New perspectives in Arakelov geometry

Caterina Consani and Matilde Marcolli

Abstract

In this paper we use Connes’ theory of spectral triples to provide a unified view, via non-commutative geometry, of the archimedean and the totally split degenerate fibers of an arithmetic surface.

1 Introduction.

The aim of this article is to report on some of the recent results obtained by the two authors on a non-commutative interpretation of the totally split degenerate fibers of an arithmetic surface. The results stated in the first part of the paper were announced in a talk given by the first author at the VII meeting of the CNTA in May 2002 at Montreal. Because of the survey nature of this article, we decided to include no proofs for the results stated here. For a detailed treatement of the material exposed in this note as well as for further results on this subject we refer the reader to the papers [6] and [7].

Let X be an arithmetic surface defined over Spec(ℤ) having the smooth algebraic curve X_{/ℚ} as its generic fiber. It is well known that, as a Riemann surface, X admits always a uniformization by means of a Schottky group Γ. In fact, the presence of this uniformization plays a fundamental role in the theory of the “fiber at infinity” of X described in [16] and [6]. In analogy to Mumford’s p-adic uniformization of algebraic curves (cf. [17]), the Riemann surface X_{/ℂ} can be interpreted as the boundary at infinity of a 3-manifold 𝔛_{Γ} defined as the quotient of the real hyperbolic 3-space ℍ^{3} by the action of the Schottky group Γ. The space 𝔛_{Γ} contains in its interior an infinite link of bounded geodesics. Manin gave an expression for the Arakelov Green function on X in terms of configurations of geodesics in 𝔛_{Γ}, thus interpreting this tangle as the dual graph G of the “closed fiber at infinity” of X.

In the first part of this paper we concentrate on Manin’s description of such dual graph and we exhibit the suspension flow S_{T} of a dynamical system T on the limit set of the Schottky group Γ as a geometric model for G. In particular, the first cohomology group of S_{T} determines a model of the first cohomology of the dual graph of the “fiber at infinity”. Furthermore, the first (co)homology group of S_{T} carries a natural filtration, which we use to define a (co)homology theory associated to the dynamical system T.

The major breakthrough of this construction arises from the fact that this dynamical cohomology contains a subspace isomorphic to the archimedean cohomology of [5]: the group of invariants for the action of the local monodromy “at infinity”. This result implies that the archimedean cohomology (a group retaining the correct arithmetic properties, in order to be interpreted as the cohomology of the “fiber at infinity” of X) can be identified with the cohomology of a topological space constructed in terms of geodesics in 𝔛_{Γ}.

The Cuntz-Krieger algebra associated to T acts on a completion of H^{1}(S_{T}). This algebra carries a refined information on the action of the Schottky group Γ on its limit set and it induces an action, by bounded operators, on a Hilbert space completion H_{dyn}^{1} of the dynamical cohomology. This representation extends to a representation of the C^{∗}-algebra A = (C(Λ_{Γ}) ⊗ C_{0}(ℍ^{3})) ⋊ Γ. As a non-commutative space, this algebra provides an analog, at arithmetic infinity, of the p-adic reduction maps of [16] and [17] which were realized in terms of certain finite graphs in a quotient of the Bruhat-Tits tree of PGL(2, K), when K is a finite extension of ℚ_{p} (see below).

Similarly, we find a representation, on a Hilbert space completion H_{1}^{dyn} of the dynamical homology, of a C^{∗}-algebra strongly Morita equivalent to A, namely the algebra of sections of a flat C(Λ_{Γ})-bundle over the handlebody 𝔛_{Γ}.

We introduce a Dirac operator D on the Hilbert space H_{dyn}^{1} ⊕ H_{1}^{dyn}, which agrees with the “logarithm of Frobenius” of [5] and [10] when restricted to the subspace identified with the archimedean cohomology. Using Connes’ technique of transferring spectral geometry between Morita equivalent non-commutative spaces, we obtain a dynamical spectral triple, over the non-commutative space A acting on the Hilbert space H_{dyn}^{1}⊕M⊗H_{1}^{dyn}, where M is the bimodule implementing the Morita equivalence.

This structure further enriches the geometric interpretation of the archimedean cohomology, giving it the meaning of spinors on the noncommutative manifold A, with the logarithm of Frobenius introduced in [10] in the role of Dirac operator.

Let p be a finite prime where X has totally split degenerate reduction. Then, the completion X̂_{p} at p of the generic fiber of 𝔛 is a split-degenerate stable curve over ℚ_{p} (also called a Mumford curve) uniformized by the action of a p-adic Schottky group Γ. The dual graph of the reduction of X̂_{p} coincides with a finite graph obtained as the quotient of a tree Δ_{Γ} by the action of Γ.

The curve X̂_{p} is holomorphically isomorphic to a quotient of a subset of the ends of the Bruhat-Tits tree Δ of ℚ_{p} by the action of Γ. Thus, in this setting, the Bruhat-Tits tree at p replaces the hyperbolic space ℍ^{3} “at infinity”, and the analog of the tangle of bounded geodesics in 𝔛_{Γ} is played by doubly infinite walks in Δ_{Γ}/Γ.

In analogy with the archimedean construction, we define the system (W(Δ_{Γ}/Γ), T) where T is an invertible shift map on the set W(Δ_{Γ}/Γ) of doubly-infinite walks on the graph Δ_{Γ}/Γ. The first cohomology group H^{1}(W(Δ_{Γ}/Γ)_{T} , ℤ) of the mapping torus W(Δ_{Γ}/Γ)_{T} of T inherits a natural filtration using which we introduce a dynamical cohomology group. The Cuntz-Krieger graph algebra C^{∗}(Δ_{Γ}/Γ) has a representation on a Hilbert space completion H_{dyn}^{1}(Δ_{Γ}/Γ) of the dynamical cohomology on which a Dirac operator D acts. It follows that the data (C^{∗}(Δ_{Γ}/Γ), H_{dyn}^{1}(Δ_{Γ}/Γ) ⊕ H_{dyn}^{1}(Δ_{Γ}/Γ), D) determine a spectral triple in the sense of Connes in exact analogy with the archimedean case. We refer to [7] for a description of the arithmetic significance of this construction.

