Dual graph and noncommutative geometry

The bulk space corresponding to the conformal boundary X(C) is given by the hyperbolic handle-body YΓ. As in the case of the BTZ black hole, it is possible to interpret these real hyperbolic 3-manifolds as analytic continuations to Euclidean signature of Minkowskian black holes that are global quotients of a-dS2+1. This is not just the effect of the usual rotation from Minkowskian to Euclidean signature, but amore refined form of analytic continuation which is adapted to the action of a Schtokky group, and which is introduced by Kirill Krasnov [92], [93] in order to deal with this class of space-times.

The above given formula for the Green functiong(·,·) in terms of oriented distances of geodesics as inYΓ gives the explicit bulk and boundary correspon- dence of the holography principle for this class of space times, so that each term in the Bosonic field propagator forX_Cis expressed in terms of geodesics in the EuclideanKrasnov black hole asYΓ =H³/Γ.

The trivial multiplicative valuation on K is defined by w(a) = 1 for any nonzeroa∈K.

A non-Archimedean,multiplicative variationwis said to be discrete ifw(K^∗) as a subgroup ofR^∗by (mv 2) is isomorphic toZ.

Check thatw(0) =c^−v(0) =c^−∞= 0. Also,

w(ab) =c^−v(ab)=c^{−v(a)−v(b)}=c^−v(a)c^−v(b)=w(a)w(b),

w(a+b) =c^−v(a+b)≤c^{−min{v(a),v(b)}}≤max{w(a), w(b)} ≤w(a) +w(b).

Thevaluedgroup ofwis defined to be the group of allw(a) for any nonzero a∈K. A multiplication valuationw onK is said to be Archimedeanif for any nonzero a, b∈K, there is n∈Zsuch thatw(na)≥w(b). Otherwise, it is said to be non-Archimedean (n-A).

IfwonK is Archimedean, then there is an embeddingσof K intoC, and w is equivalent to the pull back as |σ(a)|. If w on K (with c > 1) is non- Archimedean, then define v(a) = −log_bw(a) with b > 1, which is a rank 1 (additive) variation. The converse also holds.

Withwa multiplicative valuation (with c >1), define a metric on a field K byd(a, b) =w(a−b) fora, b∈K.

Note that d(a, b) = 0 if and only ifw(a−b) = 0 if and only if a−b= 0.

As well,d(b, a) =w(b−a) =w(−1)w(a−b) =d(a, b). Moreover,

d(a, b) =w(a−b) =w(a−c+c−b)≤w(a−c) +w(c−b) =d(a, c) +d(c, b).

IfK is complete with respect to this metric, wis said to be complete. If not so, there is the unique completionK ofKsuch that K is an extension of K with w as an extension ofw, so that K is dense in K and K is complete with respect to w.

Thep-adic completion K_p ofK is defined to be the completion ofK with respect to thep-adic completion. In particular, ifKis a algebraic number field, K_p is said to be thep-adic algebraic number field.

IfR=Zandp= (p) =pZfor a primep, the correspondingp-adic valuation is defined as w or v. Also, the p-adic number field Qp is defined to be the completion of Q with respect to the p-adic valuation. Any nonzero element a∈Qp as ap-adic number has the form

a= ∞ n=r

anpⁿ, an∈Z/pZ, r∈Z, ar= 0,

so that v(a) = r. The valuation ring of v on Qp is denoted as Zp of p-adic integers.

May check that v(ab) =v(

∞ n=r

anpⁿ ∞ m=s

bmp^m) =v(

∞ n=r

∞ m=s

anbmp^n+m) =r+s=v(a) +v(b),

v(a+b) =v(

∞ n=r

anpⁿ+ ∞ m=s

bmp^m)≥max{r, s}=max{v(a), v(b)}.

Schottky-Mumford curves. LetK be a given finite extension ofQp and let O in K its ring of integers, andm in O themaximal ideal, and k=O/m the residue field, that is a finite field of cardinalityq= card(O/m).

