Fitting ideals of Jacobian groups of graphs

The Jacobian group of a graph is a finite abelian group through which we can study the graph in an algebraic way. When the graph is a finite abelian covering of another graph, the Jacobian group is equipped with the action of the Galois group. In this paper, we study the Fitting ideal of the Jacobian group as a module over the group ring. We also study the corresponding question for infinite coverings. Additionally, this paper includes module-theoretic approach to Iwasawa theory for graphs.


Introduction
For a (finite) graph X, let Jac(X) denote its Jacobian group, which is also known as the sandpile group.It is known that Jac(X) is a finite abelian group.See §2 for basic notions about graphs and their Jacobian groups.
Let Y /X be an abelian covering of connected graphs and Γ its Galois group.Then the Jacobian group Jac(Y ) is naturally equipped with a Z[Γ]-module structure, which is the main theme of the present paper.More concretely, we focus on its Fitting ideal (see §A.1 for the definition).The main result of this paper gives a complete description of the Fitting ideal Fitt Z[Γ]/(N Γ ) (Jac(Y )/N Γ Jac(Y )).
Here, N Γ denotes the norm element of Γ. See §1.1 for the concrete statement.This result can be regarded as an analogue of a result of Atsuta and the author [1] in number theory, as explained in Remark 1.3 below.
We will also study infinite abelian coverings in a sense in §6.In this case, we obtain a complete description of the Fitting ideal of the associated Jacobian group itself, without dividing by the Jacobian group of the base graph.
The study of infinite coverings of graphs is morally an analogue of Iwasawa theory in number theory.Recently such an analogue of Iwasawa theory for graphs is developed (e.g., Gonet [5] and a series of work of Vallières including [14]).The present work is directly inspired by this situation surrounding graph theory and Iwasawa theory.Moreover, in §8, as a digression, we give concise proofs of the analogues of the Iwasawa class number formula and of Kida's formula.Those results are already proved by others, but the author hopes that this makes Iwasawa theory for graphs more accessible for those who are familiar with techniques in Iwasawa theory.
1.1.Main result for finite coverings.Now we state the main result.Let Y /X be an abelian covering of connected graphs and Γ its Galois group.In §2, we will introduce an element Z Y /X ∈ Z[Γ], which is related to the (equivariant) Ihara zeta function via the threeterm determinant formula (see §5.3).
Let us take a decomposition Γ = ∆ 1 × • • • × ∆ s , where ∆ l is a cyclic group of order n l ≥ 1 for each 1 ≤ l ≤ s.For each l, we fix a generator σ l of ∆ l and define τ l , Then ν l is the norm element of ∆ l .The norm element of Γ, defined by N Γ = γ∈Γ γ, satisfies We also define an element D l ∈ Z[∆ l ] by The main result is the following.The proof will be given in § §3-4.
Theorem 1.1.We have Here the right hand side, which is literally an ideal of Z[Γ], is regarded as an ideal of Z[Γ]/(N Γ ) via the natural projection.
Remark 1.2.In the theory of Euler systems in number theory, the element D l is often called the Kolyvagin derivative operator and plays an important role.It seems to be interesting that this element is useful in computation that is not directly related to the theory of Euler systems.Indeed, as far as the author is aware, it is the first appearance of the Kolyvagin derivative operator in computation of Fitting ideals.
In Proposition 3.5, we will show an isomorphism Jac(X) ≃ N Γ Jac(Y ).Therefore, Theorem 1.1 can be viewed as a description of the Fitting ideal of Jac(Y )/ Jac(X).Unfortunately, determining Fitt Z[Γ] (Jac(Y )) itself seems to be out of our reach.
Remark 1.3.Theorem 1.1 can be regarded as an analogue of a result of Atsuta and the author [1] on the Fitting ideal of class groups in number theory.Let us roughly review the result.Let L/K be a finite abelian extension of number fields such that K is totally real and L is a CM-field.We consider the T -ray class group Cl T L of L, where T is an auxiliary finite set of finite primes of K. Then in [1], (assuming the relevant equivariant Tamagawa number conjecture), we gave a complete description of the Fitting ideal Fitt Z[Gal(L/K)] − (Cl T,− L ), where the superscript (−) − denotes the minus component with respect to the complex conjugation.The plus component seems to be out of our reach.This is analogous to the situation in Theorem 1.1.
1.2.Idea of proof.In number theory, there are already a number of studies on the Fitting ideals of arithmetic modules.In particular, in the field of Iwasawa theory, the Fitting ideals of Iwasawa modules have been studied by Greither-Kurihara (e.g., [7], [8] with Tokio, etc.).
In [11], the author introduced the "shift theory" of Fitting ideals, which we will review in §A.2.It is a useful technique to compute the Fitting ideals of arithmetic modules.Roughly speaking, the technique has two steps: (1) we study the arithmetic module and obtain a description of the Fitting ideal using an explicit algebraic factor, and (2) we compute the algebraic factor in a purely algebraic way.
The first applications of this theory in [11] were to Iwasawa theory, i.e., infinite extensions of number fields.Then Atsuta and the author [1] made an application to class groups of number fields as explained in Remark 1.3.
In this paper, we again employ the shift theory as the basic technique to prove both Theorem 1.1 and the result for infinite coverings.Therefore, we follow the two steps explained above.See Theorems 3.4 and 4.1 for the first and second steps, respectively.
The algebraic factor to be computed in Theorem 4.1 seems to be a new one as discussed in Remark 1.2.Moreover, the proof of the formula is not easy, and it is the most technical part in this paper.We employ a combinatoric method (using graph theory), though the problem is purely algebraic.Note that Greither-Kurihara-Tokio [8] also applied graph theory to compute Fitting ideals.
On the other hand, the algebraic factor to be computed for the infinite covering case does not seem to be very new.Indeed, it can be computed by using a formula that has been obtained in [1].However, a relatively minor issue is that in this paper we deal with commutative rings that are not necessarily noetherian.Therefore, we have to generalize the shift theory, which was developed only over noetherian rings in [11].This is explained in §A.2.

