Ryukyu Math. J., 31(2018), 7-164

A locally commentative, deformative
learning in the arithmetic
noncommutative geometry as ﬂower
gardening variation
Takahiro Sudo
Dedicated to Professor Takashi Ito on his 60th birthdaywith gratitude and respect
Abstract
We would like to study noncommmutative geometry, related to Arithmetics in some sense. For this purpose forced, we review and study thelecture notes by Marcolli, on the arithmetic noncommutative geometry.

MSC 2010: Primary 46-01, 46-02, 46L05, 46L51, 46L53, 46L55, 46L80,
46L85, 46L87, 46L89, 19K14, 19K56, 81Q10, 81Q35, 81R15, 81R40, 81R60,
81T75, 11G18, 11M06, 11R37, 11R42.
Keywords: Noncommutative geometry, C*-algebra, K-theory, spectral triple,
zeta function, modular group, modular curve, quantum statistical mechanics,
Bost-Connes system, KMS state, class ﬁeld theory, Schottky, hyperbolic, arithmetic, archimedean, Arakelov geometry, Green function, black hole, L-function,
L-factor, algebraic variety, Shimura variety, Cuntz-Krieger algebra.

1OverturePreface. Just following, almost along the story of Matilde Marcolli [114], as
nothing but a running commentary, as purpose of background learning, we (asbeginners) would like to study what is growing and harvested (at that time) on
the rather mysterious noncommutative world (as a vast ﬁeld), related to Arithmetics, as a back to the past for a return to the future (BPRF) ﬂower gardening
variation as named. All necessary things are to be considered until the end is
to be watched. Our local understanding the details (in part) may be shallow
or somewhat limited. With some considerable eﬀort within the time limited
for publication, we would like to make some additional or deformed, comments,
corrections, descriptions, (partial) proofs, or remarks, on the contents, to be or
not to be self-contained.
Received November 30, 2018.

－ 7 －In particular, as a note we use the symbols · · · (texts)
as indicating
either personal comments, computations to be checked, or etc. As well, some
additional remarks starting with Remark · · · (texts)  in the text below arealso made from suitable related references to be included. All the details could
not be contained in these cases. Some notations as well as style√ are (slightly)
changed from the original ones by our taste. As usual, let i = −1. Denote by
Q, R, and C the ﬁelds of rational, real, and complex numbers, respectively.
Once upon a time and space, there is certainly or simply, a sort of synchronized improvement based on the achievement of Marcolli, like a decorated ﬁeld
on the high splended hill. The variant ﬂower garden is now open!
Table 1: Contents ♣Section & Subsection
1 Overture
1.1 The space-algebra-geometry dictionary
1.2 Noncommutative spaces encountered
1.3 Spectral triples appeared
1.4 Why NCG?
2 Noncommutative modular curves
2.1 Modular curves
2.2 The noncommutative boundary of modular curves
2.3 Limiting modular symbols
2.4 Hecke eigenforms
2.5 Selberg zeta function
2.6 The modular complex and K-theory of C ∗ -algebras
2.7 Chaotic cosmology as Intermezzo
3 Quantum statistical mechanics and Galois theory
3.1 Quantum statistical mechanics
3.2 The Bost-Connes system
3.3 Noncommutative geometry and the Hilbert 12th problem
3.4 The GL2 system
3.5 Quadratic ﬁelds
4 Noncommutative geometry at arithmetic inﬁnity
4.1 Schottky uniformization
4.2 Dynamics and noncommutative geometry
4.3 Arithmetic inﬁnity as archimedean primes
4.4 Arakelov geometry and hyperbolic geometry
4.5 Qunatum gravity and black holes as Intermezzo
4.6 Dual graph and noncommutative geometry
4.7 Arithmetic varieties and L-factors
4.8 Archimedean cohomology
5 The vista around
References－ 8 －Section 1. Let us start by recalling several preliminary notions of noncommutative geometry, following Connes [27].
Section 2. Let us describe how various arithmetice properties of modular
curves can be viewed as their noncommutative boundaries as noncommutative
spaces or algebras. The main references are Manin-Marcolli [112] and Marcolli[113], [115].
Section 3. This section includes an account of the work of Connes andMarcolli [41] on the noncommutative geometry for commensurability classes of
Q-lattices. Also included is the relation of the noncommutative geometry forthese classes to the Hilbert 12th problem of explicit class ﬁeld theory as wellas the results of Connes, Marcolli, and Ramachandran [48] on the constructionof a quantum statistical mechanical system that recovers the explicit class ﬁeld
theory of imaginary quadratic ﬁelds. As well, included is a brief explanation on
the real multiplication program of Manin [109], [110] and the problem of real
quadratic ﬁelds.
Section 4. This section contains the noncommutative geometry for theﬁbers at arithmetic inﬁnity of varieties over number ﬁelds, based on Consani
and Marcolli [58], [59], [60], [61], [63]. Also contained is a detailed account of theformula of Manin for the Green function of Arakelov geometry for arithmetic
surfaces, based on Manin [106] as well as a proposed physical interpretation ofthis formula, as in Manin and Marcolli [111].
Noncommutative geometry (NG) is developed by Connes, started in theearly 1989s as in [22], [24], [27]. The NG extends tools of ordinary geometry
to treat spaces obtained as certain quotients by some equivalence relations, forwhich the usual ring of functions on quotients deﬁned as functions invariant withrespect to the equivalence relation becomes too trivial to capture the informationon the structure of points of the quotient space, and so to deﬁne a noncommutative algebra or space with generators corresponding to coordinates of the space,remembering the structure of the equivalence relation, and analogous to noncommuting variables in quantum mechanics. These quantum spaces deﬁned so
extend the Gel’fand -Naimark correspondence between locally compact, Hausdorﬀ spaces and commutative C ∗ -algebras of all continuous, complex-valued
functions on the spaces (vanishing at inﬁnity) by dropping the commutativity
hypothesis. Namely, a noncommutative (NC) (topological) space is deﬁned to
be a noncommutative C ∗ -algebra.
Such quotient spaces are abundant in nature. For instance, they arise byfoliations as equivalence relations. As well, noncommutative spaces arise naturally in number theory and arithmetic geometry. The ﬁrst connection between
noncommutative geometry and number theory is obtained by Bost and Connes[13], in which exhibited is an interesting noncommutative space or algebra withremarkable arithmetic properties related to the class ﬁeld theory. This also involves the phenomena as spontaneous symmetry breaking in quantum statisticalmechanics related to the Galois theory. That space can be viewed as the NC
space of 1-dimensional Q-lattices up to scale, modulo the equivalence relationof commensurability (Connes-Marcolli [41]). As well, the NC space is closely－ 9 －related to the noncommutative space by Connes [29] to obtain a spectral realization of zeros of the Riemann zeta function. In fact, this NC space is that ofcommensurability classes of 1-dimensional Q-lattices, but with the scale factor
also taken into account.
The deeper connections between noncommutative geometry and number theory are obtained by Connes and Moscovici [54], [55] on the modular Hecke algebras. It is shown that the Rankin-Cohen brackets as an important algebraic
structure on modular forms (Zagier [159]) have a natural interpretation in termsof the Hopf algebra of the transverse geometry of foliations with codimension 1,
in the sense of NG. The modular Hecke algebras, in which products and actionof Hecke operators on modular forms are naturally combined, can be viewed
as the holomorphic part of the NC algebra of coordinates on the space of commensurability classes of 2-dimensional Q-lattices, constructed by Connes andMarcolli [41].
On the other hand, the classiﬁcation of noncommutative 3-spheres is obtained by Connes and Dubois-Violette [35], [36], [37], in which the corresponding
moduli space has a ramiﬁed cover by a noncommutative nilmanifold, where theNC analog of the Jacobian of this covering map is expressed naturally in termsof the Dedekind eta function, as an interesting case of number theory within
NG. As another such case, calculated by Connes [31] the explicit cyclic cohomology Chern character of a spectral triple on SUq (2), also deﬁned by Chakraborty
and Pal [17].
More other instances of NG spaces that arise in the context of number theory
and arithmetic geometry can be found in the noncommutative compactiﬁcationof modular curves, as Connes, Douglas, and Schwarz [34], Manin and Marcolli[112]. This NC space is again related to the NG of Q-lattices. In fact, it canbe seen as a stratum in the compactiﬁcation of the space of commensurability
classes of 2-dimensional Q-lattices (CM [41]).
Another context in which NG provides a useful tool for arithmetic geometryis given by the description of the totally degenerate ﬁbers at arithmetic inﬁnity ofarithmetic varieties over number ﬁelds, analyzed by Katia Consani and Marcolli[58], [59], [60], [61].
The present text is based on the lecture notes [114] on the lectures given by
Marcolli, in 2002 to 2004, may be used later on.
In the lectures, contained as the main focus is the noncommutative geometry of modular curves (MM [112]) and of the archimedean ﬁbers of arithmetic
varieties ([61]). Also included is a theory of the NG space of commensurability
classes of 2-dimensional Q-lattices ([41]). In particular, conveyed is the general
picture less than the details of the proofs of the speciﬁc results. Though proof
are almost not included in the (original) text, the reader may ﬁnd references tothe relevant literature, such as [41], [48], [61], and [112].
From the foreword by Manin as backward. (Edited, changed for short.)
Let us be imagining C ∗ -algebras as coordinate rings, and making bridges tocommutative geometry via crossed products of C ∗ -algebras, to involve noncommutative Riemannian geometry, with algebraic tools as K-theory and cyclic－ 10 －

cohomology for C ∗ -algebras and their subalgebras.
Let us be studying various aspects of uniformization by the classical modular group as well as Schottky groups. The modular group acts on the complex
half plane, partially compactiﬁed by cusps as rational points of the boundary
projective line over C. The action at irrational points is viewed as the noncommutative boundary as NC spaces, entering NG. Moreover, as contents mentioned
as follows:
Schottky uniformization provides a visualization of Arakelov geometry atarithmetic inﬁnity.
Gauss characterising regular polygons that can be constructed using only
ruler and compass.
In the complex plane, roots of unity of degree n form vertices of a regularn-gon.
Being constructible as the maximal Galois extension Q of Q, to calculate
the Galois group Gal(Q/Q).
Gauss Galois group GalAb (Q/Q) in Bost-Connes symmetry breaking andGauss statistics of continued fractions in the chaotic cosmology models.

1.1

The space-algebra-geometry dictionary

There is a dictionary relating concepts of ordinary geometry to the corresponding counterperts in noncommutative geometry (cf. [27] as well as [154]).
Table 2: The S-A-G overview dictionary, as translatedSpace as Geometry
Measure spaces:(X, B Borel structure)
Dynamical system (X, B, Γ)
by a Borel action of a group ΓTopological spaces:(X, O topology) ⊂ (X, B)(compact or not);Dynamical system (X, O, Γ)by homeomorphisms
Diﬀerential geometry
Smooth (compact) manifold M ;Dynamical system (M, Γ)
Riemannian geometry
Riemann metric d with
(Dirac) diﬀerentail operator D
More speciﬁed geometry
with more ﬁne structure

Algebra as Geometry
von Neumann algebras:
Classical L∞ (X, B)
Quantum vN crossed product as
M = L∞ (X)  Γ by automorphismsC∗ -algebras:
Classical C(X) (or C0 (X))
in L∞ (X) (unital or not);Quantum C ∗ -crossed product as
A = C(X)  Γ ⊂ M
Smooth dense ∗-subalgebras:
Classical C ∞ (M ) ⊂ C(M );Quantum A = C ∞ (M )  Γ ⊂ A
Noncommutative geometry
Spectral triple (A with A, H, D) on
Hilbert space H with D derivation
More analogous algebras
with more suitable characters－ 11 －

1.2

Noncommutative spaces encountered

The way to assign the algebra of coordinates to a quotient space X = Y / ∼ ofa space Y by an equivalence relation ∼ may be explained as follows.
• One way to consider functions f on Y with f (a) = f (b) for a ∼ b in Y .
If each equivalence class as a subset of Y is dense in Y and if f is continuous
on Y and constant c on an equivalence class, then f ≡ c1 on Y .
• The other way to consider functions on the graph of the equivalence relation.
A noncommutative algebra is obtained by the second description, with aconvolution product determined by the groupoid law of the equivalence relation,
like the usual product of the C ∗ -algebra of n × n matrices over C as⎛⎞
f11 · · · f1n
n

⎜ ..
⎟
.
n
.
.
.
f ∗ g = fg = ⎝ .
fik gkj )ni,j=1 ∈ Mn (C).
.
. ⎠ (gij )i,j=1 = (
f1n

···

fnn

k=1

Even though the two functional notions for quotients are not the same, butthey may be related by Morita equivalence, as a suitable notion as a (stable)
isomorphism between NC spaces. However, the second notion is the only one
to allow to make sense of (NC) geometry for the general quotient spaces.
Example 1.1. Refer to [30]. Let Y = [0, 1]×{0, 1} with the equivalence relationas points (x, 0) ∼ (x, 1) for any x ∈ (0, 1) the open interval. Then X = Y / ∼ is
identiﬁed with a topological union with the quotient topology as{(0, 0), (0, 1)} ∪ (0, 1) ∪ {(1, 0), (0, 1)}.
The ﬁrst way implies C([0, 1]) (not C, corrected), but the commutative C ∗ algebra of all continuous, complex-valued functions on the closed interval [0, 1].
On the other hand, the second way implies the continuous C ∗ -algebra bundle
over [0, 1] with ﬁbers M2 (C) on (0, 1) and C2 at 0 and 1, which can be written
as a C ∗ -algebra extension E as a NC algebra
0 → SM2 (C) = C0 ((0, 1)) ⊗ M2 (C) → E → C2 ⊕ C2 → 0,
where each C2 is identiﬁed with the diagonal part of M2 (C).

Note that there is analogy between the idea of preserving the information of the equivalence relation in the description of quotient spaces and theGrothendieck theory of stacks in algebraic geometry.
Morita equivalence. In NC geometry, the proper notion as isomorphisms ofC ∗ -algebras is provided by Morita equivalence of C ∗ -algebras, or (equivalent)
their stable isomorphisms. Namely, for (separable) C ∗ -algebras A and B, they
are stably isomorphic if A ⊗ K ∼= B ⊗ K, where K = K(H) is the C ∗ -algebrasof all compact operators on a (separable) Hilbert space H.
C ∗ -algebras A and B are (strongly) Morita equivalent if there is a A-B
bimodule M, which is a left Hilbert A-module with an A-valued inner product－ 12 －

·, ·A , and is a right Hilbert B-module with an B-valued inner product ·, ·B ,such that (corrected)(1) A = M, MA and B = M, MB ,(2) ξ1 , ξ2 A ξ3 = ξ1 ξ2 , ξ3 B , ξ1 , ξ2 , ξ3 ∈ M,(3) aξ, aξB ≤ a2 ξ, ξB ,ξb, ξbA ≤ b2 ξ, ξA ,for a ∈ A, b ∈ B, ξ ∈ M, so that A and B act on M from the left and right, asbounded operators.
We may assume that M = ACB for a C ∗ -algebra C such that
ACA ⊂ A ⊂ C

and BCB ⊂ B ⊂ C.

Then AM ⊂ M and MB ⊂ M. Also, we may assume that x, yA = xy ∗ ∈ A
and x, yB = x∗ y ∈ B for x, y ∈ M. Because
ACBB∗ C∗ A∗ ⊂ MM∗ ⊂ A

and B∗ C∗ A∗ ACB ⊂ M∗ M ⊂ B.

Then, for any x, y, z ∈ M,
x, yA z = xy ∗ z = xy, zB .
As well, for any a ∈ A, b ∈ B, ξ ∈ M,
aξ, aξB = ξ ∗ a∗ aξ ≤ ξ ∗ a∗ a1ξ = a2 ξ, ξB
∗ ∗

∗

∗

ξb, ξbA = ξbb ξ ≤ ξbb 1ξ = b ξ, ξA ,For instance, we may let2

in B+ ,
in A+ .

 


C C
0 0
A 0
0 0
0 B
C C


0 A C B ≡ M
≡ M.
=
0
0


ACB ≡

Then
MM∗ =



M (M )∗
0



0
0

⊂A

and M∗ M =

00
0 (M )∗ M⊂ B.

This picture quite explains the meaning of Morita equivalence. Namely thetriangle corners!
In particular, C ∼= C ⊕ 0n = A and Mn (C) ∼= 0 ⊕ Mn (C) = B as diagonalsums in Mn+1 (C) are Morita equivalent since
ACB = (C ⊕ 0n )Mn+1 (C)(0 ⊕ Mn (C))




C
Cn
C 0
0
0
=
0 0n
(Cn )t Mn (C)0 Mn (C)


n
0 C
= M,
=
0 0n
MM∗ ∼
= (Cn )∗ Cn ∼= Cn (Cn )t ∼= C and M∗ M ∼= Mn (C),with Cn as a row vector space and (Cn )t as a column vector space.

－ 13 －

The tools de noncommutative geometry.
Table 3: The tool box
Name (tool as invariant)K-theory,
Kasparov KK-theory
Hochschild (co)homology,
Connes (cyclic) cohomology
Homotopy theory
as quotients
Metric structure,Diﬀerential structure
Real, Spin structure,
Geometric, Analytic structure

Use (limited)
Topological spaces, Vector bundles,
C ∗ -algebras, Extensions
Topological spaces, Function algebras,C ∗ -algebras, subalgebras
Topological spaces,
Baum-Connes assembly maps
Manifolds, Dirac operator,
Spectral triples
Real, Spin manifolds,
Characteristic classes, Zeta functions

As an important note, please handle these tools with the greatest care!

1.3

Spectral triples appeared

We are going to take a (moment) look at the analogue in the NC world ofRiemann geometry, provided by Connes. Spectral triples as a generalization of
the structure of a Riemannian manifold play a crucial role in noncommutative
geometry. In the usual context of Riemann geometry, the deﬁnition of theinﬁnitesimal element ds on a smooth spin manifold can be expressed in terms of
the inverse D−1 of the classical Dirac operator D. The theory of spectral triples(ST) is motivated by this key remark. In particular, the geodesic distance
between two points of the manifold is deﬁned in terms of D−1 (cf. [27, VI]).
Example 1.2. (Added). The geodesic distance formula for a Riemann spin
manifold M is given byd(x, y) = sup{|f (x) − f (y)| f ∈ C ∞ (M ), [D, f ] ≤ 1},dwhere D is the Dirac operator, deﬁned as ds, so that [D, f ] = MDf the multi2
plication operator on the Hilbert space L (M ), because[D, f ]g = [D, Mf ]g = (DMf − Mf D)g= D(f g) − f Dg
= (Df )g + f Dg − f Dg = MDf g,

f, g ∈ C ∞ (M ),and C ∞ (M ) is dense in L2 (M ). For more details, may refer to [150].
The tripled(C(M ), L2 (M ), D =)ds－ 14 －

is the typical example of spectral triples in the commutavie case. Indeed, theC ∗ -algebra C(M ) of all continuous, complex-valued functions on a compact(Riemann spin) manifold M contains the norm dense ∗-subalgebra C ∞ (M ) of
all smooth diﬀerentiable functions on M , on which as a dense domain in C(M ),dthe (Dirac) diﬀerential operator ds
is certainly deﬁned. The representation of2
C(M ) on L (M ) is given as multiplication operators. As a generalization of the commutative case to the noncommutative case,given by Connes is the following (cf. [27], [28], [50]):Deﬁnition 1.3. A (Connes) spectral triple (SpTr) is deﬁned to be a triple(A, H, D) of a C ∗ -algebra A, a C ∗ -algebra representation ρ : A → B(H) theC ∗ -algebra of all bounded operators on a separable Hilbert space H, and a(unbounded, Dirac) operator D, such that
(1) D = D∗ as self-adjointness, with its spectrum in R.
(2) For any λ ∈ R, the resolvent (inverse) operator (D − λ1)−1 is a compact
operator on H.
(3) For any a ∈ A = A∞ a dense involutive (or ∗-) subalgebra A, thecommutator [D, ρ(a)] is a bounded operator on H.
Remark. (Edited). The above property (2) can be regarded as a generalizationof the ellipticity property of the standard Dirac operator on a compact manifold.
In the classical case of Riemann manifolds, the property (3) is equivalent to the
Lipschitz condition, which is weaker than the smoothness condition. 
Indeed, the geodesic distance formula for M is converted as the Lipschitz
norm for f ∈ C ∞ (M ) as|f (x) − f (y)|df=  ∞ = MDf  = [D, f ].
d(x, y)dsx=y∈ML(f ) ≡ sup

Then the boundedness of L(f ) with f ∞ ≤ 1 is equivalent to that of theoperator [D, ·].
The basic geometric structure encoded by the theory of spectral triples is
Riemann geometry. But in more reﬁned cases such as Kähler geometry, theadditional structure can be encoded as additional symmetries. For instance, inthe case where the algebra involves the action of the Lefschetz operator of acompact Kähler manifold, can be encoded the information on the Kähler form,
at the cohomological level (cf. [61], [63]).
As the relations among NG, arithmetic geometry (AG) and number theory(NT), the interesting feature of spectral triples (SpTr) is that we have an associated family of zeta functions (deﬁned as the trace for the operator |D|−z )and the theory of volumes and integration, which is related to special values ofthese zeta functions.
Volume form. Based on [27], [28]. May refer to [150] as well. A Connes
spectral triple (A, H, D) is said to be n-summable, or of dimension n if |D|−nis an inﬁnitesimal (or a compact operator) of order one, in the sense that the
eigenvalues λk (|D|−n ) of |D|−n satisfy the estimate λk (|D|−n ) = O(k −1 ) (k →－ 15 －

∞), which means 0 ≤ λk (|D|−n ) ≤ M
k for some constant M > 0 and any k large
enough.
Consider a positive compact operator T with decreasing eigenvalues λj (T ) ≥k−1
0 with multiplicities counted such thatj=0 λj (T ) = O(log k) (k → ∞)as a (logarithmic) divergence, equivalently, with the sequence boundedenss
k−11
j=0 λj (T ) ≤ M for some M > 0 and any k large enough, as the (inlog kverse) logarithmic (possible) divergence. The Dixmier trace trω (T ) of such
T ∈ L1,∞ (H) ⊂ B(H), containing a trace class operator T ∈ L1 (H) such thatT 1 = tr(|T |) =∞λj (|T |) < ∞,j=0

is deﬁned to be
k−1
k−1
1 
1 
λj (T ) ≡ ω({λj (T )}k ),trω (T ) = lim
ω log klog k j=0
j=0

where ω is a certainly chosen functional in C b (N)∗ the dual space of C b (N) theC ∗ -algebra of all bounded complex sequences, with several properties, extending
the usual limit for convergent sequences.
Namely, lim ≤ limω = ω as the limit extension. Note that functionals (onNC spaces or algebras) are viewed as (noncommutative) integrations.
The Dixmier trace is also deﬁned on compact operators with order one as
inﬁnitesimals.
The reason for this is implied by that the Euler constant γ as the limit
lim (1 +k→∞

1
1+ · · · + − log k) = γ
2kexists, with 12 ≤ γ < 1.
An operator T in the deﬁnition domain of the Dixmier trace(s) is said to bemeasurable if the D-trace trω (T ) does not depend on the choice of the limit
functional ω = limω .
The operator |D|−n on a noncommutative space A generalizes the notion ofa volume form on a usual space. The (noncommutative) (total) volume for A
or A (n-summable) is deﬁned to be
V = trω (|D|−n ) ≡ |A| or |A|.
More generally, for an element a of the algebra D generated by A and [D, A],
its (normalized) integration with respect to the volume form |D|−n is deﬁned
as1
1
a = trω (a|D|−n ).
VV
The usual notion of integration on a Riemann spin manifold M can berevovered in this context by the formulannn
f dv = 2n−[ 2 ]−1 π 2 nΓ( )trω (Mf |D|−n ),2M－ 16 －

where D is the classical Dirac operator on M associated to the metric that
determines the volume form dv, and Mf is the multiplication operator acting
on the Hilbert space of all square integrable (L2 ) spinors (or sections of a certain
bundle) on M (cf. [27], [97]).
Zeta functions. For a spectral triple (SpTr) (A, H, D), the zeta function
associated to the Dirac operator D is deﬁned to beζD (z) = tr(|D|−z ) =tr(p(λ, |D|))λ∈σp (|D|)

1,λzwhere p(λ, |D|) denotes the orthogonal projection on the eigenspace for an engenvalue λ ∈ σp (|D|) the point spectrum in the spectrum σ(|D|) of |D|. If |D|
is compact, then σ(|D|) = σp (|D|).
As an important result in the theory of SpTr ([27, IV Proposition 4]), thetotal volume is related to the residue of the zeta function at s = 1 by theformulaV = ress=1 tr(|D|−s ) = lim (s − 1)ζD (s).
s→1+0Moreover, there is a family of (extended) zeta functions associated to aspectral triple (A, H, D), containing ζD . For any a ∈ D the domain of D, thetwo zeta functions with respect to a and D are deﬁned asζa,D (z) = tr(a|D|−z ) =ζa,D (s, z) =tr(a · p(λ, |D|))λ∈σp (|D|)

tr(a p(λ, |D|))λ∈σp (|D|)

1
,
λz

and

1
.
(s − λ)z

These zeta functions are related to the heat kernel exp(−t|D|) by Mellin transform as1
Γ(z)1ζa,D (s, z) =Γ(z)∞ζa,D (z) =0∞

tz−1 θa,D (t)dt,and

tz−1 θa,D,s (t)dt,

0

where
θa,D (t) =λ∈σp (|D|)θa,D,s (t) =λ∈σp (|D|)

1tr(a p(λ, |D|)) tλ = tr(a e−t|D| )e1tr(a p(λ, |D|)) t(s−λ) .
eUnder suitable hypothesis on the asymptotic expansion of θa,D,s (cf. [107, 2
Theorems 2.7 and 2.8]), the two zeta functions admit a unique analytic continuation (cf. [50]). There is an associated regularized determinant in the sense－ 17 －

of Ray-Singer (cf. [138]):ddet (s) = exp − ζa,D (s, z)|z=0 .
∞,a,D
dz


The family of zeta functions ζa,D (z) also provides a reﬁned notion of dimension for a spectral triple (A, H, D), which is called its dimension spectrum andis deﬁned to be a subset Σ = Σ(A, H, D) of C such that all the zeta functions
ζa,D (z) for any a ∈ D extend holomorphically to C \ Σ.
Examples of SpTr with dimension spectrum not contained in R can be constructed by Cantor sets.
Index map. As well, spectral triples (A, H, D) determine an index map. In∗
fact, the√ D = D self-adjoint has a polar decomposition 2D = F |D|, where|D| = D∗ D ≥ 0 positive and F is a sign operator with F = id the identity
map on H. Following [27], [28], deﬁne a cyclic cocycle τ as
τ (a0 , a1 , · · · , an ) = tr(a0 [F, a1 ] · · · [F, an ]),where n is the dimension of the SpTr. For n even, the trace tr(·) should be
replaced by a super (or Dixmier) trace, as usual in index theory. This cocylc
has pairs with the K-theory groups of A, with K0 (A) in the even case and with
K1 (A) in the odd case, and deﬁnes a class of τ in the cyclic cohomology HC n (A)for A. That class is said to be the Chern character of the SpTr (A, H, D).
In the case where the spectral triple (A, H, D) has discrete dimension spectrum Σ, there is a local formula for the cyclic cohomology Chern character,
analogous to the local formula for the index in the space case. This is obtainedby producing Hochschild representatives as (cf. [28], [50]):ϕ(a0 , a1 , · · · , an ) = trω (a0 [D, a1 ] · · · [D, an ]|D|−n ).
Inﬁnite dimensional geometry. The main diﬃculty in constructing speciﬁc examples of spectral triples is in producing a (Dirac-like) operator D withbounded commutators with elements of A, and also a non-trivial index map.
In general, noncommutative spaces do not admit ﬁnitely summable spectral
triples. For instance, the operator |D|−p is of trace class for some p > 0. Obstructions of such spectral triples to exist are analyzed by Connes [26]. It is thenuseful to consider a weaker notion. Namely, the θ-summablve spectral triples
2
come up and satisﬁes the property as tr(e−tD ) < ∞ for any t > 0. Such spectral triples may be thought of as inﬁnite dimensional noncommutative geometry.
There are some examples of spectral triples that are not ﬁnitely summable but
θ-summable, because of the growth rate of multiplicities of eigenvalues of D or
D2 .
Spectral triples and Morita equivalence. Let (A, H, D) be a spectral triple.
Assume that A is Morita equivalent to a C ∗ -algebra B, implemented by a B-A
bimodule M, which is a ﬁnite projective (left and right) Hilbert B-A module.
The spectral triple (A, D, H) can be transfered to another by using this Morita
equivalence as follows.

－ 18 －

First consider the (left) A-module (corrected) Ω1D generated by the elements
a1 [D, a2 ] for any a1 , a2 ∈ A. Deﬁne a connection as ∇ : M → M ⊗A Ω1D alinear mapping by requiring that∇(ξa) = (∇ξ)a + ξ ⊗ [D, a],ξ ∈ M, a ∈ A,

as a derivation rule. Also require thatξ1 , ∇ξ2 A − ∇ξ1 , ξ2 A = [D, ξ1 , ξ2 A ]as compatibility of ∇ with [D, ·].
For instance, as a possible computation, for ξ1 = b1 c1 a1 and ξ2 = b2 c2 a2
in M = BCA for a C ∗ -algebra C,ξ1 , ∇ξ2 A − ∇ξ1 , ξ2 A = b1 c1 a1 , b2 c2 a2 ⊗ a1 [D, a2 ] − b1 c1 a1 ⊗ a1 [D, a2 ], b2 c2 a2 
= a∗1 c∗1 b∗1 b2 c2 a2 ⊗ a1 [D, a2 ] − [D, a2 ]∗ (a1 )∗ ⊗ a∗1 c∗1 b∗1 b2 c2 a2= ξ1 , ξ2 A a1 [D, a2 ] − [D, a2 ]∗ (a1 )∗ ξ1 , ξ2 A= Dξ1 , ξ2 A − ξ1 , ξ2 A D = [D, ξ1 , ξ2 A ],where the compatibility requires those equalities by choosing those dashed elements.
There is an induced spectral triple(B, M ⊗A H, ∇ ⊗∼ D),
where the action of B on M ⊗A H a Hilbert space is deﬁned asb(ξ ⊗A η) = (bξ) ⊗A η,b ∈ B, ξ ∈ M, η ∈ H,

and the induced Dirac operator ∇ ⊗∼ D by the connection ∇ is deﬁned as(∇ ⊗∼ D)(ξ ⊗A η) = ξ ⊗ D(η) + (∇ξ)η ∈ M ⊗A H.
Note that we need to assume that the connection ∇ is Hermitian, because
commutators [D, a] for a ∈ A may be non-trivial, and hence 1 ⊗ D would not
be well deﬁned on M ⊗A H.
K-theory for C ∗ -algebras. The K-theory groups for C ∗ -algebras, as in ahomology theory for C ∗ -algebras, are important invariants that capture information on the topology of C ∗ -algebras as noncommutative spaces, as like thatthe topological K-theory groups for spaces, as a cohomology theory for space,capture information on the topological of spaces (cf. [11], [154]).
The K0 -group K0 (A) of a (untal) algebra A is deﬁned to be an abeliangroup of suitable equivalence classes of idempotents of matrix algebras over A
with suitable operations. For a (unital) C ∗ -algebra A, K0 (A) is deﬁned similarly
by self-adjoint idempotents, i.e., projections.
Two projections p and q of a C ∗ -algebra A are (Murray-) von Neumann(vN) equivalent, denoted as p ∼ q if there is a partial isometry v ∈ A such thatp = v ∗ v and q = vv ∗ .

－ 19 －

Any idempotent p = p2 in A may be replaced with a projection pp∗ (1 − (p −p ) ) (?), preserving the von Neumann equivalence.
For, z ∈ C, q(z) = |z|2 (1 − |z − z|2 )−1 = q(z) is a projection? If so, forz ∈ R, |z|2 = |z|4 . Thus, z = 0 or 1. Hence, q(z) is a projection on {0, 1}. But
the functional calculus can be applied to what? As another possible candidate,as an attempt, may set q = −(p − p∗ )2 = (p − p∗ )∗ (p − p∗ ). Then q = q ∗ , and∗ 2 −1q 2 = (p − p∗ )4 = (p2 − pp∗ − p∗ p + (p∗ )2 )2= (p(1 − p∗ ) + p∗ (1 − p))2 = (p(1 − p∗ ))2 + (p∗ (1 − p))2
= (p(p∗ − 1))2 + (p∗ (p − 1))2 = p(p∗ − 1) + p∗ (p − 1) = q,
provided that (p(p∗ − 1))2 = p(p∗ − 1) and (p∗ (p − 1))2 = p∗ (p − 1). This holds
if (pp∗ p − 2p)(1 − p∗ ) = 0 and (p∗ pp∗ − 2p∗ )(1 − p) = 0, which holds if p isa partial isometry with p(1 − p∗ ) = 0 as being orthogonal. Note as well that
1 + q = 1 + (p − p∗ )∗ (p − p∗ ) ≥ 1, so that 1 + q is invertible.
For p, q projections of matrix algebras over A, they are stably equivalent
if there is a projection r of a matrix algebra over A such that diagonal sumsp⊕r ∼ q⊕r. Deﬁne K0 (A)+ to be the monoid (or additive semi-group) of (stably
or vN) equivalent classes of projections of matrix algebras over A (unital), with
operation as [p] ⊕ [q] = [p ⊕ q]. Let K0 (A) be the Grothendieck group of K0 (A)+The K1 -group K1 (A) of a unital (algebra or) C ∗ -algebra A is deﬁned tobe the inductive limit of quotient groups
lim GLn (A)/GLn (A)0 ,−→GLn (A) → GLn+1 (A),induced byx → x ⊕ 1 the diagonal sum,

where GLn (A) is the group of invertible n × n matrices over A and GLn (A)0 is
the connected component of GLn (A) with the identity matrix, with the multiplicative operation as [u] · [v] = [uv] = [u ⊕ v], so that K1 (A) is abelian even if
the quotients GLn (A)/GLn (A)0 are not.
However, the K-theory groups for C ∗ -algebras are not always computableor determined, but certainly computed in many important cases. On the other
hand, there are two fundamental developments in noncommutative geometry,as faced with diﬃculty in determining K-theory groups at certain cases such as
irrational rotation C ∗ -algebras (cf. [140]).
The ﬁrst development is the theory of cyclic cohomology, introduced byConnes (cf. [24], [25], [27]), to provide cocycles that pairs with K-theory groups
as Chern characters. The second is the K-theory geometrically reﬁned by BaumConnes (BC2000) [9], to produce the assembly map
μ : K ∗ (X, G) → K∗ (C0 (X)  G)
for ∗ = 0, 1, from the geometric K-theory for a space X (non-compact, ornot) with an action by a group G to the analytic K-theory for the C ∗ -algebra
crossed product C0 (X)  G (which may be viewed as the G-equivariant Ktheory K∗G (C0 (X)), for instance, when G is a compact group). The BaumConnes conjecture says that the assembly map is an isomorphism between the(topological) K-theory for spaces and the (topological) K-theory for C ∗ -algebras.

－ 20 －

It says that at the topological level, such spaces or dynamical systems
may be identiﬁed with the corresponding crossed product C ∗ -algebras as NC
spaces.
For more details on the BC conjecture, for instance, may refer to [149].

1.4

Why NCG?
Table 4: The reasonable, sensible dictionary for why NCG?

Sense
1
2
3
4

5
6

Space geometry
Untouchable boundaryin (quotient) spaces
Dynamical systems
by homeomorphisms
Coordinates as variables
Space invariants such as
K-theory and (co)homology
with Chern character
Classical mechanics
Visual type in a sense

Noncommutative geometry
Part in one, complete quantum
boundary as C ∗ -algebras
Crossed product C ∗ -algebras
by automorphims (or endo-)
Operators as NC variables
C ∗ -algebra analogues such as
K(K)-theory and (cyclic) cohomology
with Connes character
Quantum (statistical) mechanics
Deformed type in some sense

The speciﬁc reasons are explained as in the following.
As one direction in which noncommutative geometry is applied to number
theory, so featured is the explicit class ﬁeld theory. It is shown by Bost andConnes [13] that the Galois theory of the cyclotomic ﬁeld QAb is live naturally
as symmetries of zero temperature equilibrium states of a quantum statistical mechanical system (QSMS). Moreover, it is shown by Connes and Marcolli(CM) that the similar noncommutative space with a natrural time evolution
gives rise to zero temperature equilibrium states whose symmetries are given by
the automorphisms of the modular ﬁeld. Furthermore, it is shown by CM andRamachandran that these constructions ﬁt into a possible theory of noncommutative Shimura varieties, and exhibited is a related QSMS that recovers the
explicit class ﬁeld theory for imaginary quadratic ﬁelds. It opens up a concrete
possibility that noncommutative geometry may be able to deal with the real
quadratic ﬁelds or other cases where the Hilbert 12th problem of explicit classﬁled theory (ECFT) is not yet fully understood from a number theoretic pointof view. The connections between the H12 problem for real quadratic ﬁelds
and NG are considered by Manin [109] and [110]. Described later on are theseresults as well as perspectives.
As another direction in which noncommutative geometry may yield contributions to number theory and arithmetic geometry, so featured are the Lfunctions. It is Connes [29] who ﬁrst brings its possibility to light on the spec-－ 21 －

tral realization of zeros of the Riemann zeta function and of L-functions with
Grossencharakter. This work provides a new approach to the generalized Riemann hypothesis through a Selberg trace formula for the action of the idèle classgroup on a noncommutative space obtained as a quotient of the space of adèles.
The noncommutative space involved in this construction is closely related to the
Bost-Connes system, up to a duality.
As a diﬀerent direction, it is shown by Consani and Marcolli that the techniques in NG may help describing the geometry of the ﬁbers at inﬁnity of arithmetic varieties and the corresponding Gamma factors contributing to the Lfunctions. Described later on are these results.
These approaches may lead to some other results on L-functions of arithmetic
varieties.
As another striking example as the NG leading to new conceptual explanations or viewing on classical number theoretic objects, Connes and Moscovici[54], [55] give the modular Hecke algebras that recover and extend the structures like the Rankin-Cohen brackets of modular forms [159], in terms of Hopf
algebra symmetries of noncommutative spaces and their Hopf cyclic cohomology. In the similar spirit, the results of Manin and Marcolli discussed soonlater recover arithmetic properties of modular curves from the corresponding
noncommutative boundary, which opens the possibility of similar constructions
on the boundary of other moduli spaces of arithmetic signiﬁcance.
This subject may be still a building site in rapid development. The results
collected and the perspectives may present a uniﬁed picture that outlines theshape of the emerging landscape, so as to be called arithmetic noncommutavie
geometry (ANCG). To be continued, so far as now.

2

Noncommutative modular curves

The results of this section are mostly based on Yuri Manin and Matilde Marcolli[112], with additional material from Marcolli [113] and [115] (not at hand).
Included are necessary preliminary notions on modular curves based on Manin[103] and those on noncommutative tori based on Connes [22] and Rieﬀel [140].

2.1

Modular curves

Let G be a ﬁnite index subgroup of the (projective) (elliptic) modular group
Γ = P SL2 (Z) = SL2 (Z)/{±12 }, where 12 is the 2×2 identity matrix in GL2 (C)or SL2 (Z), with [12 ] = 1Γ the unit class. Denote by XG the quotient space G\H2
as the modular curve, where H2 is the real 2-dimensional hyperbolic plane,
namely the upper half plane {z ∈ C | Im(z) > 0} = R × R+ with the metricds2 = |dz|2 (Im(z))−2 . Equivalently, we may identify H2 with the Poincaré disk
D = {z ∈ C | |z| < 1} with the metric ds2 = 4|dz|2 (1 − |z|2 )−2 .
Proof. (Added). May refer to [151]. Let C+ = C∪{∞} ≈ S 2 be the Riemann (orreal 2-dimensional) sphere. Any element of GL2 (C) deﬁnes the linear fractional－ 22 －

transformation of C+ asfg (z) = g(z) =

az + ba bz=,c dcz + d⎧a⎪⎨ c if ac = 0,z ∈ C and g(∞) = 0
if a = 0,⎪⎩∞ if c = 0.

Note that if c = 0, then fg (z) = ad z + db , and if c = 0, thenfg (z) =a (ad − bc)c−2.
−cz + dcAs well, suppose that g ∈ SL2 (Z) with ad − bc = 1. If c = 0, then eithera = d = 1 or a = d = −1. If c = 0, thenRe(z) + dc − iIm(z)
1
1
=
=
.
d
d
z+ c
Re(z) + c + iIm(z)|z + dc |2
It then follows that if Im(z) > 0, then Im(fg (z)) > 0, where g ∈ SL2 (Z).
If g, h ∈ GL2 (C), then fgh = fg ◦ fh and (fg )−1 = fg−1 .
If g ∈ SL2 (Z), then −g ∈ SL2 (Z) and fg = f−g . Thus, we may deﬁne
f[g] = fg for [g] ∈ P SL2 (Z) = SL2 (Z)/{±12 }.
Note as well that for z = x + iy,
ds2 = |dz|2 (Im(z))−2 = (dx + idy)(dx − idy)(Im(z))−2 = (dx2 + dy 2 )(Im(z))−2= ((Im(z))−2 ⊕ (Im(z))−2 )(dx, dy)t , (dx, dy)t  ≡ QH2 (dz)on H2 . And on the Poincaré disk D,
ds2 = 4|dz|2 (1 − |z|2 )−2 = (dx2 + dy 2 )4(1 − |z|2 )−2= (4(1 − |z|2 )−2 ⊕ 4(1 − |z|2 )−2 )(dx, dy)t , (dx, dy)t  ≡ QD (dz).
The Cayley transform is the holomorphic isomorphism from H2 to D deﬁnedby
Cl(z) =


z−i
1 −i
z=
∈ D.
1 i
z+i

Im(z)
2
2
If fact, for z ∈ C, 1 − |Cl(z)|2 = 4|z+i|
2 . Hence Cl(H ) = D, Cl(−H \ {−i}) =cD the complement of the closure of D, and Cl(R) = S 1 the unit 1-sphere (orcircle).
As a note, (Cl)∗ Cl = 2 ⊕ 2 the diagonal sum, and4|z + i|4(x2 + (y + 1)2 )2==.
(1 − |Cl(z)|2 )24 Im(z)24 Im(z)2(The end of the proof.)－ 23 －

Remark. (Added). May refer to [116]. Any conformal mapping between
the unit open disk D in C is given as w = ε(z − z0 )(1 − z0 z)−1 with |ε| = 1,|z0 | < 1, so that |dw|(1 − |w|2 )−1 = |dz|(1 − |z|2 )−1 which is called as Poincaré’s
diﬀerential invariant. The disk D with line element ds = |dz|(1 − |z|2 )−1 asanother Poincaré metric is a Lobachevskiı̆’s non-Euclidean space.
In fact, we have 1 − |w|2 = 1 − |z − z0 |2 |1 − z0 z|−2 . In particular, if z0 = 0,then 1 − |w|2 = 1 − |z|2 .

2Let ϕ : H → XG be the quotient map. Denote by P the coset space Γ/G(which may be trivial). Then XG can be identiﬁed with the quotient Γ\(H2 ×P).
It says that the element Γ(z, hG) ⊃ (Gz, hG) is identiﬁed with ϕ(z) =Gz.
The action of P SL2 (Z) by fractional linear transformations is written interms of the generators: for z = x + iy ∈ H2 ,




1
−x + iy
1 1
0 −1
S(z) = − =and T (z) = z + 1 =
z
z=
0 1
1 0
z
|z|2
as inversion (with respect to both y-axis and the unit circle) and translation(along x-axis), with [S]2 = [S 2 ] = [−12 ] in P SL2 (Z). In particular, S(i) = i,S(iy) = iy −1 , as well as S(eiθ ) = ei(π−θ) for θ ∈ R.
Also, P SL2 (Z) can be identiﬁed with the free product group Z2 ∗ Z3 , withgenerators σ = [S] and τ = [ST ], where




−1 −1
0 −1
2
, (ST )3 = −12 .
ST =, (ST ) =1
0
1 1
The tessellation of the hyperbolic plane H2 by fundamental domains for theaction of P SL2 (Z) (by S and T ) can be illustrated but not included.
May let G = Γ = P SL2 (Z). For instance, Γ{i} contains n + i and −n+in2 +1
for n ∈ Z, and more.
Remark. (Added). May refer to [116]. A group Γ acting on a locally compact
Hausdorﬀ space X is said to be a discontinuous group if the action of Γ on
X is properly discontinuous in the sense that for any compact subset S of X,
there are only ﬁnitely many elements γ ∈ Γ such that γS ∩ S = ∅.
A fundamental domain for a discontinuous group Γ acting on X is deﬁned
to be a complete representative set F for the quotient space Γ \ X such thatΓF = X and γF ∩ F = ∅ for any γ ∈ Γ \ {1Γ }.
The decomposition X = ∪γ∈Γ γF as a union is said to be the tessellation
by a fundamental domain F .
The case of the Riemann sphere C+ . The group Auta (C+ ) of analytic automorphisms on C+ consists of complex, special, linear fractional transformations
fg for g ∈ SL2 (C), and is identiﬁed with the semi-simple Lie group P SL2 (C).
Since C+ is compact, a discontinuous group on C+ is a ﬁnite group, viewed asa ﬁnite subgroup of the motion group SO(3) of the real 2-dimensional sphere,
and hence either a cyclic group or a regular polyhedral group.
The case of the disk D identiﬁed with H2 . The group Auta (H2 ) consists ofreal, special, linear fractional (Möbius) transformations fg for g ∈ SL2 (R), and－ 24 －

is identiﬁed with the semi-simple Lie group P SL2 (R) = G. The group G actson H2 transitively and its stabilizer group for i ∈ H2 is P SO(2) = K, so that
H2 ∼= G/K. Since K is compact, a discontinuous group Γ on H2 is given by adiscrete subgroup of G.
Moreover, the action of Γ on H2 extends analytically to C+ ≈ S 2 . A limit
point of Γ is deﬁned to an accumulation point of C+ for the orbit Γz for somez ∈ C+ and is contained in R ∪ {∞} = R+ ≈ S 1 the unit sphere. The set Lof limit points of Γ is either one or two points, or an inﬁnite set, and is eitherR ∪ {∞} or nowhere dense (in other words, L is a border set in the sense thatcits complement L is dense in R+ or C+ .
A fundamental (Dirichlet) domain F for the action of Γ is deﬁned to be a
(curved) polygon with boundary composed by (curved) edges as arcs (geodesics
in H2 ) or line segments, orthogonal to the real axis, so that equivalent edgesof the polygon F and their relations correspond to generators and relations of
Γ. The number of edges of F is ﬁnite if and only if Γ is ﬁnitely generated. In
this case, Γ is said to be a Fuchsian group. A Fuchsian group is a ﬁnitely
generated, discrete subgroup of P SL2 (R). Also, a Kleinian group is deﬁned to
be a discrete subgroup of P SL2 (C).
A Fuchsian group Γ is said to be of the ﬁrst kind if L = R ∪ {∞} ≈ P1 (R)the real 1-dimensional projective line, and be of the second kind if otherwise.
Being of the ﬁrst kind is equivalent to that Γ\H2 has volume ﬁnite. A pointx ∈ R+ is said to be a (parabolic) cusp if the stabilizer group Γx consists ofparabolic transformations such as T and its conjugates. On the other hand, aﬁxed point z ∈ H2 under Γ is said to be en elliptic point since the stabilizer
group Γz is a ﬁnite cyclic groups of elliptic transformations with respect to K.
The (elliptic) modular group Γ = SL2 (Z) is a Fuchsian group of the ﬁrst
+
1
+
kind.
√ The set of cusps of Γ is Q ∪ {∞} = Q ≈ P (Q) in L = R . The point
i = −1 is an elliptic point for Γ. 
Table 5: The simply connected Riemann surfaces X
XSpace nameDeﬁnition formula
Analytic Auta (X)Discrete subgroups

Elliptic
Riemann sphere
C+ = C ∪ {∞}P SL2 (C)
KleinianParabolicGauss plane
Complex C
SDP C  C∗
Discrete az + b

Hyperbolic
Open unit disk
D = {z ∈ C | |z| < 1}
Real P SL2 (R)Fuchsian(SDP = semi-direct product (group)).
Example 2.1. (Edited). An example of ﬁnite index subgroups G of Γ is givenby the congruence subgroups Γ0 (N ) of Γ:a bΓ0 (N ) = {∈ Γ | c ≡ 0 mod N }.
c d

－ 25 －

−1k
A fundamental domain for Γ0 (N ) is given by F and ∪N
k=0 ST (F ), where F isa fundamental domain for Γ.
Note that


−1
0 −1
k
ST z =
z=
.
1 k
z+k

As well, in Γ0 (N ),

 
 

a
b
a
b
ab + bdaa + bk  N=
kN d
k  N d(ka + dk  )N kb N + dd

−1 

a
b
d
−b
and
=
.

kN d
−kN a
The quotient space XG has the structure as a non-compact Riemann surface.
This has a natural algebro-geometric compactiﬁcation, by adding the cusp points
as points at inﬁnity. The space of cusp points is identiﬁed with the quotient
G\P1 (Q) ∼= Γ\(P1 (Q) × P),
where P1 (R) ⊃ P1 (Q) ≈ Q+ . Thus, deﬁne the compactiﬁcation of XG as
XG = G\(H2 ∪ P1 (Q)) ∼= Γ((H2 ∪ P1 (Q)) × P).
Remark. (Added). The union H2 ∪ P1 (Q) may be replaced by H 2 = H2 ∪ Q ∪{i∞} (cf. [145, 3.2]). 
The modular curve XΓ for Γ = SL2 (Z) is the moduli space of elliptic curves,
with the point τ ∈ H2 parameterizing the lattice Λ = Z ⊕ τ Z in C and thecorresponding elliptic curve uniformized by Eτ = C/Λ.
The unique cusp point corresponds to the degeneration of the elliptic curveto the cylinder C∗ , when τ goes to ∞ in H2 .
The other modular curves, obtained as quotients XG by congruence subgroups, can also be interpreted as moduli spaces of elliptic curves with levelstructure. Namely, for elliptic curves E = C/Λ, this is an additional information on the torsion points N1 Λ/Λ ⊂ QΛ/Λ for some level N .
Example 2.2. (Edited). The principal congruence subgroups Γ(N ) of Γ =P SL2 (Z) consists of matrices M ∈ Γ such that M ≡ 12 mod N . Points in themodular curve Γ(N )\H2 classify elliptic curves Eτ = C/Λ, with Λ = Z + Zτ ,τtogether with a basis { N1 , N} for the torsion subgroup N1 Λ/Λ. The projection
from XΓ(N ) to XΓ forgets the extra structure.
In the case of the groups Γ0 (N ), points in the quotient space Γ0 (N )\H2
classify elliptic curves, together with a cyclic subgroup of Eτ of order N . This
extra information is equivalent to an isogeny ϕ : Eτ → Eτ  , where the cyclic
group is the kernel ker(ϕ). Recall that an isogeny is a morphism ϕ : Eτ → Eτ 
2such that ϕ(0) = 0. These are implemented by the action of GL+
2 (Q) on H .
2Namely, Eτ and Eτ  are isogenious if and only if τ and τ in H are in thesame orbit of GL+
2 (Q).

－ 26 －

Modular symbols. Given two points or cusps α, β ∈ H2 ∪ P1 (Q) ≡ H2 , thereal homology class [α, β]G ∈ H1 (XG , R) (corrected) as a modular symbol isdeﬁned asβ[α, β]G =ϕ∗ (ω),ϕ∗ (ω) = ω ◦ ϕ,

ϕ : H2 → XG = G\H2

αas a functional with variable ω holomorphic diﬀerentials (of the ﬁrst kind) on
XG , and the integration of the pull back to H2 is along the geodesic arc connecting α and β.
The functionals above as modular symbols in H1 (XG , R) (for α, β cusps)
are in fact in (rational) H1 (XG , Q) (by Manin [103] and Drinfel’d [72]).
The modular symbols satisfy the following additivity and invariance properties:[α, β]G + [β, γ]G = [α, γ]G

and [gα, gβ]G = [α, β]Gg ∈ G.