It is well known that a curveX over a finite extensionK of Qp, which is k-split degenerate for k the residue field, admits a p-adic uniformization by a p-adic Schottky group Γ acting on the Bruhat-Tits tree ΔK.

TheBruhat-Tits(BT) tree ΔK is obtained by considering freeO-modules M of rank 2, with the equivalence relation that M1 ∼M2 if there is λ∈ K^∗ such that M1 =λM2. The set Δ⁰_K of vertices of the BT tree consists of the equivalence classes [M] under the relation∼. The distance between two classes of the set Δ⁰_K is defined byd([M1],[M2]) =|l−k|, where ifM1⊃M2, then

M1/M2∼=O/m^l⊕ O/m^k for somel, k∈N.

To formthe BT tree, an edge of the set Δ¹_K of edges is defined to be a (directed or not) arrow from [M2] to [M1] ifM2⊂M1 withd([M1],[M2]) = 1. Then the BT tree ΔK = (Δ⁰_K,Δ¹_K) becomes a connected, locally finite tree with q+ 1 edges departing from each vertex.

The groupP GL2(K) acts on ΔK (fromthe left) transitively by isometries.

The Bruhat-Tits tree ΔK is the analog of the real 3-dimensional hyperbolic spaceH³at the infinite primes. The set of ends of ΔK is identified withP¹(K), just like thatP¹(C) =∂H³ in the case at infinity.

Ap-adicSchottkygroup Γ is defined to be a discrete subgroup ofP GL2(K) which consists of hyperbolic elements γ in K, for which the eigenvalues have different valuation, and Γ is isomorphic to a free group withggenerators. Denote by ΛΓ,Kthe limit set inP¹(K), which is the closure of the set of points ofP¹(K) that are fixed points of some γ ∈ Γ\ {1}. As in the case at infinity, it holds that card(ΛΓ,K)<∞ if and only if Γ = (γ)^Z for someγ ∈Γ, in the genus one case. Denote by ΩΓ(K) the complement of P¹(K) in ΛΓ,K, as the domain of discontinuity of Γ.

In the case of genus more than 1, the quotient XΓ,K = ΩΓ(K)/Γ is a Schottky-Mumfordcurve, withp-adc Schottky uniformization. In the case of genus 1, it is a Mumford (elliptic) curve, with the Jacobi-Tate uniformization.

(No original figure).

A path in the Bruhat-Tits tree ΔK which is infinite in both directions, with no back-tracking, is said to be anaxis of ΔK. Any two points z1, z2 ∈P¹(K) uniquely define an axis connectingz1andz2 as endpoints in∂ΔK. The unique axis of ΔK whose ends are the fixed points of a hyperbolic elementγ is said to be theaxisofγ. The elementγacts on its axis as a translation. Denote by Δ_Γ the smallest subtree of ΔΓ containing all the axes of elements of Γ (whichmay be called as the axistree).

The subtree Δ_Γ is Γ-invariant, with ΛΓ,K as the set of ends. The quotient Δ_Γ/Γ is a finite graph, which is the dual graph of the closed fiber of the mini- mal smooth model over O ofXΓ,K (as ak-split degenerate semi-stable curve).

With no figures, there are coorrespondences among the special (closed) fibers as curves, the dual graphs such as two circles a, bas edges attached with a point

or an edge c, and the axis trees Δ_Γ generated by {a, b} or {a, b, c}, for all the possible cases ofmaximal degenerations special for genus 2.

For eachn≥0, also consider the subtree or subgraph ΔK,n of the BT tree ΔK defined by setting the set of vertices and the set of edges as

Δ⁰_K,n={vertex v∈Δ⁰_K|d(v,Δ_Γ)≡inf{d(v, v)|v∈(Δ_Γ)⁰} ≤n}, whered(v, v) is the distance on Δ⁰_K, withv= [M], v= [M], and

Δ¹_K,n={edgew∈Δ¹_K|sources(w), ranger(w)∈Δ⁰_K,n}. In particular, ΔK,0= Δ_Γ.

