-prime elements. The structure theorem of finitely generated $\Lambda $ -modules asserts that there is a pseudo-isomorphism (that is, homomorphism such that kernel and cokernel are pseudo-null) from $M$ to a (unique) module $\bigoplus _{i=1}^t \Lambda /\langle f_i^{e_i}\rangle $ , where $f_i\in \Lambda $ are irreducible; see [Reference Neukirch, Schmidt and WingbergNSW08, Proposition 5.1.7].

Viewing $\mathrm {Jac}_{\Lambda }(X_\infty )$ as a $\mathbb {Z}_p[\![ T_1,\ldots , T_d ]\!] $ -module and using the structure theorem, we can define the characteristic ideal ${\mathrm {char}}_{\Lambda }(\mathrm {Jac}_{\Lambda }(X_\infty ))$ . In other words,

$$ \begin{align*} {\mathrm{char}}_{\Lambda}(\mathrm{Jac}_{\Lambda}(X_\infty)) = \langle f_{X}(T_1, \ldots, T_d) \rangle, \end{align*} $$

where $f_X(T_1, \ldots , T_d)$ is an element of $\Lambda $ and is a generator of the characteristic ideal (determined up to a unit); it is called the characteristic element of $\mathrm {Jac}_{\Lambda }(X_\infty )$ .

3.1 Construction

Let $X$ be a finite and connected graph. Suppose that the voltage group $G$ is finite of size $N$ . Label the elements in $G$ as $\tau _0, \ldots , \tau _{N-1}$ .

To get the vertices of $Y$ , consider $N$ copies of each vertex $x \in V (X)$ and label them as $x_{\tau _0}, \ldots , x_{\tau _{N-1}}$ . Therefore, the derived graph $Y$ has $N \vert {X}\vert $ many vertices. Suppose that $x \to x'$ is a directed edge in $X$ with voltage assignment $\alpha (e_{x,x'}) = \tau \in G$ . For every $\tau _i$ in $G$ , there is an edge in $Y$ joining $x_{\tau _i}$ to $x^{\prime }_{\tau _i \cdot \tau }$ . This process is then repeated for every $x\in V(X)$ .

When $G$ is infinite (as it is in our case), we can repeat the same construction formally.

3.2 Intuition

A voltage graph takes $\vert {G}\vert $ -many copies of the base graph $X$ stacked vertically above each other. Then, for each directed edge $x \to x'$ with voltage $\tau $ , we do the following: for each sheet indexed by $\sigma \in G$ , we join the vertex $x_{\sigma }$ to the vertex $x^{\prime }_{\sigma \cdot \tau }$ . So, if we ‘look at the layer of sheets from above’, we see vertex $x$ joined to vertex $x'$ exactly $\vert {G}\vert $ -many times.

The main goal of this section is to provide a sufficient condition for the derived graph of a planar graph to be planar. It is easy to construct examples of planar graphs and voltage assignments such that the derived graph is not planar. We provide an example as follows.

Example 3.1. Let $X$ be a bouquet graph with two loops, that is, a graph consisting of a single vertex with two loops $e_1,e_2$ . Let $G=(\mathbb {Z}/5\mathbb {Z},+)$ and define $\alpha (e_1)=1$ , $\alpha (e_2)=2$ . Let $Y=X(G,\alpha )$ . Then, $Y$ is the complete graph on $5$ vertices and is not planar.

We explain the above theorem via two explicit examples. To make the following drawings easier to read, we draw only one half of the edges. An arrow from $A$ to $B$ marked with $\tau $ means that the edge from $A$ to $B$ has voltage assignment $\tau $ . It is understood from the context that the edge in the opposite direction has voltage assignment $\tau ^{-1}$ .

Example 3.3. Let the voltage group $G=\{\mathbf {1},\tau \}$ . Consider the finite planar graph $X$ where exactly one edge (namely $AB$ ) has a nontrivial voltage assignment $\alpha (e_{AB}) = \tau $ .

Here, $e_0 = e_{AB}$ . In this case, $X'$ is the following graph:

In the case where $G=\mathbb {Z}/2\mathbb {Z}$ , the graph $Y'$ has the following shape:

The derived graph $Y=X_1 = X(\mathbb {Z}/2\mathbb {Z}, \alpha )$ in this case is a 6-gon. We draw this graph in two ways for ease of visualization.

If we consider the group $G=\mathbb {Z}/4\mathbb {Z}$ instead, we obtain a 12-gon:

Example 3.4. We consider two different voltage groups $G_1=\mathbb {Z}/2\mathbb {Z}$ and $G_2=\mathbb {Z}/4\mathbb {Z}$ . Once again, there is a unique edge (which we call $e_0$ ) with nontrivial voltage assignment, that is, $\alpha (e_0) = \tau $ , where $\tau $ is used to denote the generator of both $G_1$ and $G_2$ . The tower of derived graphs $Y= X_1 = X(\mathbb {Z}/2\mathbb {Z}, \alpha )$ and $Y=X_2 = X(\mathbb {Z}/4\mathbb {Z}, \alpha )$ is drawn as

The next result no longer requires the voltage assignment to be unramified. In other words, the derived graph we obtain is a (ramified) branched cover.

Example 3.6. We once again start with the same graph $X$ as in Example 3.3.