Indeed, we haveβγϕ∗ (ω) +αgββϕ∗ (ω),αβϕ∗ (ω) =gαγϕ∗ (ω) =βω◦ϕ◦g =αω ◦ ϕ.
αBy additivity, (for cusps) it is suﬃcient to consider modular symbols of theform [0, α], with α ∈ Q.
Because [α, β]G = [α, 0]G + [0, β]G = −[0, α]G + [0, β]G .
Moreover, (detailed)[0, α]G =


n 
n


pk−1 pk,=
qk−1 qk G

k=1
n


=−

k=1

k=1

gk (i∞)ϕ∗ ω = −gk (0)−1
pk qk−1
pk−1 qk−1

n


ϕ∗ ω

[gk (0), gk (i∞)]G

k=1

where any rational α has the ﬁnite, continued fraction expansion
α = [a0 , · · · , an ] = a0 +1
a1 + ···+1 1,anand (detailed)
pk−1 (α) pk (α)∈ GL2 (Z),qk−1 (α) qk (α)
pk−1 z + pk
pk
, (z ∈ C+ ), gk (0) =,gk (z) =qk−1 z + qk
qk
gk = gk (α) =－ 27 －

gk (∞) = gk (i∞) =

pk−1
qk−1

with

pk
pk (α)≡= [a0 , a1 , · · · , ak ] (0 ≤ k ≤ n)qk
qk (α)

the successive, reduced fractional approximations for α = pqnn .
Remark. (Added). May recall the following from [116]. For any real number
α, let α = α + b11 with a0 = α (Gauss symbol) the largest integer no morethan α, and b1 > 1 if α = α. Deﬁne inductively
bj = aj +1
bj+1

and aj = bj ,j ≥ 1.

Then α has the unique simple continued fraction, denoted as α = [a0 , a1 , · · · ].
If α is a rational number, then α = [a0 , a1 , · · · , an ] for some n. For 0 ≤ k ≤ n,
we may set
pk (α)
pk[a0 , a1 , · · · , ak ] =≡qk (α)
qk
the k-th approximate (reduced) fractional for α.
The following recursive formulae hold:pk (α) = ak pk−1 (α) + pk−2 (α) and qk (α) = ak qk−1 (α) + qk−2 (α),
for k ≥ 0, where we set p−2 (α) = 0, p−1 (α) = 1, q−2 (α) = 1, and q−1 (α) = 0.
0 +1
For instance, p0 (α) = a0 , q0 (α) = 1, and pq11 = aa11p1+0, and
p2
a2 (a1 a0 + 1) + a0
a2 p1 + a0=.
=
q2
a2 a1 + 1
a2 q1 + q0
It then follows that
pk qk−1 − pk−1 qk = (−1)k+1 ,


Hence

p
det k−1
qk−1

pk
qk

k ≥ −1.

= (−1)k .

For instance,
p1 q0 − p0 q1 = (a1 a0 + 1) − a0 a1 = 1,
p2 q1 − p1 q2 = (a2 p1 + a0 )q1 − p1 (a2 q1 + q0 ) = a0 a1 − (a1 a0 + 1) = −1.

In the classical theory of the modular symbols of [103] and [121] (cf. also [80],[120]), the cohomology classes obtained from cusp forms are evaluated against
the relative homology classes given by modular symbols. Namely, given a cusp
1form Φω on H2 , obtained as a pull back Φω = ϕ∗ (ω) dz
under the quotient map
2
ϕ : H → XG (corrected), we deﬁne the intersection numbers
gs (i∞)Δω (s) =Φω (z)dz,
gs (0)－ 28 －

gs G = s ∈ P1 (Q).

These intersection numbers can be interpreted in terms of special values of Lfunctions associated to the automorphic form which determines the cohomology
class.
That is rephrased in cohomological terms as follows, following [121]. Denoteby i and ρ the elliptic points, deﬁned as the orbits i = Γi and ρ = Γρ2πiπi
respectively, where ρ = e 3 (with ρ = e 3 , corrected, but which may beexchanged if necessary). Let
I = ϕ(i) = G\i

and R = ϕ(ρ) = G\ρ

be the image in XG of the elliptic points.
B
Set HA= H1 (XG \ A, B; Z) the relative (singular) homology group for
disjoint A and B in XG , where either of which may be empty sets.
B
A
There is the (intersection) pairing • : HA
× HB
→ Z.
The modular symbols [g(0), g(i∞)]G for gG ∈ P = Γ/G deﬁne classes in H cp
and generate it by Manin [103], where cp (as well as {cp}) denotes the set ofcusps.
For σ and τ the generators of P SL2 (Z) with σ 2 = 1 and τ 3 = 1, set PI =
σ\P and PR = τ \P.
R
There is an isomorphism Z|P| ∼.
= Hcp∪I
There are exact sequences of groups (cf. Remark below detailed and edited):
πR
0 → Hcp −−−−→ Hcp−−−R−→ Z|PR | → Z → 0

and

R
R
−−−I−→ Hcp→ 0.
0 → Z|PI | −−−−→ Z|P| ∼= Hcp∪IπR∼ HR
The image πI (x∼ ) ∈ Hcp
of an element x∼ = s∈P λs s in Z|P| =cp∪Iwith coeﬃcients λs ∈ Z represents an element x ∈ Hcp if and only if the image
πR (πI (x∼ )) = 0 in Z|PR | .
As proved in [121], for s = gG ∈ P = Γ/G, the intersection pairing • :H cp × Hcp → Z gives, for x∼ = s∈P λs s in Z|P| , with x ∈ Hcp ,[g(0), g(i∞)]G • x = λs − λσs ,σ ∈ P SL2 (Z), σ 2 = 1.

Thus, may write the intersection number as a function Δx : P → R deﬁned byΔx (s) = λs − λσs .
Remark. (Added). As obtained in [121], there are group isomorphisms:cp∪IZΓ/G ∼= HR

R
and ZΓ/G ∼,= Hcp∪I

which are induced by sending gG to ϕ(gγ) ≡ [g]0 , with [1]0 = I, and gG toϕ(gδ) ≡ [g]0 , with [1]0 = R respectively, where γ is the geodesic between i and
i∞, and δ is between ρ to a point z0 in H2 with 0 < Re(z0 ) < 12 and |z0 | > 1,
to −z0 , and to ρ. The intersection pairingcp∪IR• : HR× Hcp∪I→Z－ 29 －

is given by
1 if gG = hG,[g] • [h]0 =0 otherwise.
0

Moreover,H0 (I, Z) ∼= Z|I| ∼= Z|PI |and H0 (R, Z) ∼= Z|R| ∼= Z|PR | .

There are long exact sequences of relative homology groups (modiﬁed):j∗∅
0 −−−−→ Z|PI | −−−−→ Hcp∪I = Hcp∪I
−−−−→

⏐

⏐
1-1j∗


0 −−−−→ Z|PI | −−−−→

0 ←−−−−R∼
Hcp∪I= Z|P|⏐⏐∂
 ∗πI =j∗−−−−→−−−−→ 0R
Hcp−−−−→ 0
⏐
⏐π =∂
 R ∗

onto

←−−−− Z|PR | = H0 (R)Z∅
Hcp = Hcp⏐⏐j∗ 1-1

Z|PR | = H0 (R)as well as
0 ←−−−− Z|PI | ∼= H0 (I) ←−−−−



q∗

H0 ({cp} ∪ I)

⏐∂
⏐ ∗

πI

i

∗
←−−
−− H0 ({cp}) ←−−−− 0

⏐∂
⏐ ∗

j∗

0 ←−−−−

Z|PI |
←−−−− H cp∪I = H∅cp∪I ←−−−− H cp = H∅cp ←−−−− 0⏐⏐j∗ ⏐onto
onto⏐π R =j∗

0 ←−−−−

Z|PI |←−−−−

cp∪I ∼ |P|
HR
=Z

⏐
⏐

0 −−−−→Z−−−−→

Z|PR |

j∗

←−−−−

cp
HR

⏐
⏐

←−−−− 0Z|PR |


Remark. Recall from [116] the following. Let X be a topological space andq ∈ Z non negative. Denote by Sq (X) the free abelian group with basis as theset of all singular q-dimensional simplexes of X, where a singular q-simplexof X is a continuous map f from the q-dimensional standard simplex Δq to X,
where
q

Δq = {(t = (tj )qj=0 ∈ Rq+1 | tj ≥ 0,
tj = 1}.
j=0

In particular, Δ0 = {1}, Δ1 ≈ [0, 1], and Δ2 ≈ D. Thus,
Sq (X) = Sq (X, Z) = {σ =l

nk fk | fk : Δq → X, nk ∈ Z, l ∈ N}.

k=1－ 30 －

The boundary operator ∂q : Sq (X) → Sq−1 (X) (with S−1 (X) = 0) is deﬁnedbyq
∂q σ =(−1)k σ ◦ εk , where εk : Δq−1 → Δq
k=0

is deﬁned by εk ((tj )q−1
j=0 )= (t0 , · · · , tk−1 , 0, tk , · · · , tq−1 ). It then holds that∂q−1 ◦ ∂q = 0, so that the singular chain complex S(X) = {Sq (X), ∂q }q isdeﬁned. The integral singular homology (group) for X is deﬁned to beH∗ (X) = {Hq (X)}q = ⊕∞q=0 Hq (X),
where Hq (X) = Hq (X, Z) = Zq (X)/Bq (X), where Zq (X) = {c ∈ Sq (X) ∂q (c) =0} the group of q-singular cycles and Bq (X) = {c ∈ Sq (X) c ∈ ∂q+1 (Sq+1 (X))}
the group of q-boundary cycles. Two elements c and c ∈ Sq (X) are said to behomologous if c − c ∈ Bq (X), and then their homology classes [c] = [c ] ∈Hq (X).
In particular, ∂1 f (1) = f (0, 1) − f (1, 0) for f ∈ S1 (X). Hence, f ∈ Z1 (X)
if and only if the image of f is a closed curve (or a cycle) in X. For f ∈ S2 (X),
∂2 f (t0 , t1 ) = f (0, t0 , t1 ) − f (t0 , 0, t1 ) + f (t0 , t1 , 0) ∈ B1 (X),
where the union of the domains of these terms is the boundary of Δ2 , and(∂1 ◦ ∂2 )f (1) = (f (0, 0, 1) − f (0, 1, 0)) − (f (0, 0, 1) − f (1, 0, 0))+ (f (0, 1, 0) − f (1, 0, 0)) = 0.
If g : X → Y is a continuous map, then it induces the chain map from S(X)
to S(Y ) as well as the induced homomorphism g∗ : H∗ (X) → H∗ (Y ).
The homology (group or theory) for X has homotopy invariance in the sensethat if X and Y are homotopy equivalent as a space, then H∗ (X) ∼= H∗ (Y ).
If X is contractible, then H0 (X) = Z and Hq (X) = 0 for q ≥ 1.
If X is path-connected, then H0 (X) = Z. 
Remark. (Added). Further recall from [116] the following. Let X be a topological space and A be a subset of X. The singular chain complex S(A) ={Sq (A), ∂q } of A is viewed as a subcomplex of S(X) = {Sq (X), ∂q }. The quotient singular chain complex S(X)/S(A) = {Sq (X)/Sq (A), ∂q } is then deﬁned,
where∂q : Sq (X)/Sq (A) → Sq−1 (X)/Sq−1 (A)is induced by the boundary map ∂q . The relative singular homology (group)H∗ (X, A) of such a pair (X, A) is deﬁned to be the homology of the quotientcomplex S(X)/S(A).
If f : (X, A) → (Y, B) is a continuous map of topological pairs such thatf : X → Y is continuous and f (A) ⊂ B, then it induces the homomorphismf∗ : H∗ (X, A) → H∗ (Y, B).

－ 31 －

In particular, there is the following group homomorphism:
j∗

−−−−→

B
HA
1 ∪A2




B
HA
1



j∗

H1 (XG \ (A1 ∩ A2 ), B) −−−−→ H1 (XG \ A1 , B)
induced by j the inclusion map for a pair.
There is the exact sequence of the chain complexes for a pair (X, A) asqi

0 → S(A) −−−−→ S(X) −−−−→ S(X)/S(A) → 0.
It induces the following long exact sequence of homology for a pair (X, A):
∂

j∗

i

∂

i

· · · −−−∗−→ Hq (A) −−−∗−→ Hq (X) −−−−→ Hq (X, A) −−−∗−→ Hq−1 (A) −−−∗−→ · · ·where i : A → X and j : (X, ∅) → (X, A) are inclusion maps, and ∂∗ [c] = [∂c].
In particular, there is the following exact sequence:j∗∂

H1 (XG \ A) −−−−→ H1 (XG \ A, B) −−−∗−→






∅
HA = HA−−−−→

H0 (B)⏐∂⏐ ∗−−−−→ H B = H∅B

B
HA



The modular complex. For x, y ∈ H2 , denote by x ∩ y the oriented geodesicarc connecting x and y. Let σ and τ be the generators of P SL2 (Z) ∼= Z2 ∗ Z3
as a free product group.
Deﬁnition 2.3. The modular complex is deﬁned to be the cell complex C ={Cj , ∂j }2j=0 deﬁned as follows. The 0-dimensional cells of C0 are given by thecusps (cp) in G\P1 (Q) and the elliptic points I and R. The 1-dimensional cellsof C1 are given by the oriented geodesic arcs
G\(Γ · (i∞ ∩ i)) and G\(Γ · (i ∩ ρ)),πiwhere Γ· means the orbit under the action of Γ, and ρ = e 3 . The 2-dimensional
cells of C2 are given by classes of G\Γ · E, where E is the polygon with verticesi, ρ, 1 + i, and 1 + i∞ and sides (or edges) as the corresponding geodesic arcs.
The boundary operator ∂2 : C2 → C1 is deﬁned by sendinggE → g(i ∩ ρ) + g(ρ ∩ (1 + i)) + g((1 + i) ∩ i∞) + g(i∞ ∩ i),g ∈ Γ,and the boundary map ∂1 : C1 → C0 is deﬁned by sendingg(i∞ ∩ i) → (g · i) − (g · i∞)

and g(i ∩ ρ) → (g · ρ) − (g · i),

and in general, g(x ∩ y) = (g · y) − (g · x), and ∂0 : C0 → C−1 = 0.

－ 32 －

It then follows by cancellation that ∂j−1 ◦ ∂j = 0 for j = 2, 1. We may
denote the homology of the modular (cell) complex as H∗cl (XG ).
That gives a cell decomposition of H2 adapted to the action of P SL2 (Z) andcongruence subgroups.
It is shown by Manin [103] that
Proposition 2.4. The (ﬁrst, only non-trivial) homology of the modular (cell)complex is isomorphic to the ﬁrst homology of X G :H1cl (XG ) ∼= H1 (XG ).
B
The (ﬁrst complementary) relative homology such as HA= H1 (XG \A, B)
can be derived from the corresponding relative versions of the modular cellcomplex.
Indeed, we have Z[{cp}] = C0 /Z[R ∪ I]. Hence, the (modiﬁed) quotientcomplex is induced:∂∼∂

0 → C2 −−−2−→ C1 −−−1−→ Z[{cp}] = C∼
0 → 0,
where ∂1∼ = q ◦ ∂1 : C1 → C0 → Z[{cp}] with q : C0 → C0 /Z[R ∪ I] the quotient
map. It then computes the relative homology H1 (XG , R ∪ I).
Note that the kernel ker(∂1 ) is contained in ker(∂1∼ ), and hence H1cl (XG )is contained in H1cl (XG )∼ the homology corresponding to the modiﬁed complex
as above. There is also the following exact sequence:
j∗

i

∂

H1 (R ∪ I) −−−∗−→ H1 (XG ) −−−−→ H1 (XG , R ∪ I) −−−∗−→ H0 (R ∪ I)
The cycles (in ker(∂1∼ )) are given by Z[P], as combinations of elementsg(i ∩ ρ) with g ranging over representatives of P = Γ/G, and by the elements⊕ag (g(i∞) ∩ g(i)) satisfying
ag g(i∞) = 0. In fact, these can be represented
as relative cycles in (XG , R ∪ I).
Similarly, there is the subcomplex∂
0 → Z[P] ∼= ZΓ/G −−−−→ Z[R ∪ I] → 0,with Z[P] generated by the elements g(i∩ρ). It computes the homology H1 (XG \{cp}).
There is the following sequence:
∗
R
R∪I ∼
H1 (XG \ {cp}) = Hcp −−−−→ Hcp−−−−→ Hcp
= Z[P] −−−−→ H0 (R ∪ I).

j∗

j∗

∂

1 to 1

The relative homology
H1 (XG \ {cp}, R ∪ I) ∼= Z[P]
is generated by the relative cycles g(i ∩ ρ), with g ranging over representativesof P.

－ 33 －

There is a long exact sequence of relative homology (edited as)
0 → Hcp




−−−−→

R∪I ∼
Hcp
= Z[P](β ∼ ,β ∼ )

R
I
−−−
−−
→ H0 (R) ⊕ H0 (I) → Z → 0




j∗
∂
H1 (XG \ {cp}) −−−−→ H1 (XG \ {cp}, R ∪ I) −−−∗−→ H0 (R ∪ I) ∼= Z[PI ] ⊕ Z[PR ](with that Z ∼= H0 (XG \ {cp}) = H0 (XG ) at the right corner in the ﬁrst line,
because XG is path-connected, since so is H2 ).
cp
R∪I
The pairing between Hcp
and HR∪I
gives the identiﬁcation of Z[P] and
Z|P| , obtained by identifying the elements of P with the corresponding delta
functions.
It says that the identiﬁcation holds:
Z|P| ! (λgj G )gj G,1≤j≤k × (0)hG ⇔kλgj G χgj G ∈ Z[P],χgj G = gj G ∈ P,

j=1

where χgj G is the characteristic function on P = Γ/G at gj G, which may be
identiﬁed with the left regular representation (λΓ/G usually denoted so) of Γ/Gevaluated at gj G, which is also identiﬁed with the class gj G ∈ P.
The long exact sequence above can be rewritten as
0 → H cp −−−−→
⏐
⏐
∂∗ 

cp
Z|P| ∼= HR∪I(βR ,βI )−−−−−→ Z|PI | ⊕ Z|PR | → Z → 0



j∗

∂

H0 ({cp}) ←−−∗−− H1 (XG \ {R ∪ I}, cp) ←−−−−

H1 (XG \ {R ∪ I})

In order to understand more explicitly the map (βR , βI ), we give the quivalent algebraic formulation of the modular complex as follows (cf. [103], [121]).
R∪I ∼Again, the homology group Hcp= Z[P] is generated by the images in
2
XG = G\H of the geodesic segments gγ0 ≡ g(i ∩ ρ), with g ranging over achosen set of representatives of the coset space P = Γ/G.
cp ∼ |P|
May identify the dual basis δs of HR∪I
= Z with the images in XG of thepaths gη0 , where the path η0 is given by the geodesic arcs connecting i∞ to z0 ,z0 to τ z0 , and τ z0 to 0. It then holds that the intersection pairing (corrected)
[gγ0 ] • [hη0 ] = 1 if gG = hG,
cpR∪I• : Hcp × HR∪I → Z,
[gγ0 ] • [hη0 ] = 0 if gG = hG.
By exactness in the long exact sequence above, it holds that H cp ∼= ker(βR , βI ),which is also obtained by the identiﬁcation of the modular symbol [g(0), g(i∞)]G
with [gη0 ]. Therefore, the relations imposed on the generators δs by vanishingunder βI correspond to the relations δs ⊕δσs (or σs if s = σs), and the vanishingunder βR gives another set of relations δs ⊕ δτ s ⊕ δτ 2 s (or δs if s = τ s).

－ 34 －

Remark. (Added). May recall from [116] the following (modiﬁed). There is
the non-degenerate bilinear intersection form between homology groups:
Q : Hq (M, M1 ; Λ) × Hn−q (M, M2 ; Λ) → Λ,
where M is assumed to be an oriented connected n-dimensional manifold andΛ is a unital commutative ring, and Hq (M ) = 0 if q not in 0 ≤ q ≤ n. Thatintersection form is obtained by composing D and ∩∼ as follows. The PoincaréLefschetz (PL) duality theorem (or Poincaré duality theorem when theboundary ∂M = ∅) says that there is a duality isomorphism D between homology and relative cohomology groups
D : H q (M, ∂M ; Λ) ∼= Hn−q (M ; Λ),which implies that there is an isomorphism D between relative, homology andcohomology groups
D : H q (M, M1 ; Λ) ∼= Hn−q (M, M2 ; Λ),
where M1 and M2 are (n − 1)-dimensional manifolds such that ∂M = M1 ∪ M2
and M1 ∩ M2 = ∂M1 = ∂M2 . The homology group Hn (M, ∂M ) is isomorphic
to Z, and is generated by the fundamental homology class [M ] of M , which ismapped to [∂M ] under the boundary map ∂n : Hn (M, ∂M ) → Hn−1 (∂M ). The
PL duality D is deﬁned by D(ξ) = ξ ∩ [M ].
The cap (or intersection) product ∩ for a topological space X is deﬁned as∩ : H q (X; Λ) × Hp+q (X; Λ) → Hp (X; Λ),(σi ◦ ε) ⊗ v(σi ◦ ε )λi ∈ Sp (X) ⊗ Λv∩c=i(corrected) for v ∈ Hom(Sq (X), Λ), c = i σi ⊗ λi ∈ Sp+q (X) ⊗ Λ, where ε :Δp → Δp+q and ε : Δq → Δp+q are deﬁned by ε(t0 , · · · , tp ) = (t0 , · · · , tq , 0, · · · , 0)and ε (tp , · · · , tp+q ) = (0, · · · , 0, tp , · · · , tp+q ).
With p = n − q, the duality D is induced by∩ : H q (M, ∂M ; Λ) × Hn (M, ∂M ; Λ) → Hn−q (M, Λ).
In particular, if q = n and p = n − q = 0, then
∩ : H n (M, ∂M, Λ) × Hn (M, ∂M, Λ) → H0 (M, Λ) ∼= Λ,
so that H n (M, ∂M, Λ) ∼= Λ. As well, if p = 0, then we can deﬁne∩∼ : H q (M, Λ) × H n−q (M, ∂M ; Λ) → H0 (M, Λ) ∼=Λwith H n−q (M, ∂M, Λ) ∼= Hq (M, Λ) by the PL duality. Therefore, it is obtainedthat
Q = (∩∼ ) ◦ (D−1 , D−1 ).

－ 35 －

Also, if q = 1 and p = 0, then
∩ : H 1 (X; Λ) × H1 (X; Λ) → H0 (X; Λ) ∼=Λwhen X is connected, with H 1 (X, ∂X; Λ) ∼= H1 (X; Λ) by D if X is a 2dimensional manifold. The composed map ∩ ◦ (D−1 , id) in this case is justequal to • the intersection product. Remark. (Added as in [116]). The n-dimensional homotopy group πn (X),with some x ∈ X as the base point, is deﬁned to be the set of all homotopy
classes of continuous maps from (I n , ∂I n ) to (X, x), with I = [0, 1] and ∂I n theboundary of I n .
The n-dimensional relative homotopy group πn (X, A), with x ∈ A ⊂X, is deﬁned to be the set of all homotopy classes of continuous maps from(I n , ∂I n , J n−1 ) to (X, A, x), where J n−1 is the closure of ∂I n \ (I n−1 × {0}) in∂I n . If n ≥ 2, then it is a group, and if n ≥ 3, then it is abelian.
There is the Hurewicz homomorphism τ from the relative homotopy group
πn (X, A) to the homology group Hn (X, A; Z), deﬁned as τ ([f ]) = f∗ (εn ), where
εn is the generator of Hn (I n , ∂I n ).
If X is a path-connected space, then the fundamental (or Poincaré) group
π1 (X) of X has the quotient abelian group π1 (X)/[π1 (X), π1 (X)] isomorphic
to H1 (X, Z), by Hurewicz.
The Hurewicz isomorphism theorem says that
τ : πn (X, A) ∼= Hn (X, A) and Hi (X, A) = 0,i ≤ n − 1,for (X, A) (n − 1)-connected and n-simple, in the respective senses that
π0 (A) = π0 (X) = πi (X, A) = 01≤i≤n−1and that X and A are path-connected and the action of π1 (A) on πn (X, A) is
trivial. 

2.2

The noncommutative boundary of modular curves

The main idea that bridges between the algebro-geometric theory of modular
curves and noncommutative geometry consists of replacing P1 (Q) in the classicalcompactiﬁcation, which gives rise to a ﬁnite set of cusps, with P1 (R). This
substitution canon be done naively, since the quotient G\P1 (R) is ill behaved
as a topological space, as G does not act on P1 (R) discretely.
When we regard the quotient Γ\P1 (R), or more generally, Γ\(P1 (R) × P), asa noncommutative space, it becomes a geometric object that is rich enough torecover several aspects of the classical theory of modular curves. In particular,
it makes sense to study in terms of the geometry of such spaces the limiting
behavior for certain arithmetic invariants deﬁned on modular curves when τ →θ ∈R\Q
Moduli of noncommutative elliptic curves. The boundary Γ\P1 (R) of
the modular curve Γ\H2 , viewed as a noncommutative space, has a modular－ 36 －

interpretation, as observed originally Connes-Douglas-Schwarz [34]. In fact, we
can think of the quotient spaces of the circle S 1 by the actions of irrationalrotations 2πnθ for n ∈ Z by irrational numbers θ, as the noncommutative 2tori and as particular degenerations of the classical elliptic curves, which areinvisible to ordinary algebraic geometry. The quotient space Γ\P1 (R) classiﬁes
the noncommutative tori up to Morita equivalence ([22] and [140]) and completes
the moduli space Γ\H2 of the classical elliptic curves. Thus, from a conceptual
point of view, it is reasonable to think of Γ\P1 (R) as the boundary of Γ\H2 ,when we allow points as elliptic curves in the classical moduli space to havenon-classical degenerations to noncommutative tori.
Noncommutative tori are viewed in a sense as a prototype example of noncommutative spaces, so as to need the full range of techniques of noncommutative geometry (cf. [22] and [24]). Noncommutaive tori are irrational rotation
algebras as C ∗ -algebras. Recall some basic properties of noncommutative tori,
which justify the claim that the C ∗ -algebras behave like a noncommutaive version of elliptic curves. For this, may follow [22], [27], and [140].
Irrational rotation and Kronecker foliation
Deﬁnition 2.5. The (irrational) rotation C ∗ -algebra Aθ for a given θ ∈ R(irrational) is deﬁned to be the universal C ∗ -algebra C ∗ (u, v), generated by two
unitary operators u and v, subject to the commutation relation vu = e2πiθ uv(corrected as exchanged).
The C ∗ -algebra Aθ can be realized as a C ∗ -subalgebra of bounded operators on the Hilbert space H = L2 (S 1 ), with the circle S 1 identiﬁed with R/Z.
For a given θ ∈ R, consider two operators u and v, which act on a complete
orthonormal basis {en (= e2πint )}n∈Z (t ∈ [0, 1]) of H asuen = en+1 (= e2πit en ) and ven = e2πinθ en ,n ∈ Z.

Note thatuen , uem  = en+1 , em+1  = δn+1,m+1 = δn,m = en , em ,ven , vem  = e2πinθ en , e2πimθ em  = e2πi(n−m) δn,m = en , em .
It then follows that u∗ u = 1 and v ∗ v = 1. Also, for any ξ =j ξj ej ∈ H,

u∗ en , ξ = en , uξ = ξn−1 = en−1 , ξ,v ∗ en , ξ = en , vξ = e−2πinθ ξn = e−2πinθ en , ξ,and hence u∗ en = en−1 and v ∗ en = e−2πinθ en . Similarly, it follows that uu∗ = 1
and vv ∗ = 1. Moreover, for any ξ, η ∈ H, check that
e2πiθ uven , η = e2πiθ ue2πinθ en , η = e2πi(n+1)θ ηn+1 ,vuen , η = ven+1 , η = e2πi(n+1)θ ηn+1 .

－ 37 －

The irrational rotation C ∗ -algebra can be described in more geometric termsby the foliation on the usual commutative 2-torus T2 lines with an irrationalslope. On T2 = R2 /Z2 , consider the Kronecker foliation given by θdx = dy(corrected), for x, y ∈ R/Z.
Indeed, integrating the diﬀerential equation implies that Lc : y = θx + c
with c ∈ R/Z as constants. Therefore, T2 = #[c]∈(R/Z)/θZ Lc .
The space X of leaves Lc is described as X = R/(Z + θZ) ∼= S 1 /θZ. Thisquotient is ill behaved as a topological space, hence it can not be well describedby ordinary geometry.
A transversal to the foliation is given for instance by the choice T = T×{0},with T ∼= S1 ∼= R/Z. Then the noncommutative 2-torus T2θ is obtained ([22],[27]) asT2θ = {(fab )(or just fab ) | a, b ∈ T in the same leaf} ∼= Aθ
(in a suitable sense), where (fab ) (or just fab ) may be identiﬁed with a (ﬁnite)n∗power seriesn∈Z bn v (in the crossed product C -algebra below, under aninterpretation as given in [150] or below), where each bn ∈ C(S 1 ) the C ∗ -algebraof all complex-valued, continuous functions on S 1 .
The noncommutative 2-torus is just deﬁned as the foliation C ∗ -algebra
corresponding to the Kronecker foliation on the 2-torus T2 = (R/Z)2 . In fact, let
L be the leaf (orbit, or line with slope θ) passing the origin (0, 0), which is dense
in T2 . Consider the set of (weighted) ﬁnite paths in L, with vertexes in T ∩ L
and edges as line segments in L between vertexes. May assign each ﬁnite pathb
between a = e2πiθa and b = e2πiθb in L ∩ T to a ﬁnite sum fa b = j=a bj v j ,where each bj ∈ C(S 1 ) may be assumed to be a weight to an edge and each v j
may correspond to an vertex. Then for instance, may identify
fm,m+1 = (αm,n en + αm,n+1 en+1 )v m + (αm+1,n en + αm+1,n+1 en+1 )v m+1
αm,n+1αm,n=αm+1,n αm+1,n+1
but the matrix multiplication is not the same as the usual one, and does containthe convolution with the action αθ below involved. For example, en vem v =en αθ (em )v 2 .
The action of Z on C(S 1 ) is given by the automorphism αθ = θ (in short)by a unitary v such thatαθ (h) = vhv −1 = h ◦ rθ−1 ,h ∈ C(S 1 ),with rθ (x) = x + θ ∈ R (mod 1). The C ∗ -algebra C(S 1 ) is generated by thefunction u(t) = e2πit for t ∈ S 1 = R/Z. Then these generators u and v (not thesame as those u and v above, or exchanged) satisfy the relation vu = e−2πiθ uv.
Because vuv −1 = u ◦ rθ−1 = e2πi(t−θ) = e−2πiθ u.
It is then obtained through this description that there is an identiﬁcation ofAθ as well as T2θ with the crossed product C ∗ -algebra C(S 1 ) rθ Z, generated－ 38 －

by unitaries u and v, subject to the relation above. It represents the quotient
space S 1 /θZ as a noncommutative space.
Degenerations of elliptic curves. An elliptic curve Eτ over C can be described as the quotient Eτ = C/(Z + τ Z) of the complex plane C by a 2dimensional lattice Λ = Z + τ Z, where we can take Im(τ ) > 0. It is also
possible to describe the elliptic curve Eq for q ∈ C∗ with q = e2πiτ and|q| = e−2πIm(τ ) < 1, in terms of its Jacobi uniformization, namely as the quotient of C∗ by the action of the group generated by a single hyperbolic element
q of D (not of P SL2 (C), corrected), so that Eq = C∗ /q Z .
The fundamental domain for the action of q Z is an annulus{z ∈ C | |q| < |z| ≤ 1}of radii 1 and |q|. The identiﬁcation of the two boundary circles is obtained via
the combination of scaling and rotation given by multiplication by q.
Now let us consider a degeneration such that q goes to e2πiθ ∈ S 1 withθ ∈ R \ Q. We can say heuristically that in this degeneration the elliptic curve
becomes a noncommutative 2-torus as Eq ⇒ T2θ ∼= Aθ in the sense that theannulus as the fundamental domains shrinks to the circle S 1 , and left is thequotient S 1 /e2πiθZ of S 1 by Z ∼= e2πiθZ by the irrational rotation by e2πiθ .
Since the quotient space is ill behaved as a space, such degenerations do
not admit a description within the context of classical geometry. However,
we may replace the quotient by the corresponding crossed product C ∗ -algebraC(S 1 )θ Z, and can consider such C ∗ -algebras as noncommutative (degenerate)
elliptic curves.
More precisely, when we consider the degeneration of elliptic curves Eτ =C∗ /q Z for q = e2πiτ , what is obtained in the limit is the suspension Aθ ⊗C0 (R) ≡SAθ = Aθ S of Aθ . In fact, as the parameter q degenerates to e2πiθ ∈ S 1 , the nicequotient Eτ = C∗ /q Z degenerates to the bad quotient Eθ = C∗ /e2πiθZ , whose
noncommutative algebra of coordinates is Morita equivalent to the suspension
C ∗ -algebra of Aθ , where z ∈ C∗ has the polar decompositon as z = |z|e2πiarg(z) ,with |z| = eρ for ρ ∈ R as the radial coordinate.
The Connes’ Thom isomorphism [23] or the Bott periodicity in K-theory forC ∗ -algebras, and the Pimsner-Voiculescu six-term exact sequence [135] imply
that the K-theory groups ofAθ ⊗ C0 (R) ∼= C0 (S 1 × R) θ,id Z ∼=M C0 (R2 ) t,id Z θ,id Z(corrected) as the noncommutative space, identiﬁed with Eθ , are obtained as
K0 (Eθ ) ∼= K1 (Aθ ) ∼= Z2 ∼= Z[u] ⊕ Z[v],2
K1 (Eθ ) ∼= K0 (Aθ ) ∼=Z ∼= Z[1] ⊕ Z[θ],where id means the trivial action, t means the translation action of Z on R, sothat C0 (R)t Z ∼= C(S 1 )⊗K, where K is the C ∗ -algebra of all compact operators
on a Hilbert space, and ∼=M means the stably (or Morita) isomorphism, and [x]means the K-theory class corresponding to an element x of Aθ (in this case).

－ 39 －

This is again compatible with the identiﬁcation of Eθ = Aθ S (rather than Aθ )as degenerations of elliptic curves.
Moreover, the Hodge ﬁltration on the H 1 of an elliptic curve and the equivalence between the elliptic curve and its Jacobian have analogs for the noncommutative 2-torus Aθ in terms of the ﬁltration on HC ev (corrected) induced by
the pairing (not inclusion of) with K0 (cf. [24, p. 132-139 (up) and p. 348355 (down)] and [30, XIII] missing). By the Bott periodicity or the Connes’
Thom isomorphism, these appear again on the HC od = HC 1 in the case of thenoncommutative elliptic curve Eθ = SAθ .
The point of view as degenerations is sometimes a useful guideline. For
instance, we can study the limiting behavior of arithmetic invariants deﬁned on
the parameter space of elliptic curves as well as on modular curves, in the limit
when τ goes to θ ∈ R \ Q. An instance of this type of result is obtained astheory of limiting modular symbols of [112], which is reviewed in this chapter.
Remark. (Added). Let Aθ denote the dense (smooth) subalgebra of Aθ generated by u and v. As in [24, Theorem 53] (or [27, III, 2.β]), for any θ ∈ Q thatsatisﬁes a Diophantine condition,
H ev (Aθ ) ≡ lim Hλ2n (Aθ ) ∼= H 2 (Aθ ) ∼= HC 2 (Aθ ) ≡ Hλ2 (Aθ ) ∼= C[S(τ )] ⊕ C[ϕ],
−→
2n−1
od
3
1
1
∼
∼
∼
H (Aθ ) ≡ lim Hλ(Aθ ) = HC (Aθ ) = Hλ (Aθ ) = H (Aθ , A∗θ ) ∼= C[ϕ1 ] ⊕ C[ϕ2 ],−→where in general, for A a locally convex topological algebra, H ∗ (A) is deﬁned to
be HC ∗ (A)⊗HC ∗ (C) C as the inductive limit of the groups Hλn (A), with HC ∗ (C)identiﬁed with a polynomial ring C[σ] with the generator σ of degree two, with
HC 2n (C) = C and H 2n−1 (C) = 0, and there is the long exact sequence of theConnes’ cyclic cohomology HC = Hλ and the Hochschild cohomology H:
I=i∗

−−−
H 1 (Aθ , A∗θ ) ←−∼
=
⏐
⏐
B 0
S

Hλ1 (Aθ )Hλ0 (Aθ )−−−−−→

C[tr]−−−−−→ C[S(tr)] ⊕ C[ϕ]A◦(σ#)Hλ2 (Aθ )SA◦(σ∪)B←−−−−
0

H 2 (Aθ , A∗θ )



B=A◦B

I

0
−−−−−−→
H 2 (Aθ , A∗θ ) −−−−−−→Hλ1 (Aθ )0
inclusion∗
⏐

⏐

S ∼
=


I

−−−∗−→iC

Hλ3 (Aθ )where it holds that
HC 1 (Aθ ) ∼= H 1 (Aθ , A∗θ ) ∼= H 1 (Aθ , A∗θ )/im(I ◦ B)
and HC 0 (Aθ ) = C for any θ ∈ Q, and τ is the canonical trace on Aθ deﬁned as
sending τ (α1) = α and otherwise zero, and ϕj (x0 , x1 ) = τ (x0 δj (x1 )), andϕ(x0 , x1 , x2 ) =1τ (x0 (δ1 (x1 )δ2 (x2 ) − δ2 (x1 )δ1 (x2 )))2πifor xj ∈ Aθ , where δj are the basis derivations of Aθ .

－ 40 －

If θ ∈ Q does not satisfy any Diophantine condition, then H j (Aθ , A∗θ ) for
j = 1, 2 are inﬁnite dimensional. If θ ∈ Q, then H j (Aθ , A∗θ ) for 0 ≤ j ≤ 2 areinﬁnite dimensional. If θ ∈ Q, then H 0 (Aθ , A∗θ ) ∼= C.
On the other hand, H j (Aθ , A∗θ ) = 0 for j ≥ 3 and θ ∈ R. Hence,
I=i∗

H 2 (Aθ , A∗θ ) ←−−−− Hλ2 (Aθ )⏐⏐B 0
Hλ1 (Aθ )



C2

S

B=A◦B

I

0
−−−−−→ Hλ3 (Aθ ) −−−−−−→
H 3 (Aθ , A∗θ ) −−−−−−→Hλ2 (Aθ )0
inclusion∗A◦(σ#)
⏐


⏐


S ∼
=



S

−−−−−→A◦(σ∪)

C2

I−−−∗−→i

Hλ4 (Aθ )0

Note that it holds that
HC 2n (A) ∼= HC 0 (A)and HC 2n+1 (A) ∼=0

for any nuclear C ∗ -algebra A. If such a C ∗ -algebra A has a trace, then H ev (A) ∼=C and H od (A) ∼= 0. In particular, H ev (Aθ ) ∼= C and H od (Aθ ) ∼= 0.
As well, recall from [27, III, 1.β] that for a Hochschild cochain ϕ ∈ C n+1 (A, A∗ ),(B0 ϕ)(a0 , · · · , an ) = ϕ(1, a0 , · · · , an ) − (−1)n+1 ϕ(a0 , · · · , an , 1),with B0 ϕ ∈ C n (A, A∗ ), and A : C n (A, A∗ ) → C n (A, A∗ ) is the linear map
deﬁned as Aϕ = γ∈Γ ε(γ)ϕγ , where Γ is the group of cyclic permutations of
the set {0, 1, · · · , n}, and ϕγ (a0 , · · · , an ) = ϕ(aγ(0) , · · · , aγ(n) ).
The subspace Cλn (A) of C n (A, A∗ ) consists of the cyclic Hochschild cochains
ϕ such that ϕγ = ε(γ)ε for any γ ∈ Γ.
The subcomplex (Cλn (A), b) of the Hochschild complex (C n (A, A∗ ), b) is deﬁned to deﬁne the (Hochschild-Connes) cyclic cohomology groups Hλn (A) =HC n (A), where C n (A, A∗ ) is the space of linear maps from An to A∗ the dual
space of all linear functionals on A, and any element ϕ ∈ C n (A, A∗ ) can be
identiﬁed with a linear functional ϕ∼ : An+1 → C deﬁned byϕ∼ (a0 , a1 , · · · , an ) = ϕ(a1 , · · · , an )(a0 ).
As well, the Hochschild coboundary map b with bϕ = bϕ∼ ∈ C n+1 (A, A∗ ) isdeﬁned by(bϕ∼ )(a0 , · · · , an+1 ) = ϕ∼ (a0 a1 , a2 , · · · , an+1 )n(−1)j ϕ∼ (a0 , · · · , aj aj+1 , · · · , an+1 ) + (−1)n+1 ϕ∼ (an+1 a0 , · · · , an ).
+j=1

Moreover, the cup (or cocap) product# = ∪ : HC n (A) ⊗ HC m (B) → HC n+m (A ⊗ B),－ 41 －[ϕ] ⊗ [ψ] → [ϕ#ψ]is induced from deﬁning the cup product ϕ#ψ by (ϕ ⊗ ψ) ◦ π for ϕ ∈ C n (A, A∗ ),ψ ∈ C m (B, B∗ ), where π : Ω∗ (A ⊗ B) → Ω∗ (A) ⊗ Ω∗ (B) is the natural homomorhiphism by the universal property of the universal graded diﬀerential algebraΩ∗ (·), where Ωn (A) = Ω1 (A) ⊗A · · · ⊗A Ω1 (A), with Ω1 (A) = (A ⊕ C1) ⊗C A
the A-bimodule, with the derivation d : A → Ω1 (A) deﬁned by da = 1 ⊗ a fora ∈ A. The generator for HC 2 (C) is given by the 2-cocycle σ(1, 1, 1) = 1 for
1 ∈ C. Morita equivalence between NC 2-tori. To extend the modular interpretation of the quotient Γ\H2 as moduli of elliptic curves to the noncommutative
boundary Γ\P1 (R), it is necessary to check that points in the same orbit under
the action of the modular group P SL2 (Z) by fractional linear transformations
on P1 (R) deﬁne Morita (or stably) equivalent noncommutative 2-tori.
It is shown by Connes [22] (cf. also [140]) that the noncommutative 2-toriAθ and A− θ1 are Morita equivalent. Geometrically, in terms of the Kronecker
foliation, the Morita equivalence of T2θ and T2− 1 corresponds to changing theθchoice of the transversal from T = T × {0} to T  = {0} × T in T2 = (R/Z)2 , orto from θdx = dy to −θ−1 dx = dy.
In fact, all Morita equivalence among noncommutative 2-tori arise in thatway. Then Aθ and Aθ are Morita equivalent if and only if θ is equivalent to θunder the action of P SL2 (Z).
Constructed explicitly by Connes [22] the bimodules Mθ,θ realizing theMorita equivalence between T2θ and T2θ , withaθ + ba bθ = gθ =, g=∈ SL2 (Z) = Γ,c dcθ + ddeﬁned to be the Schwartz space S(R×Z/c) (?), with the right and left actionsof Aθ and Aθ respectively, deﬁned bycθ + d, n − 1),cnd(f v)(x, n) = e2πi(x− c ) f (x, n), and1(uf )(x, n) = f (x − , n − a),cx2πi( cθ+d−uc ) f (x, n).
(vf )(x, n) = e(f u)(x, n) = f (x −In particular, S(θ) = − θ1 , with S = 1 % −1 (transposed-diagonal sum).
Remark. (Added as in [140]). Note that Aθ ∼= Aθ as a C ∗ -algebra if and only
if θ = θ or 1 − θ (mod) 1. Also, Aθ is strongly Morita (sM) equivalent to Aθ if
and only if θ is equivalent to θ under the action of GL2 (Z), which is generated
by S  = 1 % 1 and T . In particular, S  (θ) = θ1 . Hence Aθ ∼=M A θ1 .
As a note, Aθ is stably equivalent to both of A− θ1 and A θ1 since SL2 (Z) ⊂GL2 (Z). There are inﬁnitely many sM equivalent NC tori.
By deﬁnition, two C ∗ -algebras A and B are strongly Morita equivalent if
there is an A-B equivalence (or imprimitivity) bimodule. The sM equivalent,－ 42 －

unital C ∗ -algebras are stably isomorphic, and as well each is a (full) corner pMn pof the matrix algebra Mn over the other, of a suitable size n and a projection p(not contained in any two-sided ideal). 
Other properties as NC elliptic curves. There are other ways by which theirrational rotation algebra behaves like an NC elliptic curve, most notably as in
the relation between elliptic curves and their Jacobians and some aspects of thetheory of theta functions, which recalled brieﬂy (cf. [24] and [30] missing).
The (commutative) 2-torus T2 = S 1 ×S 1 is connected, so that the C ∗ -algebraC(T2 ) does not have non-trivial projections. On the contrary, the noncommutative 2-tori Aθ contain non-trivial (Rieﬀel) projections. Indeed, it is shownby Rieﬀel [140] that for a given irrational number θ and any α ∈ (Z + Zθ) in[0, 1], there exists a projection pα ∈ Aθ such that the trace τ (pα ) = α. Also, adiﬀerent construction of projections of Aθ is given by Boca [12], with arithmetic
relevance, inasmuch as those projections that correspond to the theta functions
for noncommutative 2-tori, deﬁned by Manin [108].
The method of constructing projections of C ∗ -algebras is based on the following two steps (cf. [140] and [110] missing):(I) Suppose given a A-B bimodule M. If an element ξ ∈ M admits an
1√
2
invertible ∗-invariant square roof ξ, ξB(= ξ ∗ ξ = |ξ|), then the element ν =−1ξξ, ξB 2 (= ξ|ξ|−1 ) satisﬁes νν, νB (= ξ|ξ|−1 |ξ|−1 ξ ∗ ξ|ξ|−1 ) = ν.
(II) Let ν ∈ M be a non-trivial element such that νν, νB = ν. Then theelement p = A ν, ν(= ξ|ξ|−1 |ξ|−1 ξ ∗ ) is a projection if p = p∗ = p2 .
In the Boca construction, the elements ξ are obtained from Gaussian elements in some Heisenberg modules, in such a way that the corresponding ξ, ξB
is a quantum theta function in the sense of Manin [110](missing). An introduction to the relation between the Heisenberg groups and the theory of theta
functions is given in the third volume of the Mumford Tata Lecturres on theta[125].
Remark. (Added). As in [140], the (Rieﬀel) projection pα constructed has
the form hv ∗ + f + gv for some suitably chosen h, f, g ∈ C(T), so that f = f ∗ ,h = v ∗ g ∗ , and that (1) g(t)g(t − θ) = 0, (2) g(t)(1 − f (t) − f (t − θ)) = 0,
and (3) f (t)(1 − f (t)) = |g(t)|2 + |g(t + θ)|2 , for t ∈ R, if pα is an idempotent.
The condition (1) says that g(t) = 0 or g(t − α) = 0, (2) if g(t) = 0, thenf (t) + f (t − θ) = 1, (3) if f (t) = 0, 1, then g(t) = 0 or g(t + α) = 0. Indeed,
may assume that θ ∈ [0, 12 ]. For any 0 < ε < θ, with θ + ε < 12 , deﬁne⎧1
if t ∈ [0, ε],
⎪
εt
⎪
⎪
⎨1
if t ∈ [ε, θ]withf (t) =
1
⎪
1
−
(t
−
θ)
if
t
∈
[θ,
θ
+
ε],
⎪
ε
⎪
⎩
0
if t ∈ [θ + ε, 1],

f (t)(1 − f (t)), if t ∈ [θ, θ + ε],
and g(t) =0otherwise. － 43 －

1

f (t)dt = θ,0Remark. As well, recall from [141] the following. Assume that A and B arefull hereditary C ∗ -subalgebras of a C ∗ -algebra C. Deﬁne an A-B bimodule
M as the closed linear space ACB. Deﬁne an A-B equivalence (or imprimitivity) bimodule to be such M, equipped with an A-valued inner productand a B-valued inner product on M as a left A-module and a right B-module
respectively, deﬁned (for instance as)A x, y(= xy∗) and x, yB (= x∗ y),x, y ∈ M

such that (1) positivity, (2) symmetry, and (3) linearity hold:(1) 0 ≤ A x, x(= xx∗ ) ∈ A+ and 0 ≤ x, xB (= x∗ x) ∈ B+ ,(2) A x, y∗ = A y, x and x, y∗B = y, xB , x, y ∈ M,
(3) A ax, y = aA x, y, x, y ∈ M, a ∈ A andx, ybB = x, yB b,

x ∈ M,

x, y ∈ M, b ∈ B,

and moreover, (4) compatibility, (5) boundedness of representations of A and
B on M, and (6) density hold;(4)

A x, yz = xy, zB ,x, y, z ∈ M,(5) ax, axB ≤ a x, xB
2

xb, xbA ≤ b x, xA2(6)

in B+ ,a ∈ A, x ∈ M,

in A+ ,b ∈ B, x ∈ M,

A M, M = A and M, MB = B,andas a C ∗ -algebra.

As well, deﬁne a norm on M byx(= xC ) = A x, x(= A xx∗ ) = x, xB (= x∗ xB ).

2.3

Limiting modular symbols

Consider the action of the group Γ = P GL2 (Z) on the upper and lower halfplanes H± in the complex plane C and the modular curves deﬁned to be the±quotient spaces XG= G\H± , for G a ﬁnite index subgroup of Γ. Then thenoncommutative compactiﬁcation of the modular curves is obtained by extending the action of Γ on H± to the action on the full P1 (C) = H± ∪ P1 (R), sothat we deﬁne±
XG= G\P1 (C) = Γ\(P1 (C) × P),

with P = Γ/G.

Due to the fact that Γ does not act discretely on P1 (R), the quotient only makes
sense as a noncommutative space as the crossed product C(P1 (C)×P)Γ. Notethat P1 (R) in P1 (C) is the limit set of the group Γ, in the sense that it is the setof accumulation points of orbits of elements of Γ on P1 (C). Let us see anotherinstance of noncommutative geometry arising from the action of a group ofMöbius transformations of P1 (C) on its limit set, in the context of the geometryat the archimedean primes of arithmetic varieties.

－ 44 －

Generalized Gauss shift and dynamics. The crossed product C ∗ -algebraC(P1 (C) × P)  Γ is described as the noncommutative boundary of modular
curves. It is also able to describe the quotient space Γ\(P1 (R) × P) in thefollowing equivariant way. If Γ = P GL2 (Z), then Γ-orbits in P1 (R) are thesame as equivalence classes of points of [0, 1] under the equivalence relation
deﬁned as that for x, y ∈ [0, 1], x ∼T y if there exist n, m ∈ Z (positive) suchthat T n x = T m y, where the classical Gauss shift T of the continued fraction
expansion is deﬁned as
T x = x−1 − x−1  ∈ [0, 1],x ∈ (0, 1]

and T 0 = 0(or 1).

