Spectral triples from Mumford curves

C.Consani and M.Marcolli

Abstract We construct spectral triples associated to Schottky–Mumford curves, in such a way that the local Euler factor can be recovered from the zeta functions of such spectral triples. We propose a way of extending this construction to the case where the curve is not k-split degenerate.

1 Introduction

Let X be a curve over a finite extension K of ℚ_{p}, which is k-split degenerate, for k the residue field. It is well known that such curve admits a p-adic uniformization by a p-adic Schottky group Γ acting on the Bruhat-Tits tree Δ_{K}. We associate C^{∗}-algebras to certain subgraphs Δ of the Bruhat-Tits tree and construct corresponding dynamical cohomologies H_{dyn}^{1}(Δ/Γ) that resemble the construction at arithmetic infinity given in [6]. We introduce a Dirac operator D, which depends on the graded structure of the dynamical cohomology, in such a way that the data

(C^{∗}(Δ/Γ), H_{dyn}^{1}(Δ/Γ) ⊕ H_{dyn}^{1}(Δ/Γ), D)

give a spectral triple in the sense of Connes.

We recover, from a certain family of zeta functions associated to the spectral triple, the local Euler factor L_{p}(H^{1}(X), s) of the curve X, in the form of a regularized determinant as computed by Deninger. The advantage of this construction is that it provides, in the case we are considering, a natural geometric interpretation, in terms of dynamics of walks on the graph Δ/Γ, of the infinite dimensional cohomology theory introduced by Deninger.

We propose a way of extending the construction to the case where the curve is not k-split degenerate, by enlarging the dual graph of the special fiber by new edges, in such a way that we also obtain embeddings in the dynamical cohomology of the first cohomology group of the components of the special fiber.

2 Directed graphs

We begin by recalling some generalities about graphs that we will use throughout the paper.

A directed graph E consists of data E = (E^{0}, E^{1}, E_{+}^{1}, r, s, ι), where E^{0} is the set of vertices, E^{1} is the set of oriented edges w = {e, ϵ}, where e is an edge of the graph and ϵ = ±1 is a choice of orientation. The set E_{+}^{1} consists of a choice of orientation for each edge, namely one element in each pair {e, ±1}. The maps r, s : E^{1} → E^{0} are the range and source maps, and ι is the involution on E^{1} defined by ι(w) = {e, −ϵ}.

A morphism f of directed graphs E and Ẽ consists of maps f^{0} : E^{0} → Ẽ^{0} and f^{1} : E^{1} → Ẽ^{1} which satisfy f^{i} ∘ r = r̃∘ f^{i}, f^{i} ∘ s = s̃∘ f^{i}, and f^{i} ∘ ι = ι̃∘ f^{i}, for i = 0, 1. The morphism f is a monomorphism if the f^{i} are invertible, for i = 0, 1. This defines the automorphism group of a directed graph, which we denote Aut(E). A morphism f of directed graphs is a covering map if the f^{i} are onto, for i = 0, 1, and f^{1} gives a bijection f^{1} : s^{−1}(v) →^{∼} s̃^{−1}(f(v)) and the same with respect to the range map.

A directed graph is finite if E^{0} and E^{1} are finite sets. It is row finite if at each vertex v ∈ E^{0} there are at most finitely many edges w in E_{+}^{1} such that s(w) = v. The graph is locally finite if each vertex emits and receives at most finitely many oriented edges in E^{1}. A vertex v in a directed graph is a sink if there is no edge in E_{+}^{1} with source v. We denote by σ(E) the subset of E^{0} given by the sinks.

A juxtaposition of oriented edges w_{1}w_{2} is said to be admissible if w_{2} ≠ ι(w_{1}) and r(w_{1}) = s(w_{2}). A (finite, infinite, doubly infinite) walk in a directed graph E is an admissible (finite, infinite, doubly infinite) sequence of elements in E^{1}. A (finite, infinite, doubly infinite) path in E is a walk where all edges in the sequence are in E_{+}^{1}. We denote by W^{n}(E) the set of walks of length n, by W^{∗}(E) = ∪_{n}W^{n}(E), by W^{+}(E) the set of infinite walks, and by W(E) the set of doubly infinite walks. Similarly, we introduce the analogous notation P^{n}(E), P^{∗}(E), and P^{+}(E) for paths. We shall drop the explicit mention of the graph E in the notation for walks and paths, when no confusion arises. We denote by σ^{∗}(E) ⊂ P^{∗}(E) the set of paths that end at a sink. A directed graph is a directed tree if, for any two vertices, there exists a unique walk in W^{∗}(E) connecting them.

The universal cover Δ of a connected directed graph E, endowed with a choice of a base point v_{0} ∈ E^{0}, is a directed tree obtained by setting Δ^{0} = W^{∗}(v_{0}), the set of all walks in E that start at v_{0}, Δ^{1} = {(ω, w) ∈ W^{∗}(v_{0}) × E^{1} : r(ω) = s(w)}, r(ω, w) = ω, s(ω, w) = ωw, ι(ω, w) = (ωw, ι(w)). The fundamental group of E, with respect to the choice of base point v_{0}, is given by Γ = {γ ∈ W^{∗}(v_{0}) : r(γ) = v_{0}}.

Let G be a subgroup of Aut(E). We can then form the quotient E/G, which is also a directed graph. In particular, if Δ is the universal cover of a directed graph E and Γ is the fundamental group, with respect to the choice of a base point v_{0} ∈ E^{0}, then we have an isomorphism of directed graphs E ≃ Δ/Γ. The map Δ → E is a covering map.

Two paths ω, ω̃ in P^{+} are shift-tail equivalent, if there exists a N ≥ 1 and a k ∈ ℤ such that ω_{i} = ω̃_{i−k} for all i ≥ N. The boundary of a directed tree Δ is given by ∂Δ = (P^{+} ∪ σ^{∗})/ ∼, where the shift-tail equivalence is extended to the set of paths ending at a sink by the condition ω ∼ ω̃, for ω, ω̃ ∈ σ^{∗}, if and only if r(ω) = r(ω̃). This definition extends to a functorial notion of boundary of directed graphs, as shown in [21].

3 Schottky-Mumford curves

Throughout this chapter we will denote by p a fixed prime number and by ℚ_{p} the field of p-adic numbers. The field K will be a given finite extension of ℚ_{p}, with O ⊂ K its ring of integers, 𝔪 ⊂ O the maximal ideal and π ∈ 𝔪 a uniformizer (i.e. 𝔪 = (π)). Finally, we will denote by k the residue field k = O/𝔪. This is a finite field of cardinality q = card(O/𝔪).

Let V be a two-dimensional vector space over K. We write ℙ^{1}(K) for the set of K-rational points of ℙ_{K}^{1}, the projective line over K. The space ℙ^{1}(K) is identified with the set of K-lines passing through the origin in V. Let G = PGL(2, K) be the group of automorphisms of ℙ^{1}(K).

In this first chapter we collect some well known facts and properties on the tree of the group PGL(2, K) and on the action of a Schottky group on such tree. Detailed explanations for the statements we will recall here without proofs are contained in [14] and [18].

3.1 The tree of the group PGL(2,K)

The description of the vertices of the tree associated to PGL(2, K) is as follows. One considers the set of free O-modules of rank 2: M ⊂ V. Two such modules are equivalent M_{1} ∼ M_{2} if there exists an element λ ∈ K^{∗}, such that M_{1} = λM_{2}. The group GL(V) of linear automorphisms of V operates on the set of such modules on the left: gM = {gm | m ∈ M}, g ∈ GL(V). Notice that the relation M_{1} ∼ M_{2} is equivalent to the condition that M_{1} and M_{2} belong to the same orbit of the center K^{∗} ⊂ GL(V). Hence, the group G = GL(V)/K^{∗} operates (on the left) on the set of classes of equivalent modules.

We denote by Δ_{K}^{0} the set of such classes and by {M} the class of the module M. Because O is a principal ideals domain and every module M has two generators, it follows that

{M_{1}}, {M_{2}} ∈ Δ_{K}^{0}, M_{1} ⊃ M_{2} ⇒ M_{1}/M_{2} ≃ O/𝔪^{l} ⊕ O/𝔪^{k}, l, k ∈ ℕ.

The multiplication of M_{1} and M_{2} by elements of K preserves the inclusion M_{1} ⊃ M_{2}, hence the natural number

d({M_{1}}, {M_{2}}) = |l − k|  (3.1)

is well defined.

Definition 3.1 The graph Δ_{K} of the group PGL(2, K) is the infinite graph with set of vertices Δ_{K}^{0}, in which two vertices {M_{1}}, {M_{2}} are adjacent and hence connected by an edge if and only if d({M_{1}}, {M_{2}}) = 1.

The following properties characterize completely Δ_{K} and are well known (cf. e.g.

[14] and [18]).

Proposition 3.2 1. The graph Δ_{K} is a connected, locally finite tree with q+1 edges departing from each of its vertices.