Acknowledgment. This paper was partly written during visits of the second author to Florida State University and University of Toronto. We thank these institutions for the hospitality. The first author is partially supported by NSERC grant 72016789. The second author is partially supported by the Humboldt Foundation and the German Government (Sofja Kovalevskaya Award).

2 Notation.

Throughout this paper we will denote by K one among the following fields: (a) the complex numbers ℂ, (b) a finite extension of ℚ_{p}. When (b) occurs, we write O_{K} for the ring of integers of K, 𝔪 ⊂ O_{K} for the maximal ideal and π ∈ 𝔪 for a uniformizer (i.e. 𝔪 = (π)). We also denote by k the residue classes field k = O/𝔪.

We denote by ℍ^{3} the three-dimensional real hyperbolic space i.e. the quotient ℍ^{3} = SU(2)\PGL(2, ℂ). This space can also be described as the upper half space ℍ^{3} ≃ ℂ×ℝ^{+} endowed with the hyperbolic metric. The group PSL(2, ℂ) acts on ℍ^{3} by isometries. The complex projective line ℙ^{1}(ℂ) is identified with the conformal boundary at infinity of ℍ^{3} and the action of PSL(2, ℂ) on ℍ^{3} extends to an action on \overline{ℍ}^{3} := ℍ^{3} ∪ ℙ^{1}(ℂ). The group PSL(2, ℂ) acts on ℙ^{1}(ℂ) by fractional linear transformations.

For g ≥ 1, g ∈ ℤ, a Schottky group of rank g is a discrete subgroup Γ ⊂ PSL(2, K), which is purely loxodromic and isomorphic to a free group of rank g. The quotient space 𝔛_{Γ} := ℍ^{3}/Γ is topologically a handlebody of genus g.

We denote by Λ_{Γ}, the limit set of the action of Γ. One sees that Λ_{Γ} is contained in ℙ^{1}(K). This set can also be described as the closure of the set of the attractive and repelling fixed points z^{±}(g) of the loxodromic elements g ∈ Γ. In the case g = 1 the limit set consists of two points, but for g ≥ 2 the limit set is usually a fractal of some Hausdorff dimension 0 ≤ δ = dim_{H}(Λ_{Γ}) < 2.

We denote by Ω_{Γ} = Ω_{Γ}(K) the domain of discontinuity of Γ, that is, the complement of Λ_{Γ} in ℙ^{1}(K).

When K = ℂ, the quotient X_{/ℂ} = Ω_{Γ}/Γ is a Riemann surface of genus g and the covering Ω_{Γ} → X_{/ℂ} is called a Schottky uniformization of X_{/ℂ}. Every complex Riemann surface X_{/ℂ} admits a Schottky uniformization. The handlebody 𝔛_{Γ} can be compactified by adding the conformal boundary at infinity X_{/ℂ} to obtain \overline{𝔛}_{Γ} := 𝔛_{Γ} ∪ X_{/ℂ} = (ℍ^{3} ∪ Ω_{Γ})/Γ.

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 directed graph is finite if E^{0} and E^{1} are finite sets. It 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. 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}.

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. A directed graph is a directed tree if, for any two vertices, there exists a unique walk in W^{∗}(E) connecting them.

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 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.

Even when not explicitly stated, all Hilbert spaces and algebras of operators we consider will be separable, i.e. they admit a dense (in the norm topology) countable subset.

3 A dynamical theory at infinity.

Let Γ ⊂ PSL(2, ℂ) be a Schottky group of rank g ≥ 2. Given a choice of a set of generators {g_{i}}_{i=1}^{g} for Γ, there is a bijection between the elements of Γ and the set of all admissible walks in the Cayley graph of Γ, namely reduced words in the {g_{i}}_{i=1}^{2g}, where we use the notation g_{i+g} := g_{i}^{−1}, for i = 1, …, g.

In the following we consider the sets S^{+} and S of resp. right-infinite, doubly infinite admissible sequences in the {g_{i}}_{i=1}^{2g}:

S^{+} = {a_{0}a_{1} … a_{ℓ} … |a_{i} ∈ {g_{i}}_{i=1}^{2g}, a_{i+1} ≠ a_{i}, ∀i ∈ ℕ},  (3.1)
S = {… a_{−m} … a_{−1}a_{0}a_{1} … a_{ℓ} … |a_{i} ∈ {g_{i}}_{i=1}^{2g}, a_{i+1} ≠ a_{i}, ∀i ∈ ℤ}.  (3.2)

On the space S we consider the topology generated by the sets W^{s}(x, ℓ) = {y ∈ S|x_{k} = y_{k}, k ≥ ℓ}, and W^{u}(x, ℓ) = {y ∈ S|x_{k} = y_{k}, k ≤ ℓ} for x ∈ S and ℓ ∈ ℤ.

There is a two-sided shift operator T acting on S as the map

\begin{aligned}T ( & … & a_{−m} & … & a_{−1} & a_{0} & a_{1} & … & a_{ℓ} & … & ) = \\ & … & a_{−m+1} & … & a_{0} & a_{1} & a_{2} & … & a_{ℓ+1} & … \end{aligned}  (3.3)

Definition 3.1 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},

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

Lemma 3.2 The space S can be identified with the subshift of finite type S_{A} with the symmetric 2g × 2g matrix A given by the directed edge matrix of the Cayley graph of Γ.

The two-sided shift operator T on S of (3.3) decomposes S in a product of expanding and contracting directions, so that (S, T) is a Smale space.

The following topological space is defined in terms of the Smale space (S, T) and will be considered as a “dual graph” associated to the fiber at arithmetic infinity of the arithmetic surface X.