Namely, the equivalence relation is that of having the same tail of the continued
fraction expansion as shift-tail equivalence.
Note that x = (x−1 )−1 = (x−1  + T x)−1 for 0 < x ≤ 1.
As a simple generalization of that classical result above,
Lemma 2.6. The P GL2 (Z) = Γ-orbits in P1 (R) × P are the same as equivalence classes of points in [0, 1] × P under the equivalence relation deﬁned asthat for (x, s), (y, t), (x, s) ∼T (y, t) if there exist n, m ∈ Z (positive) such that
T n (x, s) = T m (y, t), where the shift T (by the same symbol) on [0, 1] × P generalizing the classical shift T of the continued fraction expansion is deﬁned by1−x−1  1
−1T (x, s) = (T x, T s) = ( − x ,
s).
1
0
x
Note that−x−1 
1





1
−x−1  1
a b
s≡
/G
0
1
0
c d


−x−1 a + c −x−1 b + d
=
/G.
a
b

The quotient by the equivalence relation by T (extended) is described as anoncommutative space by the C ∗ -algebra of the groupoid of the equivalence
relationG([0, 1] × P, T ) = {((x, s), m − n, (y, t)) | T m (x, s) = T n (y, t)}with objects G 0 = {((x, s), 0, (x, s)}.
In fact, for any T -invariant subset E of [0, 1] × P, we can consider theequivalence relation by T . The corresponding groupoid C ∗ -algebra denoted
as C ∗ (G(E, T )) encodes the dynamical properties of the map T on E.
Geometrically, the equivalence relation on [0, 1] × P is related to the actionof the geodesic ﬂow on the horocycle foliation on the modular curves.
Arithmetic of modular curves and noncommutative boundeary. The
result as Lemma 2.6 shows that the properties of the dynamical system in [0, 1]×
P by T can be used to describe the geometry of the noncommutative boundary－ 45 －

of modular curves. There are various types of results that can be obtained bythis method ([112], [113]), which are discussed soon in the rest of this subsection.
(1) Using the properties of this dynamical system, it is possible to recoverand enrich the theory of modular curves on XG , by extending the notion of modular symbols from geodesics connecting cusps to images (in XG ) of geodesics in
H2 connecting irrational points on the boundarry P1 (R). In fact, the irrational
points of P1 (R) deﬁne limiting modular symbols. In the case of quadratic irrationalities, these can be expressed in terms of the classical modular symbols
and recover the generators of the homology of the classical compactiﬁcation XG
by cusps. In the remaining cases, the limiting modular symbol vanishes almost
everywhere.
(2) It is possible to reinterpret Dirichlet series related to modular forms
of weight two in terms of integrals on [0, 1] of certain intersection numbers
obtained from homology classes deﬁned in terms of the dynamical system. In
fact, even when the limiting modular symbol vanishes, it is possible to associatea non-trivial (co)homology class in H1 (XG ) (corrected) to irrational points on
the boundary, in such a way that an average of the corresponding intersection
numbers give Mellin transforms of modular forms of weight two on XG .
(3) The Selberg zeta function of the modular curve can be expressed asa Fredholm determinant of the Perron-Frobenius operator associated to the
dynamical system on the boundary.
(4) Using the formulation of the boundary as the noncommutative spaceas the crossed product C ∗ -algebra, we can obtain a canonical identiﬁcation of
the modular complex with a sequence of K-theory groups of the C ∗ -algebra.
The resulting exact sequence of K-groups can be interpreted, using the orbit
description of the quotient space, in terms of the Baum-Connes assembly map
and the Connes’ Thom isomorphism.
All these show that the noncommutative space as C(P1 (R)×P)Γ considered
as a boundary stratum of C(H2 × P)  Γ (corrected) contains a part of thearithmetic information on the classical modular curve. The fact that informationon the bulk space is stored in its boundary at inﬁnity can be seen as an instanceof the physical principle of holography as bulk and boundary correspondence instring theory (cf. [111]). Discussed is the holography principle in more details
in relation to the geometry of the archimedean ﬁbers of arithmetic varieties.
Limiting modular symbols. Let γβ denote an inﬁnite geodesic in the hyperbolic half plane H with one end at i∞ and the other end at β ∈ R \ Q. Letx ∈ γβ be a ﬁxed base point, and y(τ ) be the point along γβ at a distance τas the geodesic arc length, from x towards the end β. Let [x, y(τ )]G denote thehomology class in H1 (XG ) (corrected) determined by the image of the geodesic
arc x ∩ y(τ ) in H.
Deﬁnition 2.7. The limiting modular symbol is deﬁned to be
1[x, y(τ )]G ∈ H1 (XG , R),y(τ )→β τ[[∗, β]]G ≡ lim
whenever such limit exists.

－ 46 －

The limit is independent of the choice of the initial point x as well as of the
choice of the geodesic in H ending at β, as discussed in [112]. The (modiﬁed)
notation is as introduced in [112], where ∗ as the ﬁrst variable indicates theindependence of the choice of the initial point x, the double brackets indicate
the fact that the homology class is computed as a limiting cycle.
Dynamics of continued fractions. The classical (Gauss) shift map T of thecontinued fraction is deﬁned by T (x) = x1 −  x1  ∈ [0, 1] for x ∈ [0, 1] with x = 0
and T (0) = 0 (or 1). The generalized shift on [0, 1] × P is deﬁned by1−x−1  1
−1T (x, s) = (T x, T s) = ( − x ,
s).
1
0
x
Recall the following basic notation regarding continued fraction expansion. Let k1 , · · · , kn be independent variables for n ≥ 1, and let[k1 , · · · , kn ] ≡1
pn (k1 , · · · , kn )≡,1
qn (k1 , · · · , kn )k1 + k +···
12kn

where pn and qn are polynomials with integral coeﬃcients, which can be calculated inductively from the relations
qn+1 (k1 , · · · , kn , kn+1 ) = kn+1 qn (k1 , · · · , kn ) + qn−1 (k1 , · · · , kn−1 ),pn (k1 , · · · , kn ) = qn−1 (k2 , · · · , kn ),with p0 = q−1 = 0 and q0 = 1.
For instance, [k1 ] = k11 , so that q1 = k1 = k1 q0 + q−1 and p1 = 1 = q0 .
Also, ]k1 , k2 ] = k2 kk12+1 , so that q2 = k2 q1 + q0 and p2 = k2 = q1 (k2 ).
It is obtained that
pn−1 (k1 , · · · , kn−1 ) pn (k1 , · · · , kn )[k1 , · · · , kn−1 , kn + xn ] =xn
qn−1 (k1 , · · · , kn−1 ) qn (k1 , · · · , kn )as the standard matrix notation for fractional linear transformation.
For instance,1
p0 x1 + p1
0 1
[k1 + x1 ] =
x1 .
=
=
1 k1
x1 + k1
q0 x1 + q1
As well,x2 + k2
p1 x2 + p2=
k1 x2 + k2 k1 + 1
q1 x2 + q2




0
1
k2
0 1
=
x2 =
1
1 k1
k1 k1 k2 + 1[k1 , k2 + x2 ] =


1
x2 .
k2

If α ∈ (0, 1) is an irrational number, then there is a unique sequence ofintegers kn (α) ≥ 1 such that α = limn→∞ [k1 (α), · · · , kn (α)].

－ 47 －

Indeed,1
α= 1 =α1= 1 α1  + T (α)
α +

1
1
T (α)−1 +T (T (α))

1= [ , T (α)−1 , T 2 (α)−1 , · · · ].
αMoreover, there is a unique sequence xn (α) ∈ (0, 1) such thatα = [k1 (α), · · · , kn−1 (α), kn (α) + xn (α)]
for each n ≥ 1. It is obtained (by induction) that





0
1
0
1
0
1
···
x (α).
α=1 k2 (α)
1 kn (α) n1 k1 (α)
Set[k1 (α), · · · , kn (α)] =pn (k1 (α), · · · , kn (α))
pn (α)≡.
qn (k1 (α), · · · , kn (α))
qn (α)

Also set
α = [k1 (α), · · · , kn−1 (α), kn (α) + xn (α)]

pn−1 (k1 (α), · · · , kn−1 (α)) pn (k1 (α), · · · , kn (α))=
xn (α) ≡ gn (α)xn (α)
qn−1 (k1 (α), · · · , kn−1 (α)) qn (k1 (α), · · · , kn (α))
with gn (α) ∈ GL2 (Z) with det gn (α) = (−1)n .
The Gauss shift T is then given by1
T (α) = T ([k1 (α), k2 (α), · · · ]) = T ([ , T (α)−1 , T 2 (α)−1 , · · · ])α
= [T (α)−1 , T 2 (α)−1 , · · · ] = [k2 (α), k3 (α), · · · ]in terms of the continued fraction expansion.
The properties of the generalized shift on [0, 1] × P can be used to extend
the notion of modular symbols to geodesics with irrational ends ([112]). Suchgeodesics correspond to inﬁnite geodesics on the modular curve XG which
exhibit a variety of interesting possible behaviors, from closed geodesics togeodesics that approximate some limiting cycle, to geodesics that wind arounddiﬀerent homology class exhibiting a typically chaotic behavior.
Lyapunov spectrum. A measure of how chaotic a dynamical system is, or betterof how fast nearby orbits tend to diverge, is given by the Lypunov exponent.
Deﬁnition 2.8. The Lyapunov exponent of a (diﬀerentiable) map T from[0, 1] to [0, 1] is deﬁned asλ(β) = lim

1

n→∞ n

log |(T n ) (β)| = lim= · · · = lim

1

n→∞ n

1

n→∞ n

log |T  (T n−1 (β))||(T n−1 ) (β)|

klog Πn−1
k=0 |T (T (β))|,－ 48 －β ∈ [0, 1].

The function λ(β) is T -invariant.
Indeed,1
1
log |(T n ) (T β)| = log Πnk=1 |T  (T k (β))|
n
n
1
n+1 1
=
log Πnk=0 |T  (T k (β))| − log |T  (β)|
n n+1n
→ λ(T β) = λ(β) (n → ∞).
Moreover, in the case of the classical continued fraction shift T on [0, 1], theLyapunov exponent is given by
λ(β) = 2 lim

1

n→∞ n

log qn (β),

with qn (β) the successive denominators of the continued fraction expansion forβ.
Note thatq2 (β) = 1 + k1 (β)k2 (β) = q0 (β) + q1 (β)k2 (β),
q3 (β) = k1 (β) + (1 + k1 (β)k2 (β))k3 (β) = q1 (β) + q2 (β)k3 (β),
qn (β) = qn−2 (β) + qn−1 (β)kn (β) (n ≥ 2).

andNote that if 12 < x ≤ 1, then 1 ≤ x1 < 2. Hence, T (x) = x1 − 1. Moreover,if< x ≤ n1 , then n ≤ x1 < n + 1. Thus, T (x) = x1 − n. Therefore, T is
discontinuous at n1 for n ≥ 2 and at 0, but continuous and even diﬀerentiable
otherwise, with T  (x) = − x12 .
1
n+1

In particular, it is shown as the Khintchine-Lévy theorem that for almost
all β with respect to the Lebesgue measure on [0, 1], the above limit is equal toλ0 ≡ (6 log 2)−1 π 2 . However, there is an exceptional set in [0, 1] of Hausdorﬀ
dimension dimH = 1 but with Lebesgue measure mL = 0, for elements of which,the limit deﬁning the Lyapunov exponent does not exist.
As seen later below, in some cases, the value λ(β) can be computed fromthe spectrum of the Perron-Frobenius operator of the shift T .
The Lyapunov spectrum is introduced (cf. [137]) by decomposing the unit
interval into level sets of the Lyapunov exponent λ(β). Let
Lc = {β ∈ [0, 1] | λ(β) = c ∈ R}.
These (level) sets provide a T -invariant decomposition of the unit interval as[0, 1] = (#c∈R Lc ) # {β ∈ [0, 1] | λ(β) does not exist}.
These level sets are uncountable dense T -invariant subsets of [0, 1], of varying Hausdorﬀ dimension ([137]). The Lyapunov spectrum measures how the
Hausdorﬀ dimension varies, as a function h(c) = dimH Lc .

－ 49 －

Limiting modular symbols and iterated shifts. Deﬁne a function ϕ : P = Γ/G →H cp = H1 (XG , {cp}) by ϕ(s) = [g(0), g(i∞)]G , where g ∈ Γ = P GL2 (Z) (orP SL2 (Z)) is a representative of the coset s = gG ∈ P.
Then the limiting modular symbol [[∗, β]]G as a limit is computed in thefollowing way:
Theorem 2.9. For all β ∈ Lc the level set of the Lyapunov exponent λ(β) fora ﬁxed c ∈ R, the limiting modular symbol is computed as1 ϕ ◦ T k (t0 ),n→∞ cnn[[∗, β]]G = lim

k=1

where T = T (β, t) = (T (β), T (β, t)) and t0 ∈ P.
Without loss of generality, we may consider the geodesic γβ in H × P with
one end at (i∞, t0 ) and the other at (β, t0 ), for ϕ(t0 ) = [0, i∞]G .
The argument given in [112, 2.3] is based on the fact that we may replace
the homology class deﬁned by the vertical geodesic with the class obtained by(β)
connecting the successive rational approximations pqnn(β)
to β in the continued
fraction expansion. Namely, we may replace the path x0 ∩ yn with the union ofarcs
p0
pk−1 pk
pn−1(x0 ∩ y0 ) ∪ (y0 ∩ ) ∪ · · · ∪ (∩ ) ∪ ··· ∪ (∩ yn )q0
qk−1 qk
qn−1(corrected) representing the same homology class in H1 (XG , Z), where yn ∈ H2
n−1
is the intersection of γβ and the geodesic with ends at pqn−1
and pqnn .
For instance, pq00 = 0 and pq11 = k11 = (β −1 )−1 .
The result above then follows by estimating the geodesic distance, so thatτ ∼ − log Im(y) + O(1)(y(τ ) → β)and (2qn qn+1 )−1 < Im(yn ) < (2qn qn−1 )−1 .
n−1Note that yn ∈ pqn−1∩ pqnn with Im(yn+1 ) < Im(yn ), and


1
pn+1
1  pnpn+1 
pn
and
= 
−
<β<
for any n.
2qn qn+1
2 qn
qn+1 
qn+1
qn
By taking logarithm,
log(2qn qn−1 ) < − log(Im(yn )) < log(2qn qn+1 ).
The inverse matrix gk−1 (β) with gk (β) ∈ GL2 (Z) as
−1


qk
pk−1 pk= (−1)kgk−1 (β) =qk−1 qk
−qk−1

−pk
pk−1



acts on points (β, t) ∈ [0, 1] × P as the k-th power of the shift map T , so that(−1)k+1 pk (β) (−1)k+1 qk (β)ϕ(T k t0 ) = [gk−1 (β)(0), gk−1 (β)(i∞)]G =,pk−1 (β)
qk−1 (β)
G－ 50 －

(corrected).
Ruelle and Perron-Frobenius operators. A general principle in the theoryof dynamical systems is that it is often possible to study the dynamical properties of such a map T (like ergodicity) via the spectral theory of an associatedoperator. This allows us to employ techniques of functional analysis and to
derive conclusions on dynamics.
In the case of the (generalized) shift map T on [0, 1] × P, with P = Γ/G,
associated to the map is the operator deﬁned as


 
∞

1
1
0 1
(Lσ f )(x, t) =f,t
(x + k)2σ
x+k 1 kk=1

depending on a complex parameter σ.
More generally, the Ruelle transfer operator of a (diﬀerentiable) map T isdeﬁned as(Lσ f )(x, t) =exp(h(y, s))f (y, s),(y,s)∈T −1 (x,t)

where we take h(x, t) = −2σ log |T  (x, t)|.
Check that for any (or positive) integer k and x ∈ [0, 1) with [x + k] = k,


  


 
1
0 1
−k 1
0 1
T
,
t = x + k − [x + k],t = (x, t),
1 0
1 k
x+k 1 k111==,exp(−2σ log |T  (y, s)|) = 
12σ
2σ|T (y, s)|(x + k)4σ| − y2 |1with y = x+k(so that we may remove 2 in 2σ in exp(· · · ), or replace σ with σ2
in either exp(· · · ) or Lσ , and so on).
Clearly, that operator Lσ is well suited for capturing the dynamical properties of the map T as it is deﬁned as a weighted sum over the inverse image ofeach (x, t) by T .
On the other hand, there is another operator that can be associated to a
dynamical system and which typically has better spectral properties, but is lessclearly related to the dynamics. The best circumstances arise when these two(operators) agree (or coincide) for a particular value of the parameter σ. The
other operator is called the Perron-Frobenius operator P F and is deﬁned by
the relationf (g ◦ T )dμ =(P F (f ))gdμ,[0,1]×P[0,1]×P

the integration with respect to the Lebesgue measure μ. In the case of the shift
T on [0, 1] × P we in fact have that P F = Lσ with σ = 1.
Spectral theory of the PF operator P F = L1 . In the case of the modular groupG = Γ, the spectral theory of the Perron-Frobenius operator of the Gauss shift
is studied by D. Mayer [119] (unchecked). More recently, Chang and Mayer [18](unchecked) extend the results to the case of congruence subgroups. A similar
approach is used in [112] to study the properties of the shift on [0, 1] × P.

－ 51 －

Theorem 2.10. Deﬁned on the Banach space of holomorphic functions on D×Pcontinuous up to the boundary, with D = {z ∈ Z | |z − 1| < 32 } (analyticallyextended from [0, 1] × P), under the condition as irreducibility that




0 1
∞
n
P = ∪n=0 Πj=1
(t0 ) | k1 , · · · , kn ≥ 1 ,1 kj(for some base point t0 ∈ P), the Perron-Frobenius operator P F = L1 (thegeneralized Gauss-Kuzmin operator) deﬁned as


 
∞

1
1
0 1
(L1 f )(x, s) =f,s(x + k)2
x+k 1 kk=1

for the shift on [0, 1] × P has the following properties:(1) L1 is a nuclear operator, of trace class.
(2) L1 has top eigenvalue 1, which is simple.
1(3) The eigenfunction corresponding to 1 is 1+x
up to normalization.
(4) The spectrum of L1 except 1 is contained in the ball with radius < 1.
(5) There is a complete set of eigenfunctions for L1 .
The irreducibility condition for P above is satisﬁed by congruence subgroups.
1
The function 1+x
is called Gauss density on [0, 1].
For other T -invariant subsets E ⊂ [0, 1] × P, may also consider the operators
LE,σ and P FE as the restrictions of Lσ and P F to E respectively. When the set
E has the property that P FE = LE,hE with hE = dimH E the Hausdorﬀ dimension, may use the spectral theory of the operator P FE to study the dynamical
properties of T .
In particular, the Lyapunov exponent can be derived from the spectrum of
the family of operators LE,σ as follows. Let λσ denote the top eigenvalue ofLE,σ .
Lemma 2.11. The Lyapunov exponent is given asλ(β) =dλσ
dσ

at σ = hE = dimH E, where μH a.e. in E.
The Gauss problem. Now let us deﬁne a function for n ∈ N on [0, 1] as
mn (x) = μ{α ∈ (0, 1) | xn (α) ≤ x}
valued as measure of the set, with α = [k1 (α), · · · , kn−1 (α), kn (α) + xn (α)] withxn (α) ∈ (0, 1).
The asymptotic behavior of the measures mn is known to be a famous problem on the distribution of continued fractions, formulated by Gauss. It is conjectured by Gauss thatm(x) ≡ lim mn (x) =n→∞1log(1 + x).
log 2－ 52 －

The convergence as so above is only proved by R. Kuzmin in 1928. Other
proofs are then given by P. Lévy (1929), K. Babenko (1978), and D. Mayer(1991). In the arguments used by Babenko and Mayer, used is the spectral
theory on the Perron-Frobenius operator. Of these diﬀerent arguments, only
the latter extends to the case of the generalized Gauss shift on [0, 1] × P.
2
It is evident that mn (1) = 1 and mn (0) = 0. Hence m(1) = 1 = log
log 2 and1m(0) = log
log 2 . Note also thatlog(1 + x)
=
log 2



1
0

1
dt
1+t

−1 

x
0


1
dx .
1+t

The Gauss problem can be formulated in terms of a recursive relation asmn+1 (x) = (L1 mn )(x) ≡

∞

k=1

1
m
(x + k)2 n



1
x+k


,

where the right hand side is the image of mn under the Gauss-Kuzmin (or PF)
operator L1 (in the case of the group Γ = P GL2 (Z)).
Let us deﬁne functions mn on [0, 1] × P, valued as measure of the set:mn (x, t) ≡ μ{(y, s) ∈ [0, 1] × P | xn (y) ≤ x, gn (y)−1 (s) = t}.
Or as μ{α ∈ (0, 1) | xn (α) ≤ x, gn (α)−1 (t0 ) = t} (cf. [112]).
As a consequence of the theorem above, it is obtained that
Theorem 2.12. It follows that mn (x, t) = Ln1 (1)(x, t), and the following limit
exists and equals tom(x, t) ≡ lim mn (x, t) =n→∞1
log(1 + x).
|P| log 2

Note again that mn = L1 mn−1 = · · · = Ln1 m0 at x as well as (x, t), with
m0 = 1, where x0 (α) = 0, so that m0 (x) = x. Hence m0 (x) = 1. Indeed,mn+1 (x, t) = (L1 mn )(x, t) ≡

∞

k=1

1
m
(x + k)2 n




1
0
,
x+k 1



1
(t0 ) .
kThis shows that there exists a unique T -invariant measure on [0, 1] × P. Thisis uniform in the discrete set P with the counting measure, and it is the Gauss
measure of the shift of the continued fraction expansion on [0, 1].
Two theorems on limiting modular symbols. The result of the T -invariant
measure allows to study the general behavior of limiting modular symbols.
A special role is played by limiting modular symbols [[∗, β]]G , where β is a√quadratic irrationality in R \ Q (such as p for p a prime).

－ 53 －

Theorem 2.13. Let g ∈ G be hyperbolic, with eigenvalue 0 < Λg < 1 corresponding to the attracting ﬁxed point αg− . Let Λ(g) ≡ | log Λg | (geodesic distance)and let l be the period of the continued fraction expansion of β = αg− . Then[0, g(0)]G
1  −1[gk (β)0, gk−1 (β)i∞]G .
=
Λ(g)
λ(β)l
l

[[∗, β]]G =k=1(Added). Recall from [112] (or [113]) that any hyperbolic g ∈ G has two ﬁxed
points α± , repelling and attracting, on R. Let Λ±
g be the respective eigenvalues,−
with 0 < Λ+
<
1
<
Λ
.
The
orientedgeodesicin H connecting αg− to αg+ isgg
g-invariant, and the action of g induces on it the shift by the geodesic distancelog Λ−
g . And then · · ·This shows that, in this case, the limiting modular symbols are linear combinations of classical modular symbols, with coeﬃcients in the ﬁeld generated
over Q by the Lyapunov exponents λ(β) of the quadratic irrationalities.
In terms of geodesics on modular curve, this is the case, where the (winding)geodesic has a limiting cycle, given by the closed geodesic [0, g(0)]G (no ﬁgure).
On the other hand, there is the generic case, where, contrary to the previous
example, the geodesics wind around many diﬀerent cycles in such a way thatthe resulting homology class averages out to zero over long distances, as limiting
modular symbols, as chaotic tangling and untangling (no ﬁgure).
Theorem 2.14. For a T -invariant E ⊂ [0, 1] × P, under the irreducibility
condition for E, the function Rτ (β, s) ≡ τ1 [x, y(τ )]G converges weakly to zero.
Namely, for all integrable f ∈ L1 (E, dμH ),
lim

τ →∞

Rτ (β, s)f (β, s)dμH (β, s) = 0.
EThis weak convergence can be improved to strong convergence (without f ) μH (E)almost everywhere. Thus, the limiting modular symbol satisﬁes that [[∗, β]]G ≡ 0a.e. on E.
This result depends upon the properties of the PF operator L1 and theresult on the T -invariant measure. In fact, to get the result on the weak convergence ([112]), notice that the limit computing limiting modular symbols asn1klimn→∞ cn
k=1 ϕ ◦ T (t0 ) can be evaluated in terms of a limit of iterates of
the PF operator by
1 n→∞ λ0 n
n

k=1

1 
n→∞ λ0 nn

f (ϕ ◦ T k ) = lim

lim[0.1]×P

k=1(Lk1 f )ϕ,[0.1]×P

where λ(β) = λ0 a.e. in [0, 1].
By the convergence of Lk1 1 to the density function h of the T -invariant mea-

－ 54 －


sure, so that Lk1 f converges to ( f dμ)h, it yields that[[∗, β]]G f (β, t)dμ(β, t)[0,1]×P







f dμ

=
[0,1]×P

ϕhdμ





=

[0,1]×P

f dμ[0,1]×P

1 
ϕ(s).
2|P|
s∈P

It is then checked that the sums∈P ϕ(s) = 0 since each term in the sumchanges sign under the action of the inversion σ ∈ P GL2 (Z) with σ 2 = id, but
the sum is globally invariant under σ.
The argument above can be extended to the case of other T -invariant subsets
E of [0, 1] × P, for which the corresponding PF operator LE.hE has analogous
properties (cf. [113]). The weak convergence, improved to strong convergence,
can be obtained by applying the strong law of large numbers to the randomvariables (or measurable, real-valued functions) ϕk = ϕ ◦ T k (cf. [113] for moredetails). The result plays the role of an ergodic theorem for the shift T on E,
eﬀectively.
(Added). Recall from [116] that that the strong law of large numbers holds
for random variables ϕj (on [0, 1]) means that there is a sequence {an } suchnthat n1 j=1 (ϕj − aj ) converges to zero almost surely, in the sense that1(ϕj (x) − aj ) = 0} = 1.
n→∞ n
j=1
n

μ{x ∈ [0, 1] | lim
In particular, it holds when
∞

1

j2
j=1

E[(ϕj − E[ϕj ])2 ] < ∞,with mean (or expectation) E[ϕj ] =


2.4

1
0

ϕj (x)dx and variance E[(ϕj − E[ϕj ])2 ].

Hecke eigenforms

An important question may be that what happens to modular forms at thenoncommutative boundary of the modular curves. There is a variety of phenomena in the theory of modular forms that hint to the fact that a suitableclass of modular forms survives on the noncommutative boundary. Introducedby Zagier is the term as quantum modular forms, to denote this but not yet
suﬃciently understood class of examples. Some aspects of such modular forms,
pushed to the noncommutative boundary are analyzed by [112], in the formof certain averages involving modular symbols and Dirichlet series related tomodular forms of weight two. Recall the main results in this case, as in thefollowing.
Now consider the case of congruence subgroups G = Γ0 (p) for p a prime.

－ 55 －

Let ω be a holomorphic diﬀerential (of the ﬁrst kind) on the modular curve
1
XG = G\H, with G = Γ0 (p). Let Φω = ϕ∗ (ω) dz(a cusp form) deﬁned as the∗pullback ϕ (ω) = ω ◦ ϕ under the projection ϕ : H → XG .
Let Ψ be an eigenfunction for all the Hecke operatorsHn =d−1  ndd|n b=00bdwith (p, n) = 1, such that Hn Ψ = cn Ψ for some cn . Then the L-function of ω
is given by
∞
∞

cn
(2π)s
Lω (s) = −Ψ(iy)y s−1 dy =2πiΓ(s) 0
ns
n=1(where Ψ is identiﬁed with Φω ?). Such a series on the right hand is said to bea Dirichlet series, and if cn is assumed to be a Dirichlet character, then it issaid to be the L-function assocaited to the character (cf. [20, 3.3]).
(Added). Recall from [20, 3.3] that the Gamma function is deﬁned to be∞Γ(s) =

ts e−t0dt,tRe(s) > 0.

dt
dt
0 t)
Note that d(tt0 t = t for any t0 > 0, which says that t is a Haar measure on∗
the multiplicative group R+ = {t ∈ R | t > 0}. For any n > 0, compute that∞0ts e−nt

dt
=
t

∞
0

x
n

s

e−x

dx
Γ(s)
= s ,
x
n



(x = nt).

There are known many interesting arithmetic properties of the integrals ofsuch Hecke eigenforms on modular symbols (Manin [103] uncheked).
In particular, there is the following relation between Lω and modular symbols(cf. [112, 2.2]): (σ(n) − cn )Lω (1)or i∞ ∗=
ω,b(σ(n) − cn ) 0 ϕ (ω) (corrected)
d|n b mod d [0, d ]G
(b = 1, · · · , d), where σ(n) =
q:(q,p)=1

!



1
q 2+t

d|n d, from which it is obtained thatq  ≤q,(q,q  )=1(p)Lω (2 + t)
ζ (p) (1 + t)
− (p)
ω
=
(p) (2 + t)qζζ
(2 + t)2[0, q ]G

(p)

"

i∞

ϕ∗ ω,

0

where Lω is the Mellin transform of Φω with omitted Euler p-factor, and ζ (p)is the corresponding Riemann zeta. Namely,L(p)ω (s) =

n:(n,p)=1cnns

and ζ (p) (s) =－ 56 －

n:(n,p)=11.
nsAlso,

[x,y]Gω=yxω ◦ ϕ. Note that∞Lω (1) = i

Ψ(iy)dy =
0

∞

cn
n=1

n

.

Therefore, it seems to be interesting to study the properties of integrals on
limiting modular symbols (cf. [112]) and their extensions. For this purpose,a suitable class of functions on the boundary P1 (R) × P (or just [0, 1] × P) is
introduced in [112]. Such functions have the forml(f, β) =∞f (gk (β), qk−1 (β)),

k=1where f is a complex-valued (Lévy) function deﬁned on pairs of coprime integers(q, q  ) with q ≥ q  ≥ 1 and with f (q, q  ) = O(q −ε ) (q → ∞) for some ε > 0,
and qk (β) are the successive denominators of the continued fraction expansion(β)
of β ∈ (0, 1], so that pqkk (β)→ β (k → ∞).
The reason for that choice of functions is just the following classical result
by Lévy (1929):
Proposition 2.15. For such a function f of Lévy as above,1l(f, α)dα =0q≥q  ≥1,(q,q  )=1

f (q, q  ),q(q + q  )where both the integral and the sum converge absolutely and uniformly.
May interpret the summing over pairs of successive denominators as a property that replaces modularity, at when pushed to the boundary.
Through this class of Lévy functions, it is possible to recast certain averages
related to modular symbols on XG , completely in terms of function theory on
the boundary space Γ\(P1 (R) × P). A typical result of this sort is given as thefollowing ([112]):
Proposition 2.16. (Edited). Upon choosingf (q, q  ) =q + q
q 1+t



ω,

[0, qq ]G

it is obtained that
1

!

(p)

Lω (2 + t)
ζ (p) (1 + t)
− (p)l(f, β)dμ(β) = (p)
ζ
(2
+
t)
ζ
(2 + t)2
0

"

(p)

i∞

ϕ∗ ω,

0

which expresses (or contains) the Mellin transform Lω of a Hecke eigenformΨ (or Φω ) in terms of a boundary average.

－ 57 －

Analogous results exist for diﬀerent choices of f (q, q  ), for Eisenstein series
twisted by modular symbols, double logarithms at roots of unity, and so on.
In fact that result also shows that, even when the limiting modular symbol
vanishes, non-tirivial homology classes still can be associated to the geodesics
with irrational ends.
In fact, in the case of G = Γ0 (p), for f the function chosen as in the proposition above, let us consider∞
qn+1 (β) + qn (β)
qn (β)
C(f, β) ≡0,,
qn+1 (β)1+t
qn+1 (β) G
n=1which deﬁnes a homology class in H1 (XG , {cp}, R) for Re(t) > 0 and for almost
all β, such that the integral average l(f, β) = C(f,β) ω on [0, 1] recovers Mellintransforms of cusp forms by


1

1

ω dμ(β) =0!

l(f, β)dμ(β)
0C(f,β)(p)Lω (2 + t)
ζ (p) (1 + t)−
= (p)
ζ (2 + t) ζ (p) (2 + t)2

"

i∞

ϕ∗ ω.

0

The series involved are absolutely convergent by an estimate:n
qn+1 (β) + qn (β)
qn (β)−(5+2t)nλ(β)
0,∼eϕ ◦ T k (s).
qn+1 (β)1+t
qn+1 (β) G
k=1

2.5

Selberg zeta function

Inﬁnite geodesics on XG are the images of inﬁnite geodesics on H2 × P withends in P1 (R) × P. Then they may be coded by (w− , w+ , s) with end points(w± , s) ∈ P1 (R) × P with w− ∈ (−∞, −1] and w+ ∈ [0, 1].
If w± are not cusp points, then consider inﬁnite continued fraction expansions of these end points:
w+ = [k0 , k1 , · · · , kr , · · · ]and w− = [k−1 , k−2 , · · · , k−n , · · · ],so that the corresponding geodesics are coded by (w, s) with s ∈ P, where w isa doubly inﬁnite sequencew = (· · · k−n · · · k−1 k0 k1 · · · kr · · · ) ≡ (w− )t w+ .
The equivalence relation in passing to the quotient by the group action is
implemented by the invertible double sided shift:

 

1
1
1
− w1+  1
+
−
,
s ,
−  + , −T (w , w , s) =10w+
w
w ±  w1+ 
where both the sign ± may be allowed.

－ 58 －

In particular, the closed geodesics in XG correspond to the case where theend points w± are the attractive and repelling ﬁxed points of a hyperbolic
element in the group.
The Selberg zeta function may be a suitable generating function for theclosed geodesics in XG , and is given by−(s+m)l(γ)
ZG (s) = Π∞),m=0 Πγ (1 − ewhere γ runs over primitive closed geodesics of length l(γ).
That can be expressed in terms of the generalized Gauss shift as follows:
Theorem 2.17. ([18], [112]). For G ⊂ P SL2 (Z) of ﬁnite index, the Selberg
zeta function is computed as the Fredholm determinant of the Ruelle transfer
operator:
ZG (s) = det(1 − Ls ).
There is an analogous result that it holds for ﬁnite index subgroups of
SL2 (Z), with det(1 − L2s ) involved. Also, there are generalizations to otherT -invariant subsets E, where det(1 − LE,s ) corresponds to a suitable Selberg
type zeta function, that only counts certain classes of closed geodesics.
(Added). Recall from [112] that the Selberg zeta function is deﬁned as
det(g)m
∞ZG (s) = Πm=0 Πg det 1 −ρP (g) ,N (g)s+m
where g runs over a set of representatives of GL2 (Z)-conjugacy classes of primitive hyperbolic elements of GL2 (Z), and P = GL2 (Z)/G, and ρP is the naturalrepresentation of GL2 (Z) on the space of functions on P, and for g ∈ GL2 (Z),N (g) =

2
tr(g) + d(g)2and d(g) = tr(g)2 − 4 det(g),and where an element g ∈ GL2 (Z) is said to be hyperbolic if both tr(g) and d(g)are positive, and such a hyperbolic matrix is primitive if it is not a non-trivial
power of an element of GL2 (Z). And then · · · . (Added). May also recall from [116] the following. Let Γ be a Fuchsiangroup of ﬁrst kind in SL2 (R). An element γ ∈ Γ is said to be hyperbolic if it
has two diﬀerent real eigenvalues a and a−1 with |a| > 1. Set N (γ) = a2 as thenorm of γ. If γ is hyperbolic, then so is γ n for any n ∈ N. A hyperbolic elementof Γ is said to be primitive if it is not a power of other hyperbolic elements. Le{γj } be a set of representatives of conjugacy classes of a primitive hyperbolic
element. For any character χ of Γ, the Selberg zeta function is deﬁned to be
an analytic function as an inﬁnite productχ(γj )
ZΓ (s, χ) = Πj Π∞
1
−
.
n=0
N (γj )s+n

－ 59 －

If any non-trivial element of Γ is hyperbolic, then H2 /Γ is compact andμ(H2 /Γ) = 4π(g − 1), with g the genus of H2 /Γ. In this case, the Selberg zetafunction is analytically extended to C, with genus at most 2, and with zero ats = −n of order (2n + 1)(2g − 2) for n ∈ Z non-negative, and with other zeros on
the line (or ﬁber) Re(s) = 12 , except possible ﬁnite points in (0, 1). Moreover,· · · . (May further read). .
Remark. ([116]). For real s > 1, the following inﬁnite (Riemann) sum and theinﬁnite (Euler) product are equal and converge absolutely:∞1ζ(s) ≡

nsn=1= Πp: prime

1.
1 − p1s

In particular, for m positive integers,
ζ(2m) = 22m−1 π 2m

B2m
,
2m!

For s ∈ C, the Riemann zeta function ζ(s) is a holomorphic function for Re(s) >1, and is analytically extended to C, with only pole at s = 1 of order 1. It follows
from the integral expression as

∞
∞

1−s
2
s
1
s
1
1
−1
−1
−n
πx
ξ(s) ≡ s Γ( )ζ(s) = +dx.
(x 2 + x 2
)
e
+
s 1−s
π2 2
0
n=1
The π-gamma-zeta function satisﬁes the functional equation as ξ(s) = ξ(1 − s).
The Laurent expansion at s = 1 for the zeta function is given by∞ζ(s) =1
Ck (s − 1)k ,
+γ+
s−1
k=1

where γ is the Euler constant and Ck are called the generalized Euler constants.
By the Euler product, the zeta function has no zeros for Re(s) > 1, and as
well Re(s) = 1. For Re(s) ≤ 0, the trivial zeros are given by −2m of order 1
for m positive integers. It is conjectured as the Riemann hypothesis that thenon-trivial zeros on the critical strip 0 < Re(s) < 1 are all on the critical linethat Re(s) = 12 .
For instance,
∞

1

B2
π2
1
= 2π 2 , B2 = ,
6
2!
6
n=1
∞

π
4  (−1)n−11.
ξ(2) = Γ(1)ζ(2) = =π
66 n=1 2n − 1

ζ(2) =n2=

Is there the uniqueness theorem for the zeta function?
Remark. ([116]). The Bernoulli polynomials are deﬁned by
∞

text
1
=
Bn (x)tn .
et − 1 n=0 n!

－ 60 －



Then it holds thatBn (x) =

n  

n
k=0

k

Bk (0)xn−k .

The Bernoulli numbers Bn are deﬁned to be |Bn (0)|, where B2n+1 = 0 forn ≥ 1. The Bernoulli polynomials satisfy the following properties:
Bn (x + 1) − Bn (x) = nxn−1anddBn (x) = nBn−1 (x).
dx

For instance, consider the equation at x = 0

∞
 1
t
n
t = (e − 1)
Bn (0)t
n!
n=0



t2
1
t3
= t + + + ···B0 (0) + B1 (0)t + B2 (0)t2 + · · · .
2! 3!
2!
Because of convergence and uniqueness, comparing coeﬃcients determines thatB0 (0) = 1, and 0 = 12 B0 (0) + B1 (0), so that B1 (0) = − 12 , and that 0 =
1
1
1
1

6 B0 (0) + 2 B2 (0) + 2 B1 (0), so that B2 (0) = 6 . Moreover, · · · .
Remark. ([116]). The gamma function is deﬁned to be the inﬁnite product:
x

1
1
1
Γ(x) = Π∞
1
+
,
x n=1
n
1 + nxand is also deﬁned to be the Euler integral of the second kind:∞Γ(x) =

e−t tx−1 dt,

x > 0.

0

The gamma function is determined uniquely as the solution for the diﬀerenced2
equation Γ(x + 1) = xΓ(x), with Γ(1) = 1 and dx
2 log Γ(x) ≥ 0 for x > 0. Thereis also the inﬁnite (Weierstrass) product expression:
1
x 1= xeγx Π∞
x ,n=1 1 +
Γ(x)
n en
n

where γ = limn→∞ ( k=1 k1 − log n) the Euler constant.
For s ∈ C, Γ(s) has poles of order 1 at s = −n for integers n ≥ 0.
For instance,∞

1 = Γ(1) = Π∞ 1 =e−t dt




1
1
1 N
1
1
1
1
γ ∞
N
n
e− n
=
= lim
= e Πn=1 1 +Πn=1 e Πn=1 1 +N →∞ NΓ(1)n e n1n0

1
N +1with N1 ΠN
n=1 (1 + n ) = N → 1 as N → ∞. Moreover, the other conditions forthe uniqueness of the gamma function may be checked.
－ 61 －

2.6

The modular complex and K-theory of C ∗ -algebras

The boundary P1 (R) × P with the action of Γ may be interpreted as the crossed
product C ∗ -algebra C(P1 (R) × P)  Γ as a noncommutative space. It is thenpossible to interpret some of the arithmetic properties of modular curves interms of such operator algebras. For instance, the modular symbols determine
certain elements of the K-theory of that C ∗ -algebra, and as well, the modular
complex and the exact sequence of relative homology can be identiﬁed with thePimsner six-term exact sequence for the K-theory of that C ∗ -algebra.
Now let Γ = P SL2 (Z) and G be a ﬁnite index subgroup of Γ. Let X =P1 (R) × P and Γσ = σ = Z2 and Γτ = τ  = Z3 .
The group Γ acts on the tree, with the set of edges, identiﬁed with Γ, and theset of vertices, identiﬁed with Γ/Γσ ∪ Γ/Γτ . This tree is embedded into H2 withvertices as elliptic points Γi and Γρ and edges as geodesics segments Γ(i ∩ ρ)as the dual graph of the triangulation (or compactiﬁcation) of the modular
complex.
For such a group acting on a tree, there is the Pimsner [134] (or Natsume(after Lance)) six-term exact sequence of the K-theory groups of the corresponding crossed product C ∗ -algebra as
K0 (A)

⏐
⏐

α

α∼

β∼

β

−−−−→ K0 (A  Γσ ) ⊕ K0 (A  Γτ ) −−−−→ K0 (A  Γ)⏐⏐

K1 (A  Γ) ←−−−− K1 (A  Γσ ) ⊕ K1 (A  Γτ ) ←−−−−

K1 (A)

with A = C(X) (cf. [130]).
It follows form a direct inspection of the maps in the exact diagram abovethat it contains a subsequence canonically identiﬁed with the algebraic presentation of the modular complex, compatible with the covering maps between
modular curves for diﬀerent congruence subgroups. Namely,
Theorem 2.18. (Edited). The algebraic presentation of the modular complex:
Z|P|
⏐
⏐∼
=

(βI ,βR )Z|PI | ⊕ Z|PR |⏐⏐∼=

0 → Hcp
⏐
⏐∼
=

−−−−→

0 → Hcp(βI ,βR )−−−−→ Z[P] ∼= Z[I ∪ R] −−−−−→ Z[PI ] ⊕ Z[PR ] −−−−→ Z → 0
⏐
⏐

⏐∼
⏐∼

=
=


−−−−−→∼−−−−→ Z → 0




∼

∗
∼ cp
H1 (XG \ {cp}) −−−−→ HI∪R
cp (= HR∪I ) −−−−→ H0 (I) ⊕ H0 (R) −−−−→ H0 (XG )j∗∂with PI = σ\P and PR = τ \P, is canonically isomorphic to the following
exact sequence:β∼β

0 → ker(β) → K1 (A) −−−−→ K1 (A  Γσ ) ⊕ K1 (A  Γτ ) −−−−→ im(β ∼ ) → 0
with A = C(X) = C(P1 (R) × P).

－ 62 －

A more geometric explanation for that exact sequence is given as in thefollowing. The connected Lie group P SL2 (R) can be identiﬁed with the circle (orunit sphere) bundle over the hyperbolic plane H2 = P SL2 (R)/P SO(2). Thus,
1
1
the unit tangent bundle T XG
= ∪p∈XG Tp XG
to the modular curve XG = G\H2
is given by the quotient G\P SL2 (R), where G is a ﬁnite index subgroup of the1
modular group P SL2 (Z), and Tp XG
means the unit sphere in the tangent space
Tp XG .
On the other hand, we may identify P1 (R) with B\P SL2 (R) the quotientby the group B of upper triangular 2 × 2 matrices in P SL2 (R). Thus, there isa strong Morita equivalence between the C ∗ -algebra crossed products:
C(P1 (R))  G




∼
=

−−−−→
sM

1
C(T XG
)B


‘

∼
=

C(B\P SL2 (R))  G −−−−→ C(P SL2 (R)/G)  B.
sM

The C ∗ -algebra crossed (or semi-direct) product by B is decomposed into thesuccessive crossed products by two actions by R. Therefore, the Connes’ Thom
isomorphism ([23]) implies the K-theory group isomorphism:
1
1
K∗ (C(T XG)  B) ∼)),= K∗ (C(T XG∗ = 0, 1.

1
As well, the topological K-theory group K ∗ (T XG) is related to K ∗ (XG ) by the
Gysin exact sequence.
Recall from [158] that SL2 (R) is homeomorphic to the direct productSO(2) × R2 as a topological space (Cartan-Mal’cev-Iwasawa decomposition),with SO(2) ≈ S 1 . Note that P SL2 (R) ∼= P SO(2) × R2 , with P SO(2) ∼=1
1∼
P (R) = S , andP SL2 (R)/P SO(2) ∼= R2 ∼=B∼= R  R∗+ ∼= H2 ,
as a topological space, where B ∼= R  R∗+ as a Lie group, which is a connectedsolvable Lie group deﬁned as the semi-direct product:


 t

a
e
t
t
∗
(a, e )(or (a, t)) ≡∈ B | a ∈ R, t ∈ R(or e ∈ R+ ) .
0 e−t
Note as well that
P SO(2) ∼= P1 (R) ∼
= S1
⏐
⏐


−−−−→

G\P1 (R)⏐⏐1
P SL2 (R) ∼= G\P SL2 (R)= P SO(2) × R2 −−−−→ T XG
⏐
⏐
⏐
⏐



P SL2 (R)/P SO(2) ≈ H2 −−−−→－ 63 －XG = G\H2 .

Remark. (Added). Recall from [142] the following. Let G be a locally compact(second countable) topological group and H and K be two closed subgroups.
Assume that K acts on G/H by left translation and H acts on K\G by righttranslation. Then the C ∗ -algebra crossed products are strongly Morita equivalent as∼=

C0 (G/H)  K −−−−→ C0 (K\G)  H(∼= C0 (G/K)  H)),sM

where C0 (X) is the C ∗ -algebra of all continuous C-valued functions on X vanishing at inﬁnity. 
Remark. (Added). Recall from [81] that for a ﬁber space X over a base space
B with F a ﬁber which has the same homology theory of the p-sphere S p withp ≥ 1, so that Hq (F, Z) = Z if q = 0, p and = 0 otherwise, then there are Gysin
long exact sequences in homology and cohomology theories asπ· · · → Hn (B, G) → Hn−p−1 (B, S) → Hn−1 (X, G) −−−∗−→ Hn−1 (B, G) → · · · ,π∗· · · ← H n (B, G) ← H n−p−1 (B, S) ← H n−1 (X, G) ←−−−− H n−1 (B, G) ← · · · ,where π : X → B is the projection, and S = S0 ⊗ G with S0 the Serre local
system for the p-dimensional homology in Z for the ﬁber space X → B, where
if S0 is simple, then S0 ∼= Z and S ∼= G. 
Of crusical importance in noncommutative geometry is a construction of
the classifying space BG for proper actions by Baum-Connes [9], the homotopy
quotient X ×G EG, and the assembly map:
μ : K ∗ (X ×G EG) → K∗ (C(X)  G),∗ = 0, 1

relating the topological K-theory of spaces and the analytic K-theory of C ∗ algebras. In a sense, for noncommutative spaces as C ∗ -algebras that are obtained as quotients by bad equivalence relations, the homotopy quotient isviewed as a commutative shadow, from which much crucial information on the
topology can be read (cf. [9], [25]).
In the case of G (or Γ) of XG , we have EG = H2 and the map μ is an
isomorphism, so that the Baum-Connes conjecture holds. By retracting EG =H2 to the tree of P SL2 (Z) and applying the Mayer-Vietoris sequence to vertices
and edges, obtained is the Pimsner (or Natsume) six-term exact sequence in Ktheory (for crossed product C ∗ -algebras by Γ and subgroups).

2.7

Chaotic cosmology as Intermezzo

As a momentary digression, a topic from general relativity turns out to be closely
related to the noncommutative compactiﬁcation of the modular curve XG for
G the congruence subgroup Γ0 (2) ([112]).
An important problem in cosmology seems to be understanding how anisotropy
in the early universe aﬀects the long time evolution of the space time. This problem is relevant to the study of the beginning of galaxy formation and in relating－ 64 －

the anisotropy of the background radiation to the appearance of the universe
today.
May follow [8] (unchecked) for a brief summary of anisotropic and chaotic
cosmology (cf. [7]). The simplest signiﬁcant cosmological model that presents
strong anisotropic properties is given by the Kasner metric:
ds2 = −dt2 + t2p1 dx2 + t2p2 dy 2 + t2p3 dz 2= (−1 ⊕ t2p1 ⊕ t2p2 ⊕ t2p3 )(dt, dx, dy, dz)t , (dt, dx, dy, dz)t ,∂, ∂ ) equal to thewith a Riemannian metric on R4 g = (gij )3i,j=0 = ( ∂xi ∂xj
4 × 4 diagonal matrix above (at (t, x, y, z) ∈ R4 ), where the exponents pj areconstants satisfying

p1 + p2 + p3 = 1 = p21 + p22 + p23 .
Note that (g11 , g22 , g33 ) = (t2p1 , t2p2 , t2p3 ) = (e2α1 , e2α2 , e2α3 ) (cf. [19]).
d log g3Note that for pj = d log gjj , the ﬁrst constraint
j=1 pj = 1 is just thecondition that log gij = 2αδij + βij (?) for a traceless β = (βij ) (which may
be removed, and the equations may hold by deﬁnition in that case), while the3second constraint j=1 p2j = 1 amounts to the condition that, in the Einsteinequations written in terms of α and βij ,


2 
 2
dβij
dα
8π
1
00
T +
,
=
dt
3
16π
dt




d
dβij
1
e−3α
e3α
= 8π Tij − δij Tkk ,
dt
dt
3
dβ

1
where the term T 00 is negligible with respect to the term 16π( dtij )2 , which is theeﬀective energy density of the anisotropic motion of empty space, contributing
together with a matter term to the Hubble constant.
log g
In fact, if gjj = t2pj for 1 ≤ j ≤ 3, then we have pj = 2 logjjt . The ﬁrst
constraint implies that log(g11 g22 g33 ) = 2 log t, so that or as well g11 g22 g33 = t2 .
3
The second constraint implies that j=1 (log gjj )2 = 4(log t)2 . If t2pj = e2αj ,then αj = pj log t.
Around 1970, a cosmological model as mixmaster universe is introduced byBelinskii, Khalatnikov, and Lifshitz (BKL) ([10] not checked) (cf. [19]), where
the exponents pj of the Kasner metric are allowed to depend on a parameter u(> 1), such that

p1 =
Then
3

j=1

p2j =

−u
,
1 + u + u2

p2 =

1+u
,
1 + u + u2

p3 =u(1 + u).
1 + u + u2

3
j=1 pj = 1 and as well,u2 + (1 + u)2 + u2 (1 + u)2
1 + 2u + 3u2 + 2u3 + u4
=
= 1!
2
2
(1 + u + u )1 + 2u + 3u2 + 2u3 + u4

－ 65 －

Since for ﬁxed u, the model is given by a Kasner space time, the behaviorof this universe can be approximated by a Kasner metric for certain large intervals of time. In fact, the evolution is divided into Kasner eras and each era
into cycles. During each era, the mixmaster universe goes through a volume
compression. Instead of resulting in a collapse, as with the Kasner metric, highnegative curvature resulting in a bounce as transition to a new era, which starts
again with a behavior approximated by a Kasner metric, but with a diﬀerent
value of the parameter u. Within each era, most of the volume compressionis due to the scale factors along one the space axes, while the other scale factors alternate between phases of contraction and expansion. These alternatingphases separate cycles within each era.
More precisely, we may consider metrics generalizing the Kasner metric,
which still admit SO(3) symmetry on the space like hypersurfaces and present
a singularity at t → 0. In terms of logarithmic timedΩ = −dt
dt
dt= − p1 +p2 +p3 = − ,abctt

so that Ω = − log t and Ω → +∞ as t → +0, the mixmaster universe model of
BKL admits a a discretization with the following properties.
It may holds that ((log a)2 ) = (b2 − c2 )2 − a4 (cf. [7]).
Remark. (Added). ([112]). The mixmaster universe is deﬁned as the spaceof solutions of the vacuum Einstein equations admitting SO(3) symmetry of thespace like hypersurfaces, whose metric acquires a singularity as t → +0, givenasds2 = dt2 − a(t)2 dx2 − b(t)2 dy 2 − c(t)2 dz 2
with a(t), b(t), c(t) as scale factors. A family of such metrics satisfying Einsteinequations is given by Kasner solutions, so that a(t) = tp1 , b(t) = tp2 , c(t) = tp3
3
3
such that j=1 pj = 1 = j=1 p2j . It is discovered that most of the trajectories
in the mixmaster universe exhibit a chaotic behavior as t → +0 backwards intime to the Big Bang.
(1). The time evolution is divided into Kasner eras [Ωn , Ωn+1 ] for n ∈ Z.
At the beginning of each era, we have a corresponding discrete value of theparameter un > 1 such as the parameter u of pj above.
(2). Each era, where the parameter u decreases with the (time) Ω grows, canbe subdivided into cycles corresponding to the discrete steps such as un − k fork positive 
integers. A change by −1 corresponds, after acting as a permutation
1 2 3(12)(3) =on the space coordinate, to changing sign of u, and hence
2 1 3
replacing contraction with expansion and vice versa. Within each cycle, thespace time metric is approximated by the Kasner metric with the exponents pj
with a ﬁxed value as u = un − k > 1.
(3). An era ends when, after a number of such cycles, the parameter un falls
in the range 0 < un < 1. Then the bouncing is given by the transition of u to u1 ,which gives rise to start a new series of cycles with new Kasner parameters and－ 66 －

a permutation (1)(23) =

1 2 3
1 3 2


of the space axis, in order to have again

p1 < p2 < p3 (with u > 1).
The transition formula relating the values un and un+1 of two successive
Kasner eras [Ωn , Ωn+1 ] and [Ωn+1 , Ωn+2 ] is given by un+1 = (un − un )−1 ,which is exactly the shift of the continued fraction expansion as11= un − un .
xn+1 == T xn =un+1un
The observation done previously becomes the key to a geometric descriptionof solutions of the mixmaster universe in terms of geodesics on a modular curve(cf. [112])
Theorem 2.19. Each inﬁnite geodesic on the modular curve XΓ0 (2) not endingat cusps determines a mixmaster universe.
In fact, an inﬁnite geodesic on XΓ0 (2) is the image of an inﬁnite geodesic on
H2 × P with ends in P1 (R) × P, under the quotient mapπΓ : H2 × P → Γ\(H2 × P) ∼= XG ,
where Γ = P GL2 (Z), G = Γ0 (2), and P = Γ/G ∼= P1 (F2 ) = {0, 1, ∞}. We consider the elements of P1 (F2 ) as labels assigned to the three space axes, according
to the identiﬁcation
0 = [0 : 1] → z,∞ = [1 : 0] → y,

1 = [1 : 1] → x.