We no longer require that the vertices are unramified in this extension. In fact, we assume that the vertices $B,C$ are totally ramified, that is, the corresponding inertia groups are isomorphic to $G$ at each layer. The derived graph $Y=X_1=X(\mathbb {Z}/2\mathbb {Z}, \mathcal {I}, \alpha )$ is the following:

This is easily seen to be a contraction of the hexagon obtained in Example 3.3. The vertices $B_1$ and $B_2$ (respectively $C_1$ and $C_2$ ) have been identified.

Now, when $G\simeq (\mathbb {Z}/2^2\mathbb {Z}) = \mathbb {Z}/4\mathbb {Z}$ , the derived graph $Y= X(\mathbb {Z}/4\mathbb {Z}, \mathcal {I}, \alpha )$ we obtain is the following:

This is a contraction of the cyclic 12-gon graph above in Example 3.3.

4.1 Picard group of ramified derived graphs

In this section, we prove the $d$ -dimensional analog of [Reference Gambheera and VallièresGV24, Theorem 5.4]. Let $G$ be any group, let $Y=~X(G, \mathcal {I}, \alpha )$ denote the derived graph of $X$ , and write $Y^{\mathrm {unr}} = X(G,\alpha )$ . Let

$$ \begin{align*} \iota \colon Y^{\mathrm{unr}} \longrightarrow Y \end{align*} $$

be the immersion. There is a natural surjective group morphism

$$ \begin{align*} \iota_{*} \colon \mathrm{Div}(Y^{\mathrm{unr}}) \longrightarrow \mathrm{Div}(Y). \end{align*} $$

Theorem 4.4. Let $X_\infty = X(\mathbb {Z}_p^d, \mathcal {I}, \alpha )$ denote the (ramified) derived graph of $X$ and $X_\infty ^{\mathrm {unr}} = X(\mathbb {Z}_p^d, \alpha )$ be the unramified derived graph. Denote the respective layers of the derived graph as $X_n$ and $X_n^{\mathrm {unr}}$ . Define

$$ \begin{align*} N_n={\Pr}_\Lambda^{\mathrm{unr}}(X_\infty) + (\omega_n(T_1),\ldots, \omega_n(T_d)\, \mathrm{Div}_\Lambda(X_\infty) + {\Pr}_n^{\mathrm{ram}}(X_\infty), \end{align*} $$

where $\omega _n(T_i) = (T_i+1)^{p^n} -1$ for $1\le i \le d$ . For $n \gg 0$ ,

$$ \begin{align*} \mathrm{Div}_\Lambda(X_\infty)/N_n\cong \mathrm{Pic}(X_n)\otimes \mathbb{Z}_p. \end{align*} $$

Proof. Fix a layer $X_n$ and consider the projection map

$$ \begin{align*} \pi_n \colon X_\infty \longrightarrow X_n. \end{align*} $$

This induces a surjective map on the divisors, namely,

$$ \begin{align*} \pi_n \colon \mathrm{Div}_{\Lambda}(X_\infty) \longrightarrow \mathrm{Div}(X_n)\otimes \mathbb{Z}_p. \end{align*} $$

To complete the proof, it is sufficient to show that $\pi _n^{-1}(\Pr (X_n)\, \otimes \, \mathbb {Z}_p) =N_n$ .

Consider a vertex $v\in V(X)$ . We fix a pre-image $v'=(v,1)$ of $v$ in $V(X_n^{\mathrm {unr}})$ . Associated to this $v'$ , define a principal divisor

$$ \begin{align*} P_v^n=\deg(v)(v,1)-\sum_{e\in \mathbb{E}^o_{v'}(X_n^{\text{unr}})}t(e) \in \mathrm{Div}(X_n^{\mathrm{unr}}). \end{align*} $$

Define

$$ \begin{align*} P_{v,n} = \iota_{*}(P^n_v). \end{align*} $$

A similar definition can be made over $X^{\text {unr}}_\infty $ . To simplify notation, we write $P_v$ instead of $P_{v,\infty }$ (compare with Definition 4.2). Note that $P^\infty _{v}\in \mathrm {Div}(X_\infty ^{\mathrm {unr}})$ , while $P_v = \iota _*(P_v^{\infty }) \in \mathrm {Div}(X_\infty )$ . For each vertex $v$ in $X_n$ ,

$$ \begin{align*} \pi_n(P_{v}) =P_{v,n}. \end{align*} $$

Recall that to each ramified vertex $v$ , we can associate an inertia group $I_v \subseteq \mathbb {Z}_p^d$ of rank $k = k(v)$ . Let $n\geq n_{0,v}$ , where $n_{0,v}$ is a large enough integer such that $\mathrm {rank}_p(I_v\mathcal {G}/\mathcal {G}^{p^{n_{0,v}}})=\mathrm {rank}_p(I_v)$ . Then,

$$ \begin{align*} \pi_n(\boldsymbol{\omega}_{v,n}P_{v}) & =\sum_{\vec{\mathbf{j}}}\prod_{t=1}^k\sigma_{v,t}^{j_t}P_{v,n}=\sum_{\sigma \in I_v\mathcal{G}^{p^n}/ \mathcal{G}^{p^n}}\sigma P_{v,n} \end{align*} $$

and [Reference Gambheera and VallièresGV24, Corollary 4.7] implies that $\pi _n(N_n)=\Pr (X_n)\otimes \mathbb {Z}_p$ . Therefore,

$$ \begin{align*} N_n\subset \pi_n^{-1}(\Pr(X_n)\otimes \mathbb{Z}_p) \quad \text{and}\quad \pi_n^{-1}(\Pr(X_n)\otimes \mathbb{Z}_p)=N_n+\ker(\pi_n). \end{align*} $$