2. The shortest walk in Δ_{K} connecting two vertices {M_{1}}, {M_{2}} of Δ_{K}^{0} without back-tracking has length d({M_{1}}, {M_{2}}).

3. The group G acts (on the left) transitively on Δ_{K}^{0} and it preserves the metric d.

The tree Δ_{K} is called the Bruhat–Tits tree associated to the group G = PGL(2, K).

A half-line in Δ_{K} is an infinite sequence ({M_{n}})_{n∈ℕ} of vertices of Δ_{K} without repetitions such that {M_{n}} is adjacent to {M_{n−1}} for all n. Thus, a half-line is given by a sequence M_{0} ⊃ M_{1} ⊃ … of free O-modules where M_{0}/M_{n} ≃ O/(π^{n}) for all n.

The subspace K(∩_{n∈ℕ}M_{n}) ⊂ V defines a point of ℙ^{1}(K). Conversely, given a point of ℙ^{1}(K) represented by a vector v_{1} ∈ V, choose a second vector v_{2} ∈ V such that {v_{1}, v_{2}} form a basis of V. Let M_{n} be the free O-module with basis {v_{1}, π^{n}v_{2}}. Then, K(∩_{n∈ℕ}M_{n}) defines the point of ℙ^{1}(K) we started with.

Two half-lines are said to be equivalent if they differ only by a finite number of vertices. An equivalence class of halflines is called an end of Δ_{K}. The set of ends of Δ_{K} will be denoted by ∂Δ_{K} (the “boundary” of Δ_{K}). We shall give Δ_{K} the structure of a directed graph, in such a way that this notion of boundary agrees with the one described in the previous section.

It is immediate to verify that the construction described above defines a one-to-one correspondence ∂Δ_{K} ⇄ ℙ^{1}(K) between equivalence classes of half-lines and elements of ℙ^{1}(K).

3.2 The action of a Schottky group on the tree Δ_{K}

A Schottky group Γ is a subgroup of PGL(2, K) which is finitely generated and whose elements γ ≠ 1 are hyperbolic (i.e. the eigenvalues of γ in K have different valuation). The group Γ is discrete in G = PGL(2, K), torsion free and acts freely on the tree Δ_{K}. Furthermore, one can show that Γ acts discretely at some point z ∈ ℙ^{1}(K̄).

Let Λ_{Γ} ⊂ ℙ^{1}(K) be the closure of the set of points in ℙ^{1}(K) that are fixed points of some γ ∈ Γ \ {1}. This is called the limit set of Γ. We have card(Λ_{Γ}) < ∞ if and only if Γ = (γ)^{ℤ}, for some γ ∈ Γ. We will denote by Ω_{Γ} = Ω_{Γ}(K) the set of points on which Γ acts discretely; equivalently said Ω_{Γ} is the set of points which are not limits of fixed points of elements of Γ: Ω_{Γ} = ℙ^{1}(K) \ Λ_{Γ}. This set is called the domain of discontinuity for the Schottky group Γ.

A path in Δ_{K}, infinite in both directions and with no back-tracking, is called an axis of Δ_{K}. Any two points z_{1}, z_{2} ∈ ℙ^{1}(K) uniquely define their connecting axis whose endpoints lie in the classes described by z_{1} and z_{2} in ∂Δ_{K}.

Figure 1: Two axes and the bridge B between them.

Any hyperbolic element γ ∈ Γ has two fixed points in ℙ^{1}(K). The unique axis of Δ_{K} whose ends are fixed by γ is called the axis of γ. The element γ acts on its axis as a translation.

Suppose two axes are given in Δ_{K}. Any path without back-tracking beginning on one axis and ending on the other and with no edges in common with either axis is said to be the bridge between them. A bridge may consist of a single point, else it is uniquely defined (Figure 1).

For any Schottky group Γ ⊂ PGL(2, K) there is a smallest subtree Δ′_{Γ} ⊂ Δ containing the axes of all elements of Γ. Equivalently said, Δ′_{Γ} is the maximal connected subgraph of Δ_{K} containing the axes of all elements of Γ and the bridges between them.

The set of ends of Δ′_{Γ} in ℙ^{1}(K) is Λ_{Γ}, the limit set of Γ. The group Γ carries Δ′_{Γ} into itself so that the quotient Δ′_{Γ}/Γ is a finite graph.

The graph Δ′_{Γ}/Γ has an important geometric interpretation as the dual graph of the closed fiber of the minimal smooth model over O (k-split degenerate semi-stable curve) of the algebraic curve C holomorphically isomorphic to X_{Γ} := Ω_{Γ}/Γ (cf. [18] p. 163).

Furthermore, there is a smallest tree Δ_{Γ} on which Γ acts, with vertices Δ_{Γ}^{0} ⊂ Δ_{K}^{0}, and such that the finite graph Δ_{Γ}/Γ is the dual graph of the specialization of C over O. The set Δ_{Γ}^{0} is a subset of the set of vertices of Δ′_{Γ}.

The degenerating curve C describing the analytic uniformization X_{Γ} →^{≃} C is a ksplit degenerate, stable curve. When the genus of the fibers is at least 2 - i.e. when the Schottky group has at least g ≥ 2 generators - the curves X_{Γ} are called Schottky– Mumford curves.

Figure 2: The tree Δ_{K} for K = ℚ_{2} and Δ_{L} for field extensions with f = 2 and e_{L/K} = 2

3.3 Field extensions: functoriality

Let L ⊃ K be a finite fields extension with ramification index e_{L/K} and rings of integers O_{L} and O_{K}. If M ⊂ V is a free O_{K}-module of rank 2, then M ⊗_{O_{K}} O_{L} ⊂ V ⊗_{K} L is a free O_{L}-module of the same rank. It is obvious that equivalent modules remain equivalent, hence one gets a natural embedding of the sets of vertices Δ_{K}^{0} ↪ Δ_{L}^{0}.

The isomorphism (O_{K}/𝔪^{r}) ⊗ O_{L} ≃ O_{L}/𝔪^{re_{L/K}} shows that distance would not, in general, be preserved under field extensions. To eliminate this disadvantage, one introduces on graphs Δ_{L}, for all extensions L ⊃ K, a K-normalized distance

d_{K}(\{M_{1}\}, \{M_{2}\}) = \frac{1}{e_{L/K}} d_{L}(\{M_{1}\}, \{M_{2}\}); M_{1}, M_{2} ⊂ V ⊗ L.

This way, the embedding Δ_{K}^{0} ↪ Δ_{L}^{0} becomes isometric. When L ⊃ K ramifies, e_{L/K} − 1 new vertices appear in Δ_{L}^{0} between each couple of adjacent vertices in Δ_{K}^{0} (cf. the third graph in Figure 2). Moreover, q^{f} + 1 edges each of which of length \frac{1}{e_{L/K}}, for f = \frac{1}{e_{L/K}} [L : K], depart from each vertex in Δ_{L}^{0} (cf. the second graph in Figure 2).

Because PGL(2, K) ⊂ PGL(2, L) and the natural embedding ℙ^{1}(K) ⊂ ℙ^{1}(L) is compatible with a concept of K-direction of exiting from any vertex of Δ_{K}^{0}, the construction is functorial under finite fields extensions and this process determines, for a fixed Schottky group Γ ⊂ PGL(2, K), a projective system {X_{L,Γ} : [L : K] < ∞} of Schottky-Mumford curves.

3.4 Edge orientation

We now show how to endow the graphs Δ_{Γ} and Δ_{K} with the structure of directed graph. The choice of a coordinate z ∈ ℙ^{1}(K) determines a base point ṽ_{0} in Δ_{K}. In fact, the points {0, 1, ∞} on ℙ^{1}(K) determine a unique crossroad: the unique vertex of Δ_{K} with the property that the walks without back-tracking from ṽ_{0} to the three points on ℙ^{1}(K) start from ṽ_{0} in three different directions (Figure 3).

Figure 3: The crossroad ṽ_{0} of the points {0, 1, ∞}

In order to obtain a structure of directed graph on the tree Δ_{K}, for each z ∈ ℙ^{1}(K) we consider the unique infinite chain of edges without backtracking in Δ_{K} that has initial vertex ṽ_{0} and whose equivalence class modulo shift–tail equivalence is the point z. We declare such chain of edges to be a path in P^{+}(Δ_{K}). This determines on Δ_{K} the structure of a directed graph, with ∂Δ_{K} = ℙ^{1}(K). This agrees with the boundary as defined in §3.1.

We assume here that the coordinate z ∈ ℙ^{1}(K) is chosen in such a way that the crossroad ṽ_{0} of {0, 1, ∞} is a vertex of Δ_{Γ}. Then, by the same procedure, we can regard Δ_{Γ} as a directed graph with ∂Δ_{Γ} = Λ_{Γ}.

4 C^{∗}-algebras of graphs

In this section we recall the construction of C^{∗}-algebras associated to locally finite directed graphs. We follow mostly the references [1], [10], [11], [21]. We refer the reader to the bibliography of the aforementioned articles for more information. For simplicity, we state the following results in the special case of locally finite directed graph, though the theory extends to more general directed graphs (cf. e.g. [21]).