As seen, geodesics can be coded in terms of the data (w− , w+ , s) with theaction of the shift T , where (w± , s) ∈ P1 (R) × P, with −∞ < w− ≤ −1 and
0 ≤ w+ ≤ 1. The data (w, s) with w = (w− )t w+ determines a mixmasteruniverse, with kn = un  =  x1n  in the Kasner eras, and with the transtion
between subsequent Kasner eras given by xn+1 = T xn ∈ [0, 1] and by the
permutation of axes induced by the transformation (corrected)
kn z + 1
kn 1≡ An acting on P1 (F2 ),with An z =(mod 2).
1 0
z


0 1 ∞
It is veriﬁed that this acts as the permutation
if kn is even (so
∞ 1 0


0 1 ∞
kn = 0 mod 2), andif kn is odd (so kn = 1 mod 2), that is, the
∞ 0 1




x y z
x y z
permutation
if kn is even, and the product= (12)(3) ◦
x z y
z x y
(1)(23) of the permutations if kn is odd. This is precisely what is obtained in
the mixmaster universe model by the repeated series of cycles with a Kasnerera followed by the transition to the next era.
Data (w, s) and T m (w, s) for m ∈ Z determine the same solution up to adiﬀrent choice of the initial time.

－ 67 －

There is an additional time symmetry in this model of the evolution of
mixmaster universes (cf. [8]). In fact, there is an additional parameter δn in thesystem, which measures the initial amplitude of each cycle. It is shown in [8](1+un )
that this is governed by the evolution of the parameter vn = δn+1
which
1−δn+1is subject to the transformation across cycles vn+1 = un  + v1n . By setting
yn = v1n , we obtain yn+1 = (yn +  x1n )−1 , and hence it follows that we caninterpret the evolution determined by the data (w± , s) with the shift T either
as giving the complete evolution of the u-parameter towards and away fromthe cosmological singularity, or as giving the simultaneous evolution of the twoparameters (u, v) whicle approaching the cosmological singularity.
This in turn determines the complete evolution of the parameters (u, δ, Ω),
where Ωn is the starting time of each era. For the explicit recursion as Ωn+1 =Ωn+1 (Ωn , un , δn ) (corrected), may see [8].
(Added). As in [112], with un as the initial value in that era, and δn > 0
characterizing the relative length of the era,Ωn+1 = (1 + δn kn (un +1))Ωn ,
xn

where kn = un  as the number of oscillations and xn = un − kn ∈ (0, 1).
If we put ρn = (1 − δn )δn−1 , then there is the recursion relation as ρn xn =(kn + ρn xn−1 )−1 . 
The mentioned result on the unique T -invariant measure on [0, 1] × P givenasδ(s)dxdμ(x, s) =,(3 log 2)(1 + x)
implies that the alternation of the space axes is uniform over the time evolution, namely the three axes provide the scale factor responsible for volume
compression with equal frequencies.
The Perron-Frobenius operator P F = L1 for the shift T on [0, 1] × P yields
the density of the T -invariant measure dμ above satisfying L1 f = f . The top
eigenvalue ησ of Lσ is related to the topological pressure P by ησ = exp(P (σ)).
This can be estimated numerically, using the technique of [6] and the integral
kernel operator representation of [112, 1.3].
The interpretation of solutions in terms of geodesics provides a natural way
to single out and study certain special classes of solutions on the basis of theirgeometric properties. Physically, such special classes of solutions exhibit diﬀerent behaviors approaching the cosmological singularity.
For instance, the data (w+ , s) corresponding to an eventually periodic sequence (kj )∞j=0 of some period l + 1 correspond to those geodesics on XΓ0 (2)that asymptotically wind around the closed geodesic identiﬁed with the doubly
inﬁnite sequence (aj )−∞<i∞ of period l + 1, so that aj+k(l+1) = aj for 0 ≤ j ≤ land k ∈ Z.
Physically, these universes exhibit a pattern of cycles that recurs periodicallyafter a ﬁnite number of Kasner eras.

－ 68 －

Another special class of solutions is given by the Hensley Cantor sets (cf.
[115]). These are the mixmaster universes for which there is a ﬁxed upper boundN to the number of cycles in each Kasner era, called as the controlled pulse
universes.
In terms of the continued fraction description, those solutions correspondto data (w+ , s) with w+ in the Hensley Cantor set EN ⊂ [0, 1]. The set EN
is given by all the points in [0, 1] with add digits in the continued fraction
expansion bounded by N (cf. [82]). In more geometric terms, these correspondto geodesics on the modular curve XΓ0 (2) that wander only a ﬁnite distance intoa cusp.
On the set EN , the Ruelle and Perron-Frobenius operators are given by


 
N

1
1
0 1
f
(Lσ.N f )(x, s) =,s .
(x + k)2σ
x+k 1 kk=1

This operator still has a unique invariant measure μN , whose density satisﬁes
Lσ,N f = f , withσ
6
1
72 log N+ O( 2 )= dimH (EN ) = 1 − 2 −2
π N
π4 N 2N
the Hausdorﬀ dimension of the Cantor set EN . Moreover, the top eigenvalue
ησ of Lσ,N is related to the Lyapunov exponent by
λ(β) =dησ |σ=dimH (EN ) ,dσ

for μN -almost all β ∈ EN .
A consequence of the characterization of the time evolution in terms of the
dynamical system on [0, 1] × P is that we can study global properties of suitable
moduli spaces of mixmaster universes. For instance, the moduli space for time
evolutions of the u-parameter approaching the consmological singularity as Ω →∞ is given by the quotient of [0, 1] × P by the action of the shift T .
Similarly, when we restrict to special classes of solutions, such as the Hensley
Cantor set EN , we can consider the moduli space (EN × P)/T modulo theaction of T . In this example, the dynamical system by the shift T acting on
EN × P is a subshift of ﬁnite type, and the resulting noncommutative space
is a Cuntz-Krieger C ∗ -algebra [66], generated by partial isometries with somerelations, in an interesting class of C ∗ -algebras. Another such C ∗ -algebra plays
a fundamental role in the geometry at arithmetic inﬁnity, as the topic of thenext section.
In the example of the mixmaster universe dynamics on the Hensley Cantor
sets, the shift T is described by the Markov partition
AN = {(k, t), (l, s) ∈ {1, · · · , N } × P | Uk.t = Uk × {t} ⊂ T (Ul,s )},1
where each Uk = [ k+1, k1 ] ∩ EN are the clopen subsets of EN where the local
inverses of T are deﬁned. This Markov partition determines the matrix AN =(A(k,t),(l,s) ) with entries 1 if Uk,t ⊂ T (Ul,s ) and zero otherwise.

－ 69 －

Proposition 2.20. The 3×3 sub-matrices Ak,l = (A(k,t),(l,s) )s,t∈P of the matrix
AN are of the form either
⎞
⎞
⎛
⎛
0 0 1
0 0 1
Ak,2m = ⎝0 1 0⎠ or Ak,2m+1 = ⎝1 0 0⎠ .
1 0 0
0 1 0


0 1
s = t,
This is because the condition Uk,t ⊂ T (Ul,s ) can be written as
1 l
together with the fact that the transformationby the same 2 × 
2 matrix acts


0 1 ∞
0 1 ∞
1
on P (F2 ) as the permutation
when l is even, and
∞ 1 0
1 ∞ 0
if l is odd.
As a noncommutative space associated to the Markov partition as well as
the matrix AN we may consider the Cuntz-Kriger C ∗ -algebra OAN , which isdeﬁned to be the universal C ∗ -algebra generated by partial isometries Sk,t such
that


∗
∗
∗
Sk,t Sk,t= 1 and Sl,s
Sl,s =A(k,t),(l,s) Sk,t Sk,t.
(k,t)∈{1,··· ,N }×P(k,t)

Topological invariants of this C ∗ -algebra reﬂect dynamical properties of the shift
T . (No ﬁgure provided).
(Added). As mentioned in [115], the irreducibility for the matrix AN corresponds to that the associated directed graph is strongly connected in the sensethat any two vertices are connected by an oriented path of edges. Since thematrix AN has the form⎞⎛
M M ··· M⎜L L ··· L⎟⎟⎜
AN = ⎜M M · · · M ⎟
⎠
⎝
..
..
..
.
. ···
.
the irreducibility for AN follows from that for A2 with N = 2. it then follows
that the matrix AN is aperiodic in the sense that one is the period, deﬁned tobe the greatest common divisor of the lengths of the closed directed paths. 

3Quantum statistical mechanics and Galois theory

This section is a brief review of Alain Connes and Marcolli ([41], [44]) on quantum statistical mechanics of Q-lattices, and of CM with Ramachandran ([48]as well as [49]) on the construction of a quantum statistical mechanical systemthat fully recovers the explicit class ﬁeld theory for imaginary quadratic ﬁelds
in such a way similar to what the Bost-Connes system does for the explicit classﬁeld theory of Q of rational numbers. Also included at the end of this chapter is－ 70 －

a brief review of Manin [110] on real quadratic ﬁelds and noncommutative tori
with real multiplication.
As the main notion in this section, we consider commensurability classes ofK-lattices,√ where K denotes either the rational ﬁeld Q or an imaginary quadraticﬁeld Q( −d) for d some positive integer. Begin with the case of Q.
Deﬁnition 3.1. An (n-dimensional) Q-lattice in Rn the real n-dimensional
Euclidean space is deﬁned to be a pair (Λ, ϕ) of a lattice Λ in Rn , that is,a cocompact free abelian subgroup of Rn of rank n, together with a systemof labels of its torsion points given by a homomorphism of abelian groups:
ϕ : Qn /Zn → QΛ/Λ.
A Q-lattice is said to be invertible if ϕ is an isomorphism.
Two Q-lattices (Λj , ϕj ) for j = 1, 2 are said to be commensurable, denotedas (Λ1 , ϕ1 ) ∼ (Λ2 , ϕ2 ) if QΛ1 = QΛ2 and ϕ1 = ϕ2 mod Λ1 + Λ2 .
May check that commensurability indeed deﬁnes an equivalence relation
among Q-lattices.
Proof. (Added). If QΛ1 = QΛ2 and QΛ2 = QΛ3 , then QΛ1 = QΛ3 . If ϕ1 = ϕ2
mod Λ1 + Λ2 and ϕ2 = ϕ3 mod Λ2 + Λ3 , then ϕ1 = ϕ3 mod Λ1 + Λ3 . Because
the mod Λi + Λj may be replaced with the mod Λi or Λj . (It makes sense.)
The interesting aspect of this equivalence relation is that the quotient provides another typical case that is described through noncommutative geometry.
For this it is crucial to consider non-invertible Q-lattices. In fact, most (?) Qlattices are not comensurable to an invertible one, while two invertible Q-lattices
are commensurable if and only if they are equal.
Example 3.2. (Added). For instance, let n = 1. Then Λ = kZ for some positive integer k, with some ϕ = kρ : Q/Z → QΛ/Λ for some ρ ∈ Hom(Q/Z, Q/Z),
identiﬁed with
Hom(Q/Z, T) = (Q/Z)∧ = lim(Z/nZ),←−p(which is the proﬁnite completion of Z, denoted as Z , and in fact is isomorphic
to Πp,k Zpk the direct product of indecomposable ﬁnite cyclic groups Zpk =Z/pk Z over prime numbers p and positive integers k (cf. [139, 2.9])), where we
may identify Q/Z with Q∩[0, 1) mod 1 and QΛ/Λ = Q∩[0, k) mod k. Note that
Q(kZ) = Q(lZ) if and only if k = l. If so, kZ + lZ = kZ ⊂ Z. A Q-lattice in R isinvertible if and only if ρ ∈ Aut(Q/Z). (It seems that most (or non-degenerate)
Q-lattices are invertible.) As a ﬁgure, as a 2-dimensional Q-lattice (2Z × 2Z, ϕ = 2(idZ × idZ )) as
..

.
···
···
..
.

..
.
•
•
..
.

..
.
•
•
..
.

..

.
···
···
..
.

..

.
···
ϕ
−−−−→ · · ·
···
..
.

－ 71 －

..
.
•
◦
•
..
.

..
.
◦
◦
◦
..
.

..
.
•
◦
•
..
.

..

.
···
···
···
..
.

(This seems to be invertible in the sense of that deﬁnition, but should be noninvertible as an element of End(Q2 /Z2 ).)In the following, we denote by Ltn the set of commensurability classesof n-dimensional Q-lattices in Rn . The topology on this set is encoded in anoncommutative algebra of coordinates, as a C ∗ -algeba denoted by C ∗ (Ltn ).
We discuss in some detail the case of n = 1 corresponiding to the Bost-Connes
system and the case of n = 2, considered in ([41], [44]). In these cases, we
further consider the spaces of commensurability classes of 1 and 2-dimensional
Q-lattices up to scaling, namely the respective quotients Lt1 /R∗+ and Lt2 /C∗ .
The corresponding C ∗ -algebras denoted by C ∗ (Lt1 /R∗+ ) and C ∗ (Lt2 /C∗ ) areendowed with a natural time evolution as a one-parameter family of automorphisms. Thus, may consider the C ∗ -algebras as quantum statistical mechanicalsystems , and look for equilibrium states, depending on a thermo-dynamic parameter β as inverse temperature. The interesting connection to arighmetics
arises from the fact that the action of symmetries of the system on equilibrium
states at zero temperature can be described in terms of Galois theory. In the 1dimensional case as the Bost-Connes system, this corresponds to the class ﬁeld
theory of Q, namely the Galois theory of the cyclotomic ﬁeld Qcyc . In the twodimensional case of the Connes-Marcolli system, that picture is more elaborate
and involves the automorphisms of the ﬁeld of modular functions.

3.1

Quantum statistical mechanics

In classical statistical mechanics, a state (of the system) is a probability measure
μ on the phase (metric) space X (such as a point measure or a point in X),
that assigns to each observable (function) f as a μ-measurable functionon X,
its expectation value in the form of an average as the integration X f dμ.
In particular, for a Hamiltonian system, the macroscopic thermo-dynamic
behavior is described via the Gibbs canonical ensemble. This is the normalized(Gibbs) measure deﬁned as
dμG =

1 −βH
dμL ,
e
Z

e−βH dμL

by Z =
X

1
with a thermo-dynamic parameter β = kT
, for T temperature and k the Boltzmann constant (which we can set equal to 1 suitably), where μL is the Liouvillemeasure (or a certain Lebesgue measure locally) on a manifold X.
For instance, a Hamiltonian system is given as   ∂H
 
 
d x
(x, y)
0 1
Hx
∂y
=
=
,
−1
0
Hy
dt y
− ∂H
(x,
y)
∂x

where H is a real-valued, C 2 -class function on R2 (or R2n ) (cf. [152]).
A (continuous) quantum statistical mechanical system (QSMS) consists ofthe data of an algebra of observables as a C ∗ -algebra A, together with a time
evolution given as a 1-parameter family of ∗-automorphisms σt ∈ Aut(A) for t ∈R. We may refer to the triple (A, σ, R) as a C ∗ -dynamical system, which deﬁnes－ 72 －

the crossed product C ∗ -algebra A σ R. These data describe the microscopic
quantum mechanical evolution of the system. (The discrete version of QSMS
may be deﬁned by replacing R with Z.)
The macroscopic thermo-dynamical properties are encoded in the equilibrium states of the system, depending on the inverse temperature β = T1 . While
the Gibbs measure in the classical case is deﬁned in terms of the Hamiltonian
and the symplectic structure on the phase space, the notion of equilibrium statein the quantum statistical mechanical setting only depends on the algebra of observable (operators) and its time evolution, and does not involve any additional
structure such as the symplectic structure or the approximation by regions ofﬁnite volume.
Recall the notion of states on C ∗ -algebras, which can be viewed as probability measures on noncommutative spaces.
Deﬁnition 3.3. A state on a unital C ∗ -algebra A is deﬁned to be a linear
functional ϕ : A → C such that normalization ϕ(1) = 1 and positivity ϕ(a∗ a) ≥0 for any a ∈ A are satisﬁed.
For a non-unital (or unital) C ∗ -algebra A, the normalization condition may
be replaced by the uniform norm ϕ = 1, where
ϕ =

sup

|ϕ(x)|.

x∈A,x≤1

Remark. (Added). As obtained in [127], a positive linear functional on a C ∗ algebra is bounded. A bounded linear functional on a C ∗ -algebra is positive if
and only if its uniform norm is equal to the limit (or the sup above) with respect
to an (or any) approximate unit. 
Before giving the general deﬁnition of equilibrium states via the KMS condition, we consider a simple case of a system with ﬁnitely many quantum degrees
of freedom, to see what equilibrium states are. In such a case, the algebra of observable operators is given by a C ∗ -algebra A on a Hilbert space H (or Cn ) (suchas K(H) the C ∗ -algebra of compact operators, or Mn (C)), and the time evolution as an R-action by ∗-automorphisms on A is deﬁned as σt (a) = eitK ae−itK
for a ∈ A and t ∈ R, where K is a (positive) self-adjoint operator on H suchthat exp(−βK) is of trace class for any β > 0, which may have its inverse
unbounded, and does converge to either zero or the projection to its kernel as
β → ∞ if K is positive, by functional calculus.
May check that for any a, b ∈ A and t, s ∈ R,
σt (ab) = eitK abe−itK = eitK ae−itK eitK be−itK = σt (a)σt (b),σt (a∗ ) = eitK a∗ e−itK = (eitK ae−itK )∗ = (σt (a))∗ ,σt+s (a) = eitK eisK ae−isK e−itK = (σt ◦ σs )(a).
These conditions are the reason why it has such a form and K should be selfadjoint. The set T(H) = L1 (H) of all trace class operators on H is a self-adjoint
ideal in B(H) the vN C ∗ -algebra of all bounded operators on H and is contained－ 73 －

in K(H). Indeed, for any a ∈ B(H) and b ∈ T(H), |tr(ab)| ≤ atr(|b|) and1b ≤ tr((|b| 2 )2 ) = tr(|b|). Note that B(H) ∼= T(H)∗ by this trace duality.
For such a system, the analog of the (Gibbs) measure above, as the Gibbs
equilibrium state is deﬁned as
ϕK (a) =1
tr(ae−βK ),Z

Z = tr(exp(−βK)),a ∈ A,

with tr(·) the canonical trace and Z as the normalization factor.
If A is unital, ϕK (1) = 1. Also, for any a ∈ A,
ϕK (a∗ a) =

β
β
β
β
1
1
tr(a∗ ae− 2 K e− 2 K ) = tr(e− 2 K a∗ ae− 2 K ) ≥ 0.
Z
Z

Moreover,ϕK (aσiβ (b)) =11
tr(ae−βK beβK e−βK ) = tr(bae−βK ) = ϕK (ba)ZZ2(cf. [16, II, 5.3]). Note also that if A = Mn (C) ∼= Cn , then for a, b ∈ A,

tr(ab∗ ) =

n 
n


aik bik = (aij ), (bij )Cn2 .

i=1 k=1

The Kubo-Martin-Schwinger (KMS) condition describing equilibrium statesof more general quantum statistical meachanical systems generalizes such a state
ϕK beyond the range of temperatures where exp(−βK) is of trace class (cf. [16,II, 5.3], [77], [78]).
Deﬁnition 3.4. Let (A, σ, R) be a continuous C ∗ -dynamical system. A state ϕon A is said to satisfy the KMS condition at inverse temperature 0 < β < ∞,called a KMSβ state, if for any a, b ∈ A, there exists a holomorphic function
fa,b (z) on the strip 0 < Im(z) < β, which is continuous and bounded on theclosure of the strip, such that for any t ∈ R,
fa,b (t) = ϕ(aσt (b)) and fa,b (t + iβ) = ϕ(σt (b)a).
(In particular, ϕ(aσiβ (b)) = ϕ(ba).)
The KMS∞ states on A are deﬁned as weak limits of KMSβ states as
ϕ∞ (a) = lim ϕβ (a),β→∞a ∈ A.

The deﬁnition of KMS∞ states is stronger than the one often adopted in
the literature, which simply uses the existence of such bounded holomorphic
functions ga,b (z) on the upper half plane such that ga,b (t) = ϕ(aσt (b)) only.
This notion, which may be called the ground states of the system, is weaker
than the notion of KMS∞ states given above.
Also, equivalently, a state ϕ on A is said to be a σ ground state if−iϕ(a∗ δ(a)) ≥ 0 for any a in the domain of the inﬁnitesimal generator δ ofσ, deﬁned by δ(a) = [iK, a] (cf. [16, II, 5.3]).

－ 74 －

For example, in the simplest case where the time evolution is trivial by theidentity map, all states are ground states by trivial functions, while only tracialstates are KMS∞ states. As another advantage, for any 0 < β ≤ ∞, the KMSβstates form a weakly compact, convex set Cβ (with the weak∗ topology), hence
we can consider the set Eβ of all extremal points of Cβ as equilibrium states ofA.
If ϕ, ψ ∈ Cβ with holomorphic fa,b and ga,b respectively and s ∈ [0, 1],
then tϕ + (1 − t)ψ ∈ Cβ with tfa,b + (1 − t)ga,b holomorphic. If a net {ϕλ } ⊂ Cβ
with fa,b,λ holomorphic converges weakly to ψ, namely, ϕλ (a) → ψ(a) for anya ∈ A, then ψ ∈ Cβ as well with holomorphic limit.
Remark. (Added). It is known (cf. [153, 5.1]) that for ϕ ∈ Cβ , it holds thatϕ ∈ Eβ if and only if the associated GNS representation πϕ of A is a factor
representation in the sense that the von Neumann algebra generated by πϕ (A)is a factor, namely has its center trivial. 
These provide a notion of points for the underlying noncommutative space.
This becomes especially useful in connection to an arithmetic structure speciﬁedby an arithmetic subalgebra of A, on which the KMS∞ states are evaluated.
This plays a key role in the relation between the symmetries of the system andthe action of the Galois group on states of E∞ on the subalgebra.
Symmetries. An important role in quantum statistical mechanics is playedby symmetries. Typically, symmetries of such a C ∗ -algebra A (or its subalgebra) compatible with the time evolution induce symmetries the set Eβ of theequilibrium states of A at diﬀerent temperatures. Especially important are thephenomena of symmetry breaking. In such cases, there is a global underlying
group G of symmetries of the algebra A, but in certain ranges of temperature,
the choice of an equilibrium state ϕ breaks the symmetry to a smaller stabilizer
subgroup Gϕ = {g ∈ G | g ∗ ϕ = ϕ} of G and ϕ under the induced action of G
on states. Various systems are known to exhibit either no, one, or more phasetransitions. As a typical situation in physical systems, there is the unique KMS
state for all values of the temperature T above a certain critical temperature Tc(so β = T1 < T1c = βc ). This corresponds to a chaotic phase such as randomly
distributed spins in a ferromagnet. When the system cools down and reaches
the critical temperature, the unique equilibrium state branches oﬀ into a larger
KMSβ set (β > βc ) and the symmetry is broken by the choice of an extremal
state in Eβ .
The case of Lt1 /R∗+ gives rise to a system with a single phase transition([13]), while in the case of Lt2 /C∗ the system has multiple phase transitions([41]).
An important point is that we need to consider bothe symmetries by automorphisms and by endomorphisms as well.
• [Automorphisms]. A subgroup G of the automorphism group Aut(A) of aC ∗ -algebra A (or its subalgebra) is said to be compatible with the R-action
σt ∈ Aut(A) (or σ-compatible, for short) if g ◦ σt = gσt = σt g for any g ∈ G
and t ∈ R. Then there is an induced action of G on the sets Cβ of KMS statesand in particular on the sets Eβ .

－ 75 －

Check that for any g, h ∈ G and a, b ∈ A and ϕ ∈ Cβ ,(gh)∗ ϕ(a) = ϕ((gh)−1 (a)) = ϕ(h−1 g −1 (a)) = g ∗ (h∗ ϕ)(a),g ∗ ϕ(aσt (b)) = ϕ(g −1 (a)g −1 (σt (b))) = ϕ(g −1 (a)σt (g −1 (b))) = fg−1 (a),g−1 (b) (t),g ∗ ϕ(σt (b)a) = ϕ(g −1 (σt (b))g −1 (a)) = ϕ(σt (g −1 (b))g −1 (a)) = fg−1 (a),g−1 (b) (t + iβ).
Also, if ϕ ∈ Eβ , then assume that g ∗ ϕ = tψ +(1−t)ρ for some ψ, ρ ∈ Cβ and 0 <t < 1. Then ϕ = t(g −1 )∗ ψ + (1 − t)(g −1 )∗ ρ with (g −1 )∗ ψ, (g −1 )∗ ρ ∈ Cβ . Thus,
ϕ = (g −1 )∗ ψ = (g −1 )∗ ρ. Hence g ∗ ϕ = ψ = ρ ∈ Eβ (cf. [127, Appendix]).
The adjoint, group Z action Ad(u) on A by a unitary u ∈ A is deﬁned as an
inner ∗-automorphism of A as Ad(u)(a) = uau∗ for a ∈ A. If σt (u) = u, thenwe may say that Ad(u) is σ-invariant. If so, the subgroup Ad(u) generated byAd(u) is compatible with the R-action σt , and it acts trivially on the sets Cβ of
KMS states.
Note that for any a ∈ A and t ∈ R,
Ad(u)(σt (a)) = uσt (a)u∗ = σt (uau∗ ) = σt Ad(u)(a).
Also, if ϕ ∈ Cβ , thenAd(u)∗ ϕ(aσt (b)) = ϕ(Ad(u)−1 (aσt (b))) = ϕ(Ad(u∗ )(a)σt (Ad(u∗ )(b)))= fu∗ au,u∗ bu (t),Ad(u)∗ ϕ(σt (b)a) = ϕ(Ad(u)−1 (σt (b)a)) = ϕ(σt (Ad(u∗ )(b))Ad(u∗ )(a))= fu∗ au,u∗ bu (t + iβ),both of which should be equal to fa,b (t) and fa,b (t + iβ) respectively, if trivial(?) (or if trivial up to such inner automorphisms).
• [Endomorphisms]. Let ρ be a ∗-homomorphism (or ∗-endomorphism) froma C ∗ -algebra A (or its subalgebra) to itself. Assume that ρ is σ-compatible inthat sense. If A is unital, then ρ(1) = ρ(1)ρ(1)(= ρ(1)∗ ) is an idempotent (anda projection) of A. Set p = ρ(1). If ϕ ∈ Eβ is an extremal KMS state on A suchthat ϕ(p) = 0, then there is a pull back ρ∗ ϕ deﬁned well as ρ∗ ϕ = ϕ(p)−1 ϕ ◦ ρ.
Note that 1 ≥ ϕ(p) = ϕ(p∗ p) > 0. Hence ϕ(p) = 1 if and only if ϕ(1 − p) =0. Check that for any a, b ∈ A,ρ∗ ϕ(1) = ϕ(p)−1 ϕ(p) = 1,
ρ∗ ϕ(aσt (b)) = ϕ(p)−1 ϕ(ρ(a)σt (ρ(b))) = fϕ(p)−1 ρ(a),ρ(b) (t),ρ∗ ϕ(σt (b)a) = ϕ(p)−1 ϕ(σt (ρ(b))ρ(a)) = fϕ(p)−1 ρ(a),ρ(b) (t + iβ)
since ϕ ∈ Cβ . Hence ρ∗ ϕ ∈ Cβ . Suppose now that ρ∗ ϕ = tψ1 + (1 − t)ψ2 with
ψ1 , ψ2 ∈ Cβ and 0 < t < 1. Then for any a ∈ A,
ϕ(ρ(a)) = tϕ(p)ψ1 (a) + (1 − t)ϕ(p)ψ2 (a).
But ϕ(p)ψj may not be a state on A since ϕ(p)ψj (1) = ϕ(p). If both arestates, then ϕ(p) = 1. It seems that ρ∗ ϕ ∈ Eβ is provided only that ρ is an∗-automoprhism of A.

－ 76 －

Now let s be an isometry of a unital A, namely, s∗ s = 1 ∈ A. Supposethat ss∗ = p and s is compatible with the time evolution in the sense thatσt (s) = λit s for some λ > 0. .
Note that λit = eit log λ ∈ T the 1-torus.
The adjoint, semigroup N action Ad(s) on A is deﬁned as an inner endomorphism of A as Ad(s)(a) = sas∗ for a ∈ A. Note that Ad(s)n = Ad(sn ) forn ∈ N. If the isometry s is σ-invariant in the sense above, then the semi-group
generated by Ad(s) is compatible with the time evolution in the following sense.
Check that for any a ∈ A,
Ad(s)(σt (a)) = σt (sas∗ ) = σt (Ad(s)(a)).
For any ϕ ∈ Cβ , then Ad(s)∗ ϕ ∈ Cβ is deﬁned as ρ∗ ϕ with ρ = Ad(s). But,
as checked above, this induced action is trivial (?) (or trivial up to such innerautomorphisms).
In deﬁning the induced action on states by endomorphisms, it is necessary to
be careful. In fact, there are cases where for KMS∞ states ϕ, it only holds thatϕ(p) = 0, yet it is still possible to deﬁne an interesting action by endomorphisms
by a procedure of warking up and cooling down. For this to work, we need
suﬃciently favorable conditions as that the warming up (in temperature T =1β ) map deﬁned bywβ (ϕ)(a) = tr(e−βK )−1 tr(πϕ (a)e−βK )gives a homeomorphism wβ : E∞ → Eβ for β suﬃciently large. (That e−βK may
be identiﬁed with the similar on the representation space of πϕ .) Then deﬁne
the induced action by(ρ∗ ϕ)(a) = lim (ρ∗ wβ (ϕ))(a),β→∞ϕ ∈ E∞ , a ∈ A

as the cooling down (in temperature) limit.
Check that wβ (ϕ)(1) = 1, andwβ (ϕ)(aσt+h (b)) − wβ (ϕ)(aσt (b))
hσt+h (b) − σt (b) −βK−βK −1e) tr(πϕ (a)πϕ)= tr(ehand thus the limit as h → t exists if σt (b) is analytic under the GNS representation πϕ , or if b is σ-analytic (cf. [153, 5.1]). The same is applied forwβ (ϕ)(σt (b)a). Another diﬃculty may be to ﬁnd such a holomorphic pathfa,b (t + is) with 0 ≤ s ≤ β between those analytic functions on the boundary at both s = 0 and s = β. But, we can take


β−s
s
wβ (ϕ)aσt (b) + σt (b)aββ
as a possible easy choice for the path. Masaka! (in Japanese, similar to saying
Incredible!).

－ 77 －

3.2

The Bost-Connes system

May refer to [48] for the following geometric point of view on the Bost-Connes(BC) system.
We use the notation Sm(G, X) = G(Q)\(G(Af )×X) for Shi-mura varietiespp(cf. [122]), where Af = Z ⊗Q denotes the ﬁnite adèles of Q, with Z = lim Zn ,←−Zn = Z/nZ. The simplest case is that for G = GL1 , so that
Sm(GL1 , {±1}) = GL1 (Q)\(GL1 (Af ) × {±1}) = Q∗+ \A∗f .
Lemma 3.5. The above quotient as a Shi-mura variety parameterizes the invertible, 1-dimensional Q-lattices, up to scaling.
In fact, note that a 1-dimensional Q-lattice can be written as (Λ, ϕ) =(λZ, λρ) for some λ > 0 and somepρ ∈ Hom(Q/Z, Q/Z) = lim Zn ≡ Z .
←−Thus, the set of all 1-dimensional Q-lattices, up to scaling can be identiﬁedpwith Z . The invertible 1-dimensional Q-lattices correspond to the (invertible)p,∗elements of Z , which in turn is identiﬁed with GL+
1 (Q)\GL1 (Af ).
The quotient Sm(GL1 , {±1}) can be thought of geometrically as the Shimuravariety associated to the cyclotomic tower (cf. [48]). This is the tower V ofarithmetic varieties over V = Spec(Z), with Vn = Spec(Z[ζn ]), where ζn isa primitive n-th root of unity. The group AutV (Vn ) of deck transformations
is GL1 (Z/nZ), so that the group of deck transformations of the tower is the
projective limit:p
AutV (V) = lim AutV (Vn ) = GL1 (Z ).
←−nIf, instead of invertible 1-dimensional Q-lattices up to scale, we consider
the set Lt1 /R∗+ of commensurability classes of all 1-dimensional Q-lattices up toscale, and ﬁnd a noncommutative version of the Shimura variety Sm(GL1 , {±1}).
as well as of the corresponding cyclotomic tower. This is described as the quotient
Smnc (GL1 , {±1}) ≡ GL1 (Q)\(Af × {±1}) = GL1 (Q)\(A◦ /R∗+ ),where A◦ ≡ Af × R∗ denotes the adèles of Q with invertible archimedean component. The corresponding noncommutative algebra of coordinates is givenby the C ∗ -algebra crossed productB1 ≡ C0 (Af )  Q∗+ ∼= (C(Z ) ⊗ C0 (Q))  Q∗+pby the group Q∗+ (cf. [95]). Obtained is an equivalent description of the noncommutative space Smnc (GL1 , {±1}), starting from the description of 1-dimensional
Q-lattices as (Λ, ϕ) = (λZ, λρ). The commensurability relation is implemented
by the action of N× = Z>0 bypf (n−1 ρ) ρ ∈ nZ ,αn (f )(ρ) =0otherwise.

－ 78 －

Thus, the noncommutative algebra of coordinates of the space of commensurability classes of 1-dimensional Q-lattices up to scaling can be identiﬁed with
the C ∗ -algebra crossed productA1 ≡ C ∗ (Q/Z) α N×
by the semi-group N× , where there is a C ∗ -algebra isomorphism aspC ∗ (Q/Z) ∼= C((Q/Z)∧ ) ∼= C(Z ),
pwhere Z is identiﬁed with the Pontrjagin dual (Q/Z)∧ = Hom(Q/Z, T) of theabelian group Q/Z. In fact, the C ∗ -algebra A1 is Morita equivalent to theC ∗ -algebra B1 (cf. [95]).
As shown in Laca and Raeburn [96] (added), the semi-group C ∗ -algebra
crossed product A1 is isomorphic to the Bost-Connes Hecke C ∗ -algebra H1
considered in [13], which is obtained as a Hecke algebra for an inclusion Γ0 ⊂ Γof a pair of groups, with (Γ0 , Γ) = (SZ , SQ ), where SC denotes the solvable
ax + b group with coeﬃcients in C. The pair in this case satisﬁes that the left
Γ0 -orbits for any element of Γ/Γ0 are ﬁnite, and the same holds for right orbits
on the left coset. The ration of the lengths of left and right Γ0 -orbits determinesa canonical time evolution on the C ∗ -algebra (cf. [13]).
Remark. (Added). Recall from Laca [95] the following. The group C ∗ dynamical system (C0 (Af ), Q∗+ , β) is the minimal automorphic dilation of thep(Ore) semi-group C ∗ -dynamical system (C(Z ), N× , α), so that
A1 = C(Z ) α N× ∼= χZp (C0 (Af ) β Q∗+ )χZp = χZp B1 χZp ,p

where βr (f )(a) = f (r−1 a) for a ∈ Af and r ∈ Q∗+ = (N× )−1 N× , and thepcharacteristic function χZp on Z that is compact and open in Af , belongs to
C0 (Af ) the C ∗ -algebra of all continuous functions on Af vanishing at inﬁnity,
where Af is the locally compact ring of ﬁnite adeles:
Πp Qp ⊃ Af = {(ap ) | ap ∈ Zp for all but ﬁnitely many primes}.
Moreover,

H1 ∼= qZp C ∗ (Af  Q∗+ )qZq ∼= A1 ,

where C ∗ (Af  Q∗+ ) is the group C ∗ -algebra of the semi-direct product groupAf  Q∗+ , and qZq is identiﬁed with the projection of the group C ∗ -algebra∼C ∗ (Af ) corresponding to χZq under the Fourier transform, where A∧
f = Af asa self-duality.
Such an action β dilates the action α if βn ◦ i = i ◦ αn for n ∈ N× , wherepi : C(Z ) → C0 (Af ) is the inclusion map. Such a group C ∗ -dynamical system byp
β is said to be minimal if the union ∪n∈N× βn−1 (i(C(Z ))) is dense in C0 (Af )×βQ∗+ . If these conditions are satisﬁed, the system by β is said to be the minimal
automorphic dilation of the system by α.
A semi-group S is embeddable into a group G = S −1 S if and only if S is an
Ore semigroup, that is, if S is a cancellative semigroup such that Sk ∩ Sl = ∅for any k, l ∈ S (cf. [95]). － 79 －

Remark. (Added). As in Laca [94], the group inclusion of the ax + b-groups
with coeﬃcients in Z and Q (as well as R) is
 

 


 ∗
 ∗
x
1 Z
Q+ Q
R+ R
⊂ SR =
act on.
⊂ SQ =
SZ =
0
1
0
1
1
0 1
Note that SZ ∼= Z  1 and SQ ∼= Q  Q∗+ as a semi-direct product group,
so that the quotient group is also isomorphic to
SQ /SZ ∼= (Q/Z)  Q∗+ .
It then follows that the group C ∗ -algebra of SQ /SZ is isomorphic to the C ∗ algebra crossed products by the group Q∗+ (extending the action α of N× ):p
C ∗ (SQ /SZ ) ∼= C ∗ (Q/Z)  Q∗+ ∼= C(Z )  Q∗+p⊃ H1 ∼= C(Z ) α N× = A1 .

The Morita equivalence between two C ∗ -algebras is just like the equivalence
between two projections (corresponding to respective units). Masaka!
The C ∗ -algebra H1 of the Bost-Connes system has an explicit presentation interms of certain operators of two types. The ﬁrst type consists of phase (unitary)
operators e(r), parameterized by elements r ∈ Q/Z. These phase operators canbe represented on the Fock space generated by occupation numbers |n as theoperators deﬁned as e(r)|n = α(ζrn )|n, where we denote by ζab = ζba (corrected)
the abstract roots of unity generating Qcyc and by α : Qcyc → C an embeddingthat identiﬁes Qcyc with the subﬁeld of C generated by the concrete roots ofunity.
Recall from [74, 3.1.2] the bracket (bra-ket) notation by Dirac as follows.
ξm , Aξn  ≡ m|A|n,
where the vectors ξm , ξn (as states) in the inner product with A an (observable)
operator on a (Fock) Hilbert space may be identiﬁed with the bra ξm = mand the ket ξn  = n, respectively, and as well, Aξn = A|n. The Fock spaceis deﬁned to be a Hilbert space obtained as the inﬁnite direct sum of tensor
products ⊗n H for n ∈ N of a certain Hilbert space H, with the vacuum state
generating the orthogonal complement of the direct sum (cf. [150]).
Those phase operators are used in the theory of quantum optics and opticalcoherence to model the phase quantum-mechanically (cf. [100], [102] both not
at hand), as well as to model the phasors in quantum computing. They arebased on the choice of a certain scale N at which the phase is discretized (Noﬁgure). Namely, the quantized optical phase is deﬁned as a state




N

m
1
|θm,N  = evN ,vN = √|n
N +1
N + 1 n=0=√N1α(ζ Nmn)|n,+1
N + 1 n=0

－ 80 －



mn

ζ Nmn
= e2πi N +1+1

where vN is a superposition of occupation states as above (cf. [96, 2.4]). It isthen necessary to ensure that the results are consistent over changes of scale.
The removed ﬁgure is a picture of phasors with Z6 -discretization in theplane, given and parameterized as⎛1
1
1⎜1 e2πi 6⎜⎜1 e2πi 26
⎜
⎜ ..
..
⎜.
.
⎜
⎜1 e2πi k6
⎝
..
..
.
.

1
2πi 26e4e2πi 6..
.
2ke2πi 6
..
.

···
···
···
..
.

e
10
e2πi 6
..
.

···
..
.

e2πi 6..
.

12πi 565k⎞
1 ···1 · · ·⎟⎟
1 · · ·⎟⎟.. . . ⎟⎟.
.
⎟1 · · ·⎟⎠.. . .
.
.

The other operators that generate the Bost-Connes Hecke C ∗ -algebra H1
can be thought of as implementing the changes of scales in the optical phases ina consistent way. These operators are isometries μn , parameterized by positive
integers n ∈ N× = Z>0 . The changes of scale are described by the action of μn
on e(r) as
Ad(μn )e(r) = μn e(r)μ∗n =

1 
e(s),
n ns=rn ∈ N× , r ∈ Q/Z.

For instance, check that for [x] ∈ Q/Z, with 0 ≤ x < 1 assumed,
3
1
1
1
1
[ ] = 2[ ] = 2[ + 2 ] = 2[ ],
2
4
2 2
4
1
2
1
1
1
1
[ ] = 3[ 2 ] = 3[ + 2 ] = 3[ + 2 ].
3
3
3 3
3 3
In addition to that compatibility condition, the phase operators e(r) andisometries μn satisfy other relations as below. For any n, k ∈ N× and r, s ∈ Q/Z,μ∗n μn = 1,e(0) = 1,μk μn = μkn , (kn = k ∨ n as a lattice or L.C.M.)e(r)∗ = e(−r), e(r)e(s) = e(r + s).

These and that relations give such a presentation of the Hecke C ∗ -algebra H1of the BC system ([13], [94]). It then follows that the Bost-Connes Hecke C ∗ algebra H1 is isomorphic to the semi-group C ∗ -crossed product A1 .
Those relations say that the map N× ! n → μn is viewed as an isometricrepresentation of the semi-group N× and that the map Q/Z ! r → e(r) isviewed as a unitary representation of the group Q/Z, and these representations
become a covariant representation of both, to extend to deﬁne such a semi-groupC ∗ -algebra crossed product by taking a C ∗ -algebra (norm) completion.
In terms of that explicit presentation, the time evolution is deﬁned as theform
σt (μn ) = nit μn = eit log n μn and σt (e(r)) = e(r).

－ 81 －

The space Smnc (GL1 , {±1}) = GL1 (Q)\A◦ /R∗+ can be compactiﬁed by replacing A◦ by A, as in [29], to give the quotient
Smnc (GL1 , {±1}) = GL1 (Q)\A/R∗+ .
This compactiﬁcation consists of adding the trivial lattice (with a possibly nontrivial Q-structure).
The dual space to Smnc (GL1 , {±1}), under the duality of type II and type III
factors, introduced by Connes (thesis), is a principal R∗+ -bundle over Smnc (GL1 , {±1}),whose noncommutative algebra of coordinates is obtained as the C ∗ -algebra
crossed product of the C ∗ -algebra corresponding to Smnc (GL1 , {±1}) by thetime evolution σt as an R-action.
Namely, that is, and contains(C(A) β Q∗+ ) σ R ⊃ (C0 (Af ) β Q∗+ ) σ R = B1 σ R
as a closed ideal.
The space obtained in this way is the space of adèle classes as the principalR∗+ -bundle:
Lt1 = GL1 (Q)\A⏐⏐(bundle)Lt1 /R∗+ = GL1 (Q)\A/R∗+←−−−− R∗+(ﬁber)(base space)which gives the spectral realization of zeros of the Riemann zeta function ζ, asin [29]. The passing to this dual space corresponds to considering commensurability classes of 1-dimensional Q-lattices (without up to scaling).
Consequently, we may denote asC ∗ (Lt1 /R∗+ ) = C ∗ (GL1 (Q)\A/R∗+ ) = C(A) β Q∗+ = C ∗ (Smnc (GL1 , {±1}))
and C ∗ (Lt1 ) = C ∗ (GL1 (Q)\A) = (C(A) β Q∗+ ) ×σ R.
As wellC ∗ (Smnc (GL1 , {±1})) = C ∗ (GL1 (Q)\A◦ /R∗+ ) = C0 (Af )  Q∗+ .
Moreover, we may denote as well thatC ∗ (Af β Q∗+ ) = C0 (Af ) β Q∗+ = B1 ⊂ C ∗ (Lt1 /R∗+ ),C ∗ ((Q/Z) α N× ) = C ∗ (Q/Z) α N× = A1 ⊂ B1 ,both of which are reduced from C ∗ (Lt1 /R∗+ ), so may be denoted as B1 =Cr∗ (Lt1 /R∗+ ) or A1 = Cr∗ (Lt1 /R∗+ ).
Structure of KMS states. The Bost-Connes Hecke C ∗ -algebra H1 ∼= A1 thesemigroup C ∗ -algebra crossed product of Laca and Raeburn has irreducible
representations on the Hilbert space H = l2 (N× ). These are parameterized－ 82 －

p,∧p

p

by elements α ∈ Z= GL1 (Z ) = (Z )−1 . Any such element α deﬁnes ancycembedding α : Q→ C of the subﬁeld Qcyc of C generated by the roots ofunity (by the same symbol), and the corresponding representation has the form
πα (e(r))ξk = α(ζrk )ξk = e2πirk ξkand πα (μn )ξk = ξnk

for r = [r] = r + Z ∈ Q/Z and k ∈ N× , where {ξk }k∈N× is the canonical basis
for l2 (N× ), with ξk = χk the characteristic function at k (cf. [96, 2.4]).
The Hamiltonian implementing the time evolution by the action α of N×
is given by Hξk = (log k)ξk . Thus, the partition function of the Bost-Connes
system is the Riemann zeta function
Z(β) = tr(e−βH ) =

∞

1
k=1

kβ

= ζ(β).

It is shown by Bost and Connes [13] that KMS states have the following
structure, with a phase transition at β = 1.
Theorem 3.6. (Edited). Consider on A1 = H1 with σ the time evolution.
• In the range 0 < β ≤ 1, there is a unique KMSβ state ϕβ . Its restriction
to Q[Q/Z] in C ∗ (Q/Z) has the form
1 − pβ−1a,
ϕβ (e( )) = b−β Πp prime, p|bb
1 − p−1
where the product corresponds to the prime factorization for b. Moreover, eachϕβ is associated to the hyper-ﬁnite factor of type III1 , that is, R∞ of ArakiWoods.
• For 1 < β ≤ ∞, the set Eβ of extremal KMSβ states can be identiﬁed withp
(Z )∗ , which has a free and transitive action by (itself) the group also induced
by some automorphisms of A1 . The extremal KMSβ state ϕβ,α correspondingp
to α ∈ (Z )∗ has the form on C ∗ (Q/Z)
∞11  α(x)n−βHϕβ,α (x) =.
) =tr(πα (x)eζ(β)
ζ(β) n=1 nβThere states are associated to the type I∞ factor, that is, B(H).
• At β = ∞, the Galois group G = Gal(Qcyc /Q) acts on the values of KMS∞
states ϕ ∈ E∞ on a certain arithmetic subalgebra A1,Q of A1 . These states
have a property that ϕ(A1,Q ) ⊂ Qcyc and that the class ﬁeld theory isomorphismpp
θ:G∼= (Z )∗ intertwines the Galois action on values with the action of (Z )∗ bysymmetries, namely, γϕ(x) = ϕ(θ(γ)x) for any ϕ ∈ E∞ , γ ∈ G, and x ∈ A1,Q ,where the arithemetic subalgebra A1,Q can be taken as an algebra over Q,
generated by the operators e(r) and μn , μ∗n for r ∈ Q/Z and n ∈ N× .
Remark. (Added). As mentioned in [13], the critical temperature is at T = 1.
At lower temperature for β > 1, the phases of the BC system are parameterized－ 83 －

by all possible embedding of Qcyc into C. The Galois group G acts naturally
as a group of automorphisms of A1 = H1 commuting with the time evolution σ
and the spontaneous symmetry breaking occurs for β > 1.
As shown in [41], that Q-algebra A1,Q can also be obtained as the algebra
generated by μn , μ∗n for n ∈ N× and by homogeneous functions of weight zeroon 1-dimensional Q-lattices obtained as a normalization of the functions
1by covolume.
ξk,a (Λ, ϕ) =yky∈Λ+ϕ(a)Namely, consider the functions
ck ξk,a , where c(Λ) is proportional to the covol√
ume |Λ| and satisﬁes 2π −1c(Z) = 1.
The choice of such an arithmetic subalgebra of A1 corresponds to endowing
the noncommutative space (as A1,Q ) with an arithmetic structure. The subalgebra corresonds to the rational functions and the values of KMS∞ states atelements of this subalgebra should be thought of as values of rational functions
at the classical points of the noncommutaitve space (cf. [48]).