Write $\text {Aug}(I_{v_l})$ to denote the augmentation ideal of $I_{v_l}$ . The arguments in [Reference Gambheera and VallièresGV24, Section 5] now show

$$ \begin{align*}\mathrm{Div}_\Lambda(X_\infty)=\Lambda^r\oplus \bigoplus_{l=r+1}^s \Lambda/\text{Aug}(I_{v_l}), \end{align*} $$

where $s$ is the number of vertices in $X$ and $r$ is the number of unramified vertices. Taking appropriate projections,

$$ \begin{align*} \mathrm{Div}(X_n)\cong \mathbb{Z}_p[\mathcal{G}/\mathcal{G}^{p^n}]^r\oplus \bigoplus_{l=r+1}^s \mathbb{Z}_p[\mathcal{G}/I_{v_l}\mathcal{G}^{p^n}]. \end{align*} $$

Therefore, for $n\gg 0$ ,

$$ \begin{align*} \ker(\pi_n)=(\omega_n(T_1),\ldots,\omega_n(T_d))\mathrm{Div}_\Lambda(X_\infty). \end{align*} $$

In particular, $\ker (\pi _n)\subset N_n$ and $\pi _n^{-1}(\Pr (X_n)\otimes \mathbb {Z}_p)=N_n$ .

Even though the module $\Pr _n^{\mathrm {ram}}(X_\infty )$ is necessary to describe $\mathrm {Pic}(X_n)\otimes \mathbb {Z}_p$ as a quotient of $\mathrm {Div}_\Lambda (X_\infty )$ , it does not play a role when we take projective limits as the following corollary points out. In view of the notation introduced previously,

$$ \begin{align*} \mathrm{Pic}_\Lambda(X_\infty) = \mathrm{Pic}(X_\infty) \otimes_{\mathbb{Z}_p[\mathcal{G}]} {\Lambda}. \end{align*} $$

Corollary 4.5. With notation as above,

$$ \begin{align*} \mathrm{Pic}_\Lambda(X_\infty)\cong \mathrm{Div}_\Lambda(X_\infty)/{\Pr}^{\mathrm{unr}}_\Lambda(X_\infty). \end{align*} $$

Fix $v\in V(X)$ . Let $w_{v,\infty }^{\mathrm {unr}}$ denote the vertex $(v, {\mathbf {1}}_{\mathcal {G}})$ in $X_\infty ^{\mathrm {unr}} = X(\mathcal {G},\alpha )$ . Let $\Delta $ be the operator on $\mathrm {Div}_\Lambda (X_\infty ^{\mathrm {unr}})$ given by

$$ \begin{align*} \Delta w^{\mathrm{unr}}_{v,\infty}= \begin{cases} P_v^{\mathrm{unr}} &\text{ if } v \text{ is unramified in } X_\infty/X,\\ (\sigma_{v,1}-1)w_{v,\infty}^{\mathrm{unr}} &\text{ if } v \text{ is ramified in } X_\infty/X \text{ and } \mathrm{rank}(I_{v})=1,\\ w_{v,\infty}^{\mathrm{unr}} &\text{ if } v \text{ is ramified in } X_\infty/X \text{ and } \mathrm{rank}(I_{v})>1, \end{cases} \end{align*} $$

where

$$ \begin{align*} P^{\mathrm{unr}}_v =\deg(v)(v,1)-\sum_{e\in \mathbb{E}^o_{v}(X)}(t(e),\alpha(e))\in \mathrm{Div}(X^{\mathrm{unr}}_\infty). \end{align*} $$

In previous references, the operator $P_v^{\mathrm {unr}}$ is often written as $\mathcal {L}_{\Lambda }^{\mathrm {unr}}(-)$ and is called the Laplacian operator. This notation is consistent with how $P_v$ was expressed in Definition 4.2; only now, $I_{t(e)}$ is trivial. Note that this divisor was denoted by $P_v^{\infty } \in ~\mathrm {Div}(X_\infty ^{\mathrm {unr}})$ above.

Theorem 4.7. Let $X_\infty = X(\mathbb {Z}_p^d, \mathcal {I}, \alpha )$ denote the (ramified) derived graph of $X$ . With this notation,

$$ \begin{align*} \mathrm{char}_\Lambda(\mathrm{Pic}_\Lambda(X_\infty))=(\det(\Delta)). \end{align*} $$

Proof. As before, write $X_\infty ^{\mathrm {unr}} = X(\mathbb {Z}_p^d, \alpha )$ to denote the unramified derived graph. Define

$$ \begin{align*} {\Pr}^{\prime}_\Lambda(X_\infty^{{\mathrm{unr}}}) = \Lambda\text{-submodule of } \mathrm{Div}_{\Lambda}(X_\infty^{\mathrm{unr}}) \text{ generated by } \{P_{v_l}^{\mathrm{unr}}\mid 1\le l\le r\}. \end{align*} $$