Definition 3.3 The mapping torus (suspension flow) of the dynamical system (S, T) is defined as

S_{T} := S × [0, 1]/(x, 0) ∼ (T x, 1)  (3.4)

3.1 (Co)homology of S_{T}.

In this paragraph we give an explicit description of the (co)homology H^{1}(S_{T} , ℤ).

The shift T acting on S induces an automorphism of the C^{∗}–algebra of continuous functions C(S). With an abuse of notation we still denote it by T. Consider the crossed product C^{∗}–algebra C(S) ⋊_{T} ℤ. This is a suitable norm completion of C(S)[T, T^{−1}] with product (V ∗ W)_{k} = ∑_{r∈ℤ} V_{k} · (T^{r}W_{r+k}), for V = ∑_{k} V_{k}T^{k}, W = ∑_{k} W_{k}T^{k}, and V ∗ W = ∑_{k}(V ∗ W)_{k}T^{k}.

The K–theory group K_{0}(C(S) ⋊_{T} ℤ) is described by the co–invariants of the action of T (cf. [2],[18]).

Theorem 3.4 The cohomology H^{1}(S_{T} , ℤ) satisfies the following properties

1. There is an identification of H^{1}(S_{T} , ℤ) with the K_{0}-group of the crossed product C^{∗}-algebra for the action of T on S,

H^{1}(S_{T} , ℤ) ≅ K_{0}(C(S) ⋊_{T} ℤ).  (3.5)

2. The identification (3.5) endows H^{1}(S_{T} , ℤ) with a filtration by free abelian groups F_{0} ↪ F_{1} ↪ ⋯ F_{n} ↪ ⋯, with rankF_{0} = 2g and rankF_{n} = 2g(2g − 1)^{n−1}(2g − 2) + 1, for n ≥ 1, so that

H^{1}(S_{T} , ℤ) = \underrightarrow{lim}_{n} F_{n}.

The following result computes the first homology of S_{T}.

Proposition 3.5 The homology group H_{1}(S_{T} , ℤ) has a filtration by free abelian groups K_{N},

H_{1}(S_{T} , ℤ) = \underrightarrow{lim}_{N} K_{N},  (3.6)

with

K_{N} = rank(K_{N}) = \{ \begin{matrix}(2g − 1)^{N} + 1 & N even \\ (2g − 1)^{N} + (2g − 1) & N odd \end{matrix}

The group H_{1}(S_{T} , ℤ) can also be written as

H_{1}(S_{T} , ℤ) = ⊕_{N=0}^{∞}R_{N}

where R_{n} is a free abelian group of ranks R_{1} = 2g and

R_{N} = rank(R_{N}) = \frac{1}{N} ∑_{d|N} μ(d) (2g − 1)^{N/d},

for N > 1, with μ the Möbius function. This is isomorphic to a free abelian group on countably many generators.

The ℤ-modules F_{n} are obtained as quotients P_{n}/δP_{n−1}, for δ(f) = f − f ∘ T, where P = ∪_{n}P_{n} is the module of continuous ℤ-valued functions on S that depend only on “future coordinates”. The ℤ-modules K_{N} are generated by all admissible words a_{0} … a_{N} such that the word a_{N}a_{0} is also admissible.

Combining Theorem 3.4 with Proposition 3.5, we can compute explicitly the pairing of homology and cohomology for S_{T}.

Proposition 3.6 Let F_{n} and K_{N} be the filtrations defined, respectively, in Theorem 3.4 and Proposition 3.5. There is a pairing

⟨·, ·⟩ : F_{n} × K_{N} → ℤ ⟨[f], x⟩ = N · f(x̄),  (3.7)

with x = a_{0} … a_{N}. Here the representative f ∈ [f] is a function that depends on the first n + 1 terms a_{0} … a_{n} of sequences in S, and x̄ is the truncation of the periodic sequence \overline{a_{0} … a_{N}} after the first n terms. This pairing descends to the direct limits of the filtrations, where it agrees with the classical cohomology/homology pairing

⟨·, ·⟩ : H^{1}(S_{T} , ℤ) × H_{1}(S_{T} , ℤ) → ℤ.  (3.8)

3.2 Dynamical (co)homology.

We define the dynamical cohomology H_{dyn}^{1} as the graded vector space given by the sum of the graded pieces of the filtration of H^{1}(S_{T}), introduced in Theorem 3.4. These graded pieces Gr_{p} are considered with coefficients in the p-th Hodge–Tate twist ℝ(p).

Similarly, we define the dynamical homology H_{1}^{dyn} as the graded vector space given by the sum of the terms in the filtration of H_{1}(S_{T}), introduced in Proposition 3.5. These vector spaces are again considered with twisted ℝ(p)-coefficients. The pair

H_{dyn}^{1} ⊕ H_{1}^{dyn}  (3.9)

provides a geometric setting, defined in terms of the dynamics of the shift operator T, which contains a copy of the archimedean cohomology of [5] and of its dual.

Definition 3.7 Let H^{1}(S_{T} , ℝ) = lim_{n} F_{n}, for a filtration F_{n} as in Theorem 3.4, with real coefficients. Let Gr_{n} = F_{n}/F_{n−1} be the corresponding graded pieces, with Gr_{0} = F_{0}.

1. We define the dynamical cohomology as

H_{dyn}^{1} := ⊕_{p≤0}gr_{2p}^{Γ}H_{dyn}^{1},  (3.10)

where we set

gr_{2p}^{Γ}H_{dyn}^{1} := Gr_{−p} ⊗_{ℝ} ℝ(p)  (3.11)

with ℝ(p) = (2π\sqrt{−1})^{p}ℝ.

Furthermore, we define the graded subspace of H_{dyn}^{1}

V := ⊕_{p≤0}gr_{2p}^{Γ}V

where gr_{2p}^{Γ}V is generated by the elements (2π\sqrt{−1})^{p}χ_{−p+1,k}, for χ_{n,k} := [χ_{S^{+}(w_{n,k})}] ∈ Gr_{n−1}.