1.3.
Organization of this paper.In §2, we introduce basic notions about graphs.Then in §3, we reduce the proof of Theorem 1.1 to an algebraic problem.The algebraic problem is solved in §4.
After preliminaries in §5 on voltage graphs and their derived graphs, in §6 we state and prove the result for infinite coverings.
In §7, we observe the self-duality of the Jacobian groups, which is a contrast to the analogue in number theory.In §8, we explain the module-theoretic approach to Iwasawa theory for graphs.Both § §7 and 8 can be read independently.Finally in §A, we review the definition of the Fitting ideals and the shift theory.

Graphs and their Jacobian groups
In this section, we introduce basic notions about graphs and their Jacobian groups.In this paper, graphs are always assumed to be finite.We allow graphs to have multi-edges and loops (so one may call them multigraphs).More precisely, we define graphs as follows, using Serre's formalism [18, Chapter I, §2.1].Definition 2.1.A graph X consists of a finite set V X of vertices, a finite set E X of edges, an automorphism of E X denoted by e → e, and two maps s X , t X from E X to V X (often abbreviated to s = s X and t = t X ), satisfying the following: • For any e ∈ E X , we have e = e and e = e, that is, the automorphism e → e is an involution of E X without fixed points.• For any e ∈ E X , we have s(e) = t(e) and t(e) = s(e).
Each element e ∈ E X is regarded as an edge from s(e) to t(e), and e is regarded as the inverse of e.Let us write E X for the quotient set of E X obtained by identifying e and e.Then we have a canonical two-to-one map from E X to E X , so their cardinalities satisfy which is the set of edges that start from v.
This formalism involving E X and the involution will be useful for introducing voltage graphs in §5.On the other hand, we can regard X as an undirected (multi-)graph through the quotient set E X .The notions that we will introduce in this section are essentially defined for the associated undirected graph structure.
In this paper, for simplicity, we usually deal with connected graphs.Being connected is not an essential assumption since the general case can be easily deduced.
In the rest of this section, let X be a connected graph.
Definition 2.2.We define the divisor group Div(X) as the free Z-module on the set V X , namely, Div(X) = Let us write deg X : Div(X) → Z for the Z-homomorphism that sends [v] to 1 for any v ∈ V X .We define Div 0 (X) as the kernel of deg X .
We obviously have an exact sequence for v ∈ V X , and L X = D X − A X .The presentation matrices of L X , A X , and D X (with respect to the basis {[v]} v∈V X ) are called the Laplacian matrix, the adjacency matrix, and the degree matrix, respectively.
It is easy to see deg X •L X = 0, that is, the image of L X is contained in Div 0 (X).Therefore, we can make the following definition.Definition 2.4.We define the Jacobian group Jac(X) and the Picard group Pic(X) as the cokernels Jac(X) = Cok(L X : Div(X) → Div 0 (X)) and Pic(X) = Cok(L X : Div(X) → Div(X)).
Therefore, by (2.1), we have an exact sequence A fundamental property of the Jacobian group is the following.
Theorem 2.5 (Kirchhoff's matrix tree theorem).The Jacobian group Jac(X) is a finite Z-module.In fact, the cardinality of Jac(X) is equal to the number of spanning trees of X.
Let us observe the following homological description of the Jacobian group.
Lemma 2.6.Let ι X : Z → Div(X) be the Z-homomorphism that sends is a complex that is acyclic except for the right Div(X), at which the homology group is isomorphic to Jac(X).
Proof.We only have to check Im(ι X ) = Ker(L X ); the other assertions are clear.It is easy to see that L X • ι X = 0, that is, Im(ι X ) ⊂ Ker(L X ).Thanks to Theorem 2.5, the Z-ranks of Im(ι X ) and Ker(L X ) are the same.Moreover, the definition of ι X implies that Im(ι X ) is a saturated submodule of Div(X) (i.e., the cokernel of ι X is Z-torsion-free).These observations imply Im(ι X ) = Ker(L X ).Now we quickly recall the notion of Galois coverings of connected graphs (see Terras [19,Chapter 13] or [17, §2.1] for the details).A covering Y /X of connected graphs means that we are given a covering map π : Y → X, which is surjective and locally isomorphic.The degree of Y /X is defined as #(π −1 (v)) for any choice of v ∈ V X .The Galois group Γ of Y /X is defined as the automorphism group of Y that respects the covering map.We say that a covering Y /X is Galois (or normal) if the order #Γ of Γ is equal to the degree of Y /X.We say Y /X is abelian if Γ is abelian.Definition 2.7.Let Y /X be a Galois covering of connected graphs with Galois group Γ.It is easy to see that Div(Y ) is a free Z[Γ]-module and the endomorphism L Y on Div(Y ) is a Z[Γ]-homomorphism.In case Γ is abelian, we define By the definitions of Pic(Y ) and of Fitting ideals, we have when Γ is abelian.In §5.3, we will see a relation between Z Y /X and the Ihara zeta function.