We want to emphasize that $\Pr ^{\mathrm {unr}}_\Lambda (X_\infty )$ was defined as a $\Lambda $ -submodule of $\mathrm {Div}_\Lambda (X_\infty )$ generated by $\{P_{v_l} \mid 1 \le l \le r\}$ in Definition 4.2, which is different from $\Pr ^{\prime }_\Lambda (X_\infty ^{{\mathrm {unr}}})$ . The two definitions coincide if $X_\infty = X_\infty ^{\mathrm {unr}}$ . Note that the quotient $\mathrm {Div}_\Lambda (X_\infty ^{\mathrm {unr}})/\Pr ^{\prime }_\Lambda (X_\infty ^{\mathrm {unr}})$ has $\Lambda $ -rank $s-r$ . Also, recall we have the surjection

$$ \begin{align*} \iota_* : \mathrm{Div}_{\Lambda}(X_\infty^{\mathrm{unr}}) \longrightarrow \mathrm{Div}_{\Lambda}(X_\infty). \end{align*} $$

Under this map, the pre-image of $\Pr _{\Lambda }^{\mathrm {unr}}(X_\infty )$ is precisely $\ker (\iota _*)+\text {Pr}^{\prime }_\Lambda (X_\infty ^{\mathrm {unr}})$ . Now, it follows immediately from Corollary 4.5 that

$$ \begin{align*} \mathrm{Pic}_\Lambda(X_\infty) \cong \mathrm{Div}_\Lambda(X_\infty)/\text{Pr}^{\mathrm{unr}}_\Lambda(X_\infty) =\mathrm{Div}_\Lambda(X_\infty^{\mathrm{unr}})/\ker(\iota_*)+\text{Pr}^{\prime}_\Lambda(X_\infty^{\mathrm{unr}}). \end{align*} $$

Thus, it suffices to compute the characteristic ideal of $\mathrm {Div}_\Lambda (X_\infty ^{\mathrm {unr}})/\ker (\iota _*)+\text {Pr}^{\prime }_\Lambda (X_\infty ^{\mathrm {unr}})$ . Consider the exact sequence

$$ \begin{align*} \begin{aligned} 0&\to \mathrm{Im}(\Delta)/\ker(\iota_*)+\text{Pr}^{\prime}_\Lambda(X_\infty^{\mathrm{unr}})\to \mathrm{Div}_\Lambda(X_\infty^{\mathrm{unr}})/\ker(\iota_*)\\ &\quad+\text{Pr}^{\prime}_\Lambda(X_\infty^{\mathrm{unr}})\to \mathrm{Div}_\Lambda(X^{\mathrm{unr}}_{\infty})/\mathrm{Im}(\Delta)\to 0. \end{aligned} \end{align*} $$

To prove the theorem, we show that the first term is pseudo-null and that the characteristic ideal of the last term is indeed $\det (\Delta )$ . We first prove the latter assertion. Note that $\mathrm {Div}_\Lambda (X^{\mathrm {unr}})$ is a free $\Lambda $ -module and $\Delta $ is a square (quadratic) matrix of the same dimension. Thus, $\mathrm {Div}_\Lambda (X^{\mathrm {unr}}_{\infty })/\mathrm {Im}(\Delta )$ is quadratically represented. Thus, its fitting ideal is given by $\det (\Delta )$ . Furthermore, for quadratically represented modules, the characteristic ideal and the fitting ideal coincide. Thus,

$$ \begin{align*} \mathrm{char}_\Lambda( \mathrm{Div}_\Lambda(X^{\mathrm{unr}}_{\infty})/\mathrm{Im}(\Delta))=\det(\Delta). \end{align*} $$

It remains to show that the first term of the above short exact sequence is pseudo-null. By definition, $\mathrm {Im}(\Delta )$ is generated by

$$ \begin{align*} &\{w^{\text{unr}}_{v,\infty}\mid v\in \{v_{r+1},\ldots,v_s\}, \ \mathrm{rank}(I_v)\ge 2\}\\ &\cup \{(\sigma_{v,1}-1)w_{v,\infty}^{\mathrm{unr}}\mid v\in \{v_{r+1},\ldots,v_s\}, \ \mathrm{rank}(I_v)=1\} \\ &\cup \{P_v^{\mathrm{unr}}\mid\!\!\text{ if } v \text{ is an unramified vertex in } X_\infty/X\}. \end{align*} $$

The last two sets are contained in $\ker (\iota _*)+\Pr ^{\prime }_\Lambda (X_\infty ^{\mathrm {unr}})$ . Thus, the quotient $\mathrm {Im}(\Delta )/\ker (\iota _*)+ \text {Pr}^{\prime }_\Lambda (X_\infty ^{\mathrm {unr}})$ is generated by

$$ \begin{align*} \{w^{\text{unr}}_{v,\infty}\mid v\in \{v_{r+1},\ldots,v_s\}, \ \mathrm{rank}(I_v)\ge 2\} \end{align*} $$

Each $w^{\text {unr}}_{v,\infty }$ is annihilated by $\text {Aug}(I_v)$ . As $\mathrm {rank}(I_v)\ge 2$ , the ideal $\text {Aug}(I_v)$ has height at least $2$ . In particular, the module $\Lambda /\text {Aug}(I_v)$ is pseudo-null and the result follows; see Section 2.6 for the definition of pseudo-null modules.