Here χ_{S^{+}(w_{n,k})} is the characteristic function of S^{+}(w_{n,k}) ⊂ S^{+} with w_{n,k} :=

a_{0}a_{1} … a_{n−1} = \underbrace{g_{k}g_{k} … g_{k}}_{n−times}.

2. The dynamical homology H_{1}^{dyn} is defined as

H_{1}^{dyn} := ⊕_{p≥1}gr_{2p}^{Γ}H_{1}^{dyn},  (3.12)

where we set

gr_{2p}^{Γ}H_{1}^{dyn} := K_{p−1} ⊗_{ℝ} ℝ(p).  (3.13)

We also define W ⊂ H_{1}^{dyn} as the graded sub-space W = ⊕_{p≥1}gr_{2p}^{Γ}W, where gr_{2p}^{Γ}W is generated by the 2g elements (2π\sqrt{−1})^{p} \underbrace{g_{k}g_{k} … g_{k}}_{p−times}.

In [6], we showed that the subspace V realizes a copy of the archimedean cohomology of [5] embedded in the dynamical cohomology.

An explicit geometric description for the space V in terms of geodesics in 𝔛_{Γ} is obtained by interpreting the characteristic function χ_{S^{+}(w_{n,k})} as the “best approximation” within P_{n} to a distribution supported on the periodic sequence of period g_{k},

f(g_{k}g_{k}g_{k}g_{k}g_{k} …) = 1 and f(a_{0}a_{1}a_{2}a_{3} …) = 0 otherwise.

Such periodic sequence g_{k}g_{k}g_{k}g_{k}g_{k} … describes the closed geodesic in 𝔛_{Γ} that is the oriented core of one of the handles in the handlebody. Thus, the subspace gr_{2p}^{Γ}V ⊂ gr_{2p}^{Γ}H_{dyn}^{1} is spanned by the “best approximations” within gr_{2p}^{Γ}H_{dyn}^{1} to cohomology classes supported on the core handles of the handlebody. In other words, this interpretation views the index p ≤ 0 of the graded structure V = ⊕_{p}gr_{2p}^{Γ}V as a measure of “zooming in”, with increasing precision for larger |p|, on the core handles of the handlebody 𝔛_{Γ}.

3.3 A dynamical spectral triple.

The Cuntz–Krieger algebra O_{A} (cf. [8] [9]) is defined as the universal C^{∗}–algebra generated by partial isometries S_{1}, …, S_{2g}, satisfying the relations

∑_{j} S_{j}S_{j}^{∗} = I  (3.14)

S_{i}^{∗}S_{i} = ∑_{j} A_{ij} S_{j}S_{j}^{∗},  (3.15)

where A = (A_{ij}) is the 2g × 2g transition matrix of the subshift of finite type (S, T), namely the matrix whose entries are A_{ij} = 1 whenever |i − j| ≠ g, and A_{ij} = 0 otherwise.

The algebra O_{A} can be also described in terms of the action of the free group Γ on its limit set Λ_{Γ} (cf. [19], [20]), so that we can regard O_{A} as a noncommutative space replacing the classical quotient Λ_{Γ}/Γ.

The action of Γ on Λ_{Γ} ⊂ ℙ^{1}(ℂ) determines a unitary representation

Γ → Aut(C(Λ_{Γ})) (T_{γ}f)(x) = f(γ^{−1}x),  (3.16)

where C(Λ_{Γ}) is the C^{∗}–algebra of continuous functions on Λ_{Γ}. Thus, we can form the (reduced) crossed product C^{∗}–algebra C(Λ_{Γ}) ⋊ Γ. It follows that O_{A} ≅ C(Λ_{Γ}) ⋊ Γ. The algebra O_{A} acts on H_{dyn}^{1} in the following way

Proposition 3.8 Let O_{A}^{alg} be the ∗–algebra over ℝ generated algebraically by the S_{i} satisfying (3.14) (3.15). The action S_{i} : F_{n} ↦ F_{n}

(S_{i}h)(a_{0}, … a_{n}) = h(g_{i}^{−1}a_{0} … a_{n−1}) · χ_{a_{0}≠g_{i}}(a_{0}, … a_{n}).

induces a representation

ρ : O_{A}^{alg} → End(H_{dyn}^{1}).

This extends to a representation

ρ : O_{A} → B(H_{dyn}^{1})  (3.17)

as bounded operators on the real Hilbert space completion H_{dyn}^{1}, where O_{A} is considered here as a real C^{∗}-algebra.

On the space H_{dyn}^{1} one can also define an action of the (real) C^{∗}-algebra (C(Λ_{Γ}) ⊗ C_{0}(ℍ^{3})) ⋊ Γ. Here C_{0}(ℍ^{3}) denotes the C^{∗}-algebra of continuous functions that vanish on the sphere at infinity ℙ^{1}(ℂ) of ℍ^{3}.

Proposition 3.9 Let x_{0} ∈ ℍ^{3} be a fixed base point. For f ∈ C_{0}(ℍ^{3}), we define an operator ρ̃(f) : F_{n} → F_{n} as follows. For h ∈ F_{n}, we set

(ρ̃(f)h)(a_{0} … a_{n}) = f((a_{0} … a_{n}) · x_{0}) · h(a_{0} … a_{n}),  (3.18)

where we identify the reduced word a_{0} … a_{n} with an element of Γ. This extends the representation of O_{A} on H_{dyn}^{1} given in Proposition 3.8 to a representation

ρ̃: (C(Λ_{Γ}) ⊗ C_{0}(ℍ^{3})) ⋊ Γ → B(H_{dyn}^{1}).  (3.19)

On the space H_{1}^{dyn} we construct an action of the C^{∗}-algebra C(Λ_{Γ}) in the following way.