3.3

Noncommutative geometry and the Hilbert 12th problem

The most remarkable airthmetic feature of the result of Bost-Connes recalled
above is the Galois action on the ground states of the BC system. First of all, thefact that the Galois action on the values of states would preserve positivity andwould give values of other states is an unusual property. Moreover, the valuesof extremal ground states on elements of the rational subalgebra generate themaximal Abelian extension QAb = Qcyc of Q. The explicit action of the Galoispgroup Gal(QAb /Q) ∼= GL1 (Z ) is given by automorphisms of the system (A1 , σt )∗(or the group C -algebra crossed product A1 σ R). Namely, the class ﬁeld
theory isomorphism intertwines the action of the idèle class group, as symmerygroup of the system, that of the Galois group, as permutations of the expectation
values of the rational observable operators.
In general, the main theorem in class ﬁeld theory provides a classiﬁcationof ﬁnite abelian extensions of a local or global ﬁeld K in terms of subgroups
of a locally compact abelian group canonically associated to the ﬁeld. This is
the multiplicative group K∗ = GL1 (K) in the local non-archimedean case, whilein the global case it is the quotient CL /DK of the idèle class group CK by theconnected component of the identity.
The Hilbert 12th problem can be formulated as the question of providing
an explicit set of generators of the maximal abelian extension Kab of a number
ﬁeld K, inside an algebraic closure K, and of the action of the Galois group
Gal(Kab /K). There is the following property that
Gal(Kab /K) ∼= Gal(K/K)ab .
The result motivating the Hilbert formulation of the explicit class ﬁeld theory problem is the Kronecker-Weber theorem, that is, the explicit class ﬁeld

－ 84 －

theory in the case of K = Q, already mentioned. In this case, the maximalabelian extension of Q can be identiﬁed with the cyclotomic ﬁeld Qcyc . Equivalently, the torsion points of the multiplicative group C∗ , as the roots of unity,
generate Qab ⊂ C.
As a remark, the only other case of number ﬁelds where such formulation iscarried out completely is that of imaginary quadratic ﬁelds where the construction relies on the theory of elliptic curves with complex multiplication and on
the Galois theory of the ﬁelds of modular functions (cf. [148] as a survey).
Remark. (Added). Now may recall from [116] the following fundamental
theorem on the class ﬁeld theory.
[The Artin reciprocity law]. For a ﬁnite Abel extension K over an algebraic
number ﬁeld K that is a ﬁnite extension over Q, there is a non-zero ideal mof the ring OK of algebric integers of K such that the reciprocity map from⊕Sm Z to Gal(K/K) maps Γm to zero, where (⊕Sm Z)/Γm ∼= Cl(m) the ray classgroup that is a ﬁnite abelian group, where there is a surjective map from ⊕Sm Z
to Cl(m), induced by Z ∼= K∗v /Ov∗ for v ∈ Sm a ﬁnite prime point of K which
does not divide m, identiﬁed with a fractional ideal relatively prime with m.
There is the maximal ideal m satisfying the condition above, called the conductor of K, so that induced is the homomorphism from Cl(m) to Gal(K/K).
[Existence theorem]. Conversely, for a non-zero ideal m of Ok , there is a
ﬁnite Abel extension Km of K such that the reciprocity map induces the isomorphism from Cl(m) to Gal(Km /K).
Such Abel extension Km for m is unique up to isomorphisms over K, called
the ray class ﬁeld of (the conductor) m. If m = OK , then Km is the absolute classﬁeld of K. A ﬁntie Abel extension over K with m as the conductor is a subﬁeldof Km . A class ﬁeld of K is deﬁned to be a subﬁeld of Km corresponding to
the image of a subgroup of Cl(m) under the reciprocity map.
For instance, if K = Q and m = nZ, then Km = Q(ζn ). If K = Km , thenCl(m) ∼= Gal(K/Q) ∼= Z∗n .
It then follows [Kronecker-Weber] that the largest Abel extension QAb of Q
is Q(ζn | n ≥ 2). 
Some generalizations of the original result of Bost-Connes to other global
ﬁelds as number ﬁelds and function ﬁelds are obtained by Harari and Leichtnam[79], P. Cohen [21], Arledge, Laca, and Raeburn [4]. A more detailed accountof various results related to the BC system and generalizations is given in thesection of [41] as futher developments. Because of such a close relation to theHilbert 12th problem, it seems to be clear that obtaining generalizations of the
Bost-Connes system to other number ﬁelds is a diﬃcult problem, and it is not
surprising that these constructions so far have not fully recovered the Galois
properties of the ground states of the BC system in the generalized setting.
As the strongest form of such a result in that direction, we can formulate
the following:
Theorem 3.7. (Conjecture, edited). Given a number ﬁeld K, we denote by
AK the ring of adèles of K and by A∗K = GL1 (AK ) the group of idèles of K.

－ 85 －

Let CK = A∗K /K∗ be the group of idèle classes of K and DK the connectedcomponent of the identity in CK . Construct a C ∗ -algebra dynamical system(AK , σ, R) and an arithmetic subalgebra AK,Q such that
(1) The (quotient) idèle class group CK /DK acts by symmetries on the system(AK , σ, R) preserving the subalgebra AK,Q .
(2) The extremal KMS∞ states ϕ ∈ E∞ evaluated at elements a of AK,Q
satisfy that ϕ(a) ∈ K the algebraic closure of K in C, and the values ϕ(a) ∈ C
for a ∈ AK,Q and ϕ ∈ E∞ generate KAb .
∼=(3) The class ﬁeld theory isomorphism θ : Gal(KAb /K) −→ CK /DK intertwines with the actions, so that for α ∈ Gal(KAb /K) and ϕ ∈ E∞ ,ϕ

AK,Q ⊂ AK −−−−→ KAb ⊂ C
⏐
⏐
⏐
⏐α
θ(α)

ϕ

AK,Q ⊂ AK −−−−→ KAb ⊂ C.
In general, the arithmetic subalgebra AK,Q need not be an involutive algebra.
The setup described above may provide a possible new approach to the explicit
class ﬁeld theory problem via noncommutative geometry.
Given a number ﬁeld K with [K, Q] = n, there is an embedding from its multiplicative group K∗ into GLn (Q). Such an embedding induces an embedding
from GL1 (AK.f ) into GLn (Af ), where AK,f = Af ⊗ K are the ﬁnite adèles of K.
That suggests that a possible strategy to approach the problem stated above
may be to ﬁrst study quantum statistical mechanical systems corresponding to
GLn -analogues of the Bost-Connes system.
The main result of [41] is the construction of such a system in the case of
GL2 and the analysis of the arithmetic properties of its KMS states. In the caseof GL1 , it is observed that the geometry of modular curves and the algebra ofmodular forms appear naturally.
As the work of [48], a quantum statistical mechanical system is constructed,
and satisﬁes all the properties listed above for the case where K is an imaginary
quadratic ﬁeld. The properties of this system are intermediate between thoseof the original BC system and those of the GL2 system of [41]. In fact, theconstruction is geometrically based on replacing the 1-dimensional Q-latticesof the Bost-Connes system with 1-dimensional K-lattices. The groupoid of thecommensurability relation is then a sub-groupoid of that of the GL2 system.
In fact, 1-dimensional K-lattices are viewed as a special case of 2-dimensional
Q-lattices with compatible notions of commensurability. Thus, by combining
the techniques of the BC system and of the GL2 system, it is possible to showthat the complex multiplication (CM) system deﬁned in that way recovers thefull picture of the explicit class ﬁeld theory of imaginary quadratic ﬁelds ([48]),as recalled below later.
There is the case with not yet a complete solution √to the explicit class ﬁeld
theory problem, such as the ﬁrst case where K = Q( d) real quadratic ﬁelds
for some positive integer d. It is natural to ask whether the approach outlined
above, based on noncommutative geometry, may provide any new information－ 86 －

on this case. There is a close relation between the real quadratic case andnoncommutative geometty, obtained by Manin ([109], [110]). Discussed below
is the possible relation between the approach and the GL2 system.
Remark. Now recall from [116] the following. Suppose that K is a number
ﬁeld. Let P = Pf # P∞ denote the set of all (ﬁnite or inﬁnite) primes (divisors)of K (as equivalence classes of valuations of K). For any p ∈ P , let Kp be thecompletion of K with respect to p (such as Qp of Q with p = pZ for p prime),
and let K∗p be the multiplicative group of Kp . Moreover, let Op the valuation
ring for a ﬁnite prime p, and let Up the group of invertibles of Op . Since Op as
an additive group is a compact open subgroup of Kp , the adèle ring AK of K
is deﬁned to be the restricted direct product of {Kp } with respect to {Op }. It
looks likeΠp∈P Kp ⊃ AK ⊃ O ≡ (Πp∈Pf Op ) # (Πp∈P∞ Kp ),with the quotient AK /O with discrete topology, to become a locally compact
group and ring, where any element {xp }, called adèle, of AK has any p component xp in Op except ﬁnitely many p component xp in Kp .
As well, since Up as a multiplicative group is a compact open subgroup ofK∗p , the idèal group JK of K is deﬁned to be the restricted direct sum of {K∗p }with respect to {Up }. It looks likeΠp∈P K∗p ⊃ JK ⊃ U ≡ (Πp∈Pf Up ) # (Πp∈P∞ K∗p ),with the similar condition as above, with any element of JK , called idèle. Asa group, JK = A∗K , but with the topology stronger than the relative topology
from the topology on AK . 

3.4

The GL2 system

In this subsection, described is the main features of the GL2 analogue of the
Bost-Connes system, according to the results of Connes-Marcolli [41].
Let us start with the same geometric approach in terms of Shimura varieties
as used above to introduce the BC system (cf. [48], [44]).
The Shimura variety associated to the tower of modular curves is describedby the adelic quotient
Sm(GL2 , H± ) = GL2 (Q)\(GL2 (Af ) × H± )
+
∗
= GL+
2 (Q)\(GL2 (Af ) × H) = GL2 (Q)\GL2 (A)/C .

Note that A = Af × R (!) the adèles of Q, and H± the union of theupper and lower half-planes in C is identiﬁed with the quotient GL2 (R)/C∗ ,n(n+1)+∗
2and H+ = H2 = H with GL+
with
2 (R)/C . In fact, GL2 (R) ≈ SO(n) × R
n = 2 as Iwasawa decomposition (cf. [158, Example 118]). Possibly, GL2 (Af )as well as GL2 (A) in the second line above may be replaced with respective+
components GL+
2 (Af ) and GL2 (A) with determinant positive.
The tower of modular curves has base V = P1 over Q and Vn = X(n) themodular curve corresponding to the principal congruence subgroup Γ(n). The－ 87 －

automorphisms of the projection Vn → V are given by GL2 (Z/nZ)/{±1}, sothat the group of deck transformations of the tower V has the formp

AutV (V) = lim GL2 (Zn )/{±1} = GL2 (Z )/{±1}.
←−The inverse limit of Γ\H over congruence subgroups Γ in SL2 (Z) gives a connected component of Sm(GL2 , H± ), while by taking congruence subgroups in
SL2 (Q), obtained is the adelic version Sm(GL2 , H± ).
The simple reason of being necessary to pass to the non-connected case is
the following. The varieties in the tower are arithmetic varieties deﬁned overnumber ﬁelds. However, the number ﬁeld typically changes along the levels of
the tower. In fact, Vn is deﬁned over the cyclotomic ﬁeld Q(ζn ). Passing to
non-connected Shimura varieties allows for the deﬁnition of a canonical modelcase where the whole tower is deﬁned over the same number ﬁeld.
The adelic quotient Sm(GL2 , H± ) parameterizes invertible 2-dimensional Qlattices up to scaling (cf. [122]). Instead of restricting to the invertible case, if
we consider commensurability classes of 2-dimensional Q-lattices up to scaling,
then we obtain a noncommutative space whose classical points are the Shimuravariety Sm(GL2 , H± ). More precisely, the following is obtained (cf. [48]):
Lemma 3.8. The space Lt2 /C∗ of commensurability classes of 2-dimensional
Q-lattices up to scaling is described as the quotient
Smnc (GL2 , H± ) ≡ GL2 (Q)\(M2 (Af ) × H± ).
Proof. (Edited). Any 2-dimensional Q-lattice (Λ, ϕ) can be written as in theform (λ(Z + Zτ ), λρ), for some λ ∈ C∗ , τ ∈ H, andpρ ∈ M2 (Z ) = Hom(Q2 /Z2 , Q2 /Z2 ).
Thus, the space Lt2 /C∗ with λ ∈ C∗ as the scale factor is given byp

M2 (Z ) × H mod Γ = SL2 (Z).
The commensurability relation is then implemented by a partially deﬁned
action of GL+
2 (Q) on that space, given as g(ρ, z) = (gρ, g(z)), where g(z) denotefractional linear transformations as an action.
Commensurability classes of Q-lattices (Λ, ϕ) in C are equivalent to isogeny
classes of pairs (E, η) of an elliptic curve E = C/Λ and an Af -homomorphismη : Q2 ⊗ Af → Λ ⊗ Af ∼= (Λ ⊗ Z ) ⊗ Q,pp

Λ ⊗ Z = lim Λ/nΛ,
←
−
n

with

Λ/nΛ = E[n] the n-torsion of E.

Since the Q-lattices are not necessarily invertible, it is not required that η bean Af -isomorphism (cf. [122]).
The commensurability relation between Q-lattices corresponds to the isogeny
equivalence of isogeny classes, where two isogeny classes (E, η) and (E  , η  ) areequivalent if there is an isogeny
g : E = C/Λ → E  = C/Λ－ 88 －

such that η  = (g ⊗ 1) ◦ η on Q2 ⊗ Af .
( Note that such g from E to E  may be identiﬁed with its lift on C sendingΛ to Λ .
)It then follows that the isogeny equivalence can be identiﬁed with the quotient of M2 (Af ) × H± by the action of GL2 (Q), deﬁned as sending (ρ, z) to(gρ, g(z)).
To associate a quantum statistical mechanical system to the space Lt2 /C∗of commensurability classes of 2-dimensional Q-lattices up to scaling, it is conp
venient to use the description of Lt2 /C∗ as M2 (Z ) × H mod Γ = SL2 (Z). Thenconsidered as a noncommutative algebra of coordinates can be the (Hecke)
convolution algebra Cc (U/Γ2 ) of compactly supported, continuous functions
on the quotient U/Γ2 of the spacepp

U = {(g, ρ, z) ∈ GL+
2 (Q) × M2 (Z ) × H | gρ ∈ M2 (Z )}by the action of Γ2 = Γ × Γ deﬁned as(γ1 , γ2 ) · (g, ρ, z) = (γ1 gγ2−1 , γ2 ρ, γ2 (z)).
Note that(γ1 γ1 , γ2 γ2 ) · (g, ρ, z) = (γ1 γ1 g(γ2 )−1 γ2−1 , γ2 γ2 ρ, γ2 γ2 (z))= (γ1 , γ2 ) · [(γ1 , γ2 ) · (g, ρ, z)].
Endow this algebra Cc (U/Γ2 ) denoted by us so with the convolution productand the involution(f1 ∗ f2 )(g, ρ, z) =f1 (gs−1 , sρ, s(z))f2 (s, ρ, z),ps∈Γ\GL+
2 (Q),sρ∈M2 (Z )f ∗ (g, ρ, z) = f (g −1 , gρ, g(z)).
Masaka, check that(f1 ∗ (f2 ∗ f3 ))(g, ρ, z) =

f1 (gs−1 , sρ, s(z))(f2 ∗ f3 )(s, ρ, z) =ps∈Γ\GL+
2 (Q),sρ∈M2 (Z )f1 (gs−1 , sρ, s(z))

ps∈Γ\GL+
2 (Q),sρ∈M2 (Z )







f2 (st−1 , tρ, t(z))f3 (t, ρ, z) =pt∈Γ\GL+
2 (Q),tρ∈M2 (Z )pp+t∈Γ\GL+
2 (Q),tρ∈M2 (Z ) st∈Γ\GL2 (Q),stρ∈M2 (Z )f1 (g(st)−1 , stρ, st(z))f2 ((st)t−1 , tρ, t(z))f3 (t, ρ, z) =


p

p

+
t∈Γ\GL+
2 (Q),tρ∈M2 (Z ) s∈Γ\GL2 (Q),sρ∈M2 (Z )f1 (gt−1 s−1 , s(tρ), s(t(z)))f2 (s, tρ, t(z))f3 (t, ρ, z) =
(f1 ∗ f2 )(gt−1 , tρ, tz)f3 (t, ρ, z) = ((f1 ∗ f2 ) ∗ f3 )(g, ρ, z),pt∈Γ\GL+
2 (Q),tρ∈M2 (Z )－ 89 －

and as well(f ∗ )∗ (g, ρ, z) = f ∗ (g −1 , gρ, g(z)) = f ((g −1 )−1 , g −1 gρ, g −1 g(z)) = f (g, ρ, z).
The time evolution on Cc (U/Γ2 ) is given by, with det(g) > 0, t ∈ R,
σt (f )(g, ρ, z) = det(g)it f (g, ρ, z) = eit log det(g) f (g, ρ, z).
Note that σt+s (f ) = σt (σs (f )). Check that σt is a ∗-automorphism on
Cc (U/Γ2 ) as follows. Its being bijective is clear. Also,σt (f1 ∗ f2 )(g, ρ, z) = det(g)it (f1 ∗ f2 )(g, ρ, z) = det(gs−1 s)it (f1 ∗ f2 )(g, ρ, z)
det(gs−1 )it f1 (gs−1 , sρ, s(z)) det(s)it f2 (s, ρ, z),=ps∈Γ\GL+
2 (Q),sρ∈M2 (Z )= (σt (f1 ) ∗ σt (f2 ))(g, ρ, z),
and as well,
σt (f )∗ (g, ρ, z) = det(g −1 )it f (g −1 , gρ, g(z)),= det(g)it f (g −1 , gρ, g(z)) = σt (f ∗ )(g, ρ, z),with det(g)it det(g −1 )it = det(1)it = 1 in T.
p
For ρ ∈ M2 (Z ), letpGρ = {g ∈ GL+
2 (Q) | gρ ∈ M2 (Z )},and consider the Hilbert space Hρ = l2 (Γ\Gρ ).
Note that Gρ may not be a group? Because if g, h ∈ Gρ , then (gh)ρ =ppg(hρ) ∈ M2 (Z ) only if g ∈ Ghρ . Also, if g ∈ Gρ , then g −1 ρ ∈ M2 (Z )?
p
A 2-dimensional Q-lattice L = (Λ, ϕ) = (ρ, z) ∈ M2 (Z ) × H2 determines a
representation of the Hecke ∗-algebra Cc (U/Γ2 ) by bounded operators on Hρ ,as setting(πL (f )ξ)(g) =f (gs−1 , sρ, s(z))ξ(s).
s∈Γ\Gρ

Check that(πL (f1 ∗ f2 )ξ)(g) ==(f1 ∗ f2 )(gs−1 , sρ, s(z))ξ(s)s∈Γ\Gρ

f1 (gs−1 t−1 , tsρ, ts(z))f2 (t, sρ, s(z))ξ(s)s∈Γ\Gρ t∈Γ\GL+ (Q),tρ∈M2 (Zp )2=f1 (g(ts)−1 , tsρ, ts(z))

s =ts∈Γ\Gρ

=





f2 ((ts)s−1 , sρ, s(z))ξ(s)s∈Γ\Gρ

f1 (g(ts)−1 , (ts)ρ, (ts)(z))(πL (f2 )ξ)(ts)s =ts∈Γ\Gρ= πL (f1 )πL (f2 )ξ(g),－ 90 －

and as wellπL (f )∗ ξ, η = ξ, πL (f )η


=
ξ(t)
f (ts−1 , sρ, s(z))η(s)t∈Γ\Gρ

=s∈Γ\Gρ

f (ts−1 , sρ, s(z))ξ(t)η(s)s∈Γ\Gρ t∈Γ\Gρ

=





f (st−1 , tρ, t(z))ξ(t)η(s)s∈Γ\Gρ t∈Γ\Gρ=

πL (f ∗ )ξ(s)η(s) = πL (f ∗ )ξ, η,s∈Γ\Gρ

with f ∗ (g, ρ, z) = f (g −1 , gρ, g(z)).
In particular, if the Q-lattice L = (Λ, ϕ) is invertible, then we have (?)
Hρ ∼= l2 (Γ\M2+ (Z)).
In this case, the Hamiltonian implementing the time evolution σt is given by
the operator asHξm = log det(m)ξm .
Thus, in the special case of invertible Q-lattices, a positive energy representation
is obtained by πL . In general, for Q-lattices not commensurable to an invertible
one, the corresponding Hamiltonian is not bounded below.
The Hecke algebra Cc (U/Γ2 ) admits the maximal C ∗ -algebra completion
A2 = C ∗ (U/Γ2 ), which may be called the maximal Hecke C ∗ -algebra, where
the maximal C ∗ -algebra norm is deﬁned to be the supremum over all representations πL for L ∈ Lt2 .
The partition function for this GL2 system as A2 is given byZ(β) =m∈Γ\M2+ (Z)

=

∞

σ(k)
k=1

kβ

1
det(m)= ζ(β)ζ(β − 1),

where σ(k) = d|k d. This suggests the fact that two phase transitions takeplace at β = 1 and β = 2, expected.
∞
Remark. As in [116], note that the zeta function ζ(s) = n=1 n−s converges
for s > 1 by L. Euler, and is holomorphic for s ∈ C with Re(s) > 1, and isanalytically extended to the complex place C as analytic continuation expects = 1 as the only pole of order 1, by Riemann. The set of components of Sm(GL2 , H± ) is given by
π0 (Sm(GL2 , H± )) = Sm(GL1 , {±1}).

－ 91 －

At the level of the classical commutative spaces, this is given by the map
det ×sign : Sm(GL2 , H± ) → Sm(GL1 , {±}),
which implies the equality above by passing to the set π0 (X) of connectedcomponents of a space X.
KMS states of the GL2 system. The main result of [41] on the structure of
KMS states of the GL2 system is the following:
Theorem 3.9. The KMSβ states of the GL2 system as A2 have the following
properties:(1) In the range β ≤ 1, there are no KMSβ states.
(2) In the range β > 2, the set Eβ of extremal KMSβ states is given by theclassical Shi-mura variety as
Eβ ∼= Sm(GL2 , H± )± added).
= GL2 (Q)\GL2 (A)/C∗ (∼This shows that the extremal KMSβ states at suﬃciently low temperature
T = β1 are parameterized by the invertible 2-dimensional Q-lattices L, as ϕβ,L .
These extremal KMSβ states obtained are explicitly expressed as
ϕβ,L (f ) =

1
Z(β)


m∈Γ\M2+ (Z)1
f (1, mρ, m(z)),
det(m)β

where L = (ρ, z) is an invertible 2-Q-lattice.
Its being linear is clear. Also, ϕβ,L (1) = 1. But C ∗ (U/Γ2 ) should be
non-unital. It holds that ϕβ,L  = 1. Its being positive follows from
ϕβ,L (f ∗ ∗ f ) =
1
Z(β)
=
1
Z(β)

1
Z(β)


m∈Γ\M2+ (Z)m∈Γ\M2+ (Z)m∈Γ\M2+ (Z)1
det(m)β1det(m)β1(f ∗ ∗ f )(1, mρ, m(z)) =det(m)βf ∗ (s−1 , smρ, sm(z))f (s, mρ, m(z))
ps∈Γ\GL+
2 (Q),smρ∈M2 (Z )f (s, mρ, m(z))f (s, mρ, m(z)) ≥ 0.
ps∈Γ\GL+
2 (Q),smρ∈M2 (Z )－ 92 －

Hence, ϕβ,L is a state on A2 . For its being KMSβ , observe that
ϕβ,L (f ∗ σt (h)) =

1
Z(β)




m∈Γ\M2+ (Z)f (s−1 , smρ, sm(z)) det(s)it h(s, mρ, m(z)),

ps∈Γ\GL+
2 (Q),smρ∈M2 (Z )ϕβ,L (σt (h) ∗ f ) =1
det(m)β

1
Z(β)





m∈Γ\M2+ (Z)1
det(m)βdet(s−1 )it h(s−1 , smρ, sm(z))f (s, mρ, m(z)).
ps∈Γ\GL+
2 (Q),smρ∈M2 (Z )Again, a possible choice of the holomorphic path between those may be givenasβ−ssϕβ,L (f ∗ σt (h) + σt (h) ∗ f ), 0 ≤ s ≤ β.
ββ
As the temperature T = β1 rises, and β → 2 + 0 from above, all the diﬀerent
phases of the system merge (into one state). This is a strong evidence for a fact
(or guess?) that in the intermediate range 1 < β ≤ 2, the system have only asingle KMSβ state.
The symmetry group of A2 = C ∗ (U/Γ2 ) including both automorphisms andendomorphisms can be identiﬁed with the groupp

GL2 (Af ) = GL+
2 (Q)GL2 (Z ),pwhere the group GL2 (Z ) acts by automorphisms of A2 , so thatθγ (f )(g, ρ, z) = f (g, ργ, z).
pThese symmetries are related to the group AutV (V) = GL2 (Z )/{±1} of decktransformations of coverings of modular curves.
Note that
θγ1 γ2 (f )(g, ρ, z) = f (g, ργ1 γ2 , z)
= θγ2 (f )(g, ργ1 , z) = θγ1 (θγ2 (f ))(g, ρ, z).
The novelty of the GL2 system with respect to the BC case is that GL+
2 (Q)acts as well by endomorphisms of A2 , so thatpf (g, ρ det(m)−1 m, z) ρ ∈ mM2 (Z ),θm (f )(g, ρ, z) =0
otherwise.

－ 93 －

Check thatpf (g, ρ det(m1 m2 )−1 m1 m2 z) ρ ∈ m1 m2 M2 (Z ),θm1 m2 (f )(g, ρ, z) =0
otherwisepθm2 (f )(g, ρ det(m1 )−1 m1 , z) ρm1 ∈ m2 M2 (Z ),=0otherwisepθm1 (θm2 (f ))(g, ρ, z) ρ ∈ m1 M2 (Z ),=0otherwise.
(But if so, does such happen?)
The subgroup Q∗ of GL2 (Af ) acts by inner endomorphisms, and hence thegroup S of symmetries of the set Eβ of extremal KMSβ states on A2 has theform S = Q∗ \GL2 (Af ).
In the case of the set E∞ of KMS∞ states on A2 , deﬁned as weak limits,deﬁning the action of GL+
2 (Q) on the set is more subtle. In fact, the action
θ deﬁned above does not directly induce a non-trivial action on E∞ . However,there is a non-trivial action induced by the action as ρ on the sets Eβ of KMSβstates for suﬃciently large β. This action on ϕ ∈ E∞ is obtained by the warming
up and cooling down procedure, as given before. That is,(ρ∗ ϕ)(a) =tr(πϕ (ρ(a))e−βK ),→∞ tr(πϕ (ρ(1))e−βK )

lim
−1

β=T

a ∈ A2 .

Finally, the Galois action on the extremal KMS∞ states on A2 is describedby the following result of [41]:Theorem 3.10. There exists an arithmetic subalgebra A2,Q of the unbounded
multiplier algebra M ub (A2 ) of the C ∗ -algebra A2 , such that the following holds:
For ϕ∞,L ∈ E∞ with L = (ρ, τ ) generic, the values on arithmetic elementsof A2,Q satisfy ϕ∞,L (A2,Q ) ⊂ Fτ , where Fτ is the embedding of the modular ﬁeld
F into C given by evaluation at τ ∈ H.
The values of ϕ∞,L (A2,Q ) generate Fτ .
There is an isomorphism∼=

θϕ∞,L : Gal(Fτ /Q) −−−−→ Q∗ \GL2 (Af ),that intertwines the Galois action on the values of the KMS∞ states with theaction of symmetries, so that for γ ∈ Gal(Fτ /Q), the diagram commutes
θϕ∞,L (γ)

A2,Q ⊂ M ub (A2 )⏐⏐ϕ∞,L 

−−−−−−→

A2,Q ⊂ M ub (A2 )
⏐
⏐ϕ∞,L


ϕ∞,L (A2,Q ) ⊂ Fτ ⊂ C

−−−−→

γ

ϕ∞,L (A2,Q ) ⊂ Fτ ⊂ C.

－ 94 －

Recall that the modular ﬁeld F in the statement above is deﬁned to be theﬁeld of modular functions over QAb , namely that is the union of the ﬁelds FNof modular functions of level N rational, over the cyclotomic ﬁeld Q(ζn ), that1
1
is, such that the q-expansion in powers of q N = e N 2πiτ has all coeﬃcients in
1
2πi
Q(e N ).
It has explicit generators given by the Fricke functions ([146], [98]). Let p
be the Weierstrass p-function. It gives the parameterization of the elliptic curveby the quotient C/(Z + Zτ ):y 2 = 4x3 − g2 (τ )x − g3 (τ ),z → (1, p(z; τ, 1), p (z; τ, 1)) = (1, x, y)(corrected). Then the Fricke functions are homogeneous functions of 1-dimensionallattices of weight zero of the form:
fv (τ ) = −27 35

g2 (τ )g3 (τ )p(λτ (v); τ, 1)Δ(τ )parameterized by v ∈ Q2 /Z2 , where Δ = g23 − 33 g32 (= 0) is the discriminant(up to a scalar) for the quadratic equation with respect to the cubic equation
4x3 − g2 x − g3 = 0, and λτ (v) = v1 τ + v2 .
Remark. (Added). Recall from [98] the following. A lattice L in C is a Z-rank
two subgroup of C, such that L = Zw1 + Zw2 and C = Rw1 + Rw2 . May assume
w1
1
that Im( ww2 ) > 0, that is, w2 ∈ H. An elliptic function f with respect to L is ameromorphic function on C which is periodic mod L, so that f (z + w) = f (z)for any z ∈ C and w ∈ L. An elliptic function can be viewed as a meromorphic
function on the 2-torus C/L. The Weierstrass function is deﬁned to be


1
1
1
p(z) = 2 +− 2 ,z(z − ω)2ωω
where the sum is taken over the set of all non-zero periods for elliptic functions
mod L, so that w = nw1 + mw2 for (n, m) ∈ Z2 \ {(0, 0)}.
The Laurent series expansion of p(z) is given byp(z) =∞
1+(2n + 1)s2n+2 (L)z 2n ,z 2 n=1

sm (L) =

 1
.
ωm

ω=0

By diﬀerentiating term by term,∞1p (z) = − 3 +(2n + 1)(2n)s2n+2 (L)z 2n−1 .
z
n=1

If we set g2 (L) = 60s4 (L) and g3 (L) = 140s6 (L), then it holds thatp (z)2 = 4p(z)3 − g2 (L)p(z) − g3 (L).

Remark. (Added). As in [116], the Cardano formula is as follows. To solve
a0 x3 + a1 x2 + a2 x + a3 = 0, ﬁrst put b1 = 9a0 a1 a2 − 2a31 − 27a20 a3 and b2 =－ 95 －

a21 − 3a0 a2 . Next solve t2 − b1 t + b32 = 0 to have solutions t+ and t− . Then thesolution for the ﬁrst equation is obtained as
3

3

2
2
−a1 + ζ3 t±+ ζ32 t∓.
3a0

In the case of 4x3 − g2 x − g3 = 0, we haveb1 = 27 · 42 g3

b2 = 3 · 4g2 .

The discriminant for the above quadratic equation of t is given byb21 − 4b32 = 36 44 g32 − 33 44 g23 = 33 44 (27g32 − g23 ) = −33 44 Δ.
Hencet± = 27 · 42 g3 ±#
−33 44 Δ = 3 · 42 9g3 ± 3(g23 − 27g32 ) .
3

3

1
2
2
(ζ3 t±+ ζ32 t∓).
The solution of the cubic equation is obtained as 12
For generic τ of L = (ρ, τ ) such that j(τ ) is transcendental, evaluation of
the modular functions at the point τ ∈ H gives an embedding of Fτ into C.
There is a corresponding isomoprhicm ττ from Gal(Fτ /Q) onto Aut(F ). The
isomorphism θϕ∞,L of Gal(Fτ /Q) onto Q∗ \GL2 (Af ) is given by

θϕ∞,L (γ) = ρ−1 θτ ρ,

γ ∈ Gal(Fτ /Q).

In fact, the automorphism group Aut(F ) has a completely explicit description,
due to a result of Shimura [146], so that Aut(F ) can be identiﬁed with thequotient Q∗ \GL2 (Af ).
We are going to explain what is the arithmetic subalgebra A2,Q of A2 an in
the theorem above, in the following.
We consider continuous functions f (g, ρ, z) (mod Γ2 ) on the quotient U/Γ2 ,with ﬁnite support with respect to the variable g ∈ Γ\GL+
2 (Q).
pDeﬁne f(g,ρ) (z) = f (g, ρ, z), so that f(g,ρ) ∈ Cc (H). Let pN : M2 (Z ) →M2 (Z/N Z) be the canonical projection. It is said that the function f is of levelp2
N if f(g,ρ) = f(g,pN (ρ)) for any (g, ρ) ∈ GL+
2 (Q)×M2 (Z ) mod Γ . If f is of level
N , then f is determined by the functions f(g,m) ∈ Cc (H) for m ∈ M2 (Z/N Z).
Note that the invariance asf (gγ, ρ, z) = f (g, γρ, γ(z)),γ ∈ Γ, (g, ρ, z) ∈ U

implies the following identity:f(g,m)γ = f(g,m) ,γ ∈ Γ(N ) ∩ g −1 Γg(corrected as action), so that f is invariant under a congruence subgroup.

－ 96 －

Note that(gγ, ρ, z)γ ≡ (1, γ) · (gγ, ρ, z) = (1gγγ −1 , γρ, γ(z)) = (g, γρ, γ(z)).
Also, with γ = g −1 γ  g with γ  ∈ Γ,
f(g,m)γ (z) = f (gγ −1 , γm, γ(z)) = f (gg −1 (γ  )−1 g, γm, γ(z))= f (g, γm, γ(z)) = f (gγ, m, z) = f (γ  g, m, z)= f (g, m, z) = f(g,m) (z)(mod Γ × Γ).
Deﬁne elements f of the arithmetic subalgebra A2,Q of A2 = C ∗ (U /Γ2 ), as
characterized by the following properties (1) to (3):(1) The support of f (g, ρ, z) with respect to g in Γ\GL+
2 (Q) is ﬁnite.
(2) Each f has a ﬁnite level N with f(g,m) ∈ F for m ∈ M2 (ZN ).
(3) The following cyclotomic condition is satisﬁed:


p ∗
u 0
f(g,α(u)m) = cyc(u)f(g,m) , diagonal g ∈ GL+
(Q),
u
∈
(Z
)
,
α(u)
=
,
2
0 1
where cyc : (Z )∗ → Aut(F ) denotes the action of the Galois group (Z )∗ ∼=1
1
Gal(QAb /Q) on the coeﬃcients of the q-expansion in powers of q N = e N 2πiτ .
If we take only the ﬁrst two conditions (1) and (2) above, it would followthat the algebra A2,Q contains the cyclotomic ﬁeld QAb ⊂ C, but this would
prevent the existence of states satisfying the desired Galois property. In fact,
if the subalgebra contains scalar elements in Qcyc , then the sought of Galois
property would not be compatible with the C-linearity of states. The cyclotomic
condition then forces the spectrum of the elements of A2,Q to contain all Galois
conjugates of any root of unity that appears in the coeﬃcients of the q-series,
so that elements of A2,Q can not be scalars.
The algebra A2,Q deﬁned by the properties above is a subalgebra of theunbounded multiplier algebra of A2 , which is globally invariant under the groupof symmetires Q∗ \GL2 (Af ).
p

3.5

p

Quadratic ﬁelds

The ﬁrst essential step in the direction of generalizing the Bost-Connes system
to other number ﬁelds and exploring a possible approach to the explicit classﬁeld theory problem via noncommutative geometry is the construction of a QSM
system that√recovers the explicit class ﬁeld theory for imaginary quadratic ﬁelds
as K = Q( −d) for d a positive integer.
Presented in this section is such a construction based on the work of Connes,
Marcolli, and Ramachadran [48]. Then discussed are some ideas about the unsolved problem of real quadratic ﬁelds, based on the real multiplication programof Manin ([109], [110]).
Imaginary quadratic ﬁelds and their lattices. Let O = Z + Zτ be thering of integers of an imaginary quadratic ﬁeld K = Q(τ ), with the imbedding－ 97 －

of K into C with τ ∈ H. Let Nm : K∗ → Q∗ denote the norm map. Then
Nm(x) = det(g) for g = qτ (x) the image under the embedding qτ : K∗ →GL2 (Q) determined by the choice of τ . The image of the embedding qτ ischaracterized by the propertyqτ (K∗ ) = {g ∈ GL+
2 (Q) | g(τ ) = τ }.
A quantum statistical mechanical (QSM) system for imaginary quadraticﬁelds K can be constructed by considering commensurability classes of 1-dimensional
K-lattices.
Deﬁnition 3.11. For K an imaginary quadratic ﬁeld, a 1-dimensional K-lattice(Λ, ϕ) is deﬁned to be a ﬁnitely generated O-submodule of C, such that Λ⊗O K ∼=K, together with a morphism of O-modules: ϕ : K/O → KΛ/Λ.
A 1-dimensional K-lattice is invertible if ϕ is an isomorphism of O-modules.
A 1-dimensional K-lattice is viewed as a 2-dimensional Q-lattice. This plays
an important role in the notion of commensurability below.
Deﬁnition 3.12. Two 1-dimensional K-lattices (Λ1 , ϕ1 ) and (Λ2 , ϕ2 ) are commensurable if KΛ1 = Kλ2 and ϕ1 = ϕ2 modulo Λ1 + Λ2 .
It turns out that two 1-dimensional K-lattices are commensurable if and only
if their underlying Q-lattices are commensurable.
We can give a more explicit description of the data of 1-dimensional Klattices. Namely, the data (Λ, ϕ) of a 1-dimensional K-lattice L are equivalentpp
to the data (ρ, s) of an element ρ ∈ Z ⊗ O ≡ O and s ∈ A∗K /K∗ , module thepaction of (O )∗ given by x·(ρ, s) = (x−1 ρ, xs). Thus, the space of 1-dimensional
K-lattices is identiﬁed withp

O ×(Op )∗ (A∗K /K∗ ).
Let A◦K = AK,f × C∗ denote the subset of adèles of K with non-trivial
archimedean component. There is an identiﬁcation of the set Lt1,K of commensurability classes of 1-dimensional K-lattices (not up to scale) with the spaceA◦K /K∗ . In turn, the set Lt1,K /C∗ of commensurability classes of 1-dimensional
K-lattices up to scaling can be identiﬁed with the quotientp

O /K∗ = AK,f /K∗ ,pwhere the left hand side stands for the equivalence classes of elements of O
module the equivalence relation given by the partially deﬁned action of K∗ .
Then noncommutative algebras of coordinates associated to those quotient
spaces can be introduced. The procedure is analogous to that in the BostConnes case and the GL2 case. Consider the convolution algebra Cc (Lt1,K )and its C ∗ -algebra completion C ∗ (Lt1,K ), given as the groupoid algebra andC ∗ -algebra of the commensurability relation. Regard the elements of these
algebras as functions f (L, L ) of pairs (L, L ) = ((Λ, ϕ), (Λ , ϕ )) ∈ Lt21,K , with－ 98 －

the convolution product induced by the equivalence relation. By construction,
this groupoid is a sub-groupoid of the groupoid of commensurability classes of
2-dimensional Q-lattices. If we take the quotient Lt1,K /C∗ by scaling, we deﬁne
A1,K as C ∗ (Lt1,K /C∗ ), which is still a groupoid algebra in this case, unlike what
happens in the GL2 case (cf. [48]). The C ∗ -algebra A1,K is unital as in the caseof the BC system.
The time evolution of the GL2 system induces the natural time evolution on
A1,K given byσt (f ) = N it f = eit log N f, N : K∗ → Q∗ .
The quantum statistical mechanical (QSM) system (A1,K , σ, R) has properties that are intermediated between the BC system and the GL2 system in somesense.
Note that invertible 1-dimensional K-lattices L = (Λ, ϕ) give rise to positiveenergy representations of A1,K on the Hilbert space l2 (GL ), where GL is the setof elements of the groupoid Lt1,K /C∗ with source L = (Λ, ϕ). The set Lt∗1,K /C∗of invertible 1-dimensional K-lattices up to scale can be identiﬁed with the idèleclass group CK /DK = A∗K,f /K∗ .
p
Recall that Af = Z ⊗ Q is the ring of ﬁnite adèles of Q, andppAK,f = Af ⊗ K = Z ⊗ Q ⊗ K ∼=Z ⊗Kpis the ring of ﬁnite adèles of K, so that A∗K,f containing (Z )∗ ⊗ K∗ is the groupof ﬁnite idèles of K, and A∗K,f /K∗ is the group of ﬁnite idèle classes of K.
For an invertible K-lattice L, the set GL can be identiﬁed with the set ofideals J of O = OK . Then the Hamiltonian implementing the time evolution
has the form as HeJ = log n(J)eJ , where n(J) denotes the norm of J. Thus,
the partition function of the system (A1,K , σ, R) is the J. W. R. Dedekind zetafunction ζK (β) of K (1877), so that



1
1
Z(β) =≡ ζK (β) = Πpn(J)β1 − n(p)−βJ ⊂ O: ideal(cf. [116]). It is expected that this is the natural generalization of the Riemannzeta function ζ(β) = ζQ (β) to other number ﬁelds.
If K = Q, then O = Z andζQ (β) =nZ ⊂ Z : ideal

∞

1
1
=
.
β
n(nZ)
nβ
n=1

The symmetry group of the Lt1,K system is the group A∗K,f of idèle classes,with the subgroup K∗ acting on A1,K by inner automorphisms, so that theinduced action on the KMS states on A1,K is given by the idèle class group
A∗K,f /K∗ = CK /DK . As in the case of the GL2 system, this group of symmetriespincludes an action by endomorphisms. Only the subgroup (O )∗ /O∗ with O∗∗
the group of units of O acts by automorphisms, while the (other) full action of－ 99 －

A∗K,f /K∗ involves endomorphisms. In particular, this shows the appearance ofthe class number of K, as in the following commutative diagram:
1 −−−−→p(O )∗ /O∗
⏐
⏐∼
=

−−−−→

A∗K,f /K∗
⏐
⏐∼
=

−−−−→

Cl(O)
⏐
⏐∼
=

−−−−→ 1

1 −−−−→ Gal(KAb /H) −−−−→ Gal(KAb /K) −−−−→ Gal(H/K) −−−−→ 1
where H is the Hilbert class ﬁeld of K, that is, the maximal abelian unramiﬁed
extension of K. The ideal class group Cl(O) is naturally isomorphic to the
Galois group Gal(H/K). The case of class number one is analogous to the BC
system as already observed (cf. [79]).
The arithmetic subalgebra A2,Q of A2 for the GL2 system induces, by restriction to the corresponding sub-grouppoid, a natural choice of a rational
subalgebra A√
1,K,Q of A1,K of the system (A1,K , σ, R) for the imaginary quadraticﬁeld K = Q( −d).
Note that, because of the fact that the variable z ∈ H in the coordinatesof the GL2 system is now only aﬀecting ﬁnitely many values, indexed by theelements of the ideal class group Cl(O), it is obtained that A1,K,Q is a subalgebraof A1,K as in the BC system case, though non involutive, and not just an algebraof unbounded multipliers of A1,K , as in the GL2 case.
With the choice of such an arithmetic subalgebra, it is obtained as a result
analogous to the theorem above that (cf. [48]):
Theorem 3.13. The system (A1,K , σ, R) has the following properties:(1) In the range 0 < β = T1 ≤ 1 there is a unique KMSβ state.
(2) For β = T1 > 1, the set Eβ of extremal KMSβ states is parameterized byinvertible 1-dimensional K-lattices up to scale, so thatEβ ∼= Lt1,K /C∗ ∼= A∗K,f /K∗ ∼= CK /DK ≡ (A∗K /K∗ )/(CK )0 ,with a free and transitive action of the idèle class group G = CK /DK of K.
(3) The set E∞ of extremal KMS∞ states, as weak limits of KMSβ states, isalso given by replacing Eβ in the parameterization above. The extremal KMS∞
states, evaluated on the arithmetic subalgebra A1,K,Q , generate the maximal Abelextension KAb . The class ﬁeld theory isomorphism θ : Gal(KAb /K) → A∗K,f /K∗intertwines the action AK,f /K∗ as symmetries of the system with the Galois
action of Gal(KAb /K) on the image of A1,K,Q under the extremal KMS∞ states
ϕ∞,L , so that
ϕ∞,L
A1,K,Q ⊂ A1,K −−−−→ KAb ⊂ C
⏐
⏐
⏐ ∗
⏐
∗
α∈Gal(KAb /K)
θ(α)∈AK,f /K
ϕ∞,L

A1,K,Q ⊂ A1,K −−−−→ KAb ⊂ C√
Now note that for K = Q(τ ), with τ = i d, the classes of 1-dimensional
K-lattices L, viewed as 2-dimensional Q-lattices L = (ρ, z), correspond to onlyﬁnitely many values of z ∈ H ∩ K. Moreover, these points no longer satisfy the－ 100 －

generic condition. In fact, for a CM point z ∈ H ∩ K, evaluation of elementsin the modular ﬁeld F at z no longer gives an embedding. In this case, theimage Fz (corrected) in C a copy of the maximal Abel extension KAb of K, sothat F → Fz = KAb ⊂ C (cf. [146]). The explicit action of the Galois group
Gal(KAb /K) is obtained through the action of automorphisms of the modular
ﬁeld F via Shimura reciprocity ([146]), as described in the following diagramwith exact rows:
1 −−−−→

A∗K,f /K∗

⏐
⏐

u

−−−
−→ Gal(KAb /K) −−−−→ 1
θ −1






u

1 −−−−→ K∗ −−−−→ GL1 (AK,f ) = A∗K,f −−−−→ Gal(KAb /K) −−−−→ 1

⏐
⏐


⏐
⏐qτ

N det ◦qτ
1 −−−−→ Q∗ −−−−→

GL2 (Af )

σ

−−−−→

Aut(F )−−−−→ 1

where qτ is the embedding determined by the choice of τ ∈ H with K = Q(τ ).
Thus, the explicit Galois action is given byu(γ)(f (τ )) = f σ(qτ (γ)) (τ ),f ∈ F, f α = α(f ), α ∈ Aut(F ).