Remark 4.8. If $X_\infty = X_\infty ^{\mathrm {unr}}$ , the above theorem asserts that

$$ \begin{align*} \mathrm{char}_{\Lambda}(\mathrm{Pic}_{\Lambda}(X_\infty^{\mathrm{unr}})) = (\det(\Delta)). \end{align*} $$

However, it was proven in [Reference Kleine and MüllerKM24, Theorem 5.2] that if $X_\infty /X$ is an unramified $\mathbb {Z}_p^d$ -extension of $X$ (obtained as a derived graph of $X$ ), then

$$ \begin{align*} (\det(\Delta)) = \begin{cases} \mathrm{char}_{\Lambda}(\mathrm{Jac}_{\Lambda}(X_\infty)) & \text{ if } d\ge 2,\\ (T)\, \mathrm{char}_{\Lambda}(\mathrm{Jac}_{\Lambda}(X_\infty)) & \text{ if } d=1. \end{cases} \end{align*} $$

Therefore, we conclude

$$ \begin{align*} \mathrm{char}_{\Lambda}(\mathrm{Pic}_{\Lambda}(X_\infty)) = \begin{cases} \mathrm{char}_{\Lambda}(\mathrm{Jac}_{\Lambda}(X_\infty)) & \text{ if } d\ge 2,\\ (T)\, \mathrm{char}_{\Lambda}(\mathrm{Jac}_{\Lambda}(X_\infty)) & \text{ if } d=1. \end{cases} \end{align*} $$

Compare also with [Reference Kleine and MüllerKM24, Remark 5.4].

Example 4.9. We give an example of an unramified cover of $X$ that can also be found in [Reference Kleine and MüllerKM24]. Here, the dashed line between $A$ and $B$ in the graph at the bottom layer has nontrivial voltage assignment, say $\tau $ .

For convenience of the reader, we draw another way to interpret the 18-gon in the third diagram above:

The characteristic polynomial of $\mathrm {Pic}_{\Lambda }(X_\infty )$ is $T^2$ for this unramified tower of coverings.

For ramified branched $\mathbb {Z}_p$ -covers $X_\infty $ , it has been pointed out [Reference Gambheera and VallièresGV24, Section 6] that $\det (\Delta ) = \det (D-B)$ , where $D, B$ were matrices introduced in Definition 2.10. We now provide examples of the same.

Example 4.10. Reconsider the branched covering from Example 3.6. We calculate the matrices

$$ \begin{align*} D = \begin{bmatrix} 2 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix}\quad \text{and}\quad B = \begin{bmatrix} 0 & 0 & 0 \\ T+1 & -T & 0 \\ 1 & 0 & -T \end{bmatrix}. \end{align*} $$

It follows that

$$ \begin{align*} \mathrm{char}_\Lambda(\mathrm{Pic}_\Lambda(X_\infty)) = \det(D-B) = \det\begin{bmatrix} 2 & 0 & 0 \\ -(1+T) & T & 0 \\ -1& 0 & T \end{bmatrix} = 2T^2. \end{align*} $$

Example 4.11. In this example, $G=(\mathbb {Z}/3\mathbb {Z}, +)$ for each subsequent layer, and the vertex $R\in V(X)$ is totally ramified whereas the vertex $U\in V(X)$ is unramified. The dashed edge has nontrivial voltage assignment.

We calculate the two matrices $D$ and $B$ using Definition 2.10,

$$ \begin{align*} D = \begin{bmatrix} 2 & 0 \\ 0 & 0 \end{bmatrix}\quad \text{and}\quad B = \begin{bmatrix} 0 & 0 \\ \tau^{-1}+1 & 1-(1+T)^{p^0} \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ \tau^{-1}+1 & -T \end{bmatrix}. \end{align*} $$

It follows that the characteristic ideal

$$ \begin{align*} \mathrm{char}_{\Lambda}(\mathrm{Pic}_{\Lambda}(X_\infty)) = \det(D-B) = \det\begin{bmatrix} 2 & 0 \\ -\tau^{-1}-1 & T \end{bmatrix} = 2T. \end{align*} $$

This example can be generalized in the following manner.

Example 4.12. Let $X$ be a cycle graph with $m$ vertices. Let $\alpha $ be a single voltage assignment with image in $\mathbb {Z}_p$ . Let $I_{v}=\mathbb {Z}_p$ for exactly one vertex $v$ and set $I_{v'}=1$ for all $v'\neq v$ . Let $X_\infty =X(\mathbb {Z}_p, \mathcal {I}, \alpha )$ and let $X_n$ be the intermediate covers. Assume that the image of $\alpha $ generates $\mathbb {Z}_p$ topologically. Then,

$$ \begin{align*} \mathrm{ord}_p(\vert \mathrm{Jac}(X_n)\vert)=\mathrm{ord}_p(m){p^n}. \end{align*} $$

Indeed, $X_n$ looks similar to the flower graph above. Instead of leaves consisting of one vertex and two edges, the leaves have $(m-1)$ vertices and $m$ edges. To count the spanning trees, we need to choose a spanning tree for each leaf, that is, we have to choose one edge that we ‘delete’. That leaves $m$ choices per leaf. As there are $p^n$ leaves, the claim follows.