Lemma 3.10 For f a real valued continuous functions on Λ_{Γ}, consider the operator

ρ(f) : (2π\sqrt{−1})^{p} a_{0} … a_{p} ↦ f(\overline{a_{0} … a_{p}}) (2π\sqrt{−1})^{p} a_{0} … a_{p}.  (3.20)

These operators define a representation of the algebra of real valued continuous functions C(Λ_{Γ}, ℝ) as bounded operators on H_{1}^{dyn},

ρ : C(Λ_{Γ}, ℝ) → B(H_{1}^{dyn}).

For a C^{∗}-algebra A, we denote by C_{b}(ℍ^{3}, A) the C^{∗}-algebra of bounded continuous functions f : ℍ^{3} → A. We extend the representation of C(Λ_{Γ}) on the space H_{1}^{dyn}, given in Lemma 3.10, to a representation of the algebra C_{b}(ℍ^{3}, C(Λ_{Γ})).

An element f ∈ C_{b}(ℍ^{3}, C(Λ_{Γ})) is a collection f_{u} of elements in C(Λ_{Γ}), which vary continuously in the parameter u ∈ ℍ^{3}, and converge to zero in the norm of C(Λ_{Γ}) as u approaches the boundary at infinity ℙ^{1}(ℂ) in ℍ^{3}.

Lemma 3.11 Let x_{0} ∈ ℍ^{3} be a fixed base point. For f ∈ C_{b}(ℍ^{3}, C(Λ_{Γ})) we set:

ρ̃(f) a_{0} … a_{p} = f_{(a_{0}...a_{p}) x_{0}}(\overline{a_{0} … a_{p}}) · a_{0} … a_{p}.  (3.21)

This extends the representation (3.20) to a representation

ρ̃: C_{b}(ℍ^{3}, C(Λ_{Γ})) → B(H_{1}^{dyn}).

Let 𝔄 be a C^{∗}-algebra with an action of Γ by automorphisms. Let E be the flat 𝔄-bundle

E = (𝔄 × ℍ^{3})/Γ ⟶ 𝔛_{Γ},  (3.22)

where we consider the diagonal action of Γ on 𝔄 × ℍ^{3}. We consider the C^{∗}-algebra C_{0}(𝔛_{Γ}, E) of continuous sections of E that vanish on the boundary at infinity X_{/ℂ} of the handlebody 𝔛_{Γ}.

Proposition 3.12 The representation of C_{b}(ℍ^{3}, C(Λ_{Γ})) defined by (3.21) induces a representation

ρ̃: C_{0}(𝔛_{Γ}, E) → B(H_{1}^{dyn}),

where E is the bundle (3.22) for 𝔄 = C(Λ_{Γ}).

We introduce the notation A := (C(Λ_{Γ}) ⊗ C_{0}(ℍ^{3})) ⋊ Γ and Â := C_{0}(𝔛_{Γ}, E). In Propositions 3.9 and 3.12 we have obtained representations

\begin{aligned}ρ̃: A & → B(H_{dyn}^{1}) \\ ρ̃: Â & → B(H_{1}^{dyn}). \end{aligned}  (3.23)

The important observation is that the algebras A and Â represent the same noncommutative space. To understand this point, we recall the following result (cf. [12]).

Proposition 3.13 Let G be a connected Lie group and K a maximal compact subgroup. For Γ ⊂ G a lattice and 𝔄 a C^{∗}-algebra with an action of Γ by automorphisms, the algebras (𝔄 ⊗ C_{0}(G/K)) ⋊ Γ and C_{0}(Γ\G/K, E) are strongly Morita equivalent. Here E is the flat 𝔄-bundle over Γ\G/K,

E = (𝔄 × G/K)/Γ ⟶ Γ\G/K,

where Γ acts diagonally on 𝔄 × G/K.

In our case, the result of Proposition 3.13 implies that the algebras A and Â in (3.23) are strongly Morita equivalent.

We consider the linear operator D on H_{dyn}^{1} ⊕ H_{1}^{dyn} defined by setting, for p ≤ 0,

D : gr_{2p}^{Γ}H_{dyn}^{1} → gr_{2p}^{Γ}H_{dyn}^{1} D : x ↦ p · x  (3.24)

and, for p ≥ 1,

D : gr_{2p}^{Γ}H_{1}^{dyn} → gr_{2p}^{Γ}H_{1}^{dyn} D : x ↦ p · x.  (3.25)

This defines an unbounded linear operator

D : H_{dyn}^{1} ⊕ H_{1}^{dyn} → H_{dyn}^{1} ⊕ H_{1}^{dyn}.  (3.26)

This operator has the correct properties of a Dirac operator.

Theorem 3.14 Let H = H_{dyn}^{1} ⊕ M ⊗Â H_{1}^{dyn}, with M a bimodule implementing the Morita equivalence between the C^{∗}-algebras Â and A of (3.23). Then (A, H, D̃) is a spectral triple, where D̃|_{H_{dyn}^{1}} = D as defined in (3.24), and D̃|_{M⊗H_{1}^{dyn}} = 1 ⊗ D, with D as defined in (3.25).

Figure 1: The graphs Δ_{Γ}/Γ for genus g = 2, and the corresponding fibers.

4 A Dynamical theory for Mumford curves.

Throughtout this chapter K will denote a finite extension of ℚ_{p} and Δ_{K} the Bruhat-Tits tree associated to G = PGL(2, K). In the following we recall few results about the action of a Schottky group on a Bruhat-Tits tree and on C^{∗}-algebras of graphs. Detailed explanations are contained in [15], [17] and [1], [13], [14], [20].

For a Schottky group Γ ⊂ PGL(2, K) there is a smallest subtree Δ′_{Γ} ⊂ Δ_{K} containing the axes of all elements of Γ. 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 that coincides with the dual graph of the closed fibre of the minimal smooth model of the algebraic curve C/K holomorphically isomorphic to X_{Γ} := Ω_{Γ}/Γ (cf. [17] p. 163). There is a smallest tree Δ_{Γ} on which Γ acts and such that Δ_{Γ}/Γ is the (finite) graph of the specialization of C. The curve C is a k-split 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 curve X_{Γ} is called a Schottky–Mumford curve.