Reduction to an algebraic problem
In this section, we describe the Fitting ideal that is concerned in Theorem 1.1 by using a shifted Fitting ideal of an explicit module.
We begin with an elementary lemma.Let Γ be a finite group and write where we set Lemma 3.1.We have We have an exact sequence of R-modules We have natural isomorphisms through which ι * is identified with the multiplication by #Γ on Z.Therefore, we obtain the lemma.
In the rest of this section, let Y /X be a Galois covering of connected graphs.We write Γ for its Galois group and use the same notation R and R as above.
The following is the key ingredient to prove Theorem 1.1.
Proposition 3.2.We have an exact sequence of finite R-modules Proof.Let us consider the exact sequence (2.2) for Y instead of X.By taking Tor R * (R, −), we obtain an exact sequence By applying Lemma 3.1, we obtain the proposition.
Let us study R ⊗ R Pic(Y ).Lemma 3.3.We have a short exact sequence where L Y denotes the homomorphism induced by L Y .
Proof.By the definition of Pic(Y ), the cokernel of L Y is isomorphic to R ⊗ R Pic(Y ), which is finite by Theorem 2.5 and Proposition 3.2.Then the injectivity of L Y follows since it is an endomorphism of a free Z-module of finite rank.
We write Z Y /X ∈ R for the image of Z Y /X ∈ R. By applying the shift theory (see §A.2), we obtain the following.Theorem 3.4.Suppose that Y /X is an abelian covering.Then we have Proof.As in §A.2, let P R be the category of finite R-modules whose projective dimensions are ≤ 1.By Lemma 3.3, we see that Therefore, the theorem follows from the definition of Fitt R (−) and Proposition 3.2.We will determine Fitt Before closing this section, we show a proposition that provides an interpretation of R ⊗ R Jac(Y ).In the course of the proof, we reproduce the exact sequence in Proposition 3.2.Proposition 3.5.We have a natural isomorphism Proof.We make use of Lemma 2.6 for both X and Y .Let us consider the following diagram involving those complexes.
Here, the injective homomorphism Div(X) ֒→ Div(Y ) is defined by sending where π : Y → X is the covering map.Then the image of this homomorphism is N Γ Div(Y ).
The vertical sequences are all exact and the lower horizontal sequence is the induced complex.
We regard this large diagram as a short exact sequence of complexes.Recall that the kernel and the cokernel of L Y are determined in Lemma 3.3.Then, taking the homology groups, we obtain an exact sequence The construction of the embedding of Jac(X) into Jac(Y ) shows that the image coincides with N Γ Jac(Y ), as claimed.Moreover, this reproduces the exact sequence in Proposition 3.2.

Algebraic result
Let Γ be any finite abelian group.Set R = Z[Γ] and R = Z[Γ]/(N Γ ).The goal of this section is to determine the factor Fitt which can be proved by a direct computation.This property is also important in the theory of Euler systems.The result is the following.
Theorem 4.1.We have Then clearly Theorem 1.1 follows from Theorems 3.4 and 4.1.The rest of this section is devoted to the proof of Theorem 4.1.
First we describe Fitt R (Z/(#Γ)Z) by using an (unshifted) Fitting ideal.Let be the augmentation ideal, which we can regard as an R-module.

By the definition of Fitt
The proof of this proposition will be given in §4.3.Before that, we establish a key technical proposition in §4.2.