In the following picture, we choose $p=3$ and draw the base graph and the first layer.

We calculate the two matrices $D$ and $B(T)$ using Definition 2.10,

$$ \begin{align*} D= \begin{bmatrix} 2 & 0 & 0 & \cdots & 0 \\ 0 & 2 & 0 & \cdots & 0 \\ 0 & 0 & 2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & 0 \end{bmatrix}_{m\times m}\quad \text{and}\quad B(T) = \begin{bmatrix} 0 & 1 & 0 & \ldots & 0 \\ 1 & 0 & 1 & \ldots & 0 \\ 0 & 1 & 0 & \ldots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots\\ \tau^{-1} & 0 & 0 & & -T \end{bmatrix}_{m\times m}. \end{align*} $$

Therefore,

$$ \begin{align*} \mathrm{char}_{\Lambda}(\mathrm{Pic}_{\Lambda}(X_\infty)) = \det(D-B) = mT. \end{align*} $$

Example 4.13. In the subsequent example, we consider the case when $\mathcal {G} = \mathbb {Z}_3^2$ , that is, the first layer is a $\mathbb {Z}/3\mathbb {Z}\times \mathbb {Z}/3\mathbb {Z}$ covering. We assume that the vertex $U$ is unramified and that the vertex $R$ is totally ramified. Recall that the voltage assignment $\alpha : \mathbb {E}(X) \to \mathbb {Z}_3 \times \mathbb {Z}_3$ . Let us denote the generators of $\mathcal {G}$ by $\sigma , \tau $ . The voltage assignment for the dashed edge is prescribed to be $\tau $ and that for the solid edge is $\sigma $ .

In the next step, we get a flower with $27$ leaves. Thus, for $X_n$ ,

$$ \begin{align*} \mathrm{ord}_p(\vert \mathrm{Jac}(X_n)\vert)=\mathrm{ord}_p(2)p^{2n}. \end{align*} $$

If we compute the characteristic ideal using the matrices

$$ \begin{align*}D = \begin{bmatrix} 2 & 0 \\ 0 & 0 \end{bmatrix}\quad \text{and}\quad B = \begin{bmatrix} 0 & 0 \\ \tau^{-1}+\sigma^{-1} & -1 \end{bmatrix}, \end{align*} $$

we obtain that the characteristic ideal is given by

$$ \begin{align*}\det(D-B)=2. \end{align*} $$

If we let $R$ only ramify at the group generated by $\tau $ , that is, $\langle \tau \rangle $ ,

$$ \begin{align*} B=\begin{bmatrix} 0&0\\\tau^{-1}+\sigma^{-1}&-(\tau-1) \end{bmatrix}. \end{align*} $$

In this case, the characteristic ideal is given by $2T$ . Here, we are identifying the variable $T$ with $\tau - 1$ .

4.2 Growth formula

In this section, the aim is to prove an analog of Iwasawa’s growth number formula for the number of spanning trees at the $n$ th layer of the $\mathbb {Z}_p^d$ -cover. For this, we need to appeal to the theory of ‘structures’ discussed in [Reference Cuoco and MonskyCM81]. For the convenience of the reader, we have included some details and repeated some proofs, but at various places, we refer the reader to the original text.

4.2.1 Abstract results on $\Lambda $ -modules

We first need to obtain some results on general $\Lambda $ -modules.

Definition 4.14. Let $M$ be a $\Lambda $ -module. A structure on $M$ consists of an integer $z$ and tuples $(I_j,M_j)\, 1\le j\le z$ , where the $M_j$ are submodules of $M$ and the $I_j$ are subgroups of $\mathcal {G}$ . For all $n$ , set

$$ \begin{align*} A_n=(\omega_n(T_1),\ldots, \omega_n(T_d))M+\sum_{j\,=\,1}^{z}\boldsymbol{\omega}_{j,n}M_j, \end{align*} $$

where $\boldsymbol {\omega }_{j,n}$ is defined as in (4-1) with respect to the subgroup $I_j$ . With this notation, define $\mathcal {M}_n=M/A_n$ .

Lemma 4.15. There is a constant $c$ such that $p^{2dn+c}\mathcal {M}_n[p^\infty ]=0$ .

Proof. Even though our definition of structures differs from that given in [Reference Cuoco and MonskyCM81], the proof of Theorem 4.5 there carries over almost verbatim. We repeat it for the convenience of the reader.

Let $N$ be the submodule generated by all the $M_{j}$ . Let $W=\Omega ^d$ , where $\Omega $ is the group of $p$ -power roots of unity. For each $\zeta \in W$ , there is a natural map $M/N\to (M/N)\otimes \mathbb {Z}_p[\zeta ]$ induced by the valuation $\Lambda \to \mathbb {Z}_p[\zeta ]$ .

For any $\Lambda $ -module $X$ , we set the notation $X_{\zeta } = X \otimes \mathbb {Z}_p[\zeta ]$ .

