Kida’s formula for graphs with ramifications

We begin with a brief review of classical Iwasawa theory. Let $p$ be a prime number. In classical Iwasawa theory, we mainly deal with a $\mathbb{Z}_{p}$-extension $K_{\infty}/K$ of number fields, that is,

where $K_{n}/K$ is a $\mathbb{Z}/p^{n}\mathbb{Z}$-extension. The so-called Iwasawa class number formula due to Iwasawa [3, Theorem 11] (see also Washington [13, Theorem 13.13]) claims

for some integers $\lambda=\lambda(K_{\infty}/K)\geq 0$, $\mu=\mu(K_{\infty}/K)\geq 0$, and $\nu$, which are called the Iwasawa invariants. Here, $\operatorname{Cl}(K_{n})$ denotes the class group of $K_{n}$ and $\operatorname{ord}_{p}(-)$ denotes the $p$-adic valuation that is normalized by $\operatorname{ord}_{p}(p)=1$.

Kida [7] obtained a formula that describes the behavior of the Iwasawa invariants when $K_{\infty}/K$ varies. To be concrete, let $G$ be a finite $p$-group and $\widetilde{K}_{\infty}/\widetilde{K}$ be another $\mathbb{Z}_{p}$-extension such that $\widetilde{K}_{n}/K$ is a $\mathbb{Z}/p^{n}\mathbb{Z}\times G$-extension for any $n\geq 0$. Then conceptually the formula claims that $\mu(K_{\infty}/K)=0$ is equivalent to $\mu(\widetilde{K}_{\infty}/\widetilde{K})=0$, in which case $\lambda(\widetilde{K}_{\infty}/\widetilde{K})$ can be described by using $\lambda(K_{\infty}/K)$ together with the ramification information in $\widetilde{K}_{\infty}/K_{\infty}$. (To be precise, the original Kida’s formula only concerns the minus components for the cyclotomic $\mathbb{Z}_{p}$-extensions of CM-fields.)

The main object in this paper is graphs, which we assume to be finite and connected in this introduction. For a graph $X$, let $\kappa(X)$ be the number of spanning trees of $X$. By Kirchhoff’s theorem, the number $\kappa(X)$ is equal to the order of the Jacobian group $\operatorname{Jac}(X)$ of $X$ (see §2.1). Note that $\operatorname{Jac}(X)$ is also known as the Picard group of degree zero, the sandpile group, and the critical group. This interpretation enables us to study $\kappa(X)$ by an algebraic investigation of $\operatorname{Jac}(X)$.

Let $X_{\infty}/X$ be an unramified (or unbranched) $\mathbb{Z}_{p}$-covering of graphs, that is, we are given a family of graphs

where $X_{n}/X$ is an unramified $\mathbb{Z}/p^{n}\mathbb{Z}$-covering. This $X_{n}$ is called the $n$-th layer of $X_{\infty}/X$. Then an analogue of the Iwasawa class number formula, due to Gonet [2, Theorem 1.1] or McGown–Valliéres [8, Theorem 6.1] (see also the author’s article [5, §8.1]), claims

for some integers $\lambda=\lambda(X_{\infty}/X)\geq 0$, $\mu=\mu(X_{\infty}/X)\geq 0$, and $\nu$.

To discuss Kida’s formula, let $G$ be a finite $p$-group. Let $\widetilde{X}_{\infty}/\widetilde{X}$ be another unramified $\mathbb{Z}_{p}$-covering such that the $n$-th layer $\widetilde{X}_{n}$ is an unramified $\mathbb{Z}/p^{n}\mathbb{Z}\times G$-covering of $X$ for any $n\geq 0$. Then the analogue of Kida’s formula, due to Ray–Valliéres [10, Theorem 4.1] (see also [5, §8.2]), says that $\mu(X_{\infty}/X)=0$ is equivalent to $\mu(\widetilde{X}_{\infty}/\widetilde{X})=0$, in which case we have

with $[\widetilde{X}_{\infty}:X_{\infty}]=\#G$. Compared to the original formula of Kida, the ramification terms disappear naturally because $\widetilde{X}_{\infty}/X_{\infty}$ is assumed to be unramified.

As we have emphasized, Iwasawa theory for graphs has been developed mainly for unramified coverings so far. In a latest paper of Gambheera–Valliéres [1], they initiated an Iwasawa theoretic study of ramified (or branched) coverings of graphs (see §2.2). As a fundamental result, for a possibly ramified $\mathbb{Z}_{p}$-covering $X_{\infty}/X$, they proved that the aforementioned analogue of the Iwasawa class number formula is still valid literally (see Theorem 4.1).

In this paper, we establish Kida’s formula for graphs with ramifications. As above, let $G$ be a finite $p$-group and we consider $\mathbb{Z}_{p}$-coverings $X_{\infty}/X$ and $\widetilde{X}_{\infty}/\widetilde{X}$ such that $\widetilde{X}_{n}/X$ is a $\mathbb{Z}/p^{n}\mathbb{Z}\times G$-covering for any $n\geq 0$. Let us stress that we allow the ramifications both in $\widetilde{X}_{\infty}/X_{\infty}$ and in $X_{\infty}/X$. The main theorem is the following:

In §2, we introduce basic notations concerning graphs. In §3, we study precise functoriality properties of Picard groups, which is essential to prove the main theorem. Then in §4, we prove the main theorem. Finally, in §5, we observe very simple examples.

We first introduce the notions of graphs and the Jacobian groups. See [1, §§2.1–2.2] or the author’s article [5, §2] for the details. We use Serre’s formalism of graphs [11, Chapter I, §2.1].