A Cuntz–Krieger family consists of a collection {P_{v}}_{v∈E^{0}} of mutually orthogonal projections and {S_{w}}_{w∈E_{+}^{1}} of partial isometries, satisfying the conditions: S_{w}^{∗}S_{w} = P_{r(w)} and, for all v ∈ s(E_{+}^{1}), P_{v} = ∑_{w:s(w)=v} S_{w}S_{w}^{∗}.

The edge matrix A_{+} of a locally finite (or row finite) directed graph is an (#E_{+}^{1}) × (#E_{+}^{1}) (possibly infinite) matrix. The entries A_{+}(w_{i}, w_{j}) satisfy A_{+}(w_{i}, w_{j}) = 1 if w_{i}w_{j} is an admissible path, and A_{+}(w_{i}, w_{j}) = 0 otherwise. The Cuntz–Krieger elements {P_{v}, S_{w}} satisfy the relation S_{w}^{∗}S_{w} = ∑ A(w, w̃)Sw̃S∗ ; w̃. The directed edge matrix of E is a #E^{1} × #E^{1} (possibly infinite) matrix with entries A(w_{i}, w_{j}) = 1 if w_{i}w_{j} is an admissible walk and A(w_{i}, w_{j}) = 0 otherwise.

There is a universal C^{∗}-algebra C^{∗}(E) generated by a Cuntz–Krieger family. If E is a finite graph with no sinks, we have C^{∗}(E) ≃ O_{A_{+}}, where O_{A_{+}} is the Cuntz-Krieger algebra of the edge matrix A_{+}. If the directed graph is a tree Δ, then C^{∗}(Δ) is an AF algebra strongly Morita equivalent to the commutative C^{∗}-algebra C_{0}(∂Δ). A monomorphism of directed trees induces an injective ∗-homomorphism of the corresponding C^{∗}-algebras.

If G ⊂ Aut(E) is a group acting freely on the directed graph E, with quotient graph E/G, then the crossed product C^{∗}-algebra C^{∗}(E) ⋊ G is strongly Morita equivalent to C^{∗}(E/G). In particular, if Δ is the universal covering tree of a directed graph E and Γ is the fundamental group, then the algebra C^{∗}(E) is strongly Morita equivalent to C_{0}(∂Δ) ⋊ Γ.

There is a gauge action of U(1) on the graph algebra C^{∗}(E) given by λ : {P_{v}, S_{w}} ↦ {P_{v}, λS_{w}}. A subset H of the set of vertices E^{0} of a directed graph is saturated hereditary if v ∈ H implies that, for all ω ∈ P^{∗}(E) with s(ω) = v, also r(ω) ∈ H, and conversely, if any ω ∈ P^{∗}(E) with s(ω) = v satisfies r(ω) ∈ H, then also v ∈ H. For a locally finite graph there is a bijective correspondence between saturated hereditary subsets of E^{0} and gauge invariant closed ideals in C^{∗}(E). In the case of a tree Δ, there is a bijection between saturated hereditary subsets of Δ^{0} and open sets in ∂Δ. This is proved for the more general (non necessarily locally finite) case in [21].

It is convenient to consider also the Toeplitz extensions (cf. [21])

0 → I_{S} → T O(E, S) → C^{∗}(E) → 0,

where S is a subset of E^{0} and I_{S} = ⊕_{v∈S^{c}}K_{v}, where K_{v} is the algebra of compact operators on a Hilbert space of dimension #(P^{∗} ∩r^{−1}(v)). The C^{∗}-algebra T O(E, S) is generated by operators {S_{w}}_{w∈E_{+}^{1}} and {P_{v}}_{v∈E^{0}} satisfying the conditions: S_{w}^{∗}S_{w} = P_{r(w)} and, for all v ∈ s(E_{+}^{1}), P_{v} ≥ ∑_{w:s(w)=v} S_{w}S_{w}^{∗}, with equality for v ∈ S.

If j : E ↪ Ẽ is an inclusion of directed graphs, the following functoriality property holds (cf. [21]): given S̃ ⊂ Ẽ, consider the set S of vertices v in E^{0} such that j(v) ∈ S̃ and the outgoing edges in Ẽ_{+}^{1} with origin at j(v) are all of the form j(w), for some w ∈ E_{+}^{1} with s(w) = v. Then this induces an injective ∗-homomorphism J : T O(E, S) → T O(Ẽ, S̃).

In particular, for a family of subgraphs E of a directed graph Ẽ, ordered by inclusions, with ∪E^{0} = Ẽ^{0} and ∪E^{1} = Ẽ^{1}, and for a choice of S̃ ⊂ Ẽ^{0} and corresponding S as above, we have

T O(Ẽ, S̃) = lim_{E} T O(E, S).  (4.1)

4.1 Reduction graphs

In the theory of Mumford curves, it is important to consider also the reduction modulo powers 𝔪^{n} of the maximal ideal 𝔪 ⊂ O_{K}, which provides infinitesimal neighborhoods of order n of the closed fiber. In the language of C^{∗}-algebras, this corresponds to the following construction.

For each n ≥ 0, we consider a subgraph Δ_{K,n} of the Bruhat-Tits tree Δ_{K} defined by setting

Δ_{K,n}^{0} := {v ∈ Δ_{K}^{0} : d(v, Δ′_{Γ}) ≤ n},

with respect to the distance (3.1), with d(v, Δ′_{Γ}) := inf{d(v, ṽ) : ṽ ∈ (Δ′_{Γ})^{0}}, and

Δ_{K,n}^{1} := {w ∈ Δ_{K}^{1} : s(w), r(w) ∈ Δ_{K,n}^{0}}.

Thus, we have Δ_{K,0} = Δ′_{Γ}. For n ≥ 1 the graph Δ_{K,n} has a non-empty set of sinks σ_{K,n} ⊂ ∂Δ_{K,n}. We have Δ_{K} = ∪_{n}Δ_{K,n}.

For all n ∈ ℕ, the graph Δ_{K,n} is invariant under the action of the Schottky group Γ on Δ, and the finite graph Δ_{K,n}/Γ gives the dual graph of the reduction X_{K} ⊗O/𝔪^{n+1}. Thus, we refer to the Δ_{K,n} as reduction graphs. They form a directed family with inclusions j_{n,m} : Δ_{K,n} ↪ Δ_{K,m}, for all m ≥ n, with all the inclusions compatible with the action of Γ.

For each reduction graph, we can consider corresponding C^{∗}-algebras C^{∗}(Δ_{K,n}) and C^{∗}(Δ_{K,n}/Γ) ≃ C^{∗}(Δ_{K,n}) ⋊ Γ (Morita equivalence).

The following result, which is a direct application of the statements on graph C^{∗}-algebras listed in §4, describes the relation between the algebras C^{∗}(Δ_{K,n}/Γ) and C^{∗}(Δ_{K}/Γ).

Lemma 4.1 We have injective ∗-homomorphisms J_{n,m} : C^{∗}(Δ_{K,n}) → C^{∗}(Δ_{K,m}). Correspondingly, if we set S_{n} = Δ_{K,n}^{0}\σ_{K,n}, we obtain

C^{∗}(Δ_{K}/Γ) = lim_{n} T O(Δ_{K,n}/Γ, S_{n}/Γ),

where T O(Δ_{K,n}/Γ, S_{n}/Γ) satisfies

0 → ⊕_{v∈σ(Δ_{K,n}/Γ)}K_{v} → T O(Δ_{K,n}/Γ, S_{n}/Γ) → C^{∗}(Δ_{K,n}/Γ) → 0.

5 Dynamics of walks on dual graphs

In this section we introduce a dynamical system associated to the space W(Δ/Γ) of walks on the quotient of a directed tree Δ by a free action of Γ. In particular, we are interested in the cases where Δ is (a certain extension of) one of the graphs Δ_{K,n} for some n ≥ 0. The same construction applies to the tree Δ_{Γ}, where this dynamical system is a subshift of finite type associated to the action of the Schottky group Γ on its limit set Λ_{Γ}, analogous to the one considered in [6].

Let V̄ ⊂ Δ be a finite subtree whose set of edges consists of one representative for each Γ-class. This is a fundamental domain for Γ in the weak sense (following the notation of [14]), since some vertices may be identified under the action of Γ.

Correspondingly, we denote by V the set of ends of all infinite paths in Δ starting at points in V̄.

We assume that, for the Γ-invariant directed tree Δ, the set V̄ has finitely many edges. This is the case for Δ_{Γ} as well as for any of the trees Δ_{K,n}.

Since, for n ≥ 1, the graphs Δ_{K,n} have sinks, in order to consider the space of doubly infinite walks on these graphs, we need to complete each walk ending at a sink to an infinite walk obtained by repeating the last word. This is equivalent to extending the graph Δ_{K,n} by adding an infinite tail to each sink. Appending tails to sinks is standard technique in the theory of graph C^{∗}-algebras, in order to reduce the general case to the easier case of graphs with no sinks. We use the notation Δ̄_{K,n} for the completed graph with infinite tails. These have an action of Γ obtained by extending the action on Δ_{K,n}, by translating the whole tail in the same way as the corresponding sink in Δ_{K,n}.