By [Reference Cuoco and MonskyCM81, Lemma 2.6], there exists a constant $c$ such that $p^cx$ annihilates $(M/N)_\zeta [p^\infty ]$ and $M_\zeta [p^\infty ]$ for all choices of $\zeta $ . Let $n_0$ be the index such that the rank of $I_{j}$ in $\mathcal {G}/\mathcal {G}^{p^n}$ is stable for all $n\ge n_0$ and all $j$ . First, consider an element $\zeta \in W$ such that $\mathrm {ord}(\zeta )>p^{n_0}$ . The evaluation $\Lambda \to \mathbb {Z}_p[\zeta ]$ sends $\boldsymbol {\omega }_{j,n}$ to zero in this case for all $j$ . Thus,

$$ \begin{align*} (\mathcal{M}_n)_{\zeta}[p^\infty]=M_\zeta[p^\infty] \end{align*} $$

is annihilated by $p^c$ .

It remains to estimate the contribution of those $\zeta \in W$ with $\mathrm {ord}(\zeta )\le p^{n_0}$ . For all $n\ge n_0$ , we define

$$ \begin{align*} \boldsymbol{\omega}_n = \boldsymbol{\omega}_n(T_1, \ldots, T_d) =\prod_{i=1}^d\frac{(T_i+1)^{p^n}-1}{(T_i+1)^{p^{n_0}}-1}. \end{align*} $$

Note that $\boldsymbol {\omega }_n$ describes the trace from $\mathcal {G}/\mathcal {G}^{p^n}$ to $\mathcal {G}/\mathcal {G}^{p^{n_0}}.$

Let $N'$ be the image of $N$ in $M_\zeta $ . (We emphasize that $N'$ is not $N_\zeta $ because taking tensor product is not left exact.) Then, the image of $\boldsymbol {\omega }_n N$ in $M_\zeta $ is $p^{(n-n_0)d}N'$ . It follows that $(M/\boldsymbol {\omega }_n N)_\zeta $ and $(M/N)_\zeta $ have the same $\mathbb {Z}_p$ -rank. Furthermore, $(M/\boldsymbol {\omega } _n N)_\zeta [p^\infty ]$ is annihilated by $p^{nd+c}$ . Additionally, we have a natural surjection

$$ \begin{align*} (M/\boldsymbol{\omega}_n N)_\zeta \longrightarrow (M/A_n)_\zeta \longrightarrow (M/N)_\zeta, \end{align*} $$

where all three terms have the same $\mathbb {Z}_p$ -rank. Therefore, $(M/A_n)_\zeta [p^\infty ]$ is annihilated by $p^{nd+2c}$ .

Summing over all $\zeta $ of order at most $p^n$ , we obtain that

$$ \begin{align*} \bigoplus _{\substack{\zeta \in W\\ \mathrm{ord}(\zeta)\le p^n}}(M/A_n)_\zeta[p^\infty] \end{align*} $$

is annihilated by $p^{nd+2c}$ . It now follows from [Reference Cuoco and MonskyCM81, Lemma 2.7] that $\mathcal {M}_n[p^\infty ]$ is annihilated by $p^{2dn+2c}$ .

4.2.2 Applications

We apply the previous result to the following module arising from our graph theoretic setting. Write

(4-2) $$ \begin{align} M= \mathrm{Div}_\Lambda(X_\infty)/{\Pr}_\Lambda^{\mathrm{unr}}(X_\infty) \simeq \mathrm{Pic}_{\Lambda}(X_\infty), \end{align} $$

where we proved the isomorphism in Corollary 4.5. Set $z=s-r$ and write $(I_j,M_j)=(I_{v_{j+r}},\Lambda P_{v_{j+r}})$ for $1 \le j \le s-r$ . Note that as $1\le j\le s-r$ , the vertices $v_{j+r}$ are the ramified ones in $X_\infty /X$ . This defines a structure in the above sense; by Theorem 4.4,

$$ \begin{align*} \mathrm{Pic}(X_n) \otimes \mathbb{Z}_p &= \mathrm{Div}_{\Lambda}(X_\infty)/{\Pr}_\Lambda^{\mathrm{unr}}(X_\infty) + (\omega_n(T_1),\ldots, \omega_n(T_d))\,\mathrm{Div}_\Lambda(X_\infty) + {\Pr}_n^{\mathrm{ram}}(X_\infty) \\ & = M/A_n. \end{align*} $$

Theorem 4.16. Fix $d\geq 2$ . Let $X_\infty $ denote the derived graph $X(\mathbb {Z}_p^d,\mathcal {I},\alpha )$ and $X_\infty ^{{\mathrm {unr}}}$ denote the derived graph $X(\mathbb {Z}_p^d,\alpha )$ . Assume that all intermediate $X_n$ are connected. Let $f$ be a generator of the characteristic ideal of $\mathrm {Jac}_\Lambda (X_\infty )$ . Then,

$$ \begin{align*} \mathrm{ord}_p(\vert \mathrm{Jac}(X_n)\vert) = \mu(f) p^{nd}+\lambda(f) np^{(d-1)n}+O(p^{(d-1)n}). \end{align*} $$

Remark 4.17. Recall that the number of spanning trees of a graph $X$ is equal to the size of the Jacobian of $X$ . The above statement can hence be rewritten in terms of $\mathrm {ord}_p(\kappa (X_n))$ .