That picture provides the intertwining between the symmetries of the quantum statistical mechanical system and the Galois action on the image of stateson the elements of the arithmetic subalgebra.
The structure of KMS states at positive temperatures T = β1 > 0 (which go
to 0 as β → ∞) is similar to the Bost-Connes system case (cf. [48] for moredetails).
Real multiplication. The next fundamental question in the direction of generalizations of the BC system to other number ﬁelds is how to approach themore diﬃcult case of real quadratic ﬁelds. Given is a brief outline of some ideas
on real multiplication of Manin [109], [110], and suggested is how the ideas may
be combined with the GL2 system described above.
Developed by Manin [110] is a theory of real multiplication for noncommutative tori, aimed at providing a setting, within noncommutative geometry, inwhich to treat the problem of abelian extensions of real quadratic ﬁelds on asomewhat similar footing as the known case of imaginary quadratic ﬁelds, forwhich the theory of elliptic curves with complex multiplication provides theright geometric setup.
The ﬁrst translation in the dictionary developed in [110] between elliptic
curves with complex multiplication and noncommutative tori with real multiplication is given by a parallel between lattices Λ and pseudo-lattices P L in
C:
(C, Lattice Λ)  (R, Pseudo-Lattice P L).
A pseudo lattice P L means the date (G, ψ) (as (Z2 , ψ : Z2 → R ⊂ C) up toisomorphism, rotation, and orientation) of a free abelian group G of rank two,

－ 101 －

with an injective homomorphism ψ : G → C, such that the image lies in anoriented real line.
This aims at generalizing the well known equivalence between the category of elliptic curves as Eτ = C/(Z + τ Z) and the category of 2-dimensionallattices as Λτ = Z + τ Z with τ ∈ H, realized by the period functor, to asetting that includes noncommutative 2-tori Aθ as a deformation of Eq withq = exp(2πiτ ) → exp(2πiθ) ∈ S 1 .
Lattices Λτ  Elliptic Curves Eτ =⇒ NC torus Aθ .
As seen previously, any 2-dimensional lattice has the form Λτ = Z + Zτ forτ ∈ C \ R, up to isomorphism, and non-trivial morphisms between such lattices
are given by the action of matrices of M2 (Z) by fractional linear transformations.
Thus, the moduli space of lattices up to isomorphism is given by the quotientof P1 (C) \ P1 (R) by P GL2 (Z).
Lattices/M2 (Z) ≈ [P1 (C) \ P1 (R)]/P GL2 (Z).
A pseudo-lattice has the form P Lθ = Z+Zθ for θ ∈ R\Q up to isomorphism,
and non-trivial morphisms of pseudo-lattices are given by matrices of M2 (Z),also acting on P1 (R) by fraction linear transformations. The moduli space ofpseudo-lattices is given by the quotient of P1 (R) by the action of P GL2 (Z).
Since the action does give rise to a bad classical quotient, the moduli spaceshould be treated as a noncommutative space.
Pseudo-Lattices/M2 (Z) ≈ P1 (R)/P GL2 (Z)  NC spaces.
As described in the previous chapter, the resulting space represents a component in the (noncommutative) boundary of the classical moduli space of elliptic
curves, which parameterizes the degenerations from lattices to pseudo-lattices,which are invisible to the usual algebro-geometric setting.
The cusp, i.e., the orbit of P1 (Q), corresponds to the degenerate case where
the image in C of the rank two free abelian group has rank one. This gives
another translation in the dictionary, regarding the moduli spaces asXΓ = H/Γ = H/P SL2 (Z)  C(P1 (R))  P GL2 (Z),with XΓ = (H2 ∪ P1 (Q))/Γ with P1 (Q) = Q+ ⊂ P1 (R) = R+ ≈ S 1 .
In the correspondence between pseudo-lattices P L and noncommutative toriAθ , the group of invertible morphisms of P L corresponds to isomorphisms of
NC tori, realized by strong Morita equivalences. In this context, a morphism is
not given as a morphism of algebras, but as a map of the category of modules,
obtained by tensoring with bimodules. The notion of Morita equivalences asmorphisms ﬁts into the more general context of correspondences for operator
algebras as in [27, V. Appendix B] as well as in the algebraic approach tononcommutative spaces of [143].
The category of noncommutative tori is deﬁned by considering as morphisms
the isomorphism classes of projective C ∗ -modules Pθ (corrected) that are the－ 102 －

ranges of projections of matrix algebras over NC tori. The functor F from NC
tori to pseudo-lattices P L is then given on objects by
F = K0 : T2θ = Aθ → (K0 (Aθ ) ∼= K0 (Aθ ), HC0 (Aθ ), τ : K0 (Aθ ) → HC0 (Aθ ))(cf. [109, 3.3], [110, 1.4]), where Aθ means the dense subalgebra of the C ∗ algebra Aθ , HC0 (Aθ ) = Aθ /[Aθ , Aθ ], with τ the universal trace, and the orientation is determined by the cone of positive elements of K0 (Aθ ). On morphisms,
the functor is given by
F = ⊗Aθ Mθ,θ : [Pθ ] → [Pθ ⊗Aθ Mθ,θ ],where Mθ,θ are the Aθ -Aθ (or Aθ -Aθ ) bimodules, constructed by Connes in[22]. A crucial point in this deﬁnition is the fact that, for noncommutative 2tori, ﬁnite projective modules are classiﬁed by the value of a unique normalized
trace (cf. [22], [140]).
The functor F as K0 on objects and as the bimodule tensor product onmorphisms is weaker than an equivalence of category. For instance, it maps
trivially all ring homomorphisms on NC tori, that act trivially on K0 . However,this correspondence is suﬃcient to develop a theory of real multiplication fornoncommutative tori, parallel to the theory of complex multiplication for elliptic
curves (as tori) (cf. [109], [110]).
For rank two lattices Λ or elliptic curves C/Λ, the typical situation is that
End(Λ) = Z, but there are exceptional lattices Λ, for which End(Λ) strictly
contains Z. In this case, there exists a complex quadratic ﬁeld K such that Λ isisomorphic to a lattice in K. More precisely, the endomorphism ring End(Λ) is
given by Z + f O, where O is the ring of integers of K and the integer f ≥ 1 iscalled the conductor. Such lattices are said to have complex multiplication. The
elliptic curve EK with EK (C) = C/O is endowed with a complex multiplication
map, given on the universal cover by x → ax for a ∈ O.
Similarly, there is a parallel situation for pseudo-lattices P L, that End(P L) Z happens when there exists a real quadratic ﬁeld K such that P L is isomorphic to a pseudo-lattice contained in K. In this case, it also holds that
End(P L) = Z + f O. Such pseudo-lattices are said to have real multiplication. The pseudo-lattices P Lθ with real multiplication correspond to values ofθ ∈ R \ Q that are quadratic irrationals. These are characterized by having
eventually periodic, continued fraction expansion. The real multiplication map
is given by tensoring with bimodules, so that in the case of θ with periodic, continued fraction expansion, there is an element g ∈ P GL2 (Z) such that gθ = θ,to which an Aθ -Aθ bimodule E can be associated.
An analogue of isogenies for noncommutative tori is obtained by consideringmorphisms of pseudo-lattices as P Lθ → P Lnθ , which correspond to a Morita
given by Aθ viewed as an Anθ -Aθ bimodule, where Anθ can be embedded intoAθ by sending unθ to unθ and vnθ to vθ .
Suppose that vnθ unθ = e2πinθ unθ vnθ in Anθ . Then, in Aθ ,= · · · = e2πinθ unθ vθ .
vθ unθ = e2πiθ uθ vθ un−1θ－ 103 －

By considering isogenies, enriched can be the disctionary between the moduli
space of elliptic curves and moduli space of Morita equivalent noncommutative
tori. This leads to consider the whole tower of modular curves parameterizing
elliptic curves with level structure and the corresponding tower of noncommutative modular curves described in the previous section (cf. [112]).
H/G  C(P1 (R) × P)  Γ,
for Γ = P GL2 (Z) or P SL2 (Z) and G a ﬁnite index subgroup of Γ, with P = G\Γthe coset space.
In the problem of constructing the maximal abelian extension of complex
quadratic ﬁelds, the method is based on evaluating at the torsion points of theelliptic curve EK a power of the Weierstrass function. This means consdiering
the corresponding values of a coordinate on the projective line EK /O∗ . The
analogous object in the noncommutative setting, replacing the projective line,
should be a crossed product of functions on K by the ax + b group with a ∈ O∗
and b ∈ O, for a real quadratic ﬁeld K (cf. [110]), that is, likeCc (K)  (O  O∗ ).
A diﬀerent method to construct abelian extensions of a complex quadraticﬁeld K is based on Stark numbers. Following [110], if f is an injective homomorphism of a free abelian group Λ of rank two to a 1-dimensional complex
vector space V , and λ0 ∈ Λ ⊗ Q, then consider the zeta functionζ(Λ, λ0 , s) =


λ∈Λ

1
.
|f (λ0 + λ)|2sNote that f on Λ may be extended to f ⊗ id on Λ ⊗ Q.
It is proved by Stark [147] that the numbers deﬁned byS(Λ, λ0 ) = exp ζ  (Λ, λ0 , 0)dζ) are algebraic units generating certain abelian extensions of K.
(as ζ  = ds
The argument in this case is based on a direct computational tool as the Kronecker second limit formula and upon reducing the problem to the theory ofcomplex multiplication. There is no known independent argument for the Stark
conjectures, while the analogous question is open for the case of real quadraticﬁelds.
For a real quadratic ﬁeld K, instead of zeta functions ζ(Λ, λ0 , s) of the form
above, the conjectural Stark units are obtained from zeta functions of the following form (as in [110]):ζ(L, l0 , s) = sgn(l0g )N (a)ssgn(l0 + l)g,|N (l0 + l)|s−1l ∈ L mod La

where l → lg means the action of the nontrivial element g in Gal(K/Q), andN (l) = llg , and the element l0 ∈ O is chosen so that the ideals a = (L, l0 ) and－ 104 －

l0 a−1 are coprime with La−1 , and the summation over l ∈ L in the formula isrestricted by taking only one representative from each coset class (l0 + l)ε, forunits ε satisfying (l0 + L)ε = l0 + L, i.e., ε ≡ 1 mod La−1 . Then the Stark
numbers are deﬁned to be
S(L, l0 ) = exp ζ  (L, l0 , 0).
Developed by Manin in [110] is an approach to the computation of the zetafunctions ζ(L, l0 , s) based on a version of theta functions for pseudo-lattices,which are obtained by averaging theta constants of complex lattices along geodesics
with ends at a pair of conjugate quadratic irrationals θ and θ in R \ Q.
This procedure ﬁts into a general philosophy, according to which recasted
can be part of the arithmetic theory of modular curves in terms of the noncommutative boundary as the C ∗ -algebra crossed product as C(P1 (R) × P)  Γ
by studying the limiting behavior as τ → θ ∈ R \ Q along geodesics, or someaveraging along such geodesics. In general, a non-trivial quantum system canbe obtained when approaching θ along a path that corresponds to a geodesic in
the modular curve that spans a limiting cycle, which is the case precisely when
θ as the end point is a quadratic irrational. An example of this type of behavior
is known as the theory of limiting modular symbols, developed by M and M[112] and [113].
Pseudo-lattices and the GL2 system.
The noncommutative space of the GL2 system
Smnc (GL2 , H± ) = GL2 (Q)\(M2 (Af ) × H± )also admits a compactiﬁcation, given by adding the boundary P1 (R) to H± , asin the noncommutative compactﬁcation of modular curves ([112]),Smnc (GL2 , H± ) = GL2 (Q)\(M2 (Af ) × P1 (C)) = GL2 (Q)\M2 (A)/C∗ ,where P1 (C) = H± ∪ P1 (R).
p
Note that A = Af × R denotes the adèles of Q, with Af = Z ⊗ Q of
the ﬁnite adèles, and that H± ≈ GL2 (R)/C∗ in P1 (C). (cf. [150]). Indeed, in
M2 (R), there is an identiﬁcation as


a b
g=
= (z, w) ∈ C2 , with z = (a, c), w = (b, d) ∈ C,c dso that g ∈ GL2 (R) if and only if z and w are linearly independent over R, andg ∈ GL2 (R) if and only if they are linearly dependent on R. In the latter case,there is t ∈ R such that w = tz, and the elements (z, tz) = z(1, t) for z ∈ C∗
correspond to R in (C2 \ {02 })/C∗ = P1 (C), so that R ∪ {0} ≈ S 1 ≈ P 1 (R),with 0 ∈ M2 (R). In the ﬁrst case, the component w can not be written as tz
for any R, but can be written as hz for some h ∈ H± .
That compactiﬁcation corresponds to adding to the space Lt2 of commensurability classes of 2-dimensional Q-lattices the set pL2 of (classes of) pseudolattices in the sense of [110], considered together with the Q-structure. It then

－ 105 －

seems that the real multiplication program of Manin may ﬁt in with the boundary of the noncommutative space of the GL2 system. The crucial question inthis respect becomes the construction of an arithmetic algebra accociated to thenoncommutaive modular curves. The results illustrated in the previous section,
regarding identiﬁes involving modular forms at the boundary of the classicalmodular curves and limiting modular symbols, as well as the still mysterious
phenomenon of quantum modular forms identiﬁed by Zagier, implies the factthat there should exist some class of objects replacing modular forms, whenpushed to the boundary.
Regarding the role of modular forms, note that, in the case of the noncommutative compactiﬁcation of modular curves given above, the dual system canbe considered. This is given as a C ∗ -algebra bundle as
Lt2 = GL2 (Q)\M2 (A).
On this dual space, modular forms appear naturally instead of modular functions, and the algebra of coordinates contains the modular Hecke algebra ofConnes-Moscovici as an arithmetic subalgebra (cf. [54], [55]).
Thus, the noncommutative (boundary) geometry of the space of Q-lattices
module the equivalence relation of commensurability provides a setting that
uniﬁes several phenomena involving the interaction between NC geometry andnumber theory. This include the Bost-Connes system, as well, the NC space
underlying the construction of the spectral realization of zeros of the Riemannzeta function as in [29], the modular Hecke algebras of [54], [55], and the noncommutative modular curves of [112].

4

Noncommutative geometry at arithmetic inﬁnity

This section is mostly based on the work of Consani and Marcolli ([58], [59], [60],[61], [63]). Proposed is a model for the dual graph of the maximally degenerateﬁbers at the archimedean places of an arithmetic surface, in terms of a noncommutative space as a spectral triple, related to the action of a Schottky groupon its limit set. This description of ∞-adic geometry provides a compatible
setting that combines the result of Manin [106] on the Arakelov Green function
for arithmetic surfaces in terms of hyperbolic geometry and for a cohomologicalconstruction of Consani [57] associated to the archimedean ﬁbers of arithmetic
varieties, related to the Deninger calculation of the local L-factors as regularized
determinants (cf. [68], [69]).

4.1

Schottky uniformization

A compact Riemann surface X of genus g is obtained by gluing each pair of the(oriented and paired) sides of a 4g-polygon X ∼ (a polygon in R2 with 4g sides)
along the orientation, as a topological space. The fundament group of X has－ 106 －

the presentation with one relation as
π1 (X) = a1 , · · · , ag , b1 , · · · , bg | Πgi=1 [ai , bi ] = 1,
where each of the 2g generators ai and bi correspond to an oriented shift (inX ∼ ) (or cycle (or rotation) in X) along an oriented pair of the sides of the−1
4g-polygon X ∼ , identiﬁed in X, and [ai , bi ] = ai bi a−1
i bi .
Recall that the fundamental group of a topological space X is deﬁned tobe the group of homotopy classes of continuous maps (or paths) f (t) from [0, 1]
to X with some base point of X as f (0) = f (1) as a loop, where the inverse
path of f (t) is f (1 − t) and the product f · g of two loops f (t) and g(t) is deﬁnedby (f · g)(t) = f (2t) for 0 ≤ t ≤ 12 and (f · g)(t) = g(2t − 1) for 12 ≤ t ≤ 1 (cf.
[116]).
In the case of g = 1, X is the real 2-dimensional torus T2 , with X ∼ ≈ [0, 1]2 ,where [0, 1] × {0} is identiﬁed with [0, 1] × {1}, and {0} × [0, 1] with {1} × [0, 1].
And π1 (X) ∼= Z2 , with a1 b1 = b1 a1 .
Indeed, a homotopy between the corresponding product functions f1 ·g1 andg1 ·f1 is given by moving graphs with changing variables continuously, according
to ﬁx the base point. It seems to be diﬃcult to construct a homotopy between
the product of multiplicative commutators and the identity, for g ≥ 2.
The parallelogram for g = 1 is the fundamental domain of the π1 (X) ∼= Z2
action on the complex plane C, so that X ≈ C/(Z + Zτ ) as an elliptic curve.
For genus at least ≥ 2, the hyperbolic plane H2 admits a tessellation by
regular 4g-polygons, and the action of the fundamental group by deck transformations is realized by the action of π1 (X) as a subgroup G of P SL2 (R) by
isometries of H2 . The compact Riemann surface X is then endowed with ahyperbolic metric and a Fuchsian uniformization as X = G\H2 .
Another, less well known, type of uniformization of compact Riemann surfaces is Schottky uniformization. Recall brieﬂy some general facts on Schottkygroups.
Schottky groups. A Schottky group of rank h is a discrete subgroup Γ of thesemi-simple Lie group P SL2 (C) = SL2 (C)/{±12 }, which is purely loxodromic
and isometric to a free group of rank h. The group P SL2 (C) acts on P1 (C) by
fractional linear transformations as
az + ba bγ(z) =, γ=∈ P SL2 (C), z ∈ C ∪ {∞},c d
cz + dwhere P1 (C) is viewed as the Riemann sphere C ∪ {∞} ≈ S 2 as the boundary(at inﬁnity) of (the compactiﬁcation of) the real 3-dimensional hyperbolic space
H3 ⊂ R3 with the Riemann metric, and P SL2 (C) is isomorphic to the group
Iso(H3 ) of isometries of H3 .
Recall from [158] thatS 2 ≈ S 3 /S 1 ≈ P 1 (C) = (C2 \ {02 })/ ∼,where z ∼ w if and only if z = λw for some λ ∈ C \ {0}. Also, H3 ≈P SL2 (C)/SU (2). As well, SL2 (C) ≈ SU (2) × R3 .

－ 107 －

An element γ ∈ P SL2 (C) with γ = 1 is said to be either parabolic, loxodromic, or elliptic, respectively, if it has either 1, 2, or inﬁnitely many ﬁxed
points in H3 ∪ P1 (C) as the compactiﬁcation as the real 3-dimensional closed
ball B3 . Only in the elliptic case, there are ﬁxed points inside H3 . In the othercases, ﬁxed points are on P1 (C) as the 2-sphere at inﬁnity.
Remark. As well, as in [116], a discrete subgroup of P SL2 (C) is said to bea Kleinian group. Any (matrix) element of P SL2 (C), not equal to the unit,
is said to be either elliptic, parabolic, or hyperbolic (or loxodromic) if thesquared trace of the matrix is less than, equal to, or more than 4, respectively. A
hyperbolic element has no ﬁxed points in H3 and has only one geodesic preservedwith two end points ﬁxed in C+ as the boundary of the compactiﬁcation of H3 .
A parabolic element has no ﬁxed points in H3 and has no geodesics preserved,
with only one ﬁxed point of C+ . A elliptic element has only one geodesic
ﬁxed pointwise, and it is a torsion element. A torsion free Kleinian group G
acts freely on H3 , so that H3 /G is deﬁned as a real 3-dimensional hyperbolic
manifold. There is a ﬁnitely generated Kleinian group with torsion, and there
is a ﬁnite index subgroup of P SL2 (C) without torsion, known as the Selberg
lemma.
For a Kleinian group G, the limit set ΛG is deﬁned to be the closure of
all element x ∈ C+ such that there is a non elliptic element g ∈ G such thatgx = x. The complement ΩG = C+ \ ΛG is said to be the region of discontinuity
for G. The group G acts properly discontinuously on H3 ∪ ΩG . If G is ﬁnitely
generated, the quotient ΩG /G becomes a Riemann surface, by Ahlfors, and(H3 ∪ ΩG )/G said to be a Kleinian manifold 
For a Schottky group Γ, the limit set ΛΓ of the action of Γ on H3 ∪ P1 (C) isdeﬁned to be the smallest non-empty closed Γ-invariant subset of P1 (C). This
limit set can also be described as the closure of the set of the attractive andrepelling ﬁxed points z ± (γ) of the loxodromic elements γ ∈ Γ.
In the case of rank h = 1, the limit set ΛΓ consists of two points. For
rank h ≥ 1, the limit set is usually a fractal of some Hausdorﬀ dimension as
0 ≤ δ = dimH (ΛΓ ) < 2.
Remark. Recall from [116] the following. A map f from a metric space (X, d)
to itself is said to be a contraction map if it has the contractive ratio:d(f (x), f (y))≡ cr(f ) < 1.
d(x, y)x=y∈XsupFor contraction maps f1 , · · · , fn on a complete metric space X, there uniquely
exists a non-empty compact subset K of X such that K = ∪nj=1 fj (K). This K is
said to be the self-similar set with respect to the contraction maps f1 , · · · , fn ,as a fractal.
For example, let X = [0, 1] and f1 (x) = x3 and f2 (x) = x+2
3 . For instance,|f1 (x) − f1 (y)|1|f2 (x) − f2 (y)|= =.
|x − y|3|x − y|－ 108 －

Thus cr(f1 ) = 13 = cr(f2 ). As well, f1 (X) = [0, 13 ] and f2 (X) = [ 23 , 1], so that
f1 (X) ∪ f2 (X)  X. Then1
2 1
] ∪ [ 2 , ],
2
3
3 3
2 2
2
1
2
f2 (f1 (X) ∪ f2 (X)) = [ , + 2 ] ∪ [ + 2 , 1].
3 3 3
3 3
Repeating inductively this process, the self-similar set with respect to these f1
and f2 is obtained as the standard (or ternary) Cantor set C in [0, 1].
If X = Rn and fj for 1 ≤ j ≤ m are similar, contraction maps, and if theopen set condition holds as that there is a non-empty open subset U of Rn suchthat ∪m
j=1 fj (U ) ⊂ U and fi (U ) ∩ fj (U ) = ∅ for i = j, then the Hausdorﬀ
dimension dimH K of K is obtained as
f1 (f1 (X) ∪ f2 (X)) = [0,m

cr(fj )dimH K = 1.

j=12
Consequently, it then follows that 3dim2H C = 1. Therefore, dimH C = log
log 3 .
No ﬁgure provided, as taken from Indra’s Pearls, by Mumford, Series, andWright [126].
Denote by ΩΓ the domain of discontinuity of a Schottky group Γ of rankh, which is the complement of the limit set ΛΓ in P1 (C). The quotient Γ\ΩΓ
becomes a Riemann surface X(C) of genus h. The covering ΩΓ → X(C) is said
to be a Schottky uniformization of X(C).
Every compact Riemann surface admits a Schottky uniformization.
Let γj for 1 ≤ j ≤ h be generators of a Schottky group Γ of rank h. Set
γj+h = γj−1 . Then there are 2h Jordan curves Ck as a circle on the 2-sphere at
inﬁnity as P1 (C), with pairwise disjoint interiors Dk as an open disk, such that
there are elements γk that are given by fractional linear transformations that
map the interior Dk of Ck to the exterior of Cj with |k − j| = h. The curves Ckgive a marking of Γ. The markings are circles in the case of classical Schottkygroups. A fundamental domain for the action of a classical Schottky group Γ
on P1 (C) is the region exterior to 2h circles.
For a compact Riemann surface X of genus g, its Schottky unformization
as the fundamental domain looks like the 2-sphere with 2g holes as open disks,each pair of which is attached along the boundary to make X. (No ﬁgure
again.)
Schottky and Fuchsian. It is noticed that the Fuchsian uniformization for
X a compact Riemann surface is the covering H2 → X as the universal cover,while the Schottky uniformization is the covering ΩΓ = ΛcΓ → X, which is far
from being simply connected, since it in fact is the complement of a Cantor set.
The relation between Fuchsian and Schottky uniformizations is given bypassing to the covering that corresponds to the normal subgorup π1 (X)of π1 (X)2
generated by half the generators a1 , · · · , ag , so thatΓ∼
= π1 (X)/

π1 (X),
2－ 109 －

with a corresponding covering map
H2 as Fuchsian
⏐
⏐
πG 

J

−−−−→

ΩΓ as Schottky⏐⏐π ΓX = Γ\ΩΓ = (π1 (X)/ π1 (X)2 )\ΩΓ .

X = G\H2 = π1 (X)\H2

At the level of moduli, there is a corresponding map between the Teichmüller
space Tg and the Schottky space Sg , which depends on 3g − 3 complex moduli.
Remark. Recall from [116] the following. An analytically ﬁnite Riemannsurface of type (g, h) is deﬁned to be a closed Riemann surface X of genus g, withh holes, obtained by removing h points from X. We may except the cases where(g, h) = (0, n) for 0 ≤ n ≤ 3 or (g, h) = (1, 0), for which its moduli space is either
trivial or classically known. Then on such X, a hyperbolic metric is deﬁned, sothat there is a Fuchsian group Γ such that X = H2 /Γ. The Teichmüller space
T(X) for X of type (g, h) is deﬁned to be equivalence classes of pairs (R, f ) of
Riemann surfaces R of type (g, h) and quasi-conformal mappings f : X → R,
where (R1 , f1 ) and (R2 , f2 ) are equivalent if f2 ◦ f1−1 is homotopic to someconformal mapping l : R1 → R2 .
f1

X −−−−→ R1
f1−1 :←
⏐
⏐
⏐
⏐
f2 
l
R2

R2

As well, the mapping class group M c(X) for X of type (g, h) is deﬁned to
the group of homotopy classes of quasi-conformal mappings from X to X. The
group M c(X) acts on T(X) properly discontinuously, as [R, f ] → [R, f ◦ k −1 ]for [k] ∈ M c(X). The quotient space T(X)/M c(X) becomes the moduli spaceMg,h of equivalence classes of conformal mappings of Riemann surfaces of type(g, h). 
Surface with boundary as simultaneous uniformization. To visualize
geometrically the Schottky uniformization of a compact Riemann surface, it
may be related to a simultaneous uniformization of the upper and lower halfplanes, that yields two Riemann surfaces with boundary, joined at the boundary.
A Schottky group Γ that is speciﬁed by real parameters so that it is contained in P SL2 (R) is said to be a Fuchsian Schottky group. (No ﬁgure.)
Viewed as a group of isometries of the hyperbolic plane H2 , or equivalently of
the Poincaré disk, a Fuchsian Schottky group Γ as G produces the quotient
G\H2 , topologically as a Riemann surface with boundary.
Namely, such a Fuchsian Schottky simultaneous uniformization looks like－ 110 －that−Ω+Γ ∪ ΩΓ⏐⏐ΓJH+ ∪ H−⏐⏐
πG=Γ 

−−−−→G\(H+ ∪ H− ) = X1 #∂X1 =∂X2 X2−Γ\(Ω+Γ ∪ ΩΓ ) = X.

A Jordan curve C (as a circle) in P1 (C) ≈ S 2 is said to be a quasi-circle forΓ if C is invariant under the action of Γ. In particular, such a curve contains thelimit set ΛΓ . It is proved by Bowen that if X = X(C) is a Riemann surface ofgenus ≥ 2 with Schottky uniformization, then there always exists a quasi-circle
for Γ.
Then the complement P1 (C) \ C is divided into two regions as Ω1 ∪ Ω2 . ForπΓ : ΩΓ → X(C) the covering map, let C ∧ = πΓ (C ∩ ΩΓ ) in X(C), which is aset of curves on X(C) that disconnect the Riemann surface in the union of two
surfaces with boundary, uniformized respectively by Ω1 and Ω2 .
There exist conformal maps αj : Ωj → Uj (≈) for j = 1, 2, such thatU1 ∩ U2 = P1 (C) \ P1 (R) ≈ S 2 \ S 1 , with U1 ≈ H+ and U2 ≈ H− in P1 (C).
Moreover, deﬁne Gj = αj Γ∼ αj−1 , where Γ∼ in SL2 (R) is the Γ-stabilizer of
each of the two connected components of P1 (C) \ C. Then each Gj ∼= Γ andGj ⊂ P SL2 (R), so that they are Fuchsian Schottky groups.
Let Xj = Uj /Gj , which are Riemann surfaces with boundary C ∧ . The
compact Riemann surface X(C) is then obtained as
X(C) = X1 #∂X1 =C ∧ =∂X2 X2
as a connected sum of X1 and X2 along the boundary.
In the case where X(C) has a real structure as an involution ι : X → X,
and the ﬁxed point set Fx(ι) of ι as X(R) is non-empty, it in fact holds thatC ∧ = X(R), and the quasi-circle is given by P1 (R).
Note that in the case of a Fuchsian Schottky group, the Hausdorﬀ dimension
dimH ΛΓ of the limit set ΛΓ is in fact bounded above by 1, since ΛΓ is containedin the rectiﬁable quasi-circle P1 (R).
Hyperbolic handle-bodies. The action of a rank h Schottky group Γ in
P SL2 (C) acting on P1 (C) by fractional linear transformations extends to an action by isometries on the real 3-dimensional hyperbolic space H3 . For a classicalSchottky group, a fundamental domain in H3 is given by the region (as an open
connected set) external to 2h half spheres over the circles (or disks) in P1 (C)(No ﬁgure).
The region (with boundary) looks like a 3-dimensional half cut ball with(smaller) 2h half cut balls removed from the cut face. Namely, a sort of ice
cream half cut ball after (smaller) 2h ice cream half cut balls are removed fromthe top reparately.
The quotient space H3 /Γ = YΓ becomes a handle-body of genus h ﬁlling(the inside of) the Riemann surface X(C) as a topological space.
The handle-body YΓ (with boundary) (as an inﬁnite connected sum of, each
which of components, makes a doughnut with handles by the quotient) is a real

－ 111 －

hyperbolic 3-manifold of inﬁnite volume, as a metric space, having X(C) as tis
conformal boundary ∂YΓ at inﬁnity.
Denote by YΓ the compactiﬁcation of YΓ , obtained by adding the conformalboundary at inﬁnity, so thatYΓ = (H3 ∪ ΩΓ )/Γ.
In the case of genus zero, we have the 2-sphere P1 (C) as the conformalboundary at inﬁnity of H3 , attached to make the unit ball in the Poincaré
model, asH3 ∪ P1 (C) ≈ B 3 .
In the case of genus one, we have a(n open) solid torus H3 /q Z , for q ∈ C∗
with |q| < 1 acting asq(z, y) = (qz, |q|y) ∈ C × (0, ∞) ≈ H3
in the upper half space model, with conformal boundary at inﬁnity as the Jacobi
uniformized elliptic curve C∗ /q Z , attached to make the closed solid torus, as(H3 /q Z ) ∪ (C∗ /q Z ).
In this case, the limit set ΛΓ consists of two points set {0, ∞}, the domainof discontinuity is C∗ as the complement of ΛΓ in P1 (C), and a fundamental
domain is the annulus{z ∈ C | |q| < |z| ≤ 1}
as the intersection of the (open) exterior of a closed disk with radius |q| and the(closed) exterior of the complement of a closed disk with radius 1.
The relation of Schottky uniformization to the usual Euclidean uniformization of the complex tori as X = C/(Z + τ Z) is given by q = exp(2πiτ ).
We have a return soon later to the case of genus one, to discuss a physicalinterpretation of the result of Manin on the Green function. However, for our
purposes, the most interesting case is when genus is more than 1. In this case,the limit set ΛΓ becomes a Cantor set with an interesting dynamics for theaction of Γ. That is nothing but the dynamics by the Schottky group on its
limit set, that does generate an interesting noncommutative space.
Geodesics in the handle-body YΓ . The hyperbolic handle-body YΓ has
inﬁnite volume, but it contains a region of ﬁnite volume, which is a deformation
retract of YΓ . This is called the convex core of YΓ and is obtained by taking thegeodesic hull of the limit set ΛΓ in H3 and then as the quotient by Γ. Identify
diﬀerent classes of inﬁnite geodesics in YΓ .
[Closed geodesics]. Since Γ is purely loxodromic, for any γ ∈ Γ, there exist
two ﬁxed points {z + (γ), z − (γ)} ⊂ P1 (C). The geodesics in H3 ∪ P1 (C) withends as two such points {z ± (γ)}, for some γ ∈ Γ, correspond to closed geodesics
in the quotient YΓ = H3 /Γ.
[Bounded geodesics]. The images in YΓ of geodesics in H3 ∪ P1 (C) having
both ends on the limit set ΛΓ are geodesics that remain conﬁned within theconvex core of YΓ .

－ 112 －

[Unbounded geodesics]. These are the geodesics in YΓ that eventually
wander oﬀ the convex core towards X(C) = ∂YΓ the conformal boundary at
inﬁnity. They correspond to geodesics in H3 ∪ P1 (C) with at least one end as apoint of ΩΓ the domain of discontinuity of Γ as ΛcΓ in P1 (C).
In the case of genus one, there is a unique primitive closed geodesic, namely
the image in the quotient of the geodesic in H3 connecting 0 and ∞. The
bounded geodesics are those corresponding to geodesics in H3 originating at 0or ∞.
The most interesting case is that of genus more than 1, where the bounded
geodesics form a complicated tangle inside YΓ . This is a generalized solenoid
as a topological space. Namely, it is locally the product space of a line (or line
segment) and a Cantor set.
A solenoid is a usual spiral coil. This is a real line or line segment as a
topological space. But it has a projection to the circle S 1 (in one direction of
the coil core), with ﬁbers a ﬁnite set as the winding number of the coil.

4.2

Dynamics and noncommutative geometry

Since the uniformizing group Γ is a free group, there is a simple way of obtaining
a coding of the bounded geodesics in YΓ . The set of such geodesics can be
identiﬁed with ΛΓ ×Γ ΛΓ , by specifying the end points in H3 ∪ P1 (C) module
the action of Γ.
Let {γj }gj=1 be a choice of generators of Γ and set γg+j = γj−1 for 1 ≤ j ≤ g.
We introduce a subshift of ﬁnite type as follows. LetS=−1{w = {ak }k∈Z ≡ · · · a−k · · · a−1 a0 a1 · · · ak · · · | ak ∈ {γj }2g
j=1 , ak+1 = ak , k ∈ Z}

be the set of doubly inﬁnite words w in the generators and their inverses, withthe admissibility condition, so that no cancellations occur in each word. Deﬁnethe map T on S as the invertible shift (to the left, one by one)
T (· · · a−2 a−1 a0 a1 a2 · · · ) = · · · a−1 a0 a1 a2 a3 · · ·so that T (w) = T ({ak }k∈Z ) = {ak+1 }k∈Z ∈ S as our notation.
Then the discrete dynamical system (S, T, Z) deﬁned above can be passed
to its suspension ﬂow, to obtain the mapping torus as
ST = (S × [0, 1])/ ∼,where (x, 0) ∈ S × [0, 1] is identiﬁed with (T (x), 1).
This quaotient space of the suspension ﬂow on S×[0, 1] or S×R (extended
periodically) may be exactly denoted as, for instance, ISIT or (SS)ST .
That space is a (generalized) solenoid as a topological space, that is, a bundle－ 113 －

over S 1 with ﬁber a Cantor set, as
S × (0, 1) −−−−→ ST = ISIT = (SS)ST ←−−−− 2Z ≈ Π∞ {0, 1}⊂⏐⏐S 1 = [0, 1]/ ∼where S × (0, 1) ≈ S × R is the suspension of S, on which T acts trivially,and 2Z is a Cantor set as a ﬁber.
Remark. As in [11], the mapping torus C ∗ -algebra for T is deﬁned as
MT = {f ∈ C([0, 1], C(S)) | f (1) = T (f (0))}
where T ∈ Aut(C(S)) as an ∗-automorphism of C(S) is identiﬁed with that
homeomorphism T on S. Then there is the following short exact sequence ofC ∗ -algebras:qi
0 → SC(S) −−−−→ MT −−−−→ C(S) → 0∼ C0 (R) ⊗ C(S) is the suspension of C(S).
∼ C0 (S × (0, 1)) =with SC(S) =Then the following six-term exact sequence is deduced:∗
K0 (SC(S)) ∼= K1 (C(S)) −−−−→ K0 (MT ) −−−−→

⏐
∂⏐

q∗

i

K1 (C(S))

K0 (C(S))⏐⏐
∂∗←−−−− K1 (MT ) ←−−−− K1 (SC(S)) ∼= K0 (C(S))

q∗iand moreover, Kj (MT ) ∼= Kj+1 (C(S) T Z) for j = 0, 1 (mod 2). This may
follow from comparing with the Pimsner-Voiculescu six-term exact sequence
given below and the Five (or Six) Lemma.
Homotopy quotient. The space ST deﬁned above has a natural interpretationin noncommutative geometry as the homotopy quotient in the sense of BaumConnes [9] of the noncommutative space given by the C ∗ -algebra crossed product
C(S) T Z describing the shift action T on the totally disconnected space S.
The noncommutative space as the C ∗ -algebra parameterizes bounded geodesics
in the handle-body YΓ . The homotopy quotient is given by S ×Z R = ST .
The K-theory of the C ∗ -algebra crossed product by Z can be computed by
the Pimsner-Voiculescu six-term exact sequence, with induced maps with ∗:K0 (C(S))
⏐⏐(1−T )∗ii(1−T )∗−−−−−→ K0 (C(S)) −−−∗−→ K0 (C(S) T Z)
δ∗
⏐
⏐


∗
K1 (C(S) T Z) ←−−−− K1 (C(S)) ←−−−−−δ∗

K1 (C(S))

where i : C(S) → C(S) T Z is the canonical inclusion, and δ = 1 − T .
Since the space S is totally disconnected, we have that K1 (C(S)) is zero and－ 114 －

K0 (C(S)) ∼= C(S, Z) of locally constant, integer valued functions on S. Therefore, the following exact sequence of groups is obtained:(1−T )∗

0 → K1 (C(S) T Z) → C(S, Z) −−−−−→ C(S, Z) → K0 (C(S) T Z) → 0,δ∗so that
K1 (C(S) T Z) ∼= ker(δ∗ ) ∼= Z,
K0 (C(S) T Z) ∼= coker(δ∗ ) ∼= C(S, Z)/im(δ∗ )with C(S, Z)/ker(δ∗ ) ∼= im(δ∗ ). In the language of dynamical systems, thekernel and cokernel are respectively the invariant and the coinvariant of theinvertible shift T (cf. [15], [131]).
In terms of the homotopy quotient, that exact sequence can be described
more geometrically in terms of the Thom isomorphism and the assembly μmap, as, for ∗ = 0, 1 (mod 2), (added as)∼=K∗+1 (MT )(∼= K∗+1 (MT ∧ )) −−−−→ K∗ (C(S) T Z)
⏐

⏐
⏐μ
∼
=
⏐
∼
=

K∗+1 (C(ST )) ∼= K ∗+1 (ST ) −−−−→
Ch

H ∗+1 (ST , Z)

where the Connes’ Thom isomorphism (or the Takai duality involving the crossed
product C ∗ -algebra by the dual action T ∧ to T ) may be used in the non-trivial
isomorphism in the ﬁrst horizontal line, and the map Ch in the second horizontal
line is the Chern character, which is also a non-trivial isomorphism because ofbeing torsion free of the K-theory groups (cf. [11]), and the assembly map in thiscase means the non-trivial isomorphism between the topological K-theory forspaces such as K ∗+1 (ST ) and the K-theory for C ∗ -algebras as K∗ (C(S) T Z).
Indeed, the Takai duality implies that
(C(S) T Z) T ∧ T ∼= C(S) ⊗ K.
It says that the dual crossed product of, the crossed product of a C ∗ -algebra A
by an action of an Abelian locally compact group G, by the dual action of thedual group G∧ is stably isomorphic to the C ∗ -algebra A. On the other hand,with Z ⊂ R acting trivially, with R/Z ∼= T,(C(S) T Z) T ∧ R ∼= MT ∧ ,where this mapping torus C ∗ -algebra MT ∧ is that of T ∧ on (C(S) T Z) T ∧ T(cf. [11]). The Connes’ Thom isomorphism for crossed product C ∗ -algebras by R
(to be assumed trivial in K-theory) implies that K∗+1 (C(S) T Z) ∼= K∗ (MT ∧ ).
It seems that the left vertical isomorphism is non-trivial as well. It may
follows from the same K-theory group diagram as for MT ∧ noncommutative as
well as C(ST ) = C((SS)ST ) commutative, with bridge by the Five (or Six)
Lemma.

－ 115 －

Consequently, we have
K1 (C(S) T Z) ∼= H 0 (ST ) ∼=Z

and K0 (C(S) T Z) ∼= H 1 (ST ).

The ﬁrst (Alexander or Čech) cohomology group H 1 (ST ) can be identiﬁed withthe homotopy group [ST , U (1)] given by the homotopy classes of mappingsf → exp(2πitf (x)) for f ∈ C(S, Z)/δ(C(S, Z)) (corrected).
Filtration. The identiﬁcation of H 1 (ST , Z) the cohomology group of ST with
the K-theory group K0 (C(S) T Z) of the crossed product C ∗ -algebra for theaction T on S endows H 1 (ST , Z) with a ﬁltration as follows.
Theorem 4.1. The ﬁrst cohomology of ST is a direct limit lim Fn of free−→abelian groups Fn of ranks rank(F0 ) = 2g and rank(Fn ) = 2g(2g − 1)n−1 (2g − 2)(corrected) for n ≥ 1, with Fn ⊂ Fn+1 for n ≥ 0.
In fact, it follows from the Pimsner-Voiculescu six-term exact sequence thatthe abelian group K0 (C(S) T Z) can be identiﬁed with the cokernel of themap 1 − T acting as f → f − f ◦ T on the Z-module C(S, Z) ∼= K0 (C(S)).
Then the ﬁltration is given by the submodules of functions depending only on
the a0 · · · an coordinates in the doubly inﬁnite words describing points in S.
Namely, it then follows that
H 1 (ST , Z) ∼= C(S, T)/δ(C(S, Z)) = P/δP,
where P denotes the Z-module of locally constant Z-valued functions that depend only on future coordinates. These functions can be identiﬁed with functions on the limit set ΛΓ , since each point in ΛΓ is described by an inﬁnite to
the right, admissible sequence in the generators γj and their inverses. Thus,P∼= C(ΛΓ , Z).
The module P has a ﬁltration by the submodules Pn of functions of the ﬁrst
n + 1 coordinates. Then rank(Pn ) = 2g(2g − 1)n .
The Z-module P0 has a basis corresponding to the set of generators andinverses of Γ. The Z-module P1 has a basis corresponding to the set of words
of S of length two.
Set F0 = P0 and Fn = Pn /δPn−1 for n ≥ 1. It can be shown that there are
induced injections from Fn to Fn+1 and that H 1 (ST ) ∼
F .
= lim
−→ n
If f = χγj the characteristic function at γj (or on all the words startingfrom γj ), then
⎧
⎪
at γj
⎨1
δf = χγj − χγj ◦ T = −1 at T −1 γj⎪⎩0otherwise
which is a function at γj γj = γj (T −1 γj ) as two points (as a0 a1 ).
If a0 a1 is the part of the corresponding admissible word of S, then it may
correspond to the function fa0 a1 = χa0 − χa1 ◦ T . Thenδ(fa0 a1 ) = fa0 a1 − fa0 a1 ◦ T = χa0 − χa0 ◦ T − χa1 ◦ T + χa1 ◦ T 2 ,－ 116 －

which should be deﬁned as fa0 a1 a1 .
If a0 a1 a2 is the part of the corresponding admissible word of S, then it may
correspond to the function
χa0 − χa0 ◦ T − χa1 ◦ T − χa2 ◦ T 2 .
Moreover, we have rank(Fn ) = θn −θn−1 (corrected), where θn is the numberof admissible words of length n + 1. All the Z-module Fn and the quotients
Fn /Fn−1 are torsion free (cf. [131]).
For instance,rank(F1 ) = rank(P1 ) − rank(P0 ) = 2g(2g − 1) − 2g = 2g(2g − 2).
As well,rank(F2 ) = rank(P2 ) − rank(P1 ) = 2g(2g − 1)2 − 2g(2g − 1) = 2g(2g − 1)(2g − 2).
Similarly,rank(F3 ) = 2g(2g − 1)3 − 2g(2g − 1)2 = 2g(2g − 1)2 (2g − 2).
Hilbert space and grading. It is convenient to consider the complex vector
space deﬁned as PC = C(ΛΓ , Z) ⊗ C and the corresponding exact sequence
computing the cohomology with C as coeﬃcients, asδ

0 → C → PC −−−∗−→ PC → H 1 (ST , C) → 0
obtained by tensoring with C of that sequence for C(S, Z) identiﬁed with P.
The complex vector space PC may be contained in the complex Hilbert space
L2 = L2 (ΛΓ , dμ), where μ is the Patterson-Sullivan measure on the limit set ΛΓ ,satisfying
γ ∗ dμ = |γ  |dimH ΛΓ dμ,with dimH ΛΓ the Hausdorﬀ dimension.

This deﬁnes such a Hilbert space L2 , together with a ﬁltration by ﬁnite
dimensional subspaces Pn ⊗ C. In this setting, it is natural to consider a corresponding grading operator, deﬁned as an inﬁnite direct sum
D=

∞


∧
nΠ∧
n = ⊕n nΠn ,

⊥
2
on ⊕∞
n=0 (Pn,C ∩ Pn−1,C ) = L ,n=0

as the inﬁnite direct sum Hilbert space, where Πn denotes the orthogonal pro⊥jection from L2 onto Pn ⊗ C = Pn,C and Π∧
n = Πn − Πn−1 , with Pn−1,C the2orthogonal complement of Pn−1,C in L , and Π−1 = 0 and P−1,C = {0}.
The Cuntz-Krieger algebra. There is a noncommutative space that encodes
the dynamics of the Schottky group Γ on its limit set ΛΓ . Consider the 2g × 2g
matrix A = (aij ) with entries of {0, 1} such that aij = 1 for |i − j| =
 g andaij = 0 otherwise. The adjacency (or admissibility, ad) matrix A corresponds
to the admissibility condition for sequences in S.

－ 117 －

For instance, if g = 2, then⎛
1 1 0
⎜1 1 1A = (aij ) = ⎜⎝0 1 1
1 0 1

⎛
⎞
γ1
1
⎜ γ2
0⎟
⎟⇔⎜
⎝ ∅
1⎠
1
γ2−1

γ1
γ2
γ1−1
∅

∅
γ2
γ1−1
γ2−1

⎞
γ1
∅ ⎟
⎟
γ1−1 ⎠
γ2−1

where the columns from the left to the right correspond to the admissible onewords in the right of γ1 , γ2 , γ1−1 , and γ2−1 , respectively.
The Cuntz-Kriger (CK) C ∗ -algebra OA associated to the ad matrix A isdeﬁned to be the universal unital C ∗ -algebra generated by (mutually orthogonal)
partial isometries sj for 1 ≤ j ≤ 2g satisfying the (CK) relations2g


sj s∗j = 1

and s∗k sk =

j=1

2g


akj sj s∗j .

j=1

The ﬁrst equation says that the range projections of sj add up to the unit.
The second says that the initial (or domain) projection of each sk is such a sumof the range projections of sj .
Recall that a partial isometry is an operator s satisfying s = ss∗ s.
A partial isometry s on a Hilbert space H is deﬁned to be an isometry onker(s)⊥ the orthogonal complement to the kernel ker(s) a closed subspace of H
(cf. [83]). Let pker(s)⊥ be the orthogonal projection from H onto ker(s)⊥ . For
ξ = x ⊕ y ∈ ker(s) ⊕ ker(s)⊥ = H,s∗ sξ, ξ = sy, sy = sy2 = y2 = y, y = pker(s)⊥ ξ, pker(s)⊥ ξ.
It then follows that s∗ s = pker(s)⊥ . Moreover, for ξ = x ⊕ y ∈ H as above,ss∗ sξ = s(s∗ sξ) = sy = sξ.
Hence, s = ss∗ s.
Conversely, if s∗ s is a projection p = p2 = p∗ , then ker(s) = ker(p), so thats is an isometry on ker(s)⊥ equal to the range of p.
As well, suppose that s = ss∗ s. Then (s∗ s)2 = s∗ (ss∗ s) = s∗ s = (s∗ s)∗ .
The Cuntz-Krieger C ∗ -algebra is related to the Schottky group by the following:
Proposition 4.2. There is a C ∗ -algebra isomoporhism
OA ∼= C(ΛΓ )  Γ.
Up to the stabilization by tensoring with the C ∗ -algebra K of compact operators, there is another crossed product description as
OA ⊗ K ∼= FA T Z,
where FA is an AF-algebra, deﬁned as an inductive limit of ﬁnite dimensionalC ∗ -algebras.

－ 118 －

It should be a nice question how to prove those isomorphisms above. The
proof is done below, as faithfully represented.
Consider the cochain complex of Hilbert spaces as the following bottom line
to up line completion diagram (edited):δ

0 −−−−→ C −−−−→ L2 (ΛΓ ) −−−∗−→ L2 (ΛΓ ) −−−−→ H 1 (ST , C) −−−−→ 0






⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
0 −−−−→ C −−−−→

δ−−−∗−→

PC

PC−−−−→ H 1 (ST , C) −−−−→ 0

where PC = P ⊗ C = C(ΛΓ , Z) ⊗ C.
Note that the diagram follows from the density of the complex vector space
PC in the L2 -space. The quotient H 1 of L2 by δ∗ L2 means the L2 -closure of
H 1.
Proposition 4.3. The Cuntz-Krieger C ∗ -algebra OA has a faithful representation on the Hilbert space L2 = L2 (ΛΓ , dμ).
The proof is as follows. Let dH = dimH (ΛΓ ) the Hausdorﬀ dimension.
Consider the operators on L2 deﬁned as, for 1 ≤ j ≤ 2g,
dH

and Tj f = |(γj−1 ) | 2 f ◦ γj−1 ,Pj f = χγj fwhere {γj }2g
j=1 are the generators of Γ and their inverses. For any γ ∈ Γ, deﬁnedH

Tγ f = |γ  | 2 f ◦ γ.
Deﬁne Si = j=1 aij Ti∗ Pj for 1 ≤ i ≤ 2g. Then Si are partial isometries on L2
satisfying the Cuntz-Krieger relations for the matrix A as the subshift T on S
of ﬁnite type. This extends to the representation of OA on L2 .
Note that Pj Pk = 0 if j = k and = Pj if j = k. Also, for f1 , f2 ∈ L2 ,2g

Tj∗ f1 , f2  = f1 , Tj f2  =f1 (γ)|(γj−1 ) |dH /2 f2 (γj−1 γ)

γ∈ΛΓ



=



|(γj−1 ) |dH /2 f1 (γj γ)f2 (η)η=γj−1 γ∈ΛΓ(with some weight in the inner product?). Check that
⎡
⎤
2g
2g
2g
2g




⎣
Si Si∗ =aij Ti∗ Pj
aik Pk Ti ⎦
i=1

i=1

j=1

i=1

j=1 k=1

k=1

⎡
⎤
⎡
⎤
2g
2g
2g 
2g
2g




⎣
⎣
aij aik Ti∗ Pj Pk Ti ⎦ =aij Ti∗ Pj Ti ⎦ .
=i=1－ 119 －

j=1

Moreover, compute that for f ∈ L2 ,Ti∗ Pj Ti f = Ti∗ (Pj Ti f ) = |(γi−1 ) |dH /2 (Pj Ti f ) ◦ γi
= |(γi−1 ) |dH /2 (χγj ◦ γi )(Ti f ) ◦ γi
= |(γi−1 ) |dH /2 |(γi−1 ) |dH /2 (χγj ◦ γi )f.
There may be more reason for this to be completed. Indeed, possibly, in thedH
deﬁnition of Tj , the factor may be replaced with |(γj−1 ) |i 2 , so thatdH

dH|(γj−1 ) |i 2 |(γj−1 ) |i 2 = exp(−idHdHlog |(γj−1 ) |) exp(ilog |(γj−1 ) |) = 1.
22

Note as well that the χγj should mean the characteristic function on all thewords starting from γj . It then follows that2g

Si Si∗ =

i=1

2g
2g 


aij (χγj ◦ γi ) =i=1 j=12g
ajj (χγj ◦ γj ) =

j=1

2g


(χγj ◦ γj ) = 1.

j=1

Note that the operation ◦γj should be the projection to the set of all the words
starting from γj .
2g
Check also that Sk∗ Sk = j=1 akj Sj Sj∗ . Indeed,Sk∗ Sk =2g
aki Pi Tk

i=1

2g


akl Tk∗ Pl =

2g
2g 


aki akl Pi Tk Tk∗ Pl

i=1 l=1l=1

Moreover, compute that for any f ∈ L2 (corrected),dH

Pi Tk Tk∗ Pl f = χγi Tk Tk∗ Pl f = χγi |(γk−1 ) |i 2 Tk∗ Pl f ◦ γk−1dHdH

= χγi |(γk−1 ) |i 2 |(γk−1 ) |i 2 Pl f ◦ γk ◦ γk−1 = χγi χγl f = Pi Pl f.
Therefore,
Sk∗ Sk =

2g
2g 


aki akl Pi Pl =i=1 l=12g

i=1

aki Pi =2g
aki Si Si∗ ,

i=1

with
Si Si∗ f =

2g

j=1

aij Ti∗ Pj Ti f =2g
aij (χγj ◦ γi )f = aii (χγi ◦ γi )f = Pi f

j=1

with aii = 1, and χγi ◦ γi = χγi . Note that the operation ◦γi should be theprojection to the set of all the words starting from γi .
The spectral triple for Schottky groups. On the direct sum H = L2 ⊕L2 ofthe Hilbert space L2 = L2 (ΛΓ ), consider the diagonal action of the CK algebra－ 120 －

OA by the representation obtained above and also the Dirac operator D deﬁned
as D = DL2 ⊕0 % D0⊕L2 :


∞
∞


0
D0⊕L2
2 = −
D=
nΠ∧
,
D
nΠ∧
, DL2 ⊕0 =0⊕Lnn
DL2 ⊕00
n=1

n=1(corrected and improved), where each Πn is the orthogonal projection with∧respect to Pn,C (n ≥ 0) of L2 = ∪∞n=0 Pn,C , with Πn = Πn − Πn−1 for n ≥ 1.
The choice of the sign in the formula above is not optimal from the point ofview of the K-homology class, determined by the triple (OA , H, D). A better
choice would be F = 1 % 1 on H. This would require in turn a modiﬁcation of|D|. The construction along these lines is considered by Marcolli, with Alina
Vdovina and Gunther Cornelissen. In this setting, the reason for the above
choice as D as in [61] is the last formula in this section relating to the logarithmof Frobenius at arithmetic inﬁnity.
Theorem 4.4. For a Schottky group Γ with dimH ΛΓ < 1 and rank h, a
non ﬁnitely summable, but θ-summable spectral triple is deﬁned as the date
(OA , H, D), for H = L2 ⊕ L2 (ΛΓ , dμ) with the diagonal action of OA by the2hfaithful representation deﬁned as Sj = k=1 ajk Tj∗ Pk for 1 ≤ j ≤ 2h, and with
the Dirac operator D as above.
The CK-algebra OA in the statement may be replaced with its dense∞smooth subalgebra OA.
The key point of this result is to show the compatibility relation between
the CK-algebra and the Dirac operator, namely that the commutators [D, a]∞
are bounded operators for any element of the involutive dense subalgebra OA
of OA , generated algebraically by the partial isometries Sj subject to the CK
relations.
This follows by estimating the norm of the commutators [D, Sj ] and[D, Sj∗ ], in terms of the Poincaré series of the Schottky group with dH < 1,such that|γ  |s , s = 1 > dH = dimH ΛΓ ,γ∈Γwhere the Hausdorﬀ dimension dH becomes the exponent of convergence of the
Poincaré series.
Compute that
 


Sj 0
Sj 0
[D, Sj ] = D
−
D
0 Sj
0 Sj


0
D0⊕L2 Sj − Sj D0⊕L2=.
DL2 ⊕0 Sj − Sj DL2 ⊕00Moreover, in particular,
Sj D0⊕L2 =

2h

k=1


ajk Tj∗ Pk

−

∞



nΠ∧
n

=−

n=1

－ 121 －∞2hnajk Tj∗ Pk (Πn − Πn−1 ).
n=1

k=1

It then follows that
[D0⊕L2 , Sj ] = −

∞
2h


n
ajk [Tj∗ Pk , Πn − Πn−1 ]
n=1

k=1

A possible estimate of the norm of this commutator is given by the upper∞bounded as 2h n=1 n, but which diverges. Thus, it is necessary to have thatthe norm of the commutators as the direct summands converges to zero at inﬁnity, of order less than O( n12 ) as n → ∞. Possibly, it is necessary to consider∞
elements of OA
which are smooth, rapidly decreasing, or compactly supported,
but with respect to (Π∧sort of basis.
n ) as a √The dimension of the (i = −1)n-th (corrected) eigenspace of D is 2g(2g −1)n−1 (2g − 2) for n ≥ 1, 2g for n = 0, and 2g(2g − 1)−n−1 (2g − 2) for n ≤ −1,
so that the spectral triple is not ﬁnitely summable, since |D|z is not of traceclass. But it is θ-summable, since the operator exp(−tD2 ) is of trace class, forall t > 0.
For instance, suppose that D(ξ ⊕ η) = 0 ⊕ 0 on H = ⊕2 L2 . Then∞∞∧∧n=1 nΠn η = 0 andn=1 nΠn ξ = 0. If we have the dimension of the kernel of D equal to 2g, the summations in the deﬁnition of D should start with2
n = 1. Moreover, note that for ξn ∈ Π∧
nL ,

 




0
−nΠ∧
inξn
ξn
ξn
n
=
= in
, and
nΠ∧
0
−iξn
nξn
−iξn
n








0
−nΠ∧
−inξn
ξn
ξn
n
=
= −in
.
nΠ∧
0
iξ
nξ
iξ
n
n
n
n
Using the description of the noncommutative space OA as the crossed product C ∗ -algebra FA T Z of an AF FA by the action of the shift T , it may be able
to ﬁnd a 1-summable spectral triple, where the dense subalgebra involved in it
should not contain any of the group elements. In fact, by the result of Connes[26], the group Γ is a free group, and hence its growth is too fast to support
ﬁnitely summable spectral triples on its group ring.