Each $e\in\mathbb{E}_{X}$ is regarded as an edge that connects $s(e)$ to $t(e)$, and $\overline{e}$ is the opposite of $e$. For each $v\in V_{X}$, we write $\mathbb{E}_{X,v}$ for the set of $e\in\mathbb{E}_{X}$ such that $s(e)=v$. Note that we allow multi-edges and loops, so our graphs may be called multigraphs. We will usually study connected graphs, which means that any two vertices can be connected by a path of edges.

It is known that $\operatorname{Jac}(X)$ is a finite abelian group. Indeed, by Kirchhoff’s theorem, the order of $\operatorname{Jac}(X)$ is equal to the number $\kappa(X)$ of spanning trees of $X$. It also follows that the kernel of $\mathcal{L}_{X}$ is a free $\mathbb{Z}$-module of rank one with a basis $\sum_{v\in V_{X}}[v]$. Therefore, we have two fundamental exact sequences

where $\iota_{X}$ sends $1$ to $\sum_{v\in V_{X}}[v]$, and

We introduce the notion of (possibly ramified) Galois coverings, basically following [1, §3.1]. We begin with the definition of morphisms of graphs. Let $X$ and $Y$ be finite graphs.

A vertex $w\in V_{Y}$ is said to be lying above $v\in V_{X}$ if $f_{V}(w)=v$. The number $m_{w}(Y/X)=m_{v}(f)$ is referred to as the ramification index of $w$. Naturally we say $w$ is unramified if $m_{w}(Y/X)=1$, and a vertex $v$ of $X$ is unramified if all vertices of $Y$ lying above $v$ are unramified. An unramified covering is by definition a covering such that any vertices are unramified.

Now we go on to the definition of Galois coverings. The general definition does not explicitly written in [1]; instead, they only deal with derived graphs associated to voltage graphs (see §2.3). In this paper, we introduce a reasonable definition of Galois coverings. We will see that actually all of them can be constructed as derived graphs (Proposition 2.11).

In other words, the action of $\Gamma$ on $\mathbb{E}_{Y}$ is free and $f$ induces an isomorphism between the quotient graph $\Gamma\backslash Y$, defined in an obvious way, and $X$.

It is easy to see that a $\Gamma$-covering $Y/X$ is actually a covering (cf. [1, Proposition 3.3(1)]). The ramification index $m_{w}(Y/X)$ coincides with the order of the stabilizer subgroup $\operatorname{Stab}_{\Gamma}(w)$ of $w$. This depends only on the vertex $v=f_{V}(w)\in V_{X}$ since the action of $\Gamma$ on $f_{V}^{-1}(v)$ is transitive. We then define the ramification index of $v\in V_{X}$ as $m_{v}(Y/X)=m_{w}(Y/X)$ by using any $w\in V_{Y}$ lying above $v$.

Finally, let us introduce the notion of infinite Galois coverings.

For instance, for $\Gamma=\mathbb{Z}_{p}$, the open subgroups are precisely $\{\Gamma^{p^{n}}\}_{n\geq 0}$, so a $\Gamma$-covering of $X$ consists of a system of graphs

such that $X_{n}/X$ is a $\mathbb{Z}/p^{n}\mathbb{Z}$-covering.

To deal with (finite or infinite) Galois coverings, it is convenient to use voltage graphs and derived graphs. See [5, §5] for the unramified cases, and [1, §4] for the ramified cases.

Firstly, we introduce the voltage graphs.

Now we introduce the derived graphs.

The group $\Gamma$ acts on the graph $X(\Gamma,\mathcal{I})$ from the left in the natural way. We also have a natural morphism $X(\Gamma,\mathcal{I})\to X$ by projections to the second components. It is quite easy to see that this morphism is indeed a $\Gamma$-covering. More generally, given a normal subgroup $H$ of $\Gamma$, setting $(X,\Gamma/H,\alpha_{H},\mathcal{I}_{H})$ as in Definition 2.8, we have an $H$-covering $X(\Gamma,\mathcal{I})\to X(\Gamma/H,\mathcal{I}_{H})$.

Now let us show that any Galois covering arises from a voltage graph. We begin with the finite case.

Let $f:Y\to X$ be a covering (Definition 2.4) between finite connected graphs. We define the degree of the covering $Y/X$ as the number

which is independent of $v\in V_{X}$ (see [1, §3.1]; here we need the connectedness of $X$). Note that, for any edge $e\in\mathbb{E}_{X}$, we have

that is, the fiber of an arbitrary edge of $X$ consists of $[Y:X]$ edges of $Y$.

As in [1, §3.2], let

be the $\mathbb{Z}$-homomorphisms such that

for each $w\in V_{Y}$.

Let $(X,\Gamma,\alpha,\mathcal{I})$ be a voltage graph with $\Gamma$ finite.

Set $\mathbb{Z}[\Gamma/I_{v}]=\mathbb{Z}[\Gamma]\otimes_{\mathbb{Z}[I_{v}]}\mathbb{Z}$, which we regard as a left $\mathbb{Z}[\Gamma]$-module. Then we may identify

as $\mathbb{Z}[\Gamma]$-modules. Here, the first isomorphism is given by $(\gamma I_{v},v)\leftrightarrow\gamma[v]$ and the second by simply omitting $[v]$ to ease the notation. Then, associated to the endomorphism $\mathcal{L}_{X(\Gamma,\mathcal{I})}$ on $\operatorname{Div}(X(\Gamma,\mathcal{I}))$, we introduce an endomorphism $\mathcal{L}_{X,\Gamma,\mathcal{I}}$ on $\bigoplus_{v\in V_{X}}\mathbb{Z}[\Gamma/I_{v}][v]$ (or $\bigoplus_{v\in V_{X}}\mathbb{Z}[\Gamma/I_{v}]$). In other words, $\mathcal{L}_{X,\Gamma,\mathcal{I}}$ is defined by a commutative diagram