The possible graphs Δ_{Γ}/Γ and the corresponding fiber for the case of genus 2 are illustrated in Figure 1.

To a locally finite directed graph one associates a C^{∗}-algebra in the following way. A Cuntz–Krieger family consists of a collection {P_{v}}_{v∈E^{0}} of mutually orthogonal

projections and a {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 Cuntz–Krieger elements {P_{v}, S_{w}} satisfy the relation S_{w}^{∗}S_{w} = ∑ A_{+}(w, w̃)Sw̃S∗ ; w̃, with A_{+} the edge matrix of the graph.

One defines 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.

In the following we consider the Bruhat–Tits tree Δ_{K} for a fixed finite extension K of ℚ_{p}, and the corresponding C^{∗}-algebra C^{∗}(Δ_{K}), which is strongly Morita equivalent to the abelian C^{∗}-algebra of complex valued functions C(ℙ^{1}(K)).

Notice that the commutative C^{∗}-algebra C^{∗}(X_{K}) of complex valued continuous functions on the Mumford curve X_{K} = Ω_{Γ}/Γ is strongly Morita equivalent to the crossed product C_{0}(Ω_{Γ}) ⋊ Γ, with C_{0}(Ω_{Γ}) a Γ-invariant ideal of C(ℙ^{1}(K)) with quotient algebra C(Λ_{Γ}). The algebra C^{∗}(Δ_{Γ}/Γ), in turn, is strongly Morita equivalent to C^{∗}(Δ_{Γ}) ⋊ Γ and to C(Λ_{Γ}) ⋊ Γ, where ∂Δ_{Γ} = Λ_{Γ}. Similarly, one sees that the algebra C^{∗}(Δ_{K}/Γ) is strongly Morita equivalent to the crossed product algebra C^{∗}(Δ_{K}) ⋊ Γ, which in turn is strongly Morita equivalent to C^{∗}(ℙ^{1}(K)) ⋊ Γ.

Thus, the graph algebra C^{∗}(Δ_{K}/Γ) is, up to Morita equivalence, an extension of the commutative C^{∗}-algebra C^{∗}(X_{K}) by the Cuntz-Krieger algebra C^{∗}(Δ_{Γ}/Γ) associated to the edge matrix of the finite graph Δ_{Γ}/Γ.

In this paragraph we introduce a dynamical system associated to the space W(Δ/Γ) of walks on the directed tree Δ on which Γ acts. In particular, we are interested in the cases when Δ = Δ_{K}, Δ_{Γ}.

For Δ = Δ_{Γ}, we obtain a subshift of finite type associated to the action of the Schottky group Γ on the limit set Λ_{Γ}, of the type that was considered in [6].

Let V̄ ⊂ Δ_{Γ} 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 [15]), since some vertices may be identified under the action of Γ. Correspondingly, V ⊂ ℙ^{1}(K) is the set of ends of all infinite paths starting at points in V̄.

Consider the set W(Δ_{Γ}/Γ) of doubly infinite walks on the finite graph Δ_{Γ}/Γ. These are doubly infinite admissible sequences in the finite alphabet given by the edges of V̄ with both possible orientations. On W(Δ_{Γ}/Γ) we consider the topology generated by the sets W^{s}(ω, ℓ) = {ω̃ ∈ W(Δ_{Γ}/Γ) : ω̃_{k} = ω_{k}, k ≥ ℓ} and W^{u}(ω, ℓ) = {ω̃ ∈ W(Δ_{Γ}/Γ) : ω̃_{k} = ω_{k}, k ≤ ℓ}, for ω ∈ W(Δ_{Γ}/Γ) and ℓ ∈ ℤ. With this topology, the space W(Δ_{Γ}/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}/Γ). We can describe again the dynamical system (W(Δ_{Γ}/Γ), T) in terms of subshifts of finite type.

Lemma 4.1 The space W(Δ_{Γ}/Γ) with the action of the invertible shift T is a subshift of finite type, where W(Δ_{Γ}/Γ) = S_{A} with A the directed edge matrix of the finite graph Δ_{Γ}/Γ.

Figure 2: The graphs Δ_{Γ}/Γ for genus g = 2, and the corresponding trees Δ_{Γ}.

We consider the mapping torus of T:

W(Δ_{Γ}/Γ)_{T} := W(Δ_{Γ}/Γ) × [0, 1]/(T x, 0) ∼ (x, 1).  (4.1)

4.1 Genus two example

In the example of Mumford–Schottky curves of genus g = 2, the tree Δ_{Γ} is illustrated in Figure 2.

In the first case in the Figure, the tree Δ_{Γ} is just a copy of the Cayley graph of the free group Γ on two generators, hence we can identify doubly infinite walks in Δ_{Γ} with doubly infinite reduced words in the generators of Γ and their inverses. The directed edge matrix is given by

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

In the second case in Figure 2, we label by a = e_{1}, b = e_{2} and c = e_{3} the oriented edges in the graph Δ_{Γ}/Γ, so that we have a corresponding set of labels E = {a, b, c, ā, b̄, c̄} for the edges in the covering Δ_{Γ}. A choice of generators for the group Γ ≃ ℤ ∗ ℤ acting on Δ_{Γ} is obtained by identifying the generators g_{1} and g_{2} of Γ with the chains of edges ab̄ and ac̄. Doubly infinite walks in the tree Δ_{Γ} are admissible doubly infinite sequences of such labels, where admissibility is determined by the directed edge matrix

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

The third case in Figure 2 is analogous. A choice of generators for the group Γ ≃ ℤ ∗ ℤ acting on Δ_{Γ} is given by abā and c. Doubly infinite walks in the tree Δ_{Γ} are admissible doubly infinite sequences in the alphabet E = {a, b, c, ā, b̄, c̄}, with admissibility determined by the directed edge matrix

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

The construction is analogous for genus g > 2, for the various possible finite graphs Δ_{Γ}/Γ. The directed edge matrix can then be written in block form as

A = ( \begin{matrix}α_{11} & α_{12} \\ α_{21} & α_{22} \end{matrix} ) ,

where each block α_{ij} is a #(Δ_{Γ}/Γ)_{+}^{1} × #(Δ_{Γ}/Γ)_{+}^{1}–matrix with α_{12} = α_{12}^{t}, α_{21} = α_{21}^{t}, and α_{11} = α_{22}^{t}.