Key proposition.
Let s ≥ 0 be an integer and let R s = Z[T 1 , . . ., T s ] be the polynomial ring in s indeterminates.Let us construct a free resolution of Z over R s by using tensor products of complexes.This, or rather the more complicated construction in §4.3 below, is inspired by [7] and is also used in [11], [1], etc.
For each 1 ≤ l ≤ s, we have an exact sequence Here, x l denotes an indeterminate, so Z[T l ]x l is a free Z[T l ]-module of rank one, and the map labeled where d 0 sends T l to 0 for any l, d 1 is determined by for 1 ≤ l < l ′ ≤ s (there are other choices of signs, but that does not matter).
We write N s (T 1 , . . ., T s ) for the presentation matrix of d 2 with respect to the basis {x l x l ′ | 1 ≤ l < l ′ ≤ s} and x 1 , . . ., x s .Note that we do not determine the order of the rows because it does not matter at all.Indeed, in the following we will sometimes choose various orders of bases that are suitable for computation.
Example 4.4.When s = 0 (resp.s = 1), by definition the source of d 2 is the zero module, so d 2 = 0 and N 0 () (resp.N 1 (T 1 )) is the empty matrix.When s = 2, we have When s = 3, we have Here, we use the order x 2 x 3 , x 1 x 3 , x 1 x 2 for the basis.When s = 4, we have Here, we use the order x 1 x 2 , x 1 x 3 , x 1 x 4 , x 2 x 3 , x 2 x 4 , x 3 x 4 for the basis.
For a matrix H, which is identified with a homomorphism between free modules, we have the associated ideal Fitt(H) defined as in Definition A.1.
The following formula plays a key role in the proof of Proposition 4.3.It is the most technical result in this paper.Proposition 4.5.We have Here, B 1 , . . ., B s are indeterminates that have no relation with T 1 , . . ., T s , i.e., we work over the polynomial ring The rest of this subsection is devoted to the proof of this proposition.The cases s = 0, s = 1 are clear.Let us suppose s ≥ 2.
Recall that the rows of N s (T 1 , . . ., T s ) are labeled x l x l ′ (1 ≤ l < l ′ ≤ s).We also attach a label y to the last row with #A = s, let N A be the s × s submatrix that is constructed by picking up the rows whose labels are in A. Then the left hand side in Proposition 4.5 is generated by det(N A ) for all such A.
For such an A, let us construct an undirected simple graph G A that has s vertices x 1 , . . ., x s so that x l x l ′ ∈ A if and only if x l and x l ′ are adjacent.Then the number of the edges of G A is either s − 1 or s; indeed, it is s − 1 if and only if y ∈ A. This construction gives a one-to-one correspondence between the set and the set { simple graph structures on the set of vertices {x 1 , . . ., x s } with s − 1 or s edges }.
We shall describe det(N A ) by using information about the associated graph G A .Claim 4.6.We have det(N A ) = 0 unless G A is a tree.
Proof.Suppose that G A is not a tree.Since the number of the edges of G A is at least s − 1, this assumption is equivalent to that G A has a cycle.In other words, by permutation of the indices, we may assume that {x 1 x 2 , x 2 x 3 , . . ., x r−1 x r , x r x 1 } ⊂ A for some 3 ≤ r ≤ s.In this case, the matrix if we use the order x 1 , x 2 , . . ., x r , * , . . ., * of the vertices and the order of the rows in A ( * denotes unspecified things).It is easy to see that the r × r matrix in the upper left has determinant 0. Therefore, the claim follows.
By Claim 4.6, we only have to deal with the case where G A is a tree.Note that then the number of the edges of G A is s − 1, i.e., y ∈ A. For each 1 ≤ l ≤ s, let deg A (x l ) ≥ 1 be the degree of x l in the graph G A .By definition, deg A (x l ) is the number of vertices that are adjacent to x l in G A .
Claim 4.7.If G A is a tree, then we have Proof.Let us write simply deg(−) instead of deg A (−).First we show that the claim follows from the following weaker claim: there exist signs ǫ 1 , . . ., ǫ s ∈ {±1} such that we have Suppose that such a family {ǫ l } l exists.By Claim 4.6, for any l = l ′ , we know that det(N A ) vanishes if we set B l = T l ′ , B l ′ = −T l , and the other B's to be zero.Therefore, Let us show the weaker claim.By the cofactor expansion of det(N A ) with respect to the final row (labeled y), we have where δ l denotes the cofactor of B l (i.e., the determinant of the submatrix obtained by eliminating the last row y and the l-th column).Here and henceforth, we ignore the sign of δ l , which does not matter for the weaker claim.Then it is enough to show that, for any fixed l, the cofactor δ l coincides with up to sign.Fix l.Let us reorder the vertices x 1 , . . ., x s as follows.We regard x l as the root of the tree G A .Recall that then the depth of each vertex x k , denoted by depth(x k ), is defined as the length of the unique path from the root x l to x k .(We set depth(x l ) = 0.) Now we reorder the vertices of the rooted graph G A as x σ(1) , x σ(2) , . . ., x σ(s) , where σ is a permutation of the set {1, 2, . . ., s} such that depth(x σ(k) ) ≤ depth(x σ(k+1) ) for every 1 ≤ k ≤ s − 1.We necessarily have σ(1) = l, but this σ is not unique in general.
For each 2 ≤ k ≤ s, let 1 ≤ k ≤ s be the index such that x σ(k) is the parent of x σ(k) .It is the unique vertex that is adjacent to x σ(k) and whose depth is less than that of x σ(k) .Note that we have 1 ≤ k < k.Now, to compute the cofactor δ l , we use the order of the vertices and the order of the edges.Then, thanks to k < k, the matrix whose determinant is δ l is lower triangular and we obtain Since x σ(k) is the parent of x σ(k) , for each 1 ≤ l ′ ≤ s, the exponent of the indeterminate T l ′ in this product is equal to the number of the children of the vertex x l ′ .If the vertex x l ′ is the root, i.e., if l ′ = l, then the number of children is equal to deg(x l ).Otherwise, i.e., if l ′ = l, the number of children is equal to deg(x l ′ ) − 1.This completes the proof of the claim.
Example 4.8.We illustrate the above proof by an example.Let s = 5 and consider A = {x 1 x 2 , x 1 x 3 , x 1 x 4 , x 2 x 5 , y}, so