For Δ = Δ̄_{K,n}, the set W(Δ/Γ) of doubly infinite walks on the graph Δ̄_{K,n}/Γ can be identified with the set of doubly infinite admissible sequences in the finite alphabet given by the edges in the fundamental domain V̄ of the graph Δ_{K,n}, with both possible orientations.

On W(Δ/Γ) we consider the topology generated by the sets W^{s}(ω, ℓ) = {ω̃ ∈ W(Δ/Γ) : ω̃_{k} = ω_{k}, k ≥ ℓ} and W^{u}(ω, ℓ) = {ω̃ ∈ W(Δ/Γ) : ω̃_{k} = ω_{k}, k ≤ ℓ}, for ω ∈ W(Δ/Γ) and ℓ ∈ ℤ. With this topology, the space W(Δ/Γ) is a totally disconnected compact Hausdorff space. The invertible shift map T, given by (T ω)_{k} = ω_{k+1}, is a homeomorphism of W(Δ/Γ).

We have just described the dynamical system (W(Δ/Γ), T) in terms of subshifts of finite type, according to the following definition:

Definition 5.1 A subshift of finite type (S_{A}, T) consists of all doubly infinite sequences in the elements of a given finite set W (alphabet) with the admissibility condition specified by a #W × #W elementary matrix,

S_{A} = {(w_{k})_{k∈ℤ} : w_{k} ∈ W, A(w_{k}, w_{k+1}) = 1},

with the action of the invertible shift (T w)_{k} = w_{k+1}.

Lemma 5.2 The space W(Δ/Γ) with the action of the invertible shift T is a subshift of finite type, where W(Δ/Γ) = S_{A} with A the directed edge matrix of the finite graph Δ/Γ.

We can consider the mapping torus of T,

W(Δ/Γ)_{T} := W(Δ/Γ) × [0, 1]/(T x, 0) ∼ (x, 1).  (5.1)

We consider the first cohomology group H^{1}(W(Δ/Γ)_{T} , ℤ), identified with the group of homotopy classes of continuous maps of W(Δ/Γ)_{T} to the circle. If we denote by C(W(Δ/Γ), ℤ) the ℤ-module of integer valued continuous functions on W(Δ/Γ), and by C(W(Δ/Γ), ℤ)_{T} the cokernel of the map δ(f) = f − f ∘ T, we obtain the following result (cf. [2], [19]).

Proposition 5.3 The map f ↦ [exp(2πitf(x))], which associates to an element f ∈ C(W(Δ/Γ), ℤ) a homotopy class of maps from W(Δ/Γ)_{T} to the circle, gives an isomorphism C(W(Δ/Γ), ℤ)_{T} ≃ H^{1}(W(Δ/Γ)_{T} , ℤ). Moreover, there is a filtration of C(W(Δ/Γ), ℤ)_{T} by free ℤ-modules F_{0} ⊂ F_{1} ⊂ ⋯ F_{n} ⋯, of rank θ_{n} − θ_{n−1} + 1, where θ_{n} is the number of admissible words of length n + 1 in the alphabet, so that we have

H^{1}(W(Δ/Γ)_{T} , ℤ) = \underrightarrow{lim}_{n} F_{n}.

The quotients F_{n+1}/F_{n} are also torsion free.

\underline{Proof.} A continuous function f ∈ C(W(Δ/Γ), ℤ) depends on just finitely many coordinates ω_{k} of ω ∈ W(Δ/Γ). In particular, this implies that, for some k_{0}, the composite f ∘ T^{k_{0}} is a function of only ‘future coordinates’ (ω_{k} with k ≥ 0). We denote by P ⊂ C(W(Δ/Γ), ℤ) the submodule of functions of future coordinates. It is then clear that we have C(W(Δ/Γ), ℤ)_{T} ≃ P/δP. We also have an identification P ≃ C(W^{+}(Δ/Γ), ℤ). This gives a filtration P = ∪_{n}P_{n}, where P_{n} is identified with the submodule of C(W^{+}(Δ/Γ), ℤ) generated by characteristic functions of W^{+}(Δ/Γ, ρ) ⊂ W^{+}(Δ/Γ), where ρ ∈ W^{∗}(Δ/Γ) is a finite walk ρ = w_{0} ⋯ w_{n} of length n + 1, and W^{+}(Δ/Γ, ρ) is the set of infinite paths ω ∈ W^{+}(Δ/Γ), with ω_{k} = w_{k} for 0 ≤ k ≤ n + 1. We have δ : P_{n} → P_{n+1}, with kernel the constant functions. We set F_{n} := P_{n}/δ(P_{n−1}), for n ≥ 1 and F_{0} = P_{0}. The inclusions P_{n} ⊂ P_{n+1} induce injections j : F_{n} ↪ F_{n+1}, j(f + δP_{n−1}) = f + δP_{n}, such that P/δP = \underrightarrow{lim}_{n} F_{n}. As ℤ-modules, both the F_{n} and the quotients F_{n+1}/j(F_{n}) are torsion free. ⋄

For more details see the analogous argument given in [19] Thm. 19 p. 62-63.

5.1 The effect of field extensions

Let L ⊃ K be a finite extension, with branching index e_{L/K} and with f = [L : K]/e_{L/K}. Then there is an embedding of the set of vertices Δ_{K}^{(0)} ⊂ Δ_{L}^{(0)}. In between every two vertices of Δ_{K}^{(0)} ⊂ Δ_{L}^{(0)} there are e_{L/K} − 1 new vertices of Δ_{L}. If in Δ_{K} every vertex has valence q + 1, then every vertex of Δ_{L} has valence q^{f} + 1.

In particular, the image of the tree Δ_{Γ} ⊂ Δ_{K} in Δ_{L} is the tree Δ_{Γ} ⊂ Δ_{L}, whereas, when considering the tree Δ′_{Γ}, we are inserting e_{L/K}−1 new vertices in between each two vertices of Δ′_{Γ} ⊂ Δ_{K}. Notice that the algebras C^{∗}(Δ′_{Γ}) and C^{∗}(Δ_{Γ}) are strongly Morita equivalent, since they both are equivalent to the commutative C^{∗}-algebra C(Λ_{Γ}). Thus we have:

Lemma 5.4 The strong Morita equivalence class of the graph C^{∗}-algebras C^{∗}(Δ′_{Γ}) and C^{∗}(Δ′_{Γ}/Γ) is independent of finite field extensions L ⊃ K.

The following example illustrates how the algebra O_{A_{+}} changes under field extensions, by showing the change in the edge matrix A_{+}. If the first directed graph shown

Figure 4: Effect of a field extension with e_{L/K} = 2.

in Figure 4 arises as dual graphs of the closed fiber for a totally split degenerate curve, then the effect on the graph of a field extension with e_{L/K} = 2 is illustrated in the second graph in Figure 4.

The edge matrix of the original graph was of the form

A_{+} = ( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) ,

while the new edge matrix becomes:

A_{+} = ( \begin{matrix}0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 \end{matrix} ) .

We are interested in understanding the effect of field extensions on the construction considered in the previous paragraph. We have the following result.

Proposition 5.5 A finite field extension L ⊃ K determines a morphism

H^{1}(W(Δ̄_{L,n}/Γ)_{T}) → H^{1}(W(Δ̄_{K,n}/Γ)_{T}),

which is compatible with the filtrations.

\underline{Proof.} In [14] it is shown that there is a canonical choice of the fundamental domains V̄ and V for the action of Γ on the Bruhat-Tits tree Δ_{K} that is functorial under a finite extensions L ⊃ K. This determines corresponding functorial choices of fundamental domains for the graphs Δ_{K,n}. We obtain this way a corresponding embedding of walk spaces

J_{L,K,n} : W(Δ̄_{K,n}/Γ) ↪ W(Δ̄_{L,n}/Γ),

which replaces each edge in a sequence ω ∈ W(Δ̄_{K,n}/Γ) (an edge in V̄ for Δ_{K,n}) with the corresponding e_{L/K} consecutive edges in the fundamental domain V̄ for Δ_{L,n}, thus obtaining an element in W(Δ̄_{L,n}/Γ). This map satisfies J_{L,K,n} ∘ T = T^{e_{L/K}} ∘ J_{L,K,n}. Thus, we obtain an induced map

J_{L,K,n,T} : W(Δ̄_{K,n}/Γ)_{T} → W(Δ̄_{L,n}/Γ)_{T^{e_{L/K}}} ,

where, for ℓ ≥ 1, W(Δ̄_{L,n}/Γ)_{T^{ℓ}} denotes the mapping torus

W(Δ̄_{L,n}/Γ)_{T^{ℓ}} ≃ W(Δ̄_{L,n}/Γ) × [0, ℓ]/(x, 0) ∼ (T^{ℓ}x, ℓ),

with a covering map π_{ℓ} : W(Δ̄_{L,n}/Γ)_{T^{ℓ}} → W(Δ̄_{L,n}/Γ)_{T}. Thus, we obtain a map

π_{e_{L/K}} ∘ J_{L,K,n,T} : W(Δ̄_{K,n}/Γ)_{T} → W(Δ̄_{L,n}/Γ)_{T} .