4.2 Cohomology of W(Δ/Γ)_{T}.

Let Δ = Δ_{Γ}. We identify the first cohomology group H^{1}(W(Δ_{Γ}/Γ)_{T} , ℤ) with the group of homotopy classes of continuous maps of W(Δ_{Γ}/Γ)_{T} to the circle. Let C(W(Δ_{Γ}/Γ), ℤ) be the ℤ-module of integer valued continuous functions on W(Δ_{Γ}/Γ), and let

C(W(Δ_{Γ}/Γ), ℤ)_{T} := Coker(δ),

for δ(f) = f − f ∘ T. The analog of Theorem 3.4 holds:

Proposition 4.2 The map f ↦ [exp(2πitf(x))], which associates to an element f ∈ C(W(Δ_{Γ}/Γ), ℤ) a homotopy class of maps from W(Δ_{Γ}/Γ)_{T} to the circle, gives an isomorphism C(W(Δ_{Γ}/Γ), ℤ)_{T} ≃ H^{1}(W(Δ_{Γ}/Γ)_{T} , ℤ). Moreover, there is a filtration of C(W(Δ_{Γ}/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}/Γ)_{T} , ℤ) = \underrightarrow{lim}_{n} F_{n}.

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

The space W(Δ_{Γ}/Γ)_{T} corresponds to a space of “bounded geodesics” on the graph Δ_{K}/Γ, where geodesics, in this setting, are just doubly infinite walks in Δ_{K}/Γ. In particular, a closed geodesic is the image under the quotient map π_{Γ} : Δ_{K} → Δ_{K}/Γ of a doubly infinite walk in the Bruhat-Tits tree Δ_{K} with ends given by the pair z^{+}(γ), z^{−}(γ) of fixed points of some element γ ∈ Γ. Similarly, a bounded geodesic is an element ω ∈ W(Δ_{K}/Γ) which is the image, under the quotient map, of a doubly infinite walk in Δ_{K} with both ends on Λ_{Γ} ⊂ ℙ^{1}(K). This implies that a bounded geodesic is a walk of the form ω = π_{Γ}(ω̃), for some ω̃ ∈ W(Δ_{Γ}/Γ). By construction, any such walk is an axis of Δ_{Γ}.

Orbits of W(Δ_{Γ}/Γ) under the action of the invertible shift T correspond bijectively to orbits of the complement of the diagonal in Λ_{Γ} ×Λ_{Γ} under the action of Γ. Thus, we see that W(Δ_{Γ}/Γ)_{T} gives a geometric realization of the space of “bounded geodesics” on the graph Δ_{K}/Γ, much as, in the case of the geometry at arithmetic infinity, we used the mapping torus of the shift T as a model of the tangle of bounded geodesics in a hyperbolic handlebody.

In [6] §4, we showed how the mapping torus S_{T} of the subshift of finite type (S, T), associated to the limit set of the Schottky group, maps surjectively to the tangle of bounded geodesics inside the hyperbolic handlebody, through a map that resolves all the points of intersection of different geodesics. In the case of the Mumford curve, where we replace the real hyperbolic 3-space by the Bruhat–Tits building Δ_{K}, the analog of the surjective map from S_{T} to the tangle of bounded geodesics is a map from W(Δ_{Γ}/Γ)_{T} to the dual graph Δ_{Γ}/Γ. Here is a description of this map.

As before, we write elements of W(Δ_{Γ}/Γ) as admissible doubly infinite sequences

ω = … w_{i_{−m}} … w_{i_{−1}}w_{i_{0}}w_{i_{1}} … w_{i_{n}} …,

with the w_{i_{k}} = {e_{i_{k}}, ϵ_{i_{k}}} oriented edges on the graph Δ_{Γ}/Γ. We consider each oriented edge w of normalized length one, so that it can be parameterized as w(t) = {e(t), ϵ}, for 0 ≤ t ≤ 1, with w̄(t) = {e(1 − t), −ϵ}. Since ω ∈ S_{A} is an admissible sequence of oriented edges we have w_{i_{k}}(1) = w_{i_{k+1}}(0) ∈ Δ_{Γ}^{(0)}.

We consider a map of the covering space W(Δ_{Γ}/Γ) × ℝ of W(Δ_{Γ}/Γ)_{T} to |Δ_{Γ}| of the form

Ẽ(ω, τ) = w_{i_{[τ]}}(τ − [τ]).  (4.2)

Here |Δ_{Γ}| denotes the geometric realization of the graph. By construction, the map Ẽ satisfies Ẽ(T ω, τ) = Ẽ(ω, τ + 1), hence it descends to a map E of the quotient

E : W(Δ_{Γ})_{T} → |Δ_{Γ}|.  (4.3)

We then obtain a map to |Δ_{Γ}/Γ|, by composing with the quotient map of the Γ action, π_{Γ} : Δ_{Γ} → Δ_{Γ}/Γ, that is,

Ē := π_{Γ} ∘ E : W(Δ_{Γ})_{T} → |Δ_{Γ}/Γ|.  (4.4)

Thus, we obtain the following.

Proposition 4.3 The map Ē of (4.4) is a continuous surjection from the mapping torus W(Δ_{Γ})_{T} to the geometric realization |Δ_{Γ}/Γ| of the finite graph Δ_{Γ}/Γ.