In this case we have deg
To consider the cofactor δ 1 of B 1 , we regard x 1 as the root.Then depth(x 2 ) = depth(x 3 ) = depth(x 4 ) = 1 and depth(x 5 ) = 2, so we use the order x 2 , x 3 , x 4 , x 5 for the vertices.We have 2 = 1, 3 = 1, 4 = 1, 5 = 2, so we use the order x 1 x 2 , x 1 x 3 , x 1 x 4 , x 2 x 5 for the edges.Then we obtain Here and in the following examples, we omit writing ± before the indeterminates since we may ignore the signs.
Similarly, to consider δ 2 , we use x 1 , x 5 , x 3 , x 4 for the vertices and x 1 x 2 , x 2 x 5 , x 1 x 3 , x 1 x 4 for the edges, and obtain To consider δ 3 , we use the orders x 1 , x 2 , x 4 , x 5 and x 1 x 3 , x 1 x 2 , x 1 x 4 , x 2 x 5 and obtain Now we return to the general case.By Claims 4.6 and 4.7, the Fitting ideal to be computed in Proposition 4.5 is generated by where G A varies all tree structures on the set of vertices {x 1 , . . ., x s }.Note that we have s l=1 It remains only to show that, conversely, the tuple (deg A (x 1 ) − 1, . . ., deg A (x s ) − 1) can be any tuple whose sum is s − 2. This assertion follows from the following.Claim 4.9.Let s ≥ 2 and let f 1 , . . ., f s be integers such that f l ≥ 0 and Then there exists a tree structure on the set of vertices x 1 , . . ., x s such that the degree of x l equals f l + 1 for all 1 ≤ l ≤ s.
Proof.We argue by the induction on s.When s = 2, we must have f 1 = f 2 = 0, and the unique tree structure satisfies the property.Suppose s ≥ 3.By f 1 + • • • + f s = s − 2, we have f l = 0 and f l ′ ≥ 1 for some l and l ′ , so we may assume that f s = 0 and f 1 ≥ 1.By the induction hypothesis, there exists a tree structure on x 1 , . . ., x s−1 such that the degree of x 1 is f 1 and the degree of x l is f l + 1 for 2 ≤ l ≤ s − 1.Then we obtain the desired graph by simply connecting x 1 and x s .This completes the proof of Proposition 4.5.We first construct a free resolution of the augmentation ideal I ⊂ R = Z[Γ] over R in a similar way as in §4.2.Note that the idea is already used in previous work such as [1].
For each 1 ≤ l ≤ s, we have an exact sequence where x l denotes an indeterminate, the map ν l sends x 2 l to ν l x l , τ l sends x l to τ l , and the map Z[∆ l ] → Z is the augmentation map.

By taking the tensor product of the complexes [Z[∆
where d 0 is the augmentation map, d 1 is determined by and d 2 is determined by Therefore, the presentation matrix of d 2 is of the form Here, N s (τ 1 , . . ., τ s ) denotes the matrix obtained by setting T l = τ l in the matrix N s (T 1 , . . ., T s ) constructed in §4.2.
We have constructed a presentation M s (ν 1 , . . ., ν s , τ 1 , . . ., τ s ) of the Z[Γ]-module I = Ker(d 0 : R → Z).Our next task is to construct a presentation of I/R(#Γ − N Γ ).For that purpose, we define elements b Proof.By using the identity (4.1), we can compute Thus we obtain the lemma.
Then Lemma 4.10 implies that To compute the Fitting ideal of this matrix, we first observe Here, a runs over the subsets of {1, 2, . . ., s} with #a = j and define a 1 , . . ., a s by requiring We can show (4.2) directly by using the identity ν l τ l = 0 (see [1, Proposition 4.7] for a similar reasoning).
Let us compute the right hand side of (4.2) by using Proposition 4.5.When j = s, we have a = {1, 2, . . ., s}, so the term is When j = s − 1, we have {1, 2, . . ., s} \ a = {l} for some 1 ≤ l ≤ s, and then the term is Finally we consider 0 ≤ j ≤ s − 2. For each a with #a = j, the term can be computed as where the first equality follows from the identity τ l ν l = 0; the second from Lemma 4.10.
It is easy to see that the ideal generated by (4.3), (4.4), and (4.5) coincides with the right hand side of the proposition (observe that, thanks to (4.3), the N Γ in (4.5) can be ignored).This completes the proof of Proposition 4.3.This also completes the proof of Theorem 1.1.