4.3

Arithmetic inﬁnity as archimedean primes

An algebraic number ﬁeld K, which is an extension of Q such that [K, Q] = n,admits n = r1 + 2r2 embeddings as α : K → C. These can be subdivided into
r1 real embeddings as K → R and r2 pairs of complex conjugate embeddings.
The embeddings α are said to be the archimedean primes of the number ﬁeld
K. The set of archimedean primes is often referred to as arithmetic inﬁnity,
as a terminology, borrowed from the case of the unique embedding as Q → R,
which is called the inﬁnite prime.
An extension of a ﬁeld K is a ﬁeld L with an inclusion K ⊂ L with K
as a subﬁeld. The extension degree [L, K] is deﬁned to be the dimension of L
as a vector space over K, since KL ⊂ L.

－ 122 －√ √For instance, [Q( 2, 3), Q] = 4 = r1 with r2 = 0. Indeed, as vector spaces,
√
√
√
√ √
Q( 2, 3) ∼= Q1 ⊕ Q 2 ⊕ Q 3 ⊕ Q 6√√ √∼ Q( 2)1 ⊕ Q( 2) 3 with
=
√ √
√
√ √
√
[Q( 2, 3), Q] = [Q( 2), Q][Q( 2, 3), Q( 2)] = 22 .
√ √As well,
There is the identity embedding as Q(√ 2,√ 3) → R.
√ √√ the√others
are√obtainedbysending
respectively
(
2,
3)
→

(−
2,
3),
(
2, − 3), or√(− 2, − 3).
√As well, [Q(i), Q] = 2 with r2 = 1. Indeed, Q(i) ∼= Q1 ⊕ Q −1, and thereis the identity embedding as Q(i) → C, and the other is obtained by sendingi → −i (cf. [129]).
A general strategy in arithmetic geometry is to adapt the tools of classical algebraic geometry to the arithmetic setting. In particular, the (spectrum)
Sp(Z) of primes over Q is the analog of the aﬃne line in arithmetic geometry.
It then becomes clear that some compactiﬁcation is necessary, at least in order to have a well behaved form of intersection theory in arithmetic geometry.
Namely, we need to pass from the aﬃne (Spec) Sp(Z) to the projective case.
The compactiﬁcation is obtained by adding the inﬁnite prime to the set of ﬁnite
primes. Then a goal of arithmetic geometry becomes developing a setting thattreats the inﬁnite prime and the ﬁnite primes in the equal footing.
More generally, for a number ﬁeld K, with OK as its ring of integers, the set
Sp(OK ) of primes is compactiﬁed by adding the set of archimedean primes, as
Sp(OK ) = Sp(OK ) ∪ {α : K → C}.
Arithmetic surfaces. Let X be a smooth projective algebraic curve deﬁned
over Q. Then, obtained by clearing denominators is an equation with coeﬃcientsin Z. This determins a scheme XZ over Sp(Z), as
XZ ⊗Z Sp(Q) = X,
where the closed ﬁber of XZ at a prime p ∈ Sp(Z) is the reduction of X
mod p. Thus, an algebraic curve viewed as an arithmetic variety becomes a2-dimensional ﬁbration over the aﬃne line Sp(Z).
Also consider reductions of X deﬁned over Z modulo pn for some primep ∈ Sp(Z). The corresponding limit as n → ∞ deﬁnes a p-adic completion of
XZ . This can be thought of as an inﬁnitesimal neighbourhood of the ﬁber at p.
The corresponding picture is more complicated at arithmetic inﬁnity, since
there is no suitable notion of reduction mod ∞, available to deﬁne the closed
ﬁber. On the other hand, there is the analog of the p-adic completion at hand.
This is given by the Riemann surface X(C) as a smooth projective algebraic
curve over C, determined by the equation of the algebraic curve X over C,
under the embedding of Q ⊂ C, as
X(C) = X ⊗Q Sp(C),－ 123 －

with the absolute value | · | at the inﬁnite prime, replacing the p-adic valuations.
Similarly, for K a number ﬁeld with [K : Q] = n and OK its ring of integers,a smooth proper algebraic curve X over K determines a smooth minimal model
XOK , which deﬁnes an arithmetic surface XOK over Sp(OK ). The closed ﬁber
Xp of XOK over a prime p ∈ OK is given by the reduction mod p.
When Sp(OK ) is compactiﬁed by adding the archimedean primes, obtained
are n = r1 +2r2 Riemann surfaces Xα (C), which are obtained from the equationdeﬁning X over K under the n embeddings α : K → C. The corresponding r1
Riemann surfaces of n have the real involution.
Then, the picture of an arithmetic surface XSp(OK ) over Sp(OK ) with inclusions is as follows:
⊂

⊂

⊃

⊂

⊃

Xp −−−−→ XOK = XSp(OK ) −−−−→ XSp(OK ) ←−−−−
⏐
⏐
⏐
⏐
⏐
⏐



⊂

p −−−−→

Sp(OK )?
⏐⏐
−−−−→ Sp(OK ) ←−−−− {α : K → C}

where there is no explicit geometric description of the closed ﬁbers as (?) over
the archimedean primes α.
An arithmetic surface over Sp(Z) of primes looks like a bundle of curves orcrossed ones over primes, whose limit at inﬁnity is what as (?). (No ﬁgure).
It is able to formally enlarge the group of divisors on the arithmetic surface
by adding formal real linear combinations asα λα Fα of irreducible closed
vertical ﬁbers Fα at inﬁnity, where the ﬁbers Fα are only treated as formal
symbols, and no geometric model of such ﬁbers is provided. A remarkable factsays that Hermitian geometry on the Riemann surfaces Xα (C) is suﬃcient tospecify the contribution of such divisors to intersection theory on the arithmetic
surface, even without an explicit knowledge of the closed ﬁber.
The main ideal of Arakelov geometry is that it is suﬃcient to work with
the inﬁnitesimal negihbourhood Xα (C) of the ﬁbers Fα , to have well deﬁned
intersection indices.
By analogy, in the case of the classical geometry of a degeneration of algebraic curves over a disk with a special ﬁber at 0, the analogous statement
would be that the geometry of the special ﬁber is completely determined by thegeneric ﬁbers. This is a strong statement as the form of the degeneration. For
instance, blowing up points at the special ﬁber may be not seen by just looking
at the generic ﬁbers. Investigating this analogy leads to expect that the ﬁber
at inﬁnity should behave like the totally degenerate case. This is the case as
the maximal degeneration, where all the components of the closed ﬁber are thesame as P1 and the geometry of the degeneration is completely encoded by thedual graph, which describes in a purely combinatorial way how the components
as P1 are joined. The dual graph has a vertex for each components of the closed
ﬁber and an edge for each double point.
The local intersection multiplicities of two ﬁnite, horizontal, irreducible divisors D1 , D2 on XOK is given by[D1 , D2 ] = [D1 , D2 ]f + [D1 , D2 ]∞ ,－ 124 －

where the ﬁrst term counts the contribution from the ﬁnite places as what
happens over Sp(OK ) and the second term is the contribution of the archimedean
primes, at the part of the intersection that happens over arithmetic inﬁnity.
While the ﬁrst term is computed in algebro-geometric terms, from the local
equations for the divisors Di at a point p, the second term is deﬁned as a sumof values of Green functions gα on the Riemann surfaces Xα (C) as ααgα (pα[D1 , D2 ]∞ = −1,β , p2,γ ),
α

β,γ

α
at points pα1,β and p2,γ of Xα (C) for 1 ≤ β ≤ [K(D1 ), K] and 1 ≤ γ ≤ [K(D2 ), K]
respectively, for ﬁnite extensions K(Dj ) of K determined by Dj for j = 1, 2,
where α = 1 for α real embeddings and α = 2 for α complex embeddings.
For a more detailed account of these notions of Arakelov geometry, may refer
to [65] and [99].
Further evidence for the similarity between the archimedean and the totally
degenerate ﬁbers comes from an explicit computation of the Green function
at the archimedean places, as derived by Manin [106], in terms of a Schottky
uniformization of the Riemann surface Xα (C). Such a uniformization has ananalog at a ﬁnite prime, in terms of p-adic Schottky groups, only in the totally
degenerate case. Another source of evidence comes from a cohomological theoryof the local factors at archimedean primes, as developed by Consani [57], inwhich shown is that the resulting description of the local factor as regularized
determinant at the archimedean primes resembles closely the case of the totally
degenerate reduction at a ﬁnite prime.
Presented soon are both results in the light of the noncommutative space asa spectral triple (OA , H, D) introduced in the previous section. As shown by
Consani and Marcolli in [58], [59], [60], and [61], the noncommutative geometryof this space is naturally related to both the result of Manin on the Arakelov
Green function and the cohomological construction of Consani.

4.4

Arakelov geometry and hyperbolic geometry

In this subsection, given is a detailed account of the result of Manin [106] on therelation between the Arakelov Green function on a Riemann surface X(C) =Γ\ΩΓ with Schottky uniformization ΩΓ = ΛcΓ in P1 (C) ≈ S 2 and geodesics inthe real 3-dimensional hyperbolic handle-body YΓ = H3 /Γ. The exposition in
the following follows the seminal paper [106] closely.
Arakelov Green function. Given a divisor A = x∈X mx (x) with support
denoted as |A| on a smooth compact Riemann surface X(C) = X, and alsoa choice of a positive real analytic 2-form dμ on X(C), the Green function
gA = gA,μ is deﬁned to be a real analytic function on X(C), uniquely determinedby the following conditions:• Laplace equation: ∂∂gA = πi(deg(A)dμ − δA ) is satisﬁed, with the δcurrent δA : ϕ → x∈X mx ϕ(x).

－ 125 －

• Singularities: gA −mx log |z| is locally real analytic, for z a local coordinate
in a neighbourhood of x.
• Normalization: X gA dμ = 0 is satisﬁed.
If B = y∈X ny (y) is another divisor, such that |A| ∩ |B| = ∅, then theexpression deﬁned as gμ (A, B) = u ny gA,μ (y) is symmetric and biadditivein A, B. In general, such an expression as gμ depends on μ, where the choiceof μ is equivalent to the choice of a real analytic Riemannian metric on X,
compatible with the complex structure. However, in the special case of degree
zero divisors as deg A = deg B = 0, the expression gμ (A, B) are conformal
invariants, denoted as g(A, B), named as the Arakelov Green function.
In the case of the Riemann sphere P1 (C) ≈ S 2 ≈ C ∪ {∞}, if wA is ameromorphic function with Div(wA ) = A, then we haveg(A, B) = log Πy∈|B| |wA (y)|ny = Re

dwA
,
γB wA

where γB is a 1-chain with boundary B.
In the case of degree zero divisors A, B on a Riemann surface of higher genus,
the formula as above for g(A, B) can be generalized, replacing the logarithmic
diﬀerential dwA /wA with wA a diﬀerential of the third kind, as a meromoprhic
diﬀerential with non-vanishing residues, with purely imaginary and residues as
mx at x, as given asg(A, B) = Re

wA .
γB

Thus, g(A, B) can be explicitly computed from a basis of diﬀerentials of thethird kind with purely imaginary periods.
Remark. Recall from [116] the following. A diﬀerential on a Riemann surface
R is a 1-form as w = udx + vdy with z = x + iy locally, with w∗ = −vdx + udy
conjugate diﬀerential, with (w∗ )∗ = −w. If dw =0, the w is said to be closed.
Then, for α ∈ H1 (R) the homology, the integral α w is deﬁned uniquely to be
the period along a 1-cycle representing the class α. If there is a function F on
R such that dF = w, then w is said to be exact. if w∗ = −iw, then w is said
to be pure. Then w = f (z)dz locally. If f is holomorphic, then w is said to beholomorphic or analytic, which is equivalent to that w is closed and pure. If f
is meromorphic (as holomorphic or with only poles), then w is meromorphic or
Abelian diﬀerential. An Abelian diﬀerential on a closed (or compact) Riemannsurface have the sum of residues equal to zero. Abelian diﬀerentials are divided
into being of the ﬁrst, second, and third kind, respectively, if f is holomorphic,
pf has zero residues for poles, and otherwise. The indeﬁnite integral p0 w is said
to be Abelian integral, where p0 is not a pole. In particular, an Abel integral
on a closed Riemann surface of genus 1 is said to be elliptic integral.
As well, a divisor on a Riemann surface R is deﬁned to be a formal sum
with coeﬃcients in Z as D = p∈R np p, where the support of D is a discrete
set of p ∈ R with nonzero np ∈ Z. Deﬁne the degree of D as deg D = p∈R np .

－ 126 －

A divisor on D on R is said to be a divisor for f or w = f (z)dz if the support of
D consists of zeros and poles of f , and if p is a zero, np is its degree (i.e., f (z) =(z − p)np h(z) locally, with h holomorphic with h(p) = 0) and if p is a pole, thenh(z)−np ≥ 1 is its degree (i.e., f (z) = (z−p)−np locally, similarly as above). On aclosed (or compact) Riemann surface of genus g ≥ 1, a divisor of a meromorphic
function is said to be principal, and a divisor of an Abelian diﬀerential is said
to be canonical. For D principal, deg D = 0. For D canonical, deg D = 2g − 2.
Cross ratio and geodesics. The basic step leading to the result of Manin[106], expressing the Arakelov Green function in terms of geodesics in the hyperbolic handle-body YΓ = H3 /Γ is a simple classical fact in hyperbolic geometry. Namely, it is the fact that the (logarithmic) cross ratio of four points in
P1 (C) ≈ S 2 ≈ C ∪ {∞} can be expressed in terms of geodesics in the interior
H3 aslog |[a, b, c, d]| = −ord(a ∗ [c, d], b ∗ [c, d]),where [a, b, c, d] = −(a − b)(c − d)(a − d)−1 (b − c)−1 is the cross ratio of points
a, b, c, d ∈ P1 (C), and ord(·, ·) means the oriented distance in H3 , and a ∗ [c, d]indicates the on the geodesic [c, d] in H3 with end points c, d ∈ P1 (C), obtained
as the intersection of [c, d] with the unique geodesic from a that intersects with[c, d] at a right angle. (No original ﬁgure).
Diﬀerentials and Schottky uniformization. The next important step inthe result of Manin [106] is to show that if X(C) = Γ\ΩΓ is a Riemann surface
with a Schottky uniformization, then obtained is a basis of diﬀerentials of thethird kind with purely imaginary periods, by taking suitable averages over thegroup Γ of expressions involving the cross ratio of points of P1 (C).
Denote by C(γ) a set of representatives of Γ/γ Z , and by C(ρ, γ) a set ofrepresentatives ρZ \Γ/γ Z , and by S(γ) the conjugacy class of γ in Γ.
Let wA be a meromorphic function on P1 (C) with divisor A = (a) − (b), suchthat the support |A| is contained in the complement of an open neighbourhoodof ΛΓ .
For a ﬁxed choice of a base point z0 ∈ ΩΓ , the series
ν(a)−(b) =d log[a, b, γz, γz0 ],with [a, b, γz, γz0 ] =γ∈Γ(a − b)(γz − γz0 ),−(a − γz0 )(b − γz)gives the lift to ΩΓ of a diﬀerential of the third kind on the Riemann surface
X(C), endowed with the choice of Schottky uniformization. These diﬀerentials
have residues ±1 at the image of a, b in X(C), and they have vanishing ak
periods, where ak , bk for 1 ≤ k ≤ g are the generators of the (ﬁrst) homology ofX(C).
Similarly, obtained are the lifts of diﬀerentials of the ﬁrst kind on X(C), by－ 127 －

considering the seriesd log[hz + (γ), hz − (γ), z, z0 ],ωγ =h∈C(γ)

with[hz + (γ), hz − (γ), z, z0 ] =(hz + (γ) − hz − (γ))(z − z0 ),−(hz + (γ) − z0 )(hz − (γ) − z)where z ± (γ) ∈ ΛΓ denote the pair of the attractive and repelling ﬁxed points ofγ ∈ Γ.
The series as ν(a)−(b) and ωγ converge absolutely on compact subsets of ΩΓ ,whenever dimH ΛΓ < 1. Moreover, they do not depend on the choice of the basepoint as z0 ∈ ΩΓ .
In particular, given {γk }gk=1 a choice of generators of the Schottky groupΓ, obtained is a basis of holomorphic diﬀerentials ωγk , satisfying ak ωγl =√2π −1δkl .
Then use a linear combination of the holomorphic diﬀerentials ωγk to correct the meromorphic diﬀerentials ν(a)−(b) in such a way that the resulting
meromorphic diﬀerentials have purely imaginary bk -periods. Let xl (a, b) denotecoeﬃcients such that the diﬀerential of the third kind deﬁned as
ω(a)−(b) = ν(a)−(b) −gxl (a, b)ωγl

l=1(corrected) have purely imaginary bk -periods. The coeﬃcients xl (a, b) satisfy
the system of equationsgxl (a, b)Rebk

l=1

ωγl = Rebk

log |[a, b, z + (h), z − (h)]|h∈S(γk )=−

ν(a)−(b) =ord(a ∗ [z + (h), z − (h)], b ∗ [z + (h), z − (h)])

h∈S(γk )(speciﬁed). Thus, it is obtained ([106], cf. also [157]) the Arakelov Greenfunction g(A, B) for X(C) with Schottky uniformization can be computed asg((a) − (b), (c) − (d)) = Re

ω(a)−(b)γ(c)−(d)[ν(a)−(b) −= Re
γ(c)−(d)ν(a)−(b) −= Re
γ(c)−(d)

=


h∈Γ

g

l=1
g


xl (a, b)ωγl ]xl (a, b)Re

l=1
g


log |[a, b, hc, hd]| −l=1

γ(c)−(d)xl (a, b)h∈S(γl )－ 128 －

ωγ llog |[z + (h), z,− (h), c, d]|.

Note that[a, b, c, d] =(a − b)(c − d)(c − d)(a − b)=
= [c, d, a, b].
−(a − d)(b − c)−(c − b)(d − a)

It is noticed that the result above seems to indicate that there is a choiceof Schottky uniformization involved as an additional datum for Arakerov geometry at arithmetic inﬁnity. However, as remarked previously, at least in thecase of archimedean primes that are real embeddings as the case of arithmetic
inﬁnity for Q, the Schottky uniformization is detemined by the real structure,by splitting X(C) along the real locus X(R), when the latter is non-trivial.
Green function and geodesics. By combining the basic formula by theoriented distance with the formula calculated above for the Arakelov Greenfunction on a Riemann surface with Shottky uniformization, we can replace eachterm appearing in the formula with the corresponding term which computes theoriented geodesic length of a certain arc of geodesic in YΓ as
ord(a ∗ [hc, hd], b ∗ [hc, hd])g((a) − (b), (c) − (d)) = −
h∈Γ

+

g

l=1

xl (a, b)

ord(z + (h) ∗ [c, d], z − (h) ∗ [c, d]).

h∈S(γl )The coeﬃcients xl (a, b) can also be expressed in terms of geodesics, using thesystem equations given above.

4.5

Quantum gravity and black holes as Intermezzo

The anti-de Sitter space a-dSd+1 of real dimension d + 1 is a highly symmetric space-time, satisfying Einstein’s equations with constant curvature R < 0.
Physically, it describes an empty space with a negative cosmological constant. In
order to avoid time-like closed geodesics, it is customary to pass to the universal
cover a-dS∼
d+1 , the boundary at inﬁnity of which (or a-dSd+1 ) is a compactiﬁcation of d-dimensional Minkowski space time Mkd . When passing to Euclidean
signature, the space a-dS∼
d+1 becomes the real d+1-dimensional hyperbolic space
Hd+1 .
The real 4-dimensional anti-de Sitter space with d = 3 is a well known
example as space-time in general relativity. The space a-dS3+1 has the form
S 1 × R3 the cylinder as a topological space, while it is realized metrically as thehyperboloid in R5 deﬁned as−u2 − v 2 + x2 + y 2 + z 2 = 1

with ds2 = −du2 − dv 2 + dx2 + dy 2 + dz 2

4
as the metric element. The universal cover a-dS∼
3+1 is R as a topological space.
In the context of quantum gravity, it is especially interesting to consider thecase of the real 3-dimensional a-dS2+1 and its Euclidean counterpart as the real
3-dimensional hyperbolic space H3 .

－ 129 －

In this case, the Minkowski space time Mkd ≈ Rd ≈ R × Rd−1 with themetric as the diagonal sum
G = (gμν ) = −1 ⊕ 1d−1 .
Note also that, with G = −12 ⊕ 13 , in the anti-de Sitter space a-dS4 ,ds2 = ds · ds = (−12 ⊕ 13 )(du, dv, dx, dy, dz)t , d(u, v, x, y, z).
The holographic principle postulates the existence of an explicit correspondence between gravity on a bulk space which is asymptotically a-dS∼
d+1 , like aspace obtained as a global quotient of a-dS∼
d+1 by a discrete group of isometries,
and ﬁeld theory on its conformal boundary at inﬁnity.
The above given, relation between the log of the cross ratio as [a, b, c, d]for a, b, c, d ∈ P1 (C) and the oriented distance ord(·, ·) in terms of geodesics in
H3 , which identiﬁes the Green function g((a) − (b), (c) − (d)) on P1 (C) withthe oriented length of a geodesic arc in H3 , can be thought of as an instanceof the holography principle, when we interpret one side as geodesic propagator on the bulk space in a semi-classical approximation, and the other side as
the Green function as the two-point correlation function of the boundary ﬁeld
theory. Note that, because of the prescribed behavior of the Green function
at the singularities given by the points of the divisor, the four-point invariantas g((a) − (b), (c) − (d)), when a and b converge to c, d respectively, gives thetwo-point correlator with a logarithmic divergence which is intrinsic and does
not depend on a choice of cut-oﬀ functions, unlike the way in regularization isoften used in the physics literature.
It is shown in [111] that the above given, (another) relation between Arakelov
Green functions and conﬁgurations of geodesics in the hyperbolic handle-bodyYΓ , proved by Manin in [106], provides in fact precisely the correspondence prescribed by the holography principle, for a class of (2+1)-dimensional space-time
known as Euclidean Krasnov black holes. These as holes include the BañadosTeitelboim-Zanelli (BTZ) black holds, as an important class of space-times in
the context of (2 + 1)-dimensional quantum gravity.
The Bañados-Teitelboim-Zanelli black hole. Consider the case of a hyperbolic handle-body as YΓ of genus one, as a solid (open) torus (as a donut),with conformal boundary at inﬁnity given by an elliptic curve.
Recall that elliptic curves can be described via the Jacobi uniformization, as
already encountered in the previous sections, in the context of noncommutative
elliptic curves. Let Xq (C) = C∗ /q Z be such a description of an elliptic curve,
where q is a hyperbolic element of P SL2 (C) with ﬁxed points as {0, ∞} on the
2-sphere as P1 (C) at inﬁnity, for H3 , so that q ∈ C∗ with |q| < 1. The action ofq on P1 (C) extends to an action on H3 = H3 ∪ P1 (C) byq · (z, y) = (qz, |q|y) ∈ C × (0, ∞) ≈ H3 .
It follows that the quotient space Yq = H3 /q Z by this action is a solid (open)
torus as a topological space, which is compactiﬁed at inﬁnity by the conformalboundary Xq (C) as ∂Yq (as a torus).

－ 130 －

The space Yq is well known in the physics literature as the Euclidean BañadosTeitelboim-Zanelli black hole, where the parameter q ∈ C∗ is written as the
form



2π(i|r− | − r+ )12
q = exp
with r±M l ± M 2 l2 + J ,=l2
where M and J are the mass and angular momentum of the rotating blackhole, and − l12 is the cosmological constant. The corresponding black hole in
Minkowskian signature may be illustrated, but no ﬁgure provided (as in [3]).
In the case of the elliptic curve as Xq (C) = C∗ /q Z , the formula of AlvarezGaumé, Moore, and Vafa [2] gives the operator product expansion of the path
integral for bosonic ﬁeld theory as −1 log |z|nn −1g(z, 1) = log |q|2 B2 ( log |q| ) |1 − z|Π∞| .
n=1 |1 − q z||1 − q zThis is in fact the Arakelov Green function on Xq (C). In terms of geodesics inthe Euclidean BTZ black hole (BH), that becomes
 


1
lγ0 (z, 1)g(z, 1) = − l(γ0 )B2+
lγ1 (0, zn ) +lγ1 (0, zn∼ ),2l(γ0 )n≥0

n≥1

where B2 (v) = v 2 − v + 16 is the second Bernoulli polynomial, and we use thenotation as that x = x ∗ [0, ∞], zn = q n z ∗ [1, ∞), and zn∼ = q n z −1 ∗ [1, ∞)in H3 as well as Xq , as in [106]. (No respective ﬁgures.) These terms describe
gravitational properties of the Euclidean BTZ BH. For instance, l(γ0 ) measures
the black hole entropy. The whole expression is a combination of geodesic
propagators.
Krasnov black holes. The problem of computing the bosonic ﬁeld propagatoron an algebraic curve XC (such as X(C) = Γ\ΩΓ ) can be solved by providing
diﬀerentials of the third kind with purely imaginary periods as
ω(a)−(b) = ν(a)−(b) −gxl (a, b)ωγl ,l=1

and thus it can be related directly to the problem of computing the Arakelov
Green function.
Diﬀerentials as above then determine all the higher correlation functions as
G(z1 , · · · , zm , w1 , · · · , wl ) =

m 
l


qk ϕ(zk , zk )ϕ(wj , wj )qj ,j=1 k=1

for qk a system of charges at positions zk interacting with charges qj at positions wj from the basic two-point correlator Gμ (a − b, z) given by the Greenfunction expressed in terms of the diﬀerentials ω(a)−(b) (z). By using a Schottky
uniformization, obtained are the diﬀerentials ω(a)−(b) as given above.

－ 131 －

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 ispossible 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 a more reﬁned form of analytic continuation which isadapted 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 function g(·, ·) in terms of oriented
distances of geodesics as in YΓ gives the explicit bulk and boundary correspondence of the holography principle for this class of space times, so that each termin the Bosonic ﬁeld propagator for XC is expressed in terms of geodesics in the
Euclidean Krasnov black hole as YΓ = H3 /Γ.

4.6

Dual graph and noncommutative geometry

The result of Manin on the Arakelov Green function and hyperbolic geometry
suggests a geometric model for the dual graph of the mysterious ﬁber at arithmetic inﬁnity. In fact, the result discussed above on the Green function has ananalog, due to Drinfel’d and Manin [73] in the case of a ﬁnite prime with a totally split ﬁber. This is the case where the p-adic completion admits a Schottky
uniformization by a p-adic Schottky group.
Remark. Record from [116] as well as [123], [128] the following. Let R be a
Dedekind ring (or a Noether integral domain) as a subring of a (quotient) ﬁeld
K(= RR∗ of R) and p its maximal ideal. For any nonzero a ∈ K, deﬁne v(a) = n
if the ideal (a) is the product of pn for n ∈ Z with an ideal, relatively prime withp (or if a = pn cb for b, c ∈ R and n ∈ Z, where p is a prime element of R and b, c
are not divided by p). Then v is a normal, discrete, non-Archimedean, additive
valuation (av) of K, which is said to be a p-adic exponential valuation. Namely,v is a map from K to G ∪ {∞}, with G an ordered additive group as Z, suchthat (av 1) v(a) = ∞ ⇔ a = 0, (av 2) v(ab) = v(a) + v(b) for nonzero a, b ∈ K,
(av 3) v(a + b) ≥ min{v(a), v(b)} for any a, b ∈ K. The valuation ring of v
is deﬁned to be the subring Rv of all a ∈ K such that v(a) ≥ 0. There is
the unique maximal ideal mv of all a ∈ K such that v(a) > 0, deﬁned as thevaluation ideal of v. The quotient ring Rv /mv is said to be the residue classﬁeld of v.
The (av 2) implies that v(−1) = v(−1) + v(1), so that v(1) = 0.
The p-adic valuation of K is a non-Archimedean (n-A), multiplicative valuation (mv) w of K deﬁned as w(a) = c−v(a) ∈ R (non-negative) for someconstant c > 1 and a p-adic exponential valuation v, satisfying that (mv 1)
w(a) = 0 ⇔ a = 0, (mv 2) w(ab) = w(a)w(b), (mv 3) w(a + b) ≤ w(a) + w(b),and moreover (n-A) w(a + b) ≤ max{w(a), w(b)}, for a, b ∈ K (and otherwise,
Archimedean).
By (mv 2), w(1) = w(1)2 in R. Hence w(1) = 1 or 0. Thus, w(1) = 1 by
(mv 1). Also, w(1) = w(−1)2 . Thus, 0 ≤ w(−1) = 1. Moreover, 1 = w(1) =w(aa−1 ) = w(a)w(a−1 ) for any a ∈ K ∗ .

－ 132 －

The trivial multiplicative valuation on K is deﬁned by w(a) = 1 for any
nonzero a ∈ K.
A non-Archimedean, multiplicative variation w is said to be discrete if w(K ∗ )as a subgroup of R∗ by (mv 2) is isomorphic to Z.
Check that w(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).
The valued group of w is deﬁned to be the group of all w(a) for any nonzeroa ∈ K. A multiplication valuation w on K is said to be Archimedean if for
any nonzero a, b ∈ K, there is n ∈ Z such that w(na) ≥ w(b). Otherwise, it issaid to be non-Archimedean (n-A).
If w on K is Archimedean, then there is an embedding σ of K into C, andw is equivalent to the pull back as |σ(a)|. If w on K (with c > 1) is nonArchimedean, then deﬁne v(a) = − logb w(a) with b > 1, which is a rank 1(additive) variation. The converse also holds.
With w a multiplicative valuation (with c > 1), deﬁne a metric on a ﬁeld
K by d(a, b) = w(a − b) for a, b ∈ K.
Note that d(a, b) = 0 if and only if w(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).
If K is complete with respect to this metric, w is said to be complete. If
not so, there is the unique completion K  of K such that K  is an extension of
K with w as an extension of w, so that K is dense in K  and K  is completewith respect to w .
The p-adic completion Kp of K is deﬁned to be the completion of K withrespect to the p-adic completion. In particular, if K is a algebraic number ﬁeld,
Kp is said to be the p-adic algebraic number ﬁeld.
If R = Z and p = (p) = pZ for a prime p, the corresponding p-adic valuation
is deﬁned as w or v. Also, the p-adic number ﬁeld Qp is deﬁned to be thecompletion of Q with respect to the p-adic valuation. Any nonzero elementa ∈ Qp as a p-adic number has the forma=∞


an pn ,an ∈ Z/pZ, r ∈ Z, ar = 0,

n=r

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

∞


n=r
∞


v(a + b) = v(

n=r

an pn

∞


bm pm ) = v(

m=s
∞


an pn +∞ ∞an bm pn+m ) = r + s = v(a) + v(b),n=r m=sbm pm ) ≥ max{r, s} = max{v(a), v(b)}.

m=s－ 133 －

Schottky-Mumford curves. Let K be a given ﬁnite extension of Qp and let
O in K its ring of integers, and m in O the maximal ideal, and k = O/m theresidue ﬁeld, that is a ﬁnite ﬁeld of cardinality q = card(O/m).
It is well known that a curve X over a ﬁnite extension K of Qp , which is
k-split degenerate for k the residue ﬁeld, admits a p-adic uniformization by ap-adic Schottky group Γ acting on the Bruhat-Tits tree ΔK .
The Bruhat-Tits (BT) tree ΔK is obtained by considering free O-modules
M of rank 2, with the equivalence relation that M1 ∼ M2 if there is λ ∈ K ∗such that M1 = λM2 . The set Δ0K of vertices of the BT tree consists of theequivalence classes [M ] under the relation ∼. The distance between two classesof the set Δ0K is deﬁned by d([M1 ], [M2 ]) = |l − k|, where if M1 ⊃ M2 , thenM1 /M2 ∼= O/ml ⊕ O/mkfor some l, k ∈ N.

To form the BT tree, an edge of the set Δ1K of edges is deﬁned to be a (directed
or not) arrow from [M2 ] to [M1 ] if M2 ⊂ M1 with d([M1 ], [M2 ]) = 1. Then the
BT tree ΔK = (Δ0K , Δ1K ) becomes a connected, locally ﬁnite tree with q + 1
edges departing from each vertex.
The group P GL2 (K) acts on ΔK (from the left) transitively by isometries.
The Bruhat-Tits tree ΔK is the analog of the real 3-dimensional hyperbolic
space H3 at the inﬁnite primes. The set of ends of ΔK is identiﬁed with P1 (K),just like that P1 (C) = ∂H3 in the case at inﬁnity.
A p-adic Schottky group Γ is deﬁned to be a discrete subgroup of P GL2 (K)which consists of hyperbolic elements γ in K, for which the eigenvalues havediﬀerent valuation, and Γ is isomorphic to a free group with g generators. Denoteby ΛΓ,K the limit set in P1 (K), which is the closure of the set of points of P1 (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 some γ ∈ 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-Mumford curve, with p-adc Schottky uniformization. In the case ofgenus 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, withno back-tracking, is said to be an axis of ΔK . Any two points z1 , z2 ∈ P1 (K)uniquely deﬁne an axis connecting z1 and z2 as endpoints in ∂ΔK . The uniqueaxis of ΔK whose ends are the ﬁxed points of a hyperbolic element γ is said tobe the axis of γ. The element γ acts on its axis as a translation. Denote by ΔΓthe smallest subtree of ΔΓ containing all the axes of elements of Γ (which may
be called as the axis tree).
The subtree ΔΓ is Γ-invariant, with ΛΓ,K as the set of ends. The quotientΔΓ /Γ is a ﬁnite graph, which is the dual graph of the closed ﬁber of the minimal smooth model over O of XΓ,K (as a k-split degenerate semi-stable curve).
With no ﬁgures, there are coorrespondences among the special (closed) ﬁbers ascurves, the dual graphs such as two circles a, b as edges attached with a point－ 134 －

or an edge c, and the axis trees ΔΓ generated by {a, b} or {a, b, c}, for all thepossible cases of maximal degenerations special for genus 2.
For each n ≥ 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
Δ0K,n = {vertex v ∈ Δ0K | d(v, ΔΓ ) ≡ inf{d(v, v  ) | v  ∈ (ΔΓ )0 } ≤ n},
where d(v, v  ) is the distance on Δ0K , with v = [M ], v  = [M  ], andΔ1K,n = {edge w ∈ Δ1K | source s(w), range r(w) ∈ Δ0K,n }.
In particular, ΔK,0 = ΔΓ .
For all n ∈ N, the subtree or subgraph ΔK,n is invariant under the action ofa p-adic Schottky group Γ on ΔK , and the quotient ΔK,n /Γ is a ﬁnite graph,
which is the dual graph of the reduction XΓ,K ⊗ O/mn+1 .
For a more 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 thefollowing:
Table 6: The dictionary between the BT tree and the Hyperbolic 3-geometry
NotionSpace
Boundary
Path (mini)Group action
Discontinuityof Γ by Γ
Solid quotientCore part
Bounded quo.

Tree geometric dynamics
BT tree graph ΔK = (Δ0K , Δ1K )P1 (K) = ∂ΔKConnecting edges in ΔK
Schottky Γ ⊂ P GL2 (K)Schottky-Mumford curve
XΓ,K = ΩΓ (K)/Γ = ΛcΓ,K /Γ
BT Graph ΔK /Γ
Axis subtree ΔΓ ⊂ ΔΓFinite dual graph ΔΓ /Γ

Hyperbolic 3GD
H 3-space H3
1
P (C) = ∂H3 ≈ S 2
Geodesics in H3
Schottky Γ ⊂ P SL2 (C)Riemann surface
XΓ = ΩΓ /Γ = ΛcΓ /ΓHandle-body YΓ = H3 /Γ
Convex core in H3
Bounded geodesics in YΓSince bounded geodesics in YΓ can be identiﬁed with inﬁnite geodesics in H3
with endpoints on ΛΓ ⊂ P1 (C) module the action of Γ, these are parameterized
by the complement of the diagonal in ΛΓ ×Γ ΛΓ . 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 shift T . Thus obtained is the following
model for the dual graph of the ﬁber at inﬁnity.
• The solenoid ST = (S×[0, 1])/ ∼ as the mapping torus for T is a geometricmodel of the dual graph of the ﬁber at inﬁnity of an arithmetic surface.
• The Cuntz-Krieger C ∗ -algebra OA as a noncommutative space, representing the algebra of coordinates on the quotient ΛΓ /Γ, corresponds to the setof vertices of the dual graph as the set of components of the ﬁber at inﬁnity,－ 135 －

while the C ∗ -algebra crossed product C(S) T Z as a noncommutative space,corresponding to the quotient ΛΓ ×Γ ΛΓ , 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 pm deﬁned by the subgraphs
ΔK,n of the BT tree ΔK in the case of Mumford curves. In fact, the reduction map corresponds to the paths connecting ends of the graph ΔK,n /Γ tothe corresponding vertices of the graph ΔΓ /Γ = ΔK,0 /Γ. Then the analog atarithmetic inﬁnity consists of geodesics in YΓ which are the images of geodesics
in H3 starting at some point x0 ∈ H3 ∪ ΩΓ and having the other end as a pointof ΛΓ . These are parameterized by the set ΛΓ ×Γ (H3 ∪ ΩΓ ). Thus, in termsof NC geometry, the reduction mod ∞ corresponds to a compactiﬁcation of thehomotopy 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 as H3 . Then the reduction mod ∞ is given by the assembly
map μ asμ : K ∗+1 (ΛΓ ×Γ H3 ) → K∗ (C(ΛΓ )  Γ), ∗ = 0, 1.
This shows that the spectral triple (OA , H, D) as a noncommutative (Riemann)space is closely related to the geometry of the ﬁber at arithmetic inﬁnity of analgebraic 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 thenext section that as proved in [61], the Dirac operator gives another importantarithmetic invariant, namely the local L-factor at the Archimedean prime.

4.7

Arithmetic varieties and L-factors

The L-function is an important invariant of arithmetic varieties. This is written
as a product of contributions from the ﬁnite primes and the archimedean primes,asΠp∈Sp(OK ) Lp (H m (X), s).
For a detailed account on the subject, may refer to [144]. Let us only try toconvey some basic ideas.
The reason why necessary is it to consider also the contribution of thearchimedean primes can be seen in the case of the aﬃne line (Spec) Sp(Z),where the Riemann zeta function is involved and written as the Euler product as∞11
ζ(s) == Πp.
sn
1 − p1sn=1That is holomorphic for Re(s) > 1 is analytically extended to a meromorphic function on C with only pole s = 1 of order 1. Namely, as the Laurent
expansion, locally, but for s ∈ C \ {1},
 n

∞

1
1
ζ(s) =
ck (s − 1)k , γ = lim+γ+− log n
n→∞
s−1
k
k=1

k=1

－ 136 －

with γ the Euler constant and ck the generalized Euler constants.
However, to have the functional equation as below, we needs to consider the
products 1
1−s 1
ζ(s)Γ( ) s = ζ(1 − s)Γ()
2 π2
2 π 1−s2included with a contribution of the archimedean prime, expressed in terms of
the Gamma function, with factorials∞

Γ(s) =
0ts
dt = (s − 1)Γ(s − 1),tetΓ(n) = (n − 1)!,

Γ(1) = 1.

An analogy with ordinary geometry suggests to think of the functional equation as a sort of Poincaré duality, which holds for a compact manifold, hence forwhich, we need to compactify arithmetic varieties by adding the archimedean
primes and the corresponding archimedean ﬁbers.
Looking at an arithmetic variety over a ﬁnite prime p ∈ Sp(OK ), the factthat the reduction lives over a residue ﬁeld of positive characteristic implies
that there is a special operator as the geometric Frobenius F rp∗ , acting on asuitable cohomology theory as étale cohomology, induced by the Frobenius
automorphism ϕp of Gal(Fp /Fp ) (as a pn -power map).
Note that Fp ∼= ∪1≤n<∞ Fpn as an algebraic closure of the ﬁnite ﬁeld Fp
with order a prime p, with [Fpn , Fp ] = n (cf. [123]).
The local L-factors of the L-function ﬁrst given as the product at ﬁnite
primes p encode the action of the geometric Frobenius F rp∗ as in the form, asin [144]
Lp (H m (X), s) = det(1 − F rp∗ N (p)−1 |H m (X, Ql )Ip )−1 ,where we consider the action of the geometric Frobenius F rp∗ on the inertia
invariants H m (X, Ql )Ip of the étale cohomology, as explained in the following.
We use the notation N for the norm map in the local L-factors (converted).
An introduction to étale cohomology as well as a precise deﬁnition of thesearithmetic structures are beyond the scope of this survey. In fact, our primaryconcern would only be the contribution of the archimedean primes to the Lfunction, where the construction is based on the ordinary de Rham cohomology.
Thus, we only give a quick and somewhat heuristic explanation of the local
L-factors. May refer to [144], [155] for a detailed and rigorous account and forthe precise hypotheses under which the following holds.
Let X = X(Q) be a smooth projective algebraic variety of any dimension
deﬁned over Q. Set X = X ⊗ Sp(Q) ≡ X(Q), where Q denotes an algebraic
closure. For l a prime, the (étale) cohomology H ∗ (X, Ql ) is a ﬁnite-dimensionalQl -vector space, satisfying
H j (X(C), C) ∼= H j (X, Ql ) ⊗ C.
The (countable!) algebraic closure
Q√
of Q may be given as the set of all√
algebraic numbers over Q (such as 2 and 3), up to a Q-isomorphism. Then

－ 137 －

Q ⊂ Q but Q  C, because there are transendental numbers in C such as e andπ (and γ?) (cf. [116], [129]).
The absolute Galois group Gal(Q/Q) = Aut(Q/Q) acts on H ∗ (X, Ql ).
Similarly, we can consider the (étale) cohomology H ∗ (X, Ql ) for X = X(K)deﬁned over a number ﬁeld K ⊃ Q, with X = X ⊗ Sp(K) ≡ X(K). For
p ∈ Sp(OK ) of maximal ideals of OK the ring of integers in K and l a prime
such that (l, q) = 1, where q is the cardinality of the residue ﬁeld OK /p at p,the (physical law of) inertia invariants are deﬁned to be
H ∗ (X, Ql )Ip ⊂ H ∗ (X, Ql )as the part of the l-adic cohomology, on which the inertia group at p acts
trivially, and the inertia group is deﬁned as the kernel of the short exact sequenceof groups with rp as the quotient map:rp

1 → Ip = ker(rp ) → Dp = {σ ∈ Gal(Q/Q) | σ(p) = p} −−−−→ Gal(Fp /Fp ) → 1.
The Frobenius automorphism ϕp of Gal(Fp /Fp ) (as a pn -power map) lifts to
ϕp ∈ Dp /Ip by the same symbol, which induces the geometric Frobenius F rp∗ =
∗
∗
Ip
(ϕ−1
p ) acting on H (X, Ql ) . Thus, the local L-factor given above can be
written equivalently as
Lp (H m (X), s) = Πλ∈sp(F rp∗ )1Ipm(1 − λq −s )dim H (XΓ )λ,I

where H m (XΓ )λp denotes the eigen-space of the Frobenius with eigenvalue λ ∈sp(F rp∗ ) the (operator) (point) spectrum.
I
I
The notation H m (XΓ )λp may better be replaced with H m (X, Ql )λp .
For our purpose, what is most important to retain from the discussion aboveis that the local L-factors depend on the data (H ∗ (X, Ql )Ip , F rp∗ ) of a vector
space and a linear operator on it, which have a cohomological interpretation.
Archimedean L-factors. Since the étale cohomology satisﬁes the compatibility as
H j (X, Ql ) ⊗ C ∼= H j (X(C), C),if we again resort to the general philosophy, according to which we can workwith the smooth complex manifold X(C) and gain information on the closed
ﬁber at arithmetic inﬁnity, then we are led to expect that the contribution of
the archimedean primes to the L-function may be expressed in terms of thecohomology H ∗ (X(C), C), or equivalently in terms of de Rham cohomology.
In fact, it is shown by Serre [144] that the expected contribution of thearchimedean primes depends on the Hodge structure as
H m (X(C)) = ⊕p+q=m H p,q (X(C))
and is again expressed in terms of Gamma functions, as in the case of the funcs
tional equation for ζ(s)Γ( 2s )π − 2 . Namely, given is the product of Gamma functions according to the Hodge numbers hp,q (as the dimension of H p,q (X(C)))－ 138 －

for the (another) local L-factor:
L(H m (X(C)), s) =

p,q
p,+
p,−
Πp<q ΓC (s − p)h Πp ΓR (s − p)h ΓR (s − p + 1)h , for the real embeddingp,qΠp,q ΓC (s − min{p, q})h ,for the complex one
where hp,± is the dimension of the ±(−1)p -eigenspace of the involution on H p,pinduced by the real structure, andΓC (s) ≡ (2π)−s Γ(s)and

ΓR (s) ≡√ −1 √ −s2
π Γ(2−1 s).

One of the general ideas in arithmetic geometry is the always seeking a uniﬁed picture of what happens at the ﬁnite and at the inﬁnite primes. In particular, it should be a suitable reformulation of the local L-factors Lp (H m (X(K)), s)
and L(H m (X(C)), s) as the determinant or the product, where both formulae
can be expressed in the similar way.
By seeking a uniﬁed description of local L-factors at ﬁnite and inﬁnite
primes, both L-factors as above are expressed as inﬁnite determinants, byDeninger in [68], [69], [70].
Recall that the Ray-Singer determinant of an operator T with pure (imaginary) point spectrum λ ∈ sp(T ) with ﬁnite multiplicities mλ is deﬁned to be



mλ
d
det(s − T ) ≡ exp − ζT (s, z)|z=0 , ζT (s, z) =∞dz(s − λ)zλ∈sp(T )as the zeta function of the operator T . Described in [107] are suitable conditions
for the convergence of these expressions in the case of the local L-factors.
It is shown by Deninger that the local L-factor Lp (H m (X), s) as the productcan be written equivalently in the form
Lp (H m (X), s)−1 = det(s − Θq ),∞for an operator s − Θq with spectrum given as
sp(s − Θq ) = {s − αλ +2πin| n ∈ Z, λ ∈ sp(F rp∗ )}log qwith multiplicities dλ and q αλ = λ.
Moreover, the (another) local L-factor L(H m (X(C)), s) at inﬁnity can be
written similarly in the form()−1
L(H m (X(C)), s) = det (2π)−1 (s − Φ)|Hm,∞

where Hm is an inﬁnite dimensional vector space and Φ is a linear operator with
spectrum sp(Φ) = Z with ﬁnite multiplicities. This operator is regarded as alogarithm of Frobenius at arithmetic inﬁnity.

－ 139 －Given the two formulae of Deninger for the local L-factors by det∞ , it isnatural to ask for a cohomological interpretation of the data (Hm , Φ), somewhat
analogous to the data (H ∗ (X, Ql )Ip , F rp∗ ).
Arithmetic surfaces: L-factor and Dirac operator. As a return, let us
now consider the special case of arithmetic surfaces, in the case of genus g ≥ 2.
At an archimedean prime, consider the Riemann surface Xα (C) with a Schottky
uniformization X(C) = ΩΓ /Γ. In the case of a real embedding α : K → C, we
can assume that the choice of a Schottky uniformization corresponds to the real
structure, obtained by cutting X(C) along the real locus X(R).
Now consider the spectral triple (OA , H, D) associated to the Schottky groupΓ acting on its limit set ΛΓ , where H = ⊕2 L2 (ΛΓ ) and OA ∼= C(ΛΓ )  Γ and


∞
∞


∧
.
nΠn % −
nΠ∧
D=
n
n=1

n=1

Since the spectral triple is not ﬁnitely summable (but θ-summable), we can not
deﬁne zeta functions of the spectral triple as in the form tr(a|D|z ). However,
we can consider the restriction of D to a suitable subspace of H (if any), on
which the trace is ﬁnite.
In particular, consider the zeta function with respect to the (Dirac) operator
D (restricted)
ζπV ,D (s, z) =tr(πV Π(λ, D))λ∈sp(D)i1(s − λ)z(i inserted), where Π(λ, D)(= p(λ, D)) is the projection on the eigenspace for an
eigenvalue λ of D, and πV is the orthogonal projection of H on the the subspace
V of 0 ⊕ L2 ⊂ H deﬁned by
Πn πV Πn =gsnj (s∗j )n

on L2 ,

j=1

where Πn are the projections onto Pn,C in L2 and are also the spectral projections of the Dirac operator D, with Π(−n, D) = Πn for n ≥ 0.
In the case of an arithmetic surface, as the interesting local factor. the ﬁrst
cohomology L(H 1 (X), s) is computed ([59], [61]) as in the following, from thezeta function ζπV ,D of the spectral triple (OA , H, D).
Theorem 4.5. The local L-factor given as L(H m (X(C)), s) above is computed
as
−1

d
L(H 1 (X(C)), s) = exp − ζπV ,(2π)−1 D ((2π)−1 s, z)|z=0.
dz
In the case of a real embedding, the same holds, with the projection πV,F∞ onto
the +1 eigenspace of the involution F∞ induced by the real structure on V.

－ 140 －

In particular, this result shows that, for the special case of arithmetic surfaces
with X(C) of genus g ≥ 2, the pair (V ⊂ L2 (ΛΓ ), D|V ) is a possible geometricconstruction of the pair (H1 , Φ) of (Hm , Φ). Moreover, the Dirac operator of
the spectral triple has an arithmetic meaning, as that it recovers the logarithmof Frobenius as D|V = Φ.
Looking more closely at the subspace V of the Hilbert space H, we seethat it has a simple geometric interpretation in terms of the geodesics in thehandle-body YΓ . Recall that the ﬁltered subspace PC = C(ΛΓ , Z) ⊗ C of L2 =
L2 (ΛΓ , dμ) describes 1-cochains on the mapping torus ST , with PC /δ∗ PC =H 1 (ST ).
The mapping torus ST is viewed as a copy of the tangle of bounded geodesics
inside YΓ . Among these geodesics there are g fundamental closed geodesics
that correspond to the generators of Γ, which correspond to geodesics in H3
connecting the ﬁxed points {z ± (γj )} for j = 1, · · · , g. Topologically, these are
the g core handles of the handle-body YΓ , which generate the homology H1 (YΓ ).
Consider the cohomology H 1 (ST ). Suppose that there are elements of1
H (ST ) supported on those fundamental closed geodesics. But this is impossible, because 1-cochains on ST are deﬁned by functions in C(S, Z), that aresupported on some clopen subsets covering the totally disconnected set S, whichcontains no isolated points. However, it is possible to choose a sequence of 1cochains on ST whose supports are much smaller clopen subsets containing theinﬁnite word in S corresponding to one of the g fundamental geodesics. The
complex ﬁnite dimensional subspace Vn = V ∩ Pn,C ⊂ Pn,C with dimC Vn = 2ggives representatives of such cohomology classes in H 1 (ST ), where we get 2g
instead of g because we take into account the two possible choices of orientation.
Thus, it gives us the cohomological interpretation of the space H1 = V ofthe pair (H1 , Φ), in the case of arithmetic surfaces. Moreover, the Schottky
uniformization also provides us with a way of expressing the cohomology as
H1 = V in terms of the de Rham cohomology H 1 (X(C)). In fact, already seen in
the calculation of the Green function, under the hypothesis that dimH (ΛΓ ) < 1,
to each generator γj of the Schottky group Γ, we can associate a holomorphic
diﬀerential ωγj on the Riemann surface X(C). Namely, the mapd log[hz + (γj ), hz − (γj ), z, z0 ]γj → ωγj =h∈C(γj )induces an identiﬁcation asV∼= ⊕n∈Z,n≥0 H 1 (X(C)).
See in the next section that, in fact, the right hand side above is a particularcase of a more general construction that works for arithmetic varieties in any
dimension and that gives a cohomological interpretation of (Hm , Φ).