Proof. As before, let $I_v$ be the inertia subgroup associated with the vertex $v$ . We arrange the ramified vertices in increasing order of the ranks of the corresponding subgroups $I_v$ . In other words, we assume that $\mathrm {rank}(I_{v_j})\le \mathrm {rank}(I_{v_{j+1}})$ for $r+1\le j\le s$ . Let $s'\ge r+1$ be the smallest index such that $\mathrm {rank}(I_{v_{j}})\ge 2$ for all $j\ge s'$ . For ramified vertices $v_j$ with inertia rank 1, we denote the topological generator of $I_{v_j}$ by $\sigma _{v_j}$ and define

$$ \begin{align*} \widetilde{T}_{j}=\sigma_{v_{j},1}-1,\quad \text{ where } r+1\le j \le s'-1. \end{align*} $$

Note that $\widetilde {T}_j$ and $T_i$ are not related to each other per se, except that they lie in the same Iwasawa algebra. Decompose $\mathrm {Div}_\Lambda (X_\infty )$ as follows:

$$ \begin{align*} \mathrm{Div}_\Lambda(X_\infty)=\mathfrak{M}_1\oplus \mathfrak{M}_2\oplus \mathfrak{M}_3, \end{align*} $$

where:

• • $\mathfrak {M}_1$ is $\Lambda $ -free and the contribution to $\mathfrak {M}_1$ comes from the unramified vertices;

• • $\mathfrak {M}_2= \displaystyle \bigoplus _{j=1}^{s'-1}\Lambda /\widetilde {T}_j$ and the contribution to $\mathfrak {M}_2$ comes from those ramified vertices for which the rank of $I_v$ is 1;

• • $\mathfrak {M}_3$ is pseudo-null and $\mathbb {Z}_p$ -free.

For $1\le i\le 3$ , we denote by $\overline {\mathfrak {M}_i}$ the image of $\mathfrak {M}_i$ in $\mathrm {Pic}_\Lambda (X_\infty )$ . Clearly (by definition/construction), $\overline {\mathfrak {M}_3}$ is pseudo-null. By [Reference Cuoco and MonskyCM81, Lemma 3.1],

$$ \begin{align*} \mathrm{rank}_p(\pi_n(\overline{\mathfrak{M}_3}))\le \mathrm{rank}_p(\overline{\mathfrak{M}_3}/(\omega_n(T_1),\ldots,\omega_n(T_d))\overline{\mathfrak{M}_3})=O(p^{n(d-2)}). \end{align*} $$

By Lemma 4.15, we conclude that

$$ \begin{align*} \mathrm{ord}_p(\vert \pi_n(\overline{\mathfrak{M}_3})[p^\infty]\vert )=O(2nd p^{n(d-2)}). \end{align*} $$

Set $M'=M/\overline {\mathfrak {M}_3}$ , where $M$ is defined as in (4-2). Define $M^{\prime }_j$ as the image of the submodule $M_j = \Lambda P_{v_{j+r}}$ in $M'$ . Set $z'=s'-r-1$ ; the tuples $(I_{v_{j+r}}, M^{\prime }_j)$ for $1\le j\le z'$ define a structure on $M'$ . Let $\mathcal {M}^{\prime }_n$ be the module constructed in Definition 4.14 associated to this structure. The structure defined on $M'$ is an admissible structure in the sense of [Reference Cuoco and MonskyCM81, Definitions 4.1 and 4.6]. Writing $f$ to denote a generator of the characteristic ideal of $M = \mathrm {Pic}_{\Lambda }(X_\infty )$ , it follows from [Reference Cuoco and MonskyCM81, Theorems 4.12 and 4.13] and using the estimate for the contribution of $\overline {\mathfrak {M}_3}$ that

$$ \begin{align*} \begin{aligned} \mathrm{ord}_p(\vert \mathcal{M}_n[p^\infty]\vert)&=\mathrm{ord}_p(\vert \mathcal{M}^{\prime}_n[p^\infty]\vert )+O(p^{n(d-1)})\\ &=\mu(f) p^{nd}+\lambda(f) np^{(d-1)n}+O(p^{(d-1)n}). \end{aligned} \end{align*} $$

We have $\mathrm {char}_{\Lambda }(\mathrm {Pic}_{\Lambda }(X_\infty )) = \mathrm {char}_{\Lambda }(\mathrm {Jac}_{\Lambda }(X_\infty ))$ since $d\geq 2$ . Indeed, there is a short exact sequence

$$ \begin{align*} 0\longrightarrow \mathrm{Jac}_\Lambda(X_\infty)\longrightarrow \mathrm{Pic}_\Lambda(X_\infty)\longrightarrow \mathbb{Z}_p\longrightarrow 0. \end{align*} $$

For $d\ge 2$ , a free $\mathbb {Z}_p$ -module of finite rank is always pseudo-null. The result follows since $M$ and $M'$ have the same characteristic ideal (because $\overline {\mathfrak {M}_3}$ is pseudo-null).

Remark 4.18. The second-named author and Kleine obtained a growth formula for unramified $\mathbb {Z}_p^d$ -covers in [Reference Kleine and MüllerKM24]. In contrast to the above result, the result in loc. cit. does not involve an error term. The main difference is that the situation of branched coverings requires the use of structures in the above sense. For unramified coverings, this difficulty does not occur which allows the authors of loc. cit. to derive an explicit formula without an error term.