Voltage graphs and derived graphs
To formulate the setup for infinite coverings, it is convenient to use the notion of voltage graphs and their derived graphs.Definition 5.1.A voltage graph (X, Γ, α) consists of a graph X, a group Γ, and a map α : E X → Γ satisfying α(e) = α(e) −1 for any e ∈ E X .We do not assume that Γ is finite or abelian unless explicitly stated.
The group Γ naturally acts on the graph X(Γ) from the left.Moreover, X(Γ) is a covering of X via the natural projection and the action of Γ respects this covering structure.
If X(Γ) is connected (so X is also connected), then X(Γ) is actually a Galois covering of X whose Galois group is Γ.Conversely, if we are given a Galois covering Y /X of connected graphs, then there exists a voltage graph structure (X, Γ, α) with Γ the Galois group such that X(Γ) and Y are isomorphic as coverings of X.In a nutshell, the notion of derived graphs covers the notion of Galois coverings of connected graphs.

Definition 5.3. We define Z[Γ]-homomorphisms
for any v ∈ V X and L X,Γ = id ⊗D X − A X,Γ , where D X : Div(X) → Div(X) is as in Definition 2.3.Note that when Γ is trivial, these maps A X,Γ and L X,Γ are naturally identified with the maps A X and L X in Definition 2.3.
The following lemma is easily proved.
where the vertical isomorphisms are defined by sending γ ⊗ [v] to [(γ, v)].In particular, Pic(X(Γ)) is isomorphic to the cokernel of the homomorphism L X,Γ .Definition 5.5.Suppose that Γ is abelian.We define Suppose that Γ is finite and abelian and that X(Γ) is connected.Then Definition 2.7 gives us an element Z X(Γ)/X ∈ Z[Γ].It is related to the element Z X,Γ simply by Z X(Γ)/X = Z X,Γ , thanks to Lemma 5.4.
For each open normal subgroup U of Γ, we have the voltage graph (X, Γ/U, α /U ), where α /U is the composite map of α and the natural projection Γ → Γ/U.Therefore, we have the associated derived graph X(Γ/U), on which Γ/U acts.
Even though X(Γ) is not defined unless Γ is finite, let us write X(Γ) to mean the family {X(Γ/U)} U .Then X(Γ) can be regarded as an infinite covering of X.For instance, suppose that Γ is isomorphic to Z p = lim ← −n Z/p n Z as a topological group.Then the open subgroups of Γ, written multiplicatively, are Γ p n for each n ≥ 0. Thus X ∞ = X(Γ) is the collection of X n = X(Γ/Γ p n ) for n ≥ 0 in this case.This family is illustrated as a tower of coverings We call such an X ∞ /X a Z p -covering.This is regarded as an analogue of Z p -extensions of number fields in Iwasawa theory; see §8 for more on this theme.
For simplicity, in the rest of this subsection, we will always assume the following: Assumption 5. 6.For any open normal subgroup U of Γ, the derived graph X(Γ/U) is connected.
As we will review in the proof of the following lemma, we have an equivalent condition for the derived graph to be connected, and that imposes a restriction to the structure of Γ as follows.
Lemma 5.7.If Assumption 5.6 holds, then the group Γ is finitely generated as a profinite group.Indeed, it is generated by Moreover, the condition that the derived graph X(Γ/U) is connected is equivalent to that the group homomorphism π 1 (X, v 0 ) → Γ/U induced by α /U is surjective (see [17,Theorem 2.11] for instance).This implies that Assumption 5.6 is equivalent to that the image of the group homomorphism π 1 (X, v 0 ) → Γ is dense in Γ.Therefore, we obtain the lemma.
In order to guarantee the exactness of inverse limits, we change the coefficient ring from Z to a compact Z-algebra Λ that is flat over Z (i.e., torsion-free as a Z-module).Fundamental examples of Λ include Ẑ and Z p for a prime number p, where Ẑ (resp.Z p ) is the profinite (resp.pro-p) completion of Z.
Remark 5.15.In §6, we will compute the Fitting ideals rather than the characteristic ideals.Compared to the characteristic ideals, the notion of Fitting ideals has roughly two advantages.One is that Fitting ideals are defined over any commutative rings.The other is that, even when the coefficient ring is a regular local ring, Fitting ideals are more refined than characteristic ideals; indeed, the Fitting ideal determines the characteristic ideal.Therefore, the main result in §6 (stated as Theorems 6.1 and 6.2) is a refinement of Theorem 5.14.

5.3.
Relation with Ihara zeta functions.In this subsection, we briefly explain the analytic aspect of Jacobian groups.As illustrated in Theorem 5.14 for instance, the structure of the Jacobian group is closely related to the element Z X,Γ .The purpose of this subsection is to explain that Z X,Γ has an interpretation using the (equivariant) Ihara zeta function.This fact may be regarded as an analogue of the conjectural relations between algebraic aspects and analytic aspects for various arithmetic objects in number theory (e.g., the Iwasawa main conjecture, the equivariant Tamagawa number conjecture, etc.).The results in this subsection are not used in the other parts of this paper.A nice reference is Terras [19].
Let (X, Γ, α) be a voltage graph such that Γ is abelian.First we define the associated zeta function (see [19,Chapters 2 and 18]).Definition 5.16.We define the (equivariant) Ihara zeta function by where P runs over primitive paths in X, [P ] denotes the rotation class of P , and ν(P ) denotes the length of P .When the path P consists of edges e 1 , . . ., e n (e i ∈ E X with n = ν(P )), we define α(P ) = α(e 1 ) • • • α(e n ).