4.8

Archimedean cohomology

Given by Consani in [57], for general arithmetic varieties in any dimension, acohomological interpretation of the pair (Hm , Φ) in the Deninger calculation－ 141 －

of the archimedean L-factors as regularized determinants. Its construction ismotivated by the analogy between geometry at arithmetic inﬁnity and the classical geometry of a degeneration over the disk. Introduced by her is a double
complex of diﬀerential forms with an endomorphism n representing the logarithm of the monodromy around the speical ﬁber at arithmetic inﬁnity, which
is modelled on a resolution of the complex of nearby cycles in the geometriccase. The deﬁnition of the complex of nearby cycles and of its resolution israther technical, on which the following construction is modelled. What is easy
to visualize it geometrically is the related complex of the vanishing cycles of ageometric degeneration. (No original ﬁgure).
But it looks like that a surface with genus two degenerates to the corresponding (2-dimensional) graph with 3 vertexes and 6 edges, which looks likethat two (solid) triangles are attached at respective vertices.
Describe the construction of [57] using the notation of [63]. And construct
the cohomology theory underlying (Hm , Φ) in the following three steps. In thefollowing, assume that X = X(C) is a complex compact Kähler manifold.
A complex manifold X with J an almost complex structure such that a
map J : T M → T M the tangent bundle over M with J 2 = −id the identity map,
and with a Riemann metric g as an Hermite metric, so that g(Jx, Jy) = g(x, y),is said to be a Kähler manifold if dω = 0, where ω(x, y) = g(Jx, y) is a 2-form(cf. [116]).
As the step 1, we begin by considering a doubly inﬁnite graded complexasC ∗ = Ω∗ (X) ⊗ C[u, u−1 ] ⊗ C[h, h−1 ],where Ω∗ (X) is the de Rham complex of diﬀerential forms on X, while u and h
are forma variables, with u of degree two and h of degree zero. On this complex,
consider the diﬀerentials√dC = hd and dC = −1(∂ − ∂),with total diﬀerential δC = dC + dC (with the exterior derivative d = ∂ + ∂as the decomposition by the Dolbeault operator ∂ with respect to a Hermite
metric). Deﬁne an inner product asα ⊗ ur ⊗ hk , β ⊗ us ⊗ ht  = α, βδr,s δk,t ,where the usual Hodge inner product of forms is given byα ∧ ∗C(β)β,

α, β =with C(η) = ip−q for η ∈ Ωp,q (X)X(corrected), where α ∧ ∗β = α, β ∗ 1 with ∗ the Hodge (complex) operator
deﬁned so, with ∗1 as the volume form (cf. [116]).
As the step 2, we use the Hodge ﬁltration asF p Ωm (X) = ⊕p +q=m,p ≥p Ωp ,q (X)－ 142 －

to deﬁne linear subspaces of C ∗ of the form
Cm,2r = ⊕p+q=m,k≥max{0,2r+m} F m+r−k Ωm (X) ⊗ ur ⊗ hk
and the Z-graded vector spaceC∗ = ⊕∗=m+2r Cm,2r .
As the step 3, we pass to the real vector space by considering T∗ = (C∗ )c=id ,where c denotes the complex conjugation.
∗
In terms of the intersection γ ∗ = F ∗ ∩ F of the Hodge ﬁltrations, we canwrite it as T∗ = ⊕∗=m+2r Tm,2r , whereTm,2r = ⊕p+q=m,k≥max{0,2r+m} γ m+r−k Ωm (X) ⊗ ur ⊗ hk .
The Z-graded complex vector space C∗ is a subcomplex of C ∗ with respect
to the diﬀerential dC . For p⊥ the orthogonal projection onto C∗ under the innerproduct deﬁned above, we deﬁne the second diﬀerential as d = p⊥ dC . Similarly,d = dC and d = p⊥ dC deﬁne diﬀerentials on the Z-graded real vector spaceT∗ since the inner product is real on real forms and induces an inner producton T∗ . Deﬁne the total diﬀerential as δ = d + d .
The real vector spaces T∗ can be described in terms of certain cutoﬀs on theindices of the complex C ∗ . Namely, deﬁneΛp,q = {(r, k) ∈ Z2 | k ≥ l(p, q, r) ≡ max{0, 2r + m,

|p − q| + 2r + m}},
2

and then identify T∗ with the real vector space spanned by α ⊗ ur ⊗ hk for(r, k) ∈ Λp,q and α = ξ + ξ with ξ ∈ Ωp,q (X). (No original ﬁgure).
But may consider the lines in the (r, k)-plane deﬁned as k = 0, k = 2r + m,
and k = r + 2−1 (|p − q| + m), all of which make a sort of the boundary of Λp,q as∗well as the corresponding subspace Tm
p,q of T , as cutoﬀs deﬁning the complex
at arithmetic inﬁnity.
The logarithmic and Lefschetz operators. The complex (T∗ , δ) has someinteresting structures given by the action of certain linear operators, as follows.
There are the (logarithmic) operators n and Φ that correspond to the logarithmof the monodromy and the logarithm of Frobenius, respectively. These are∂deﬁned as the forms n = uh and Φ = −u ∂u, satisfying[n, d ] = [n, d ] = 0and[Φ, d ] = [Φ, d ] = 0,

and hence [n, δ] = 0 and [Φ, δ] = 0, and thus they induce the corresponding
operators in cohomology.
Moreover, there is another important (Lefschetz) operator l, which corresponds to the Lefschetz operator on forms, deﬁned asl(η ⊗ ur ⊗ hk ) = (η ∧ ω) ⊗ ur−1 ⊗ hk ,－ 143 －

where ω is the Kähler form on the complex manifold X. This satisﬁes that[l, d ] = [l, d ] = 0, so that it also descends to that in cohomology.
The pairs (n, Φ) and (l, Φ) of those operators satisfy the interesting commutation relations[Φ, n] = −n and [Φ, l] = l,which can be seen as an action of the ring of diﬀerential operators
C[p, q]/q with [p, q] = pq − qp = q.
As representations of SL2 (R). As another important piece of the structureof the complex (T∗ , δ), there are two involutions deﬁned ass(α ⊗ ur ⊗ h2r+m+l ) = α ⊗ u−r−m ⊗ hl∼s (α ⊗ u ⊗ h ) = C(∗α) ∗ α ⊗ u
r

k

and

r−n+m

⊗ hk

(corrected). These maps, together with the nilpotent operators n and l, deﬁne
two representations σ L and σ R , given byσ L (a(s)) = s−n+m ,σ L (n(t)) = exp(tl),σ L (r) = in Cs∼ ,σ R (a(s)) = s2r+m ,σ R (n(t)) = exp(tn),σ R (r) = Cs,

where SL2 (R) has the Iwasawa decomposition KAN = SO(2) · R∗ · R (Compact(or Kompakt in German) · Abelian · Nilpotent) with SO(2) = P1 (R) = T = S 1 ,(partially) respectively generated by






0 1
s
0
1 t
r=
, a(s) =
,
n(t)
=
−1 0
0 s−1
0 1
for s ∈ R∗ and t ∈ R, where the matrix r is called the Weyl element.
The representation σ L extends to an action by bounded operators on theHilbert space completion of T∗ with the inner product, while the action of thesubgroup of a(s), s ∈ R∗ of SL2 (R) under the representation σ R on the same
Hilbert space is given by unbounded densely deﬁned operators.
Renormalization group and monodromy. That general structure for varieties in any dimension also has interesting connections to noncommutative
geometry. For instance, we can see that the map n, in fact does play the role ofthe logarithm of the monodromy, using an analog in our context, of the theoryof renormalization à la Connes-Kreimer [39].
In the classical case of a geometric degeneration on a disk, the monodromy
around the special ﬁber is deﬁned as the map t = exp(−2πires0 (∇)), where n =res0 (∇) is the residue at zero of the connection ∇, acting as an endomorphismof the cohomology.
Let T∗C be the complexiﬁcation of the real vector space T∗ . Consider loops
ϕμ with values in the group G = Aut(T∗C , δ), depending on a mass parameterμ ∈ C∗ . The Birkhoﬀ decomposition of a loop ϕμ is given as the multiplicative−1 +decomposition ϕμ (z) = ϕ−ϕμ (z), for z ∈ ∂Δ ⊂ P1 (C) ≈ S 2 , where Δ
μ (z)－ 144 －

is a small disk with center zero. Of the two terms in the decomposition, ϕ+μextends to a holomorphic function on Δ and ϕ−μ does to a holomorphic function
on P1 (C) \ Δ with values in G. The decomposition is normalized by requiringthat ϕ−μ (∞) = 1.
By analogy with the Connes-Kremier theory of renormalization we require
the following two properties:• The time evolution deﬁned by θt (a) = e−tΦ aetΦ (a ∈ G) as a natural
choice, given by the geodesic ﬂow associated to the Diract operator Φ, acts onloops by scaling as ϕλμ (z) = θtz ϕμ (z), for λ = et ∈ R∗+ and z ∈ ∂Δ.
−• The term ϕ−in the Birkhoﬀ decomposition is independent of theμ = ϕenergy (as mass) scale μ.
The residue of a loop ϕμ = ϕ, as in [39], is given byd − 1 −1
ϕ ( ) |z=0 (in G ?),dzz
and the beta function of renormalization is deﬁned asdβ(= β(ϕ)) = b res(ϕ)(∈ G) with b = θt |t=0 .
dt
It follows thatdb(a) = θt (a)|t=0 = [a, Φ].
dt
There is the scattering formula (as in [39]), by which ϕ− can be reconstructed from the residue. Namely, (in G) dk
1, dk =
θ−s1 (β) · · · θ−sk (β)ds1 · · · dsk .
=1+
−
ϕ (z)
zk
s1 ≥···≥sk ≥0res(ϕ) =k≥1

The renormalization group (in G) is given byρ(λ) = lim ϕ− (ε)θtε (ϕ− (ε)−1 ),ε→0

λ = et ∈ R∗+ .

Thus, we only need to specify the residue in order to have the corresponding
renormalization theory associated to (T∗C , δ). By analogy with the case of thegeometric degeneration as n = res0 (∇), it is natural to require that res(ϕ) = n.
It then follows that ([63])Proposition 4.6. A loop ϕμ valued in G = Aut(T∗C , δ) with res(ϕμ ) = n, subject
to the time evolution by scaling, and with ϕ− = ϕ−μ independent of μ, satisﬁes
ϕμ (z) = exp(z −1 μz n) with Birkhoﬀ decomposition (corrected)
ϕμ (z) =ϕ+ (z)
exp(z −1 (μz + 1)n)
=
.
−
ϕ (z)
exp(z −1 n)

Proof. (Edited). In fact, by b = [·, Φ] and res(ϕμ ) = n, we haveβ = β(ϕ) = b res(ϕ) = [n, Φ] = n
and θt (n) = et n. Hence, the scattering formula as above gives ϕ− (z) = exp(z −1 n)and the scaling property determines ϕμ (z) = exp(z −1 μz n).

－ 145 －

The (positive) part ϕ+μ (z) of the Birkhoﬀ decompostition that is regular atz = 0 satisﬁesnϕ+μ (0) = μ = exp(n log μ),withμz − 1
ez log μ − 1= log μ lim= log μ.
z→0z→0zz log μlim

The renormalization group has the form ρ(λ) = λn = exp(tn), with log λ =log et = t, which, through the representation σ R of SL2 (R), corresponds to the
horocycle ﬂow on SL2 (R), as
1 tρ(λ) = n(t) =.
0 1
The Birkhoﬀ decomposition as above gives a trivialization of a principal
G-bundle over P1 (C). Then the associated vector bundle E∗μ with ﬁber T∗C isconsidered. Moreover, a Fuchsian connection ∇μ on this bundle is given by
1d μz − 1dz,∇μ : E∗μ → E∗μ ⊗OΔ Ω∗Δ (log 0), ∇μ = n+zdz z
where Ω∗Δ (log 0) denotes forms with logarithmic poles at 0, and local gauge
potentials for ∇μ are given as−ϕ+ (z)−1log π(γ)dz +ϕ (z) + ϕ+ (z)−1 dϕ+ (z),zwith respect to the monodromy representation π : π1 (Δ∗ ) ∼= Z → G, givenby π(γ) = exp(−2πin), for γ the generator for π1 (Δ∗ ). This corresponds tothe monodromy t in the classical geometric case. This Fuchsian connection has
residue resz=0 ∇μ = n as in the geometric case.
Local factor and archimedean cohomology. It is shown by Consani [57]that the data (Hm , Φ) can be identiﬁed with (H∗ (T∗ , δ)n=0 , Φ), where H∗ (T∗ , δ)is the hyper-cohomology as the cohomology with respect to the total diﬀerential
δ of the complex T∗ , and H∗ (T∗ , δ)n=0 is the kernel of the map induced by n on∂
the cohomology. The operator Φ is induced on the cohomology by Φ = −u ∂u.
m
The kernel is isomorphic to H and called the archimedean cohomology. This
also can be viewed as a piece of the cohomology of the cone of the monodromyn. This is the complex with diﬀerential asδ −nCon(n)∗ = T∗ ⊕ T∗ [+1], D =.
0 δ
The complex Con(n)∗ inherits a positive deﬁnite inner product from T∗ ,which descends on cohomology. The representation σ L of SL2 (R) on T∗ induces
a representation on Con(n)∗ . The corresponding representationdσ L : g = sl2 (R) → End(T∗ )－ 146 －

of the Lie algebra g extends to a representation of the universal enveloping
algebra U (g) on T∗ as well as Con(n)∗ . This gives a representation in thealgebra of bounded operators on the Hilbert space completion of Con(n)∗ with
the inner product.
Theorem 4.7. The triple (U (g), H∗ (Con(n)), Φ) as a spectral triple (A, H, D)
has the following four properties:(1) Self-adjointness as D = D∗ ;1(2) Unboundedness of D2 as that (1 + D2 )− 2 is a compact operator;(3) Boundedness of [D, ·] as that the commutators [D, a] are bounded operators for any a ∈ U (g);(4) Summability as that the triple is 1+ = 1 + ε-summable for any ε > 0.
Thus, the triple (A, H, D = Φ) has most of the properties of spectral triples,
conﬁrming the fact that Φ as the logarithm of Frobenius should be thought ofas a Diract operator.
In any case, the structure is suﬃcient to consider zeta functions for thatspectral triple. In particular, the alternating products of the local L-factors at
inﬁnity can be recovered from zeta functions of the spectral triple.
Theorem 4.8. The zeta functions with respect to Φ deﬁned as (corrected)
ζa,Φ (s, z) =tr(a p(λ, |Φ|))λ∈σp (|Φ|)

1(s − λ)z

with a = σ L (r)give the formuladdet (s) ≡ exp − ζσL (r),Φ (s, z)|z=0Ldz∞,σ (r),Φ



m

m
(−1)
= Π2n
.
m=0 L(H (X), s)

Arithmetic surfaces as homology and cohomology. In the particularcase of arithmetic surfaces, there is an identiﬁcation between H∗ (Con(n)) andH∗ ⊕ (H∗ )∨ , where H∗ is the archimedean cohomology and (H∗ )∨ is its dual
under the involution s deﬁned above.
The identiﬁcationu : H1 → V ∼= ⊕n∈Z,n≥0 H 1 (X(C)) ⊂ L2
can be extended by considering a subspace W of the homology H1 (ST ) withW ∼= (H∗ )∨ . The homology H1 (ST ) can also be computed as a direct limit
limn Kn , where Kn are free abelian groups of rank (2g − 1)n + 1 for n even−→and of (2g − 1)n + (2g − 1) for n odd. The Z-module Kn is generated by theclosed geodesics represented by periodic sequences in S of period n + 1. These
need not be primitive closed geodesics. In terms of primitive closed geodesics,
H 1 (ST , Z) can be written equivalently as ⊕∞n=0 Rn , where Rn are free abeliangroups of rank
1rk(Rn ) =μ(d)rk(K nd ),n
d|n－ 147 －

with μ(d) the Möbius function satisfying d|n μ(d) = δn,1 .
The pairing between the homology H1 (ST ) = limn Kn and the cohomol−→ogy H 1 (ST ) = limn Fn is given by−→
·, · : Fn × Kn → Z,

[f ], x = n f (x).

This determines a graded subspace W in H1 (ST , Z) dual to V in H 1 (ST ). With
the identiﬁcation asu
H1 −−−−→ V ⊂ H 1 (ST )
∼
=
⏐
⏐
⏐
⏐
s∼
⊥·,·=u(H1 )∨ −−−−→ W ⊂ H1 (ST ),∼=yes we can identify the Dirac operator D on the Hilbert space H = L2 ⊕ L2
with the logarithm of Frobenius, as D|V⊕V = ΦH1 ⊕(H1 )∧ as restricted.

5

The vista aroundMay ﬁnd more details in [64] and [46], or just invoking the following vista.
Noncommutative geometry for Shi-mura varieties. The results viewed
as arising from a noncommutative version of the Shimura varieties Sm(GL1 , ±1)
and Sm(GL2 , H± ) are obtained by Bost-Connes [13], Connes-Marcolli [41], and
CM-Ramachandran [48]. The results on the noncommutative boundary of themodular curves and the modular Hecke algebras of Connes-Moscovici [54], [55]are obtained by Manin-Marcolli [112], to contribute to illustrate the NC geometry associated to the modular tower, i.e., to the Shimura variety Sm(GL2 , H± )(named as, East Mount Village, 2 Cho-me).
Then it is a natural idea to seek for quantum statistical mechanical systems QSMS associated to more general Shimura varieties, and to investigate
the arithmetic properties of low temperature KMS states.
Shimura varieties have adelic groups of symmetries as a typical property,
so that the problem of compatibility between adelic symmetries and the Galois
action on the values of zero temperature KMS states on an arithmetic subalgebra
may be formulated naturally in that context.
Generalizations of the Bost-Connes system and of the GL2 system of ConnesMarcolli [41] to a class of Shimura varieties are constructed by Eugene Ha andFrédéric Paugam (HP) [76] as well as P [132], [133]. Their approach is to consider
a more general Shimura datum (G, X), in which the passing from A∗f to Af
in the Bost-Connes, and from GL2 (Af ) to M2 (Af ) in the GL2 is analogously
achieved by considering a suitable enveloping semi-group M . To the resulting
datum (G, X, M ), associated is a groupoid in the category of stacks, whose
convolution algebra HP generalizes the algebra A2 of the GL2 system, withvariants depending on the choice of a level structure, which can be used to
avoid the stacky singularities in the groupoid. There is a natural time evolution
on the algebra HP , which is deﬁned in terms of the data of the level structure,－ 148 －

and the resulting NC system has an adèlic group of symmetries. In particular,
these results can be seen explicitly in the Hilbert module case. Also obtained
through this general method are QSMS naturally associated to number ﬁelds.
The construction agrees with that of CMR [48] for imaginary quadratic ﬁelds.
The arithmetic properties of KMS states for these systems may be investigatedor ﬁnished.
Classical Shimura varieties admit noncommutative boundaries, in the same
sense as that a noncommutative boundary of modular curves is constructed byManin-Marcolli [112]. Studied by Paugam [132], [133] are such noncommutative boundaries, described as the convolution algebras associated to the double
cosets spaces L\G(R)/P (K), where L is an arithmetic subgroup of a connected
reductive algebraic group G(Q) over Q, and P (K) is a real parabolic subgroupin G(R). Described as moduli spaces are degenerations of complex structures
on tori in multi-foliations. The point of view is from Hodge theoretic considerations.
In this general perspective, an interesting question seems to ask whether aconnection between this type of noncommutative geometry of modular curves
and the generalization of the theory of Heegner points by Darmon [67].
Sectral tiples and trace formulae. Besides the spectral triples in the context
of the archimedean factors at arithmetic inﬁnity, mentioned in the last section,
it would be interesting to seek other cases in the number theoretic context.
For example, there may be spectral triples naturally associated to the algebras of the quantum statistical mechanical systems for noncommutative Shimuravarieties considered above.
In the case of the Bost-Connes system, this should be related to the noncommutative space of the Connes spectral realization of zeros of the Riemannzeta function, and for the Connes-Marcolli GL2 system, also to the modular
Hecke algebras of Connes-Moscovici, hence to modular forms. The more casesof general Shimura varieties should be interesting to consider in this perspective.
As an important topic related to the Connes spectral realization of zeros of
the Riemann zeta function, considered is the trace formula in noncommutative
geometry [29]. As an interesting direction of development of this idea, considered is a possible extension of the trace formula to the case of local L-factorsof arithmetic varieties (cf. CCM such as [32], [33]). This approach may besuitable to merge the adelic construction of the noncommutative space of commensurability classes of Q-lattices, underlying the Connes trace formula, withthe conjectural foliated space by Deninger [69], [70], underlying an arithmetic
cohomology related to L-functions of arithmetic varieties.
Another class of spaces of arithmetic signiﬁcance, that may carry more interesting spectral triples, contains spaces like the Mumford curves (cf. [60]), that
admit p-adic uniformizations, for instance, in the case of the Cherednik-Drinfeld
theory of p-adic uniformization of Shimura curves, as well as the generalizationsof Bruhat-Tits trees with the action of p-adic Schottky groups, given by certainclasses of higher rank buildings. Constructions of this type may be related to
the Berkovich spaces, or to the Buium arithmetic diﬀerential invariants.

－ 149 －

Noncommutative algebraic geometry. Noncommutative geometry initiatedand developed by Connes is followed and considered so far, in the texts above.
But to be mentioned, there are other variants of noncommutative geometry as
the versions of noncommutative algebraic geometry, developed by Artin, Tate,and van den Bergh [5], by Manin [105], by Konsevich [90], by Rosenberg [143],
and KR [91]. May also refer to Mahanta [101] as a guided tour to the literature
limited in noncommutative algebraic geometry.
As noticed, more algebraic versions of noncommutative geometry may be
more suitable for application to problems of algebraic number theory and arithmetic algebraic geometry. Certainly expected is that an interplay between thevarious versions of NG would help push the number theoretic applications further, besides being desirable in terms of the internal development of the ﬁeld.
Connes and Dubois-Violette [35], [36], [37] open up important new perspectives
that combine these diﬀerent forms of noncommutative geometry, to allow for
the use of both analytic and algebro-geometric tools.
As an example of such an interplay, obtained is the result of Polishchuk[136], inspired by the Manin real multiplication program, where noncommutative projective varieties are naturally associated to noncommutative tori withreal multiplication.
Hopf algebra actions. Symmetries of ordinary spaces are encoded by group
actions, while symmetries of noncommutative spaces are also done by group
actions, or given by Hopf algebra actions.
In the context of relations between noncommutative geometry and number
theory, as being proved to be a powerful technique, the Holf algebraic structureof noncommutative spaces is exploited by Connes and Moscovici [54], [55] on
the modular Hecke algebras. The product of modular forms is combined withthe action of Hecke operators, to deﬁne the modular Hecke algebras, seen as theholomorphic part of the algebra of coordinates of the noncommutative space ofcommensurability classes of 2-dimensional Q-lattices discussed above, endowedwith symmetries given by the action of the Connes-Moscovici Hopf algebra H1CM
of transverse geometry in codimension 1. This Hopf algebra H1CM arises in thecontext of foliations, acting on crossed product C ∗ -algebras, and is in a familyof Hopf algebra HnCM of ransverse geometry in codimension n, obtained fromthe dual of the enveloping algebra of the Lie algebra of vector ﬁelds [52].
The action of that Hopf algebra H1CM on the modular Hecke algebras isdetermined by (1) a grading operator, corresponding to the weight of modular
forms, (2) a derivation introduced by Ramanujan, which corrects the ordinary
diﬀerentiation by a logarithmic derivative of the Dedekind η function, and (3)an operator that acts as multiplication by a form-valued cocycle on GL2 (Q)+that measures the lack of modular invariance of η 4 dz. Allowed by this is totransfer notions and results from the transverse geometry of foliations to to thecontext of modular forms and Hecke operators.
A tool useful in dealing with Hopf algebra symmetries of noncommutative
spaces, not mentioned above, is the Hopf cyclic cohomology of Hopf algebras,
introduced by Connes and Moscovici (cf. [51], [53], [56]) and as well reﬁned－ 150 －

through a systematic foundational treatment by Khalkhali and Rangipour (cf.
[89] as an overview). It can be thought of as the analog of group and Lie algebra
cohomology in noncommutative geometry.
In the case of the modular Hecke algebras, the use of Hopf cyclic cohomology
yields interesting results on modular forms. In the case of the Hopf algebra
H1CM , there are three basic cyclic cocycles, which respectively correspond to(1) the Schwarz derivative, (2) the Godbillon-Vey class, and (3) the transverse
fundamental class, in the context of transverse geometry. May refer to [150].
(1) The cocycle associated to the Schwarz derivative is realized by an inner∞
n3 q nderivation, given in terms of the Eisenstein series E4 (q) = 1 + 240 n=1 1−qn.
This cyclic cocycle has an arithmetic signiﬁcance, related to the data used in
deﬁning canonical Rankin-Cohen algebras in Zagier [159].
(2) The cocycle associated to the Godbillon-Vey class is expressed in termsof an 1-cocycle on GL2 (Q)+ with values in Eisenstein series of weight two.
(3) The cocycle associated to the transverse fundamental class gives rise toa natural extension of the ﬁrst Rankin-Cohen brackets of modular forms (cf.
[159]).
Remormalization and motivic Galois symmetry. Seen in the last section
is an application of the formalism of the Connes-Kreimer theory of perturbative
renormalization in the context of arithmetic geometry. Revealed by a closer
inspection of the CK formalism are the connections to arithmetic, in the context
of perturbative renormalization of quantum ﬁeld theories. This is investigatedby Connes and Marcolli [40], [42], [43].
Their work is started from the Connes-Kreimer formulation of perturbative
renormalization in terms of the Birkhoﬀ decomposition of loops, as in the form
ϕ+ (z)μϕμ (z) = ϕ−for μ ∈ C∗ and z ∈ S 1 , with values ϕμ (z) in the pro-unipotent(z)μLie group G(C) of complex points of the aﬃne group scheme, associated to theConnes-Kreimer Hopf algebra of Feynman graphs of CK [38]. It is shown by
CM that the scattering formula of Connes-Kreimer, as given above as the form dk
1
, dk =
θ−s1 (β) · · · θ−sk (β)ds1 · · · dsk ,
=1+
−
ϕ (z)
zk
s1 ≥···≥sk ≥0
k≥1

can be conveniently formulated in terms of iterated integrals and the time ordered exponential, or expansional. This follows from a calculation generalizing
the Birkhoﬀ decomposition given above as
ϕμ (z) =ϕ+ (z)
exp(z −1 (μz + 1)n)
=
,
−
ϕ (z)
exp(z −1 n)which gives the time ordered exponential expression as∞

ϕ− (z) = T exp(−z −1

θ−t (β)dt).
0

Once expressed in the expansional form as above, we can understand theterms in the Birkhoﬀ decomposition as solutions of certain diﬀerential equations. This leads to a reformulation of perturbative renormalization in terms of－ 151 －

the equivalence classes of certain diﬀerential systems with singularities, within
the context of the Riemann-Hilbert correspondence and the diﬀerential Galois
theory.
The nature of the singularities is determined by the two conditions on the
behavior of the terms in the Birkhoﬀ decomposition with respect to the scalingof the mass parameter, namely the condition ∂μ ϕ−
μ = 0 as the independenceof the energy scale μ in ϕ− = ϕ−
and
the
time
evolution scaling conditionμ
ϕet μ (z) = θtz ϕμ (z), as mentioned above.
Those two conditions are expressed geometrically through the notion of Gvalued equi-singular connections on a principal C∗ -bundle B over a disk Δ,
where G is the pro-unipotent Lie group of characters of the Connes-Kreimer
Hopf algebra of Feynman graphs. The equi-singularity condition is the property
that such a connection is C∗ -invariant and that its restrictions to sections of theprincipal bundle, that agree at 0 ∈ Δ, are mutually equivalent, in the sense thatthose are related by a gauge transformation by a G-valued C∗ -invariant map
regular in B. It then follows that the geometric formulation equivalent to thedata of perturbative renormalization is given by a class, up to the equivalence
given by such gauge transformations, of ﬂat equi-singular G-valued connections.
The class of diﬀerential systems deﬁned by the ﬂat equi-singular connections
can be studied by techniques of the diﬀerential Galois theory. In particular, by
this way, an underlying group of symmetries can be identiﬁed with the diﬀerential Galois group. In fact, it is shown that the category of equivalence classesof ﬂat equi-singular bundles is a neutral Tannaka category, which is equivalent
to the category of ﬁnite dimensional linear representations of an aﬃne groupscheme U ∗ = U  Gm , where U a pro-unipotent aﬃne group scheme and Gm
the multiplicative group. The renormalization group lifts canonically to a oneparameter subgroup of U , and this gives it an interpretation as a group of Galois
symmetries.
The aﬃne group scheme U corresponds to the free graded Lie algebra with
one generator in each degree n ∈ N. These corresponds to the splitting up therenormalization group ﬂow in homogeneous components by loop number. Thisaﬃne group scheme admits an arithmetic interpretation as the motivic Galois
group of the category of mixed Tate motives on the scheme of N -cyclotomic
integers, after localization at N , for N = 3 or N = 4.
The generator of the renormalization group lifted to U deﬁnes a universal
singular frame, whose explicit expression in the generators of the Lie algebra has
the same rational coeﬃcients that appear in the local index formula of ConnesMoscovici [50]. This appears as an intriguing connection between perturbative
renormalization (PR) and NG, possibly through an interpretation of the local
index formula in terms of chiral anomalies [47]. Noncommutative geometry
also appears to provide a geometric interpretation for the deformation to complex dime of dimensional regularization in perturbative quantum ﬁeld theory,
through the notions of spectral triples and dimension spectram (cf. [47]).
Remark. May recall from [116] the following. Let A be a unital commutative
ring. The spectrum Sp(A) of A is deﬁned to be the set all non-trivial prime－ 152 －

ideals of A. The Zariski topology Z on Sp(A) is deﬁned as that for any subset
B ⊂ A, the set V (B) of all prime ideals of A containing B is a closed set. A
closed point in Sp(A) corresponds to a maximal ideal of A. For any a ∈ A, letO(a) = V (a)c the complement in Sp(A). The set of all O(a) for a ∈ A is theopen basis for Z.
Denote by Aa the quotient ring of A by the multiplicative set {an | n ≥ 0}.
By associating O(a) to Aa , obtained is the sheaf A∗ over Sp(A). Then Aa is
isomorphic to Γ(O(a), A∗ ) of sections over O(a). In particular, Γ(Sp(A), A∗ ) ∼=A.
For instance, V (0) = Sp(A). Hence O(0) = ∅. Deﬁne A0 = {0}. Fora, b ∈ A, we have V ({a, b}) ⊂ V (a), with V ({a, b}) = V (a) ∩ V (b). Thus,
O(a) ⊂ O({a, b}), with O({a, b}) = O(a) ∪ O(b). There is a homomorphismfrom Aa to A0 .
The aﬃne scheme for A is deﬁned to be Sp(A) as a space with A∗ as a local
ring.
A scheme is a space X with a local ring O, which is locally an aﬃne scheme.
Namely, there is an open covering {Uj } of X such that each Uj with the local
ring O|Uj restricted to Uj is isomorphic to an aﬃne scheme as a space with alocal ring. As closing the garden, more details may be included in the next time, if
continued. There may be still some (possibly minor) mistakes found in the
texts, because of the time and eﬀort for editing, limited to the last minute.

References[1] N. Alling and N. Greenleaf, Foundations of the theory of Klein
surfaces, Lecture Notes in Math. Vol. 219, Springer (1971).
[2] L. Alvarez-Gaumé, G. Moore, and C. Vafa, Theta functions, modular
invariance, and strings, Comm. Math. Phys. 106 (1986), no. 1, 1-40.
[3] S. Aminneborg, I. Bengtsson, D. Brill, S. Holst, and P. Peldan, Black holes and wormholes in 2 + 1 dimensions, Class. Quantum
Grav. 15 (1998) 627-644.
[4] J. Arledge, M. Laca, and I. Raeburn, Semigroup crossed products
and Hecke algebras arising from number ﬁelds, Doc. Math. 2 (1997)
115-138.
[5] M. Artin, J. Tate, M. Van den Bergh. Modules over regular algebras of dimension 3, Invent. Math. 106 (1991), no. 2, 335-388.
[6] K. I. Babenko, On a problem of Gauss, Dokl. Akad. Nauk SSSR, Tom
238 (1978) No. 5, 1021-1024.
[7] J. D. Barrow, Chaos in the Einstein equations, Physical Review Letters, Vol. 46, No. 15 (1981), 963-966.

－ 153 －

[8] J. D. Barrow, Chaotic behaviour and the Einstein equations, Classical
General Relativity, Cambridge Univ. Press, (1984), 25-41.
[9] P. Baum and A. Connes, Geometric K-theory for Lie groups andfoliations, L’Enseignement Mathematique 46 (2000), 1-35.
[10] V. A. Belinskiı̆, E. M. Lifshitz and I. M. Khalatnikov, Oscillatory approach to the singular point in relativistic cosmology, 13, No.
6 (1971), 745-765.
[11] B. Blackadar, K-theory for Operator Algebras, Second Edition, Cambridge, (1998).
[12] F. P. Boca, Projections in rotation algebras and theta functions, Commun. Math. Phys., 202 (1999) 325-357.
[13] J. B. Bost and A. Connes, Hecke algebras, type III factors and phasetransitions with spontaneous symmetry breaking in number theory, Selecta Math. New Ser. 1 (1995) no. 3, 411-457.
[14] R. Bowen, Hausdorﬀ dimension of quasi-circles, Publ. Math. IHES
50 (1979) 11-25.
[15] M. Boyle and D. Handelman, Orbit equivalence, ﬂow equivalence,and ordered cohomology, Israel J. Math. 95 (1996) 169-210.
[16] Ola Bratteli and Derek W. Robinson, Operator Alagebras andQuantum Statistical Mechanics I, II, Springer (1981).
[17] Partha Sarathi Chakraborty and Arupkumar Pal, Equivariant
spectral triples on the quantum SU (2) group, K-theory 28, 107-126(2003).
[18] C. H. Chang and D. Mayer Therodynamic formalism and Selberg’s
zeta function for modular groups, Regular and chaotic dynamics 5(2000) No. 3, 281-312.
[19] David F. Chernoff and John D. Barrow, Chaos in the mixmasteruniverse, Physical Review Letters, Vol. 50, No. 2 (1983) 134-137.
[20] Henri Cohen and Fredrik Strömberg, Modular Forms, A Classical Approach, Graduate Studies in Math. 179, Amer. Math. Soc.
(2017).
[21] P. B. Cohen, A C ∗ -dynamical system with Dedekind zeta partition
function and spontaneous symmetry breaking, Journal de Théorie desNombres de Bordeaux 11 (1999) 15-30.
[22] A. Connes, C ∗ -algèbres et géométrie diﬀerentielle, C. R. Acad. Sci.
Paris, Ser. A-B 290 (1980) 599-604.

－ 154 －

[23] A. Connes, An analogue of the Thom isomorphism for crossed products of a C ∗ -algebra by an action of R, Adv. in Math. 39 (1981) no. 1,
31-55.
[24] A. Connes, Noncommutative diﬀerential geometry, Inst. Hautes
Etudes Sci., Publ. Math. 62 (1985) 257-360.
[25] A. Connes, Cyclic cohomology and the transverse fundamental classof a foliation, Geometric Methods in Operator Algebras (Kyoto 1983),
Pitman Res. Notes in Math., 123 (1986), 52-144.
[26] A. Connes, Compact metric spaces, Fredholm modules, and hyperﬁniteness, Ergod. Th. Dynam. Syst. 9 (1989) 207-220.
[27] Alain Connes, Noncommutative Geometry, Academic Press (1994).
[28] A. Connes, Geometry from the spectral point of view, Letters in Math.
Phys. 34 (1995) 203-238.
[29] Alain Connes, Trace formula in noncommutative geometry and thezeros of the Riemann zeta function, Selecta Math., New Series 5 (1999),
29-106.
[30] A. Connes, A short survey of noncommutative geometry, J. Math.
Phys. 41 (2000), no. 6, 3832-3866.
[31] Alain Connes, Cyclic cohomology, quantum group symmetries and the
local inde formula for SUq (2), J. of the Inst. of Math. Jussieu (2004) 3(1), 17-68.
[32] A. Connes, Caterina Consani, and M. Marcolli, Noncommutative geometry and motives: The thermodynamics of endomotives, Advances in Math. 214 (2007) 761-831. [math.QA/0512138].
[33] A. Connes, C. Consani, and M. Marcolli, The Weil proof and thegeometry of the adèles class space, Algebra, arithmetic, and geometry,
Vol I, Progr. Math. 269 (2009) 339-405. [math.NT/0703392].
[34] A. Connes, Michael R. Douglas, and Albert Schwarz, Noncommutative geometry and matrix theory: compactiﬁcation on tori, J.
High Energy Physics, 02 (1998) 003 (35 pp.).
[35] A. Connes and Michel Dubois-Violette, Noncommutative ﬁnitedimensional manifolds. I. Spherical manifolds and related examples,
Commun. Math. Phys. 230, 539-579 (2002).
[36] A. Connes and Michel Dubois-Violette, Moduli space and structure of noncommutative 3-spheres, Letters in Math. Phys. 66: 91-121(2003).

－ 155 －

[37] A. Connes and Michel Dubois-Violette, Noncommutative ﬁnitedimensional manifolds II. Moduli space and structure of noncommutative 3-spheres, Commun. Math. Phys. 281 (2008) no 1., 23-127.
[38] A. Connes and D. Kreimer, Renormalization in quantum ﬁeld theory and the Riemann-Hilbert problem I. The Hopf algebra structure ofgraphs and the main theorem, Comm. Math. Phys. 210 (2000), no. 1,
249-273.
[39] A. Connes and D. Kreimer, Renormalization in quantum ﬁeld theory and the Riemann-Hilbert problem II. The β-function, diﬀeomorphisms and the renormalization group, Comm. Math. Phys. 216 (2001),
no. 1, 215-241.
[40] A. Connes and M. Marcolli, Renormalization and motivic Galois
theory, Int. Math. Res. Not. (2004), no. 76, 4073-4091.
[41] A. Connes and M. Marcolli, Quantum statistical mechanics of Qlattices, From physics to number theory via noncommutative geometry,
Part I, Frontiers in Number Theory Physics and Geometry, Springer(2006) 269-347. [arXiv: math.NT/0404128].
[42] A. Connes and M. Marcolli, Renormalization, the Riemann-Hilbertcorrespondence, and motivic Galois theory, From physics to number
theory via noncommutative geometry, Part II, [arXiv: hep-th/0411114].
[43] A. Connes and M. Marcolli, Quantum ﬁelds and motives, to appear in J. Geom. and Phys. (The same as below?)[44] Alain Connes and Matilde Marcolli, Q-lattices: Quantum statistical mechanics and Galois theory (Review), Journal of Geometry
and Physics 56 (2006) 2-23.
[45] Alain Connes and Matilde Marcolli, A walk in the noncommutative garden, An Inviation to Noncommutative Geometry, (2008) 1-128.
[46] A. Connes and M. Marcolli, Noncommutative Geometry, Quantum
Fields and Motives, Colloquium Publ. 55 Amer. Math. Soc. (2008).
[47] A. Connes and M. Marcolli, Anomalies, Dimensional Regularization and Noncommutative Geometry, (unpublished?).
[48] A. Connes, M. Marcolli, and Niranjan Ramachandran, KMS
states and complex multiplication, Selecta Math. New ser. 11 (2005),
325-347. [math.OA/0501424].
[49] A. Connes, M. Marcolli, and N. Ramachandran, KMS states
and complex multiplication. (Part II), Operator Algebras - The Abel
Symposium 2004 Springer (2006) 15-60.

－ 156 －

[50] A. Connes and H. Moscovici, The local index formula in noncommutative geomerty, Geom. And Funct. Anal. 5 (1995), 174-243.
[51] A. Connes and H. Moscovici, Hopf algebras, cyclic cohomology
and the transverse index theorem, Commun. Math. Phys. 198, 199-246(1998).
[52] A. Connes and H. Moscovici, Cyclic cohomology and Hopf algebras,
Lett. Math. Phys. 48 (1999), no. 1, 97-108.
[53] A. Connes and H. Moscovici, Diﬀerentiable cyclic cohomology andHopf algebraic structures in transverse geometry, Essays on geometry
and related topics, Vol. 1, 2, Enseignement Math. 38 (2001) 217-255.
[54] A. Connes and Henri Moscovici, Modular Hecke algebras and their
Hopf symmetry, Moscow Math. J. Vol. 4, No. 1, (2004) 67-109.
[55] A. Connes and Henri Moscovici, Rankin-Cohen brackets and the
Hopf algebra of tranverse geometry, Moscow Math. J. Vol. 4, No. 1,(2004) 111-130.
[56] A. Connes and H. Moscovici, Background independent geometryand Hopf cyclic cohomology, math.QA/0505475.
[57] C. Consani, Double complexes and Euler L-factors, Compositio Math.
111 (1998) 323-358.
[58] C. Consani and M. Marcolli, Triplets spectreaux in geometried’Arakelov, C. R. Acad. Sci. Paris, Ser. I, 335 (2002) 779-784.
[59] C. Consani and M. Marcolli, New perspectives in Arakelov geometry, Proceedings of CNTA7 meeting, Montreal (2002).
[60] C. Consani and M. Marcolli, Spectral triples from Mumford curves,
International Math. Research Notices 36 (2003) 1945-1972.
[61] C. Consani and M. Marcolli, Noncommutative geometry, dynamics, and ∞-adic Arakelov geometry, Selecta Math. (N. S.) 10 (2004)
no. 2, 167-251.
[62] C. Consani and M. Marcolli, New persoectives in Arakelov geometry, Number Theory, CRM Proc. Lecture Notes 36, Amer. Math. Soc.
(2004) 81-102.
[63] C. Consani and M. Marcolli, Archimedean cohomology revisited,
Noncommutative geometry and number theory, Aspects Math. E 37
(2006) 109-140, arXiv math.AG/0407480.
[64] C. Consani and M. Marcolli, (Editors) Proceedings of the conference “Noncommutative geometry and number theory”, Bonn 2003,
Aspects Math. E 37 (2006).

－ 157 －

[65] G. Cornell and J. H. Silverman (editors), Arithmetic Geometry,
Springer (1986).
[66] J. Cuntz and W. Krieger, A class of C ∗ -algebras and topological
Markov chains, Invent. Math. 56 (1980) 251-268.
[67] H. Darmon, Rational points on modular elliptic curves, CBMS Regional Conference Ser. Math., 101, AMS (2004).
[68] C. Deninger, On the Γ-factors attached to motives, Invent. Math. 104(1991) 245-261.
[69] C. Deninger, Local L-factors of motives and regularized determinants,
Invent. Math. 107 (1992), no. 1, 135-150.
[70] C. Deninger, Motivic L-functions and regularized determinants, Motives, Proceedings of Symposia in Pure Math., Vol. 55 (1994) Part I,
707-743.
[71] C. Deninger, Some analogies between number theory and dynamicalsystems on foliated spaces, Proceedings of the ICM, Vol. I (Berlin 1998).
Doc. Math. 1998, Extra Vol. I, 163-186.
[72] V. G. Drinfel’d, Two theorems on modular curves, Funct. Anal. and
its Appl., Vol. 7, Issue 2 (1973) 155-156.
[73] V. Drinfel’d and Yu. I. Manin, Periods of p-adic Schottky groups,
J. Reine Angew. Math. 262/263 (1973), 239-247.
[74] K. Goto, T. Nishiyama, K. Yamamoto, K. Mochizuki, T.
Kanki, A. Okiji, Quantum mechanics exercising, Kyo-Ritu, (1982)(in Japanese).
[75] José M. Gracia-Bondı́a, Joseph C. Várilly, and Héctor
Figueroa, Elements of Noncommutative Geometry, Birkhäuser(2001).
[76] E. Ha and F. Paugam, Bost-Connes-Marcolli systems for Shimuravarieties. I. Deﬁnitions and formal analytic properties, IMRP 5 (2005)
237-286.
[77] R. Haag, Local Quantum Physics, Springer (1992).
[78] R. Haag, N. M. Hugenholtz, and M. Winnink, On the equilibrium
states in quantu statisﬁcal mechanics, Comm. Math. Phys. 5 (1967)
215-236.
[79] D. Harari and E. Leichtnam, Extension du phénomène de brisurespontanée de symétrie de Bost-Connes au cas des corps globaux quelconques, Selecta Math. (New Series) 3 (1997) 205-243.

－ 158 －

[80] G. Harder, General aspects in the theory of modular symbols, Seminar on number theory, Paris 1981/1982, Progr. Math. 38, Birkhäuser,(1983) 73-88.
[81] Akio Hatsutori,
Japanese).

Topological Geometry,

Iwanami,(1991) (in[82] D. Hensley, Continued fraction Cantor sets, Hausdorﬀ dimension,and functional analysis, J. Number Theory 40 (1992) 336-358.
[83] Fumio Hiai and Kenjiro Yanagi, Hilbert spaces and Linear operators, Makino bookstore, (1995) (in Japanese).
[84] I. M. Khalitnikov, E. M. Lifshitz, K. M. Khanin, L. N. Schur,
Ya. G. Sinai, On the stochasticity in relativistic cosmology, J. Stat.
Phys. 38: 1/2 (1985), 97-114.
[85] Masoud Khalkhali, Basic Noncommutative Geometry, Second edition, EMS (2013).
[86] M. Khalkhali and R. Rangipour, A new cyclic module for Hopf
algebras, K-theory 27 (2002) no. 2, 111-131.
[87] M. Khalkhali and R. Rangipour, Invariant cyclic homology, Ktheory 28 (2003) no. 2, 183-205.
[88] M. Khalkhali and R. Rangipour, Cyclic cohomology of (extended)
Hopf algebras, Noncommutative Geometry and Quantum Groups (Warsaw 2001), Banach Center Publ., 61, Polish Acad. Sci., Warsaw (2003)
59-89.
[89] M. Khalkhali and R. Rangipour, Introduction to Hopf-Cyclic Cohomology, arXiv math.QA/0503244.
[90] M. Kontsevich. Formal (non)commutative symplectic geometry, The
Gel’fand Mathematical Semiars, 1990-1992, Brikhäuser (1993) 173-187.
[91] M. Kontsevich and A. L. Rosenberg. Noncommutative smooth
spaces, The Gel’fand Mathematical Semiars, 1996-1999, Brikhäuser(2000) 85-108.
[92] K. Krasnov, Holography and Riemann Surfaces, Adv. Theor. Math.
Phys. 4 (2000) 929-979.
[93] K. Krasnov, Analytic Continuation for Asymptotically AdS 3D Gravity, gr-qc/0111049.
[94] M. Laca, Semigroups of ∗-endomorphisms, Dirichlet series, and phasetransitions, J. Funct. Anal. 152 (1998), no. 2, 330-378.

－ 159 －

[95] M. Laca, From endomorphisms to automorphisms and back: dilations
and full corners, J. London Math. Soc. 61 (2) (2000) 893-904.
[96] Marcelo Laca and Iain Raeburn, A semigroup crossed productarising in number theory, J. London Math. Soc. (2) 59 (1999) 330-344.
[97] Giovanni Landi, An Introduction to Noncommutative Spaces andTheir Geometries, LNP m 51, Springer (1997).
[98] Serge Lang, Elliptic Functions, Second Edition, GTM 112, Springer(1987).
[99] S. Lang, Introduction to Arakelov Theory, Springer (1988).
[100] R. Loudon, The Quantum Theory of Light, third edition, Oxford Univ.
Press (2000).
[101] S. Mahanta, A Brief Survey of Non-Commutative Algebraic Geometry, arXiv math.QA/0501166.
[102] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics,
Cambridge Univ. Press (1995).
[103] Yu. I. Manin, Parabolic points and zeta functions of modular curves,
Math. USSR Izvestija, vol. 6, No. 1 (1972) 19-64. Selected Papers,
World Scientiﬁc (1996) 202-247.
[104] Yu. I. Manin, p-adic automorphic functions, J. Soviet Math., 5 (1976)
279-333.
[105] Yu. I. Manin, Topics in Noncommutative Geometry, Princeton (1991).
[106] Yuri I. Manin, Three-dimensional hyperbolic geometry as ∞-adic
Arakelov geometry, Invent. Math. 104 (1991) 223-244.
[107] Yu. I. Manin, Lectures on zeta functions and motives (according to
Deninger and Kurokawa), Columbia Univ. Number Theory Seminar(NewYork 1992), Astérisque No. 228 (1995), 4, 121-163.
[108] Yu. I. Manin, Theta functions, quantum tori, and Heisenberg groups,
Lett. in Math. Phys., 56 (2001) 295-320.
[109] Yu. I. Manin, Von Zahlen und Figuren, arXiv math.AG/0201005.
[110] Yu. I. Manin, Real Multiplication and Noncommutative Geometry, The legacy of Niels Henrik Abel, Springer (2004) 685-727.
[math.AG/0202109].
[111] Yu. I. Manin and M. Marcolli, Holography principle and arithmeticof algebraic curves, Adv. Theor. Math. Phys. 3 (2001) No. 5, 617-650.

－ 160 －

[112] Yu. I. Manin and M. Marcolli, Continued fractions, modular symbols, and noncommutative geometry, Selecta Math. (N. S.) 8 (2002) no.
3, 475-521.
[113] M. Marcolli, Limiting modular symbols and the Lyapunov spectrum,
J. Number Th. 98 (2003) no. 2, 348-376.
[114] Matilde Marcolli, Arithmetic Noncommutative Geometry, Univ.
Lec. Ser. 36 AMS (2005).
[115] M. Marcolli, Modular curves, C ∗ -algebras, and chaotic cosmology,
Frontiers in Number Theory, Physics, and Geometry, II, Springer (2007)
361-372. [arXiv math-ph/0312035].
[116] M. S. J., Mathematics Dictionary, The 4th edition, Math. Soc. Japan(MSJ), Iwanami, (2007) (in Japanese).
[117] K. Matsuzaki and M. Taniguchi, Hyperbolic manifolds andKleinian groups, Oxford Univ. Press (1998).
[118] D. H. Mayer, Relaxation properties of the mixmaster universe, Phys.
Lett. A 121 (1987), no. 8-9, 390-394.
[119] D. H. Mayer, Continued fractions and related trasformations, in Ergodic theory, symbolic dynamics, and hyperbolic spaces, Oxford Univ.
Press (1991).
[120] B. Mazur, Curbes elliptiques et symboles modulaires, Séminaire Bourbaki vol. 1971/72, Exposés 400-417, LNM 317, Springer (1973) 277-294.
[121] L. Merel, Intersections sur les courbes modulaires, Manuscripta
Math., 80 (1993) 283-289.
[122] J. S. Milne, Canonical models of Shimura curves, manuscript, (2003)(available at www.jimilne.org).
[123] Yasuo Morita, An introduction to Algebra, Sho-Ka-Bo, (1987), (inJapanese).
[124] D. Mumford, An analytic construction of degenerating curves overcomplete local rings, Compositio Math. 24 (1972) 129-174.
[125] D. Mumford (with M. Nori and P. Norman), Tata lectures on theta,
III, Progress in Math., 97 Birkhäuser (1991).
[126] D. Mumford, C. Series, and D. Wright, Indra’s Pearls, The Vision of Felix Klein, Cambridge Univ. Press (2002).
[127] Gerard J. Murphy, C ∗ -algebras and Operator theory, Academic
Press, (1990).

－ 161 －

[128] M. Nagata, Commutative Field Theory, Sho-Ka-Bo (1967) (inJapanese).
[129] Masayoshi Nagata, New Mastering Algebra, Modern Math Comp.
(1984) (in Japanese).
[130] Toshikazu Natsume, On K∗ (C ∗ (SL2 (Z))) (Appendix to “K-theory
for certain group C ∗ -algebra” by E. C. Lance), J. Operator Theory 13(1985), 103-118.
[131] W. Parry and S. Tuncel, Classiﬁcation Problems in Ergodic Theory,
London Math. Soc. Lecture Notes Series 67 (1982).
[132] F. Paugam, Three examples of noncommutative boundaries of Shimuravarieties, arXiv math.AG