For alln∈N, the subtree or subgraph ΔK,n is invariant under the action of a p-adic Schottky group Γ on ΔK, and the quotient ΔK,n/Γ is a finite graph, which is the dual graph of the reductionXΓ,K⊗ O/mⁿ⁺¹.

For amore detailed account of Schottky-Mumford curves, see [104] and [124].

Model of the dual graph. The (revised) dictionary between the case of Mumford curves and the case at arithmetic infinity is now summarized as in the following:

Table 6: The dictionary between the BT tree and the Hyperbolic 3-geometry Notion Tree geometric dynamics Hyperbolic 3GD

Space BT tree graph ΔK = (Δ⁰_K,Δ¹_K) H 3-spaceH³ Boundary P¹(K) =∂ΔK P¹(C) =∂H³≈S² Path (mini) Connecting edges in ΔK Geodesics inH³ Group action Schottky Γ⊂P GL2(K) Schottky Γ⊂P SL2(C) Discontinuity Schottky-Mumford curve Riemann surface

of Γ by Γ XΓ,K= ΩΓ(K)/Γ = Λ^c_Γ,K/Γ XΓ= ΩΓ/Γ = Λ^c_Γ/Γ Solid quotient BT Graph ΔK/Γ Handle-bodyYΓ=H³/Γ

Core part Axis subtree Δ_Γ⊂ΔΓ Convex core inH³ Bounded quo. Finite dual graph Δ_Γ/Γ Bounded geodesics inYΓ

Since bounded geodesics inYΓ can be identified with infinite geodesics inH³ with endpoints on ΛΓ⊂P¹(C)module the action of Γ, these are parameterized by the complement of the diagonal in ΛΓ×ΓΛΓ. This quotient is identified with the quotient of the totally disconnected space S of admissible, doubly infinite words with generators of a free group Γ and their inverses as generating characters, by the action of the invertible shiftT. Thus obtained is the following model for the dual graph of the fiber at infinity.

•The solenoidST = (S×[0,1])/∼as themapping torus forTis a geometric model of the dual graph of the fiber at infinity of an arithmetic surface.

•The Cuntz-KriegerC^∗-algebraOAas a noncommutative space, represent- ing the algebra of coordinates on the quotient ΛΓ/Γ, corresponds to the set of vertices of the dual graph as the set of components of the fiber at infinity,

while the C^∗-algebra crossed productC(S)T Z as a noncommutative space, corresponding to the quotient ΛΓ×ΓΛΓ, gives the set of edges of the dual graph.

Moreover, given by using noncommutative geometry is a notion of reduction mod ∞, analogous to the reduction maps mod p^m defined by the subgraphs ΔK,n of the BT tree ΔK in the case of Mumford curves. In fact, the reduc- tion map corresponds to the paths connecting ends of the graph ΔK,n/Γ to the corresponding vertices of the graph Δ_Γ/Γ = ΔK,0/Γ. Then the analog at arithmetic infinity consists of geodesics inYΓ which are the images of geodesics inH³ starting at some pointx0∈H³∪ΩΓ and having the other end as a point of ΛΓ. These are parameterized by the set ΛΓ×Γ(H³∪ΩΓ). Thus, in terms of NC geometry, the reductionmod∞corresponds to a compactification of the homotopy quotient ΛΓ ×Γ H³ with H³ =EΓ and BΓ = H³/Γ = YΓ. Hence, we can view ΛΓ/Γ as the quotient of a foliation on the homotopy quotient with contractible leaves asH³. Then the reductionmod∞is given by the assembly mapμas

μ:K^∗+1(ΛΓ×ΓH³)→K_∗(C(ΛΓ)Γ), ∗= 0,1.

This shows that the spectral triple (O^A, H, D) as a noncommutative (Riemann) space is closely related to the geometry of the fiber at arithmetic infinity of an algebraic variety. Then,may ask a question that what arithmetic information is captured by the Dirac operator D of the spectral triple. Let us see in the next section that as proved in [61], the Dirac operator gives another important arithmetic invariant, namely the localL-factor at the Archimedean prime.