This induces a corresponding map in cohomology,

(π_{e_{L/K}} ∘ J_{L,K,n,T})^{∗} : H^{1}(W(Δ̄_{L,n}/Γ)_{T}) → H^{1}(W(Δ̄_{K,n}/Γ)_{T}).

To see this at the level of the filtrations of Proposition 5.3, notice that the map J_{L,K,n} also induces a restriction map

r_{L,K,n} : C(W(Δ̄_{L,n}/Γ), ℤ) → C(W(Δ̄_{K,n}/Γ), ℤ).

If we denote by P_{N}^{(L)} and P_{j}^{(K)} the respective filtrations, then we have restriction maps r_{L,K,n} : P_{je_{L/K}}^{(L)} → P_{j}^{(K)}. If we denote by δ_{ℓ} the map δ_{ℓ}(f) = f − f ∘ T^{ℓ}, then the restriction also satisfies r_{L,K,n} ∘ δ_{e_{L/K}} = δ ∘ r_{L,K,n}, hence there is an induced map

r̄_{L,K,n} : F_{j}^{(L,e_{L/K})} → F_{j}^{(K)}, where we have set F_{j}^{(L,e_{L/K})} := P_{je_{L/K}}^{(L)}/δ_{e_{L/K}}P_{(j−1)e_{L/K}}^{(L)}. An argument analogous to the one used in the proof of Proposition 5.3 shows that the F_{j}^{(L,e_{L/K})} give a filtration of

H^{1}(W(Δ̄_{L,n}/Γ)_{T^{e_{L/K}}} , ℤ) = \underrightarrow{lim}_{j} F_{j}^{(L,e_{L/K})}.

There is a corresponding map

C(W(Δ̄_{L,n}/Γ), ℤ)_{T} → C(W(Δ̄_{L,n}/Γ), ℤ)_{T^{e_{L/K}}} ,

induced by the covering π_{e_{L/K}} : W(Δ̄_{L,n}/Γ)_{T^{e_{L/K}}} → W(Δ̄_{L,n}/Γ)_{T}. On the level of filtrations this has the following description. The module P_{je_{L/K}}^{(L)} can be identified with

the span of functions in P_{j,s}^{(L)}, s = 0, …, e_{L/K} −1, where P_{j,s}^{(L)} = T^{j}(P_{j}^{(L)}). The inclusion ι : P_{j}^{(L)} ↪ P_{je_{L/K}}^{(L)} as P_{j,0}^{(L)} then satisfies δ ∘ ι = ι ∘ δ_{e_{L/K}}. We obtain an induced map

F_{j}^{(L)} → F_{j}^{(L,e_{L/K})}. Thus, the map induced by the covering π_{e_{L/K}} also preserves the filtrations, and we obtain maps F_{j}^{(L)} → F_{j}^{(K)} that induce (π_{e_{L/K}} ∘ J_{L,K,T})^{∗} on the direct limits. ⋄

5.2 Dynamical cohomology

Let Δ be a directed tree on which the Schottky group Γ acts, with the same assumptions as in the previous paragraph.

On the set of ends ∂Δ we consider a measure dμ defined by first introducing the distance function d(v) := dist(v, x_{0}), for v ∈ Δ^{0} and x_{0} the base point in Δ^{0} with respect to which the structure of directed graph on Δ is determined. Then the measure on ∂Δ is defined by assigning its value on the clopen set V (v), given by the ends of all paths in Δ starting at a vertex v, to be

μ(V (v)) = q^{−d(v)−1},

with q = card(O/m).

Proposition 5.6 This induces a measure on W(Δ/Γ), with respect to which the shift map T is measure preserving.

\underline{Proof.} Notice that, if we identify the points of V (v) with infinite paths starting at v,

V (v) = {w_{0}w_{1} … w_{n} …: s(w_{0}) = v, w_{k} ∈ (Δ)_{+}^{1}},

then the image T (V (v)) will have measure μ(T (V (v))) = μ(V (v))/q. In the case of walks starting at a vertex v, the map T scales the measure of the set of walks starting with an edge w ∈ (Δ)_{+}^{1} by a factor q^{−1} and the measure of the set of walks starting with an edge w with w̄ ∈ (Δ)_{+}^{1} by a factor q.

We can define a map from W(Δ/Γ) to V × V, by splitting each doubly infinite sequence

… w_{−n}w_{−n+1} … w_{−1}w_{0}w_{1} … w_{ℓ}w_{ℓ+1} …

into the two sequences

(w_{0}w_{1} … w_{ℓ}w_{ℓ+1} …, w̄_{−1}w̄_{−2} … w̄_{−n+1}w̄_{−n} …),  (5.2)

each of which defines a point in the fundamental domain V, if we identify admissible sequences of edges in the fundamental domain Δ/Γ with admissible sequences of edges in Δ with the condition that the first edge w_{0} (or w̄_{−1}) lies in a chosen fundamental domain V̄ of the action of Γ which contains the base point x_{0}. Then the action of T maps (5.2) to

(w_{1} … w_{ℓ}w_{ℓ+1} …, w̄_{0}w̄_{−1}w̄_{−2} … w̄_{−n+1}w̄_{−n} …),  (5.3)

hence it scales the measure on one factor by q and on the other factor by q^{−1}, so that the measure induced on W(Δ/Γ), by restricting to V × V the product measure on ∂Δ × ∂Δ, is preserved by T. ⋄

Consider the free ℤ-modules P_{n}, introduced in the proof of Proposition 5.3, of functions of at most n + 1 future coordinates. We can realize P_{n} ⊗_{ℤ} ℂ as a vector subspace of L^{2}(∂Δ, dμ). The operator δ is bounded in norm, hence the F_{n} = P_{n}/δP_{n−1} have induced norms and bounded inclusions F_{n} ⊗ ℂ ↪ F_{n+1} ⊗ ℂ, where the F_{n} are the torsion free ℤ-modules of Proposition 5.3.

Using the inner product induced from L^{2}(∂Δ, dμ), we can split F_{n} ≅ F_{n−1} ⊕(F_{n−1}^{⊥} ∩ F_{n}).

Definition 5.7 The dynamical cohomology H_{dyn}^{1}(Δ/Γ) is the norm completion of the graded complex vector space

H_{dyn}^{1}(Δ/Γ) := ⊕_{k≥0}Gr_{k}

with Gr_{n} := (F_{n} ∩ F_{n−1}^{⊥}).

We now construct a representation, by linear bounded operators, of a graph C^{∗}- algebra on this Hilbert space.

Proposition 5.8 There is a representation of the algebra C^{∗}(Δ/Γ) by bounded linear operators on the Hilbert space H_{dyn}^{1}(Δ/Γ).

\underline{Proof.} For w ∈ (Δ/Γ)_{+}^{1}, we define linear operators T_{w} of the form

(T_{w}f)(w_{0}w_{1}w_{2} …) = \{ \begin{matrix}f(ww_{0}w_{1}w_{2} …) & r(w) = s(w_{0}) \\ 0 & r(w) ≠ s(w_{0}). \end{matrix}

For v ∈ Δ/Γ^{0}, let P_{v} denote the projection

(P_{v}f)(w_{0}w_{1}w_{2} …) = \{ \begin{matrix}f(w_{0}w_{1}w_{2} …) & s(w_{0}) = v \\ 0 & s(w_{0}) ≠ v, \end{matrix}

We also define projections Π_{w} of the form

(Π_{w}f)(w_{0}w_{1}w_{2} …) = \{ \begin{matrix}f(w_{0}w_{1}w_{2} …) & w_{0} = w \\ 0 & w_{0} ≠ w. \end{matrix}

Finally, we define linear operators s_{w} and S_{w}

s_{w} := ∑_{w′} A(w, w′)T_{w}Π_{w′} and S_{w} := \sqrt{q} s_{w}.  (5.4)

These define bounded linear operators on L^{2}(∂Δ, dμ). The S_{w} are partial isometries, satisfying S_{w}S_{w}^{∗} = Π_{w}. Thus, the operators {P_{v}, S_{w}} form a Cuntz–Krieger family, since we obtain P_{v} = ∑_{s(w)=v} Π_{w} = ∑_{s(w)=v} S_{w}S_{w}^{∗} and S_{w}^{∗}S_{w} = ∑ A(w, w′)S_{w′}S_{w′}^{∗} = ∑ A(w, w′)P_{w′} = P_{r(w)}.