The fibers of the map (4.4) are explicitly described as

Ē^{−1}(w_{i}(t)) = W(Δ_{Γ})(w_{i}) × {t} ∪ W(Δ_{Γ})(w̄_{i}) × {1 − t},  (4.5)

where w_{i}(t), for t ∈ [0, 1] is a parameterized oriented edge in the graph Δ_{Γ}/Γ, and W(Δ_{Γ})(w_{i}) ⊂ W(Δ_{Γ}) consists of

W(Δ_{Γ})(w_{i}) = {ω ∈ W(Δ_{Γ}) : w_{i_{0}} = w_{i}},

for ω = … w_{i_{−m}} … w_{i_{−1}}w_{i_{0}}w_{i_{1}} … w_{i_{n}} . . ..

We now consider the case where Δ is the Bruhat-Tits tree Δ_{K}. The set of doubly infinite walks W(Δ_{K}/Γ) can be identified with the set of admissible doubly infinite sequences in the oriented edges of a fundamental domain for the action of Γ. Therefore, we obtain the following identification

W(Δ_{K}/Γ) ≃ V ×W^{∗}(Δ_{Γ}/Γ)×V ∪V ×W^{+}(Δ_{Γ}/Γ)∪W^{+}(Δ_{Γ}/Γ)×V ∪W(Δ_{Γ}/Γ), (4.6)

where we distinguish between walks that wander off from the finite graph Δ_{Γ}/Γ along one of the paths leading to an end in V, in one or in both directions, and those that stay confined within the finite graph Δ_{Γ}/Γ. Here we include in W^{∗}(Δ_{Γ}/Γ) = ∪_{n}W^{n}(Δ_{Γ}/Γ) also the case W^{0}(Δ_{Γ}/Γ) = ∅, where the walk in Δ_{K}/Γ does not intersect the finite graph Δ_{Γ}/Γ at all.

In the topology induced by the p-adic norm, ℙ^{1}(K) is a totally disconnected compact Hausdorff space, and so is W(Δ_{K}/Γ) by (4.6). Again, we consider the invertible shift map T, which is a homeomorphism of W(Δ_{K}/Γ), and we form the mapping torus

W(Δ_{K}/Γ)_{T} := W(Δ_{K}/Γ) × [0, 1]/(T x, 0) ∼ (x, 1).  (4.7)

We obtain the analog of Proposition 4.2, though in the case of Δ_{K} the ℤ-modules F_{n} will not be finitely generated.

4.3 A dynamical theory.

On ℙ^{1}(K) we consider the measure dμ as in [3]. This induces a measure on W(Δ_{K}/Γ) and on W(Δ_{Γ}/Γ), with respect to which the shift map T is measure preserving.

Consider the submodule P ⊂ C(W(Δ_{Γ}/Γ), ℤ) of functions of future coordinates where P ≃ C(W^{+}(Δ_{Γ}/Γ), ℤ). This gives a filtration P = ∪_{n}P_{n}, where P_{n} is identified with the submodule of C(W^{+}(Δ_{Γ}/Γ), ℤ) generated by characteristic functions of W^{+}(Δ_{Γ}/Γ, ρ) ⊂ W^{+}(Δ_{Γ}/Γ), where ρ ∈ W^{∗}(Δ_{Γ}/Γ) is a finite walk ρ = w_{0} ⋯ w_{n} of length n + 1, and W^{+}(Δ_{Γ}/Γ, ρ) is the set of infinite paths ω ∈ W^{+}(Δ_{Γ}/Γ), with ω_{k} = w_{k} for 0 ≤ k ≤ n + 1. The free ℤ-module P_{n} of functions of at most n + 1 future coordinates determines a vector subspace P_{n} ⊗_{ℤ} ℂ of L^{2}(ℙ^{1}(K), dμ). The operator δ of §4.2 is bounded in norm, and the F_{n} = P_{n}/δP_{n−1} have induced Hilbert norms and bounded inclusions F_{n} ⊗ ℂ ↪ F_{n+1} ⊗ ℂ, where the F_{n} are the torsion free ℤ-modules of Proposition 4.2. Using the inner product induced from L^{2}(ℙ^{1}(K), dμ), we can split F_{n} ≅ F_{n−1} ⊕ (F_{n−1}^{⊥} ∩ F_{n}).

Definition 4.4 The dynamical cohomology H_{dyn}^{1}(Δ_{Γ}/Γ) is the norm completion of the graded complex vector space

H_{dyn}^{1}(Δ_{Γ}/Γ) := F_{0} ⊕ (F_{1} ∩ F_{0}^{⊥}) ⊕ (F_{2} ∩ F_{1}^{⊥}) ⊕ ⋯ (F_{n} ∩ F_{n−1}^{⊥}) · · · .

Proposition 4.5 There is a representation of the algebra C^{∗}(Δ_{Γ}/Γ) by bounded linear operators on the Hilbert space H_{dyn}^{1}(Δ_{Γ}/Γ).

4.4 A dynamical spectral triple for a Mumford curve.

Using the representation of Proposition 4.5, we construct a spectral triple associated to a Mumford curve, where, in defining the Dirac operator we only use the graded structure of the space H_{dyn}^{1}(Δ_{Γ}/Γ).

Let H be the Hilbert space

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

We consider the algebra C^{∗}(Δ_{Γ}/Γ) acting on H diagonally.

We define an unbounded self-adjoint operator |D| on H, with the property that (λ − |D|)^{−1} is a compact operator for all λ ∉ ℝ.

Let Gr_{+,n} = 0 ⊕ Gr_{n} and Gr_{n,−} = Gr_{n} ⊕ 0. We define the Dirac operator D by setting

D|_{Gr_{+,n}} = n D|_{Gr_{−,n}} = −n − 1.  (4.8)

Proposition 4.6 The data (C^{∗}(Δ_{Γ}/Γ), H, D) determine a spectral triple in the sense of Connes [4].

The proof is completely analogous to the argument given in [6].

In [7] we recover arithmetic information such as the local L–factors of [11] from the dynamical cohomology and the spectral triple of Proposition 4.6.

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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