Let us define
By definition we have Z X,Γ (1) = Z X,Γ .The result is the following.Theorem 5.17 (Three-term determinant formula).We have In particular, ζ X,Γ (u) is a rational function.
Proof.In case Γ is trivial, the theorem is [19,Theorem 2.5].More generally, the three-term determinant formula for Artin-Ihara L-functions is proved in [19,Theorem 18.15].We can prove our theorem by imitating those proofs.Alternatively, it is possible to deduce our theorem from the formula for Artin-Ihara L-functions.Indeed, the general statement can be reduced to the case where Γ is finite and X(Γ) is connected.In that case, the statement follows by combining the formula for Artin-Ihara L-functions for all the characters of Γ.We omit the details.

Results for profinite coverings
Let (X, Γ, α) be a voltage graph such that Γ is profinite and abelian.We always suppose Assumption 5.6, so Lemma 5.7 implies Γ is finitely generated.Theorem 6.2.The fractional ideal Fitt 6.2.Proof of Theorem 6.2.First we recall a result of Atsuta and the author [1] that plays a key role in the proof.For each i ≥ 0, the i-th Fitting ideal of the matrix is described as Let us begin with constructing a free resolution of Z p over Z p [[Γ 0 ]] in a similar way as in § §4.2-4.3.Observe that we have an exact sequences Then, by taking the tensor product of complexes, we obtain an exact sequence where d 0 is the augmentation map, d 1 is determined by and d 2 is determined by Therefore, the presentation matrix of d 2 is of the form M s,t (ν 1 , . . ., ν s , τ 1 , . . ., τ s , T s+1 , . . ., T s+t ) = ).Since we have an exact sequence 0 → Cok(d 2 ) M s,t (ν 1 , . . ., ν s , τ 1 , . . ., τ s , T s+1 , . . ., T s+t ) T s+t+1 . . .
For each 0 ≤ i ≤ s + t, by an analogous consideration as (4.2), we have Here, we use the same notation as (4.2).Let us compute the right hand side of (6.1) by using Proposition 6.3.First we consider i = 0. Then by Proposition 6.3, only j = s + t contributes to the sum, which is possible only when t = 0. Therefore, the right hand side of (6.1) vanishes unless t = 0.If t = 0, the right hand side equals (ν 1 • • • ν s ) (the term for j = s and a = {1, 2, . . ., s}).
6.3.Essentially finite case.Let (X, Γ, α) be a voltage graph such that Γ is profinite abelian and we suppose Assumption 5.6.Let us fix a prime number p In this section so far, we studied the case where the order of Γ is divisible by p ∞ .In case the order of Γ is not divisible by p ∞ (but possibly infinite), we can obtain the following results, which are profinite generalizations of the results in § §3-4.We state the results without proof because it is essentially the same as in § §3-4.
Let us construct a decomposition where ∆ l is a procyclic group (whose order is necessarily not divisible by p ∞ ).We fix an open subgroup U l ⊂ ∆ l such that the order of U l is not divisible by p. Then we introduce τ l ∈ Z p [∆ l ], a positive integer n l , and ] in the same way as in §6.1.We then define Then, as an analogue of Theorem 3.4, we have . Moreover, as an analogue of Theorem 4.1, we have Fitt Here, ] is any element such that (4.1) holds.

Self-duality of Jacobian groups
The prerequisite for this section is §2.In this section, we observe a self-duality property of the Jacobian groups.This is in contrast to the corresponding story in number theory as discussed in Remark 7.3 below.
First we fix our convention about duals.For a finite Z-module M, we define its Pontryagin dual by M ∨ = Hom Z (M, Q/Z).Note that we have an alternative description M ∨ = Ext 1 Z (M, Z).For a finitely generated Z-module M, we also define its Z-linear dual by M * = Hom Z (M, Z).Both (−) ∨ and (−) * are contravariant functors.
Here, the upper sequence is the complex constructed in Lemma 2.6, and the lower is the Z-linear dual of the upper.The middle two vertical isomorphisms are (7.1).The left and right vertical isomorphisms are the natural ones.It is straightforward to prove that this diagram is commutative.
Proof.By Lemma 2.6, the upper complex is quasi-isomorphic to the finite module Jac(Y ) (located at the appropriate degree).Therefore, its Z-linear dual is quasi-isomorphic to Ext 1 Z (Jac(Y ), Z) ≃ Jac(Y ) ∨ .This proves the assertion.In a more elementary way, it is possible to deduce the assertion by splitting the upper complex to three short exact sequences and then taking their Z-linear duals (we omit the details).Now we obtain the following.Proof.This follows immediately from Lemma 7.1.
as long as Γ is commutative.This phenomenon is in contrast to the situation in number theory.As revealed in [1] (see Remark 1.3), the Fitting ideal of Cl T,− L is much more complicated than that of Cl T,−,∨ L .