The same argument given in [6], Proposition 4.19, shows that S_{w}δ = δS_{w}, so that they descend to bounded operators on H_{dyn}^{1}(Δ/Γ). ⋄

5.3 Embedding of cohomologies

Let Δ be any of the trees Δ̄_{K,n}, n ≥ 0, or Δ_{Γ}. We construct a family of embeddings of the cohomology of the dual graph Δ_{Γ}/Γ in the dynamical cohomology H_{dyn}^{1}(Δ/Γ). Let |Δ_{Γ}| denote the geometric realization of the dual graph Δ_{Γ}/Γ.

Theorem 5.9 For each N ≥ 0 there are embeddings ϕ_{N} of the cohomology of the dual graph H^{1}(|Δ_{Γ}/Γ|, ℂ) into the dynamical cohomology H_{dyn}^{1}(Δ/Γ).

\underline{Proof.} We define maps ϕ_{N} from the cohomology H^{1}(|Δ_{Γ}/Γ|, ℂ) to H_{dyn}^{1}(Δ_{K}/Γ) in the following way: let {γ_{i}}_{i=1}^{g} be a chosen set of generators of the Schottky group Γ, [γ_{i}] the corresponding homology classes in H_{1}(|Δ_{Γ}/Γ|, ℤ) = Γ/[Γ, Γ], and η_{i} the dual generators in cohomology. Let w_{i} be the finite admissible word in the edges of the directed graph V̄ (the fundamental domain for the Γ-action on Δ) that represents the generator γ_{i}, with length ℓ_{i} = |w_{i}|. Here, in the case of Δ = Δ̄_{K,n}, we first notice that the first cohomology group of the dual graph is the same as H^{1}(|Δ′_{Γ}/Γ|, ℤ), since the insertion of extra vertices does not change the topology of the graph. Since we have Δ′_{Γ} ⊂ Δ_{K,n} we obtain the w_{i} as above, as edges in the corresponding fundamental domain V̄ for Δ_{K,n}.

We then set

ϕ_{N}(η_{i}) = P_{Nℓ_{i}}^{⊥} χ_{i,N},  (5.5)

where

χ_{i,N} := χ_{W^{+}(Δ/Γ,\underbrace{w_{i} ⋯ w_{i}}_{N−times})}  (5.6)

is the characteristic function of the set W^{+}(Δ/Γ, w_{i} ⋯ w_{i}) of walks in Δ/Γ that begin with the word w_{i} repeated N-times. The elements χ_{i,N} lie in F_{Nℓ_{i}}. We denote by P_{k}^{⊥} the orthogonal projection of F_{k} onto Gr_{k}.

The elements ϕ_{N}(η_{i}) are all linearly independent in H_{dyn}^{1}(Δ/Γ), hence the ϕ_{N} give linear embeddings of H^{1}(|Δ_{Γ}/Γ|, ℂ) into H_{dyn}^{1}(Δ/Γ). ⋄

Corollary 5.10 For any finite set of distinct {N_{k}}, the set ∪_{k}{ϕ_{N_{k}}(η_{i})}_{i=1}^{g} consists of linearly independent vectors in H_{dyn}^{1}(Δ/Γ). Thus, we obtain an embedding

Φ = ⊕_{N}ϕ_{N} : ⨁_{N} H^{1}(|Δ_{Γ}/Γ|, ℂ) → H_{dyn}^{1}(Δ/Γ).  (5.7)

Notice that the map Φ of (5.7) does not preserve the graded pieces, due to the rescaling of the degrees by ℓ_{i} in (5.5). This has to be taken into account if we want to recover arithmetic information such as the local L–factors of [8] from the dynamical cohomology. For this reason, it may be necessary to blow up of some double points on the special fiber.

Lemma 5.11 It is always possible to reduce to the case where all the ℓ_{i} = ℓ, after blowing up a certain number of double points on the special fiber.

\underline{Proof.} Blowing-up a double point on the special fiber C(k), for k = O/𝔪, corresponds to introducing one extra vertex in the dual graph Δ_{Γ}/Γ. This changes by one the lengths ℓ_{i} for those generators of Γ for which the corresponding chain of edges w_{i} passes through the newly inserted vertex. ⋄

Thus, possibly after blowing up some double points, we obtain:

ϕ_{N} : H^{1}(|Δ_{Γ}/Γ|, ℂ) ↪ Gr_{Nℓ} ⊂ H_{dyn}^{1}(Δ_{K}/Γ),

for some ℓ ≥ 1.

5.4 Spectral triples

In this paragraph we show that we can associate to a given Mumford curve a family of spectral triples (A_{n}, H_{n}, D_{n}), for n ≥ −1. Each triple in this family corresponds to the choice of a graph Δ = Δ̄_{K,n}, for n ≥ 0 and Δ = Δ_{Γ} for n = −1. The Dirac operator in these spectral triples depends only on the graded structure of the space H_{dyn}^{1}(Δ/Γ).

Recall that a spectral triple (A, H, D) is a triple of a C^{∗}-algebra, with a representation in the algebra of bounded operators on the Hilbert space H, and a Dirac operator D, which is a self-adjoint unbounded operator acting on H, such that (λ − D)^{−1} is a compact operator for all λ ∉ ℝ and the commutators [a, D] are bounded operators for all a ∈ A_{0}, with A_{0} a dense involutive subalgebra of A.

Recall also that a spectral triple (A, H, D) has an associated family of zeta functions of the form

ζ_{a,|D|}(z) = T r(a|D|^{−z}) = ∑_{λ∈Spec(|D|)\{0}} T r(aΠ_{λ})λ^{−z},  (5.8)

with a ∈ A_{0} ∪ [D, A_{0}], and with Π_{λ} the spectral projection on λ ∈ Spec(|D|). The properties of these zeta functions are related to the notion of dimension spectrum for the spectral triple (the set of poles of the ζ_{a,|D|}) and to the local index formula of Connes and Moscovici. For the purpose of this paper, we will extend the zeta functions (5.8) to the case where the element a is a weak limit of certain sequences of elements in A, since, in our construction, this is the type of zeta functions that recovers the arithmetic invariant given by the local Euler factor.

There are corresponding two-variable zeta functions

ζ_{a,|D|}(s, z) = ∑_{λ∈Spec(|D|)} T r(aΠ_{λ})(s + λ)^{−z},  (5.9)

and associated regularized determinants

det_{∞,a,|D|}(s) = exp (−\frac{d}{dz} ζ_{a,|D|}(s, z)|_{z=0}) .  (5.10)

We consider the Hilbert space

H = H_{dyn}^{1}(Δ/Γ) ⊕ H_{dyn}^{1}(Δ/Γ).  (5.11)

On this space we consider the diagonal action of C^{∗}(Δ/Γ). We also introduce the notation Gr_{n,−} := Gr_{n} ⊕ 0 and Gr_{n,+} := 0 ⊕ Gr_{n}.

We define the Dirac operator D acting on H by setting

D|_{Gr_{+,n}} = n D|_{Gr_{−,n}} = −n − 1.  (5.12)

Proposition 5.12 The data (C^{∗}(Δ/Γ), H_{dyn}^{1}(Δ/Γ)⊕H_{dyn}^{1}(Δ/Γ), D) determine a spectral triple in the sense of Connes [4].

\underline{Proof.} In order to obtain a spectral triple we need to check the compatibility requirement between the Dirac operator D and the action of the algebra C^{∗}(Δ/Γ). It is sufficient to check that the commutators [D, S_{w}] are bounded operators. This follows easily since S_{w} : Gr_{±,n} → Gr_{±,n−1}, so that [D, S_{w}]f = ∓S_{w}f. The remaining properties are easily verified. ⋄

We can modify slightly the above construction, in order to take into account the scaling factor ℓ in the grading between ⊕_{N}H^{1}(|Δ_{Γ}/Γ|, ℂ) and its image under Φ in H_{dyn}^{1}(Δ/Γ).

Corollary 5.13 We modify the operator D of (5.12), by setting

D|_{Gr_{+,n}} = \frac{n}{ℓ} \frac{2π}{log q} D|_{Gr_{−,n}} = \frac{−(n + 1)}{ℓ} \frac{2π}{log q} .  (5.13)

Here q is the characteristic of the residue field k(v) and ℓ is the length of all the words representing the generators of Γ, possibly after blowing up some points on the special fiber. With this modified operator D, we still obtain a spectral triple

(C^{∗}(Δ/Γ), H_{dyn}^{1}(Δ/Γ) ⊕ H_{dyn}^{1}(Δ/Γ), D).

5.5 Local L-factor

Let X be a curve over a global field 𝕂. We assume semi-stability at all places of bad reduction. The local Euler factor at a place v has the following description ([20]):

L_{v}(H^{1}(X), s) = det (1 − F r_{v}^{∗}N(v)^{−s}|H^{1}(X̄, ℚ_{ℓ})^{I_{v}})^{−1} .  (5.14)

Here F r_{v}^{∗} is the geometric Frobenius acting on ℓ-adic cohomology of X̄ = X ⊗Spec(𝕂̄), with 𝕂̄ the algebraic closure and ℓ a prime with (ℓ, q) = 1, where q is the characteristic of the residue field k(v) at v. We denote by N the norm map. The determinant is evaluated on the inertia invariants H^{1}(X̄, ℚ_{ℓ})^{I_{v}} at v (all of H^{1}(X̄, ℚ_{ℓ}) when v is a place of good reduction).

Suppose v is a place of k(v)-split degenerate reduction. Then the completion of X at v is a Mumford curve X_{Γ}. In this case, the Euler factor (5.14) takes the following form:

L_{v}(H^{1}(X_{Γ}), s) = ∏_{λ}(1 − λq^{−s})^{− dim H^{1}(X_{Γ})_{λ}^{Iv}} = (1 − q^{−s})^{−g},  (5.15)

since the eigenvalues {λ} of the Frobenius, in this case, are all λ = 1. Here we denote by H^{1}(X_{Γ})_{λ}^{I_{v}} the eigenspaces of the Frobenius.

Deninger in [8] and [9] obtained the local factor (5.14) as a regularized determinant over an infinite dimensional cohomological theory.

In the case of Mumford curves, Deninger’s calculation can be recast in terms of the data of the spectral triples of Proposition 5.12 and of the embeddings Φ^{±} of cohomologies.

We consider the operator iD (an imaginary rotation of the Dirac operator) and the zeta functions

ζ_{a,iD,+}(s, z) := ∑_{λ∈Spec(iD)∩i[0,∞)} Tr(aΠ_{λ})(s + λ)^{−z}
ζ_{a,iD,−}(s, z) := ∑_{λ∈Spec(iD)∩i(−∞,0)} Tr(aΠ_{λ})(s + λ)^{−z}.

(5.16)

Then we have a regularized determinant

det_{∞,a,iD}(s) := exp (−ζ′_{a,iD,+}(s, 0)) exp (−ζ′_{a,iD,−}(s, 0))  (5.17)

Theorem 5.14 Let π(V) be the orthogonal projection of H_{dyn}^{1}(Δ/Γ)⊕H_{dyn}^{1}(Δ/Γ) onto the graded subspace V = Im(Φ^{−}) ⊕ Im(Φ^{+}), where we denote by Φ^{±} the maps Φ ⊕ 0 and 0 ⊕ Φ. Then the regularized determinant (5.17), with a = π(V) and D the Dirac operator of Corollary 5.13 satisfies

det_{∞,π(V),iD}(s) = L_{v}(H^{1}(X_{Γ}), s)^{−1}.  (5.18)

\underline{Proof.} When we compute the zeta functions (5.16) for the Dirac operator of Corollary 5.13, and a = π(V), we obtain

ζ_{π(V),iD,+}(s, z) = ∑_{n=0}^{∞} Tr(π(V)Π_{+,nℓ})(γ(τ + n))^{−z}
ζ_{π(V),iD,−}(s, z) = ∑_{n=0}^{∞} Tr(π(V)Π_{−,nℓ})(γ(τ − n))^{−z} − Tr(π(V)Π_{−,0})(τγ)^{−z},

for γ = \frac{2πi}{log q} and τ = \frac{log q}{2πi}s, with choice of arguments −π < arg γ(τ + n) < π, as in [9].

Furthermore, we have T r(π(V)Π_{±,nℓ}) = dim(Gr_{±,nℓ} ∩ V) = g. In fact, the space Gr_{+,nℓ} ∩ V = Im(Φ^{+}) is generated by 0 ⊕ χ_{i,n} for i = 1, …, g and Gr_{−,nℓ} ∩ V = Im(Φ^{−}) is generated by the element χ_{i,n} ⊕ 0 for i = 1, …, g.

The result then follows the calculation of the regularized determinant given in [9]. Namely, we obtain

(1 − q^{−s})^{−g} = (τγ)^{−g} exp(−gζ′_{γ}(τ, 0)) exp(−gζ′_{−γ}(−τ, 0)),

which is exactly the regularized determinant det_{∞}(s − Θ_{q}) computed in [9] for the spectrum (with multiplicity g)

Spec(s − Θ_{q}) = \{\frac{2πi}{log q} (\frac{s log q}{2πi} + n) : n ∈ ℤ\} .

⋄

It is very interesting to notice an important difference between the archimedean and non-archimedean cases. At the archimedean prime (cf. [6], [7]) the local factor is described in terms of zeta functions for a Dirac operator D. On the other hand, at the non-archimedean places, in order to get the correct normalization as in [9], we need to introduce a rotation of the Dirac operator by the imaginary unit, D ↦ iD. This rotation corresponds to the Wick rotation that moves poles on the real line to poles on the imaginary line (zeroes for the local factor) and appears to be a manifestation of a rotation from Minkowskian to Euclidean signature it ↦ t, as already remarked by Manin ([12] p.135), who wrote that “imaginary time motion” may be held responsible for the fact that zeroes of Γ(s)^{−1} are purely real whereas the zeroes of all non-archimedean Euler factors are purely imaginary. This seems to hint to the existence of a more refined construction involving Minkowskian geometry, where the rotation D ↦ iD could be interpreted as a rotation it ↦ t of an infinitesimal length element D^{−1} ∼ ic dt. A more precise treatment would require adapting the structure of spectral triple to the case of Minkowskian signature. Another piece of supporting evidence for the idea that a more refined construction should involve Minkowskian geometry comes from the cohomological construction of [5]. In fact, in [6], we only used part of the full symmetry group determined by the Lefschetz module structure, namely the part corresponding to real hyperbolic geometry, so as to match the results of [13]. The full symmetry group is SL(2, ℝ)×SL(2, ℝ), which is the isometry group not of 3-dimensional real hyperbolic geometry, but of its Minkowskian version, anti de Sitter space AdS_{2+1}. On the relation between the results of [13] and AdS_{2+1} see also [16].

5.6 AF core

In order to understand the zeta functions ζ_{π(V),D,±}(τ, z) of Theorem 5.14 in terms of the spectral triple, we still need to express the operator π(V) in terms of elements in the algebra C^{∗}(Δ/Γ).

The graph C^{∗}-algebra C^{∗}(Δ/Γ) contains an AF core given by the AF algebra obtained as the norm closure of ∪_{n}F_{n}, where the finite dimensional algebras F_{n} are given by

F_{n} = span{S_{μ}S_{ν}^{∗} : μ, ν ∈ P^{n}(Δ/Γ), t(μ) = t(ν)}.  (5.19)

Here we used the notation S_{μ} = S_{w_{1}} ⋯ S_{w_{k}}, for μ = w_{1} ⋯ w_{k}. The AF core can be identified with the fixed point algebra of the gauge action (cf. [1]),

\overline{⋃_{n} F_{n}} ≅ C^{∗}(Δ/Γ)^{U(1)}.  (5.20)

Let {w_{i}}_{i=1}^{g} be the words corresponding to the generators of Γ, which we assume all of equal length ℓ, possibly after some blow ups. Let S_{w_{i}} be the corresponding operators in C^{∗}(Δ/Γ). The operators S_{w_{i}}^{n}S_{w_{i}}^{∗}^{n} belong to the subalgebra F_{nℓ} in the AF core of C^{∗}(Δ/Γ).

Each Q_{i,n} = S_{w_{i}}^{n}S_{w_{i}}^{∗}^{n} acts on L^{2}(∂Δ, dμ) as multiplication by the characteristic function χ_{i,n} of (5.6), hence Q_{i,n} maps Gr_{±,nℓ} to itself, with range the one dimensional subspace of Gr_{±,nℓ} spanned by χ_{i,n}. Thus, the operator Q_{n} = ∑_{i} Q_{i,n} projects Gr_{±,nℓ} onto the g-dimensional subspace Gr_{±,nℓ} ∩ V.

For the Dirac operator of (5.13), we write D = ∑_{n≥0} Π_{n}λ_{n}, where with λ_{n} = 2πn/ℓ log q and Π_{n} = Π_{+,n} ⊕ Π_{−,n−1}. We then have the following result.

Proposition 5.15 the zeta function

ζ_{π(V),|D|}(z) = T r(π(V)|D|^{−z})

can be written in the form

ζ_{π(V),|D|}(z) = Tr (∑_{n>0} Q_{i,n}Π_{nℓ}λ_{n}^{−z})  (5.21)

with the Q_{i,n} in the AF core of C^{∗}(Δ/Γ).

6 Foam spaces

We now consider the local factor (5.14) in the more general case, where we drop the assumption that v is a place of k(v)-split degenerate reduction. In this case, we no longer have a p–adic uniformization of the completion of X at v as a Mumford curve, and correspondently, the local factor is no longer determined solely in terms of the combinatorics of the dual graph, but it depends essentially on extra geometric information on the nature of the degeneration. In particular, the inertia invariants H^{1}(X̄, ℚ_{ℓ})^{I_{v}} are described only partly by the cohomology of the dual graph, with the extra information provided by the cohomology of the single components of the dual fiber, which in the general case will no longer be just rational curves.