Iwasawa theory for graphs from a module-theoretic viewpoint
The prerequisite for this section is § §2 and 5.In this section, we discuss Iwasawa theory for graphs.In §8.1 (resp.§8.2), we give a short proof of an analogue of the Iwasawa class number formula (resp. of Kida's formula) for graphs.The results are not new and indeed already obtained by others; the Iwasawa class number formula is proved independently by Gonet [5] and McGown-Vallières [14], and Kida's formula is proved by Ray-Vallières [17].However, the proofs in this paper are different from the previous ones to some extent.Our observation is that those results directly follow from rather general module-theoretic propositions.An advantage of this is, for instance, that we can avoid separate discussion on the case where the Euler characteristic of the base graph is zero as in previous works.The author thinks that this method is more concise and so it is worth publishing.8.1.Iwasawa class number formula for graphs.Let us briefly review the original Iwasawa class number formula, proved by Iwasawa [10,Theorem 11].Let K ∞ /K be a Z pextension of number fields.This means that K ∞ /K is a Galois extension whose Galois group Γ is isomorphic to Z p .For each n ≥ 0, let K n be the intermediate field corresponding to Γ p n ⊂ Γ.Then K n /K is a Galois extension whose Galois group is Γ/Γ p n ≃ Z/p n Z and we obtain a tower of number fields The invariants λ(M), µ(M) are defined using the structure theorem for modules over Z p [[Γ]].We do not review the details in this paper (see, e.g., [20, §13.2] or [15, (5.1.10)]for the structure theorem and [15, (5.3.9)] for the definitions of λ-, µ-invariants).Let us just recall that we have µ(M) = 0 if and only if M is finitely generated over Z p , in which case λ(M) is equal to the Z p -rank of M. Now we consider the analogue for graphs.Let (X, Γ, α) be a voltage graph such that Γ is isomorphic to Z p .We suppose Assumption 5.6.Then, as introduced in §5.2, we have a Z p -covering X ∞ = X(Γ) of X.For each n ≥ 0, we put X n = X(Γ/Γ p n ), which is called the n-th layer of X ∞ /X.
To prove this theorem, we apply the following algebraic proposition.

[ 1 ]
R (Z/(#Γ)Z) that appeared in Theorem 3.4.4.1.Statement.First we state the result.As in §1.1, we fix a decomposition Γ = ∆ 1 × • • • × ∆ s into cyclic groups.For each 1 ≤ l ≤ s, we choose a generator σ l of ∆ l and define n l = #∆ l and elements τ l , ν l , D l ∈ Z[∆ l ] by the same formulas.Clearly we have τ l ν l = 0.A key property of D l is (4.1)

[ 1 ]
R (−), this implies the lemma.By Lemma 4.2, the proof of Theorem 4.1 is reduced to the following.
This is where the Kolyvagin derivative operators come into play.Lemma 4.10.We have s l=1 b l τ l = #Γ − N Γ .
If we have a left (resp.right) action of a group Γ on such an M, we introduce a right (resp.left) action of Γ on M ∨ or M * by (φγ)(x) = φ(γx) (resp.(γφ)(x) = φ(xγ)) for γ ∈ Γ, φ ∈ M ∨ or φ ∈ M * , and x ∈ M. Let ι be the involution on Γ defined by ι(γ) = γ −1 .If M is a left (resp.right) Γ-module, we define a right (resp.left) Γ-module M ι by M ι = M as an additive module and the group action is defined by xγ = γ −1 x (resp.γx = xγ −1 ) for γ ∈ Γ and x ∈ M. Now let Y be a connected graph equipped with a left action of a group Γ.Since the set of vertices {[w]} w∈V Y is a Z-basis of Div(Y ), we can construct the dual basis {φ w } w∈V Y of Div(Y ) * .Namely, φ w is the homomorphism characterized by φ w ([w]) = 1 and φ w ([w ′ ]) = 0 for any w ′ = w.It is easy to check that φ γw = φ w γ −1 for any w ∈ V Y and γ ∈ Γ.Therefore, we have a Z-isomorphism of left Z[Γ]-modules (7.1) Div(Y ) ≃ Div(Y ) * ,ι by sending [w] to φ w .Now let us consider the following commutative diagram (7.2) 0
Now the Iwasawa class number formula states that there exist integers λ ≥ 0, µ ≥ 0, and ν such that ord p (# Cl(K n )) = λn + µp n + ν for n ≫ 0. Here, Cl(K n ) denotes the ideal class group of K n and ord p denotes the additive p-adic valuation normalized so that ord p (p) = 1.The proof of this formula is explained in Washington[20,  §13.3].In general, associated to a finitely generated torsion Z p [[Γ]]-module M are integers λ(M) ≥ 0, µ(M) ≥ 0 that are respectively called the λ-, µ-invariants of M. The integers λ, µ in the Iwasawa class number formula are exactly the λ-, µ-invariants of the associated Iwasawa module.