More precisely, if we denote by H^{1}(X̄)_{λ}^{I_{v}} the eigenspace of the geometric Frobenius, with eigenvalue λ, we can write the Euler factor in the form

L_{v}(H^{1}(X), s) = ∏_{λ}(1 − λq^{−s})^{−d_{λ}},  (6.1)

with d_{λ} = dim H^{1}(X̄)_{λ}^{I_{v}}. Deninger’s description of the local factor as regularized determinant holds in this more general case, in the form det_{∞}(s − Θ) where Θ has spectrum \{α_{λ} + \frac{2πin}{log q}\}, with n ∈ ℤ, λ ∈ Spec(F r_{v}^{∗}), and q^{α_{λ}} = λ.

We want to modify the graphs Δ/Γ considered in the previous sections, in such a way that the corresponding dynamical cohomology will contain a linear subspace isomorphic to an infinite direct sum ⊕_{N}H^{1}(X̄, ℚ_{ℓ})^{I_{v}}, and such that the construction of the spectral triple and the derivation of the regularized determinant described in the case of Mumford curves will extend to this case to recover (6.1).

Using the exact sequence of [17] p.110-111, we obtain an identification

H^{1}(X̄, ℚ_{ℓ})^{I_{v}} ≅ H^{1}(|Δ_{Γ}/Γ|) ⊗_{σ_{ℓ}} ℚ_{ℓ} ⊕ H^{1}(X_{v}^{[0]}) ⊗_{σ_{ℓ}} ℚ_{ℓ},  (6.2)

where σ_{ℓ} : ℚ_{ℓ} → ℂ is a fixed embedding of ℚ_{ℓ} in ℂ, for a prime ℓ with (ℓ, q) = 1, and we denote by X_{v}^{[0]} the disjoint union of the components of the special fiber. In the case of k(p)-split reduction, where all components are ℙ^{1}’s, (6.2) is simply identified with H^{1}(|Δ_{Γ}/Γ|)⊗_{σ_{ℓ}} ℚ_{ℓ} as in the previous sections. The finite decomposition H^{1}(X̄, ℚ_{ℓ})^{I_{v}} = ⊕_{λ}H^{1}(X̄, ℚ_{ℓ})_{λ}^{I_{v}} in eigenvalues of the geometric Frobenius provides corresponding spaces H^{1}(|Δ_{Γ}/Γ|)_{λ} and H^{1}(X_{v}^{[0]})_{λ} of dimensions d_{λ}^{Γ} and d_{λ}^{0}, respectively, with d_{λ}^{Γ} + d_{λ}^{0} = d_{λ}.

We choose vertices x_{λ} (not necessarily distinct) of Δ_{Γ}/Γ, and attach to the vertex x_{λ} new outgoing edges w_{i,λ}^{0}, with i = 1, … d_{λ}^{0}. We denote by E_{v} the oriented graph obtained via this construction, after appending tails to all sinks.

Remark 6.1 For certain classes of examples, our graph E_{v} can be embedded as a subgraph of the “foam space” defined in [3]. The foam space is a graph F_{v} associated to the fiber X_{v} of an arithmetic surface 𝔛 over Spec(O_{𝕂}), obtained by replacing the special fiber X_{v} with an infinite series of blowups of its 𝔽_{q}-points. The graph F_{v} is the limit of the dual graphs associated to this series of blow-ups (cf. [15] §35). For this reason, we think of the graphs E_{v} as a generalization of “foam spaces”.

To our foam space E_{v}, we associate the corresponding dynamical cohomology H^{1}(E_{v}) as in the previous sections, and the graph C^{∗}-algebra C^{∗}(E_{v}). The argument of Proposition 5.12 extends to this case and gives a spectral triple

(C^{∗}(E_{v}), H^{1}(E_{v}) ⊕ H^{1}(E_{v}), D).

We now define embeddings of cohomology groups as follows. Let ω_{i,λ}, for i = 1, …, d_{λ}^{Γ} be loops of edges in Δ_{Γ}/Γ, with |ω_{i,λ}| = ℓ_{i,λ}, representing homology classes dual to a basis {η_{i,λ}^{Γ}} of H^{1}(|Δ_{Γ}/Γ|)_{λ}. Up to adding vertices to the graph Δ_{Γ}/Γ by blowing up double points in the closed fiber, we can assume that all the ℓ_{i,λ} = ℓ. Adding vertices in this way does not change H^{1}(X_{v}^{[0]}), since the components of the closed fiber that correspond to the new vertices all have trivial H^{1}.

We consider then the linear embedding

Φ_{N,λ}^{Γ} : H^{1}(|Δ_{Γ}/Γ|)_{λ} ↪ Gr_{Nℓ} ⊂ H^{1}(E_{v}),

given by

Φ_{N,λ}^{Γ}(η_{i,λ}^{Γ}) = P_{Nℓ_{λ}}^{⊥}χ_{W^{+}(E_{v},\underbrace{ω_{i,λ} ⋯ ω_{i,λ}}_{N−times})}.

We also consider the embeddings

Φ_{N,λ}^{0} : H^{1}(X_{v}^{[0]})_{λ} ↪ Gr_{Nℓ} ⊂ H^{1}(E_{v}),

Φ_{N,λ}^{0}(η_{i,λ}^{0}) = P_{N}^{⊥}χ_{W^{+}(E_{v},\underbrace{w_{i,λ}^{0} ⋯ w_{i,λ}^{0}}_{Nℓ−times})},

where the η_{i,λ}^{0} form a basis of H^{1}(X_{v}^{[0]})_{λ} and the w_{i,λ}^{0} are the corresponding oriented edges of E_{v}. Let Φ_{λ}^{Γ} = ⊕_{N}Φ_{N,λ}^{Γ} and Φ_{λ}^{0} = ⊕_{N}Φ_{N,λ}^{0}, and let Φ_{λ} = Φ_{λ}^{Γ} ⊕ Φ_{λ}^{0}. With Φ_{λ}^{±} defined as the Φ^{±} in Theorem 5.14, we denote by V_{λ} = Im(Φ_{λ}^{−})⊕Im(Φ_{λ}^{+}), and by π(V_{λ}) the corresponding orthogonal projection.

We then extend the result of Theorem 5.14 to this more general setting.

Theorem 6.2 Consider the regularized determinants (5.17), with a_{λ} = π(V_{λ}) and D the Dirac operator of Corollary 5.13 for the spectral triple (C^{∗}(E_{v}), H^{1}(E_{v})⊕H^{1}(E_{v}), D). We obtain

∏_{λ} det_{∞,π(V_{λ}),iD}(s) = L_{v}(H^{1}(X), s)^{−1}.  (6.3)

The operators π(V_{λ}) are related to the AF core of the C^{∗}-algebra C^{∗}(E_{v}) as in (5.21).

\underline{Proof.} We compute the zeta functions (5.16) for the Dirac operator of the spectral triple (C^{∗}(E_{v}), H^{1}(E_{v}) ⊕ H^{1}(E_{v}), D), with a = π(V). We obtain

ζ_{π(V_{λ}),iD,+}(s, z) = ∑_{n=0}^{∞} Tr(π(V_{λ})Π_{+,nℓ})(γ(τ_{λ} + n))^{−z}
ζ_{π(V_{λ}),iD,−}(s, z) = ∑_{n=0}^{∞} Tr(π(V_{λ})Π_{−,nℓ})(γ(τ_{λ} − n))^{−z} − Tr(π(V_{λ})Π_{−,0})(τ_{λ}γ)^{−z},

for γ = \frac{2πi}{log q}, τ_{λ} = \frac{log q}{2πi}(s − α_{λ}), and q^{α_{λ}} = λ, and with choice of arguments as in [9].

By construction, we have T r(π(V_{λ})Π_{±,nℓ}) = dim(Gr_{±,nℓ} ∩ V_{λ}) = d_{λ}, hence the left hand side of (6.3) is the regularized determinant det_{∞}(s − Θ_{q}) computed in [9], with spectrum (with multiplicities d_{λ})

Spec(s − Θ_{q}) = \{\frac{2πi}{log q} (\frac{log q}{2πi} (s − α_{λ}) + n) : n ∈ ℤ, λ ∈ Spec(F r_{v}^{∗})\} .

The proof that the operators π(V_{λ}) can be obtained as strong limits of operators in the AF core of the C^{∗}-algebra C^{∗}(E_{v}) is analogous to the one given in the case of Mumford curves. ⋄

Acknowledgment. Part of this work was done during visits of the first author to the Max Planck Institute in Bonn and of the second author to Florida State University and University of Toronto. We thank these institutions for hospitality and support. We are very grateful to Christopher Deninger for a crucial remark about normalizations. This research has been partially supported by NSERC grant 72016789 and by Humboldt Foundation and the German Government (Sofja Kovalevskaya Award).

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Caterina Consani, Department of Mathematics, University of Toronto, Canada. email: kc@math.toronto.edu

Matilde Marcolli, Max–Planck–Institut für Mathematik, Bonn Germany. email: marcolli@mpim-bonn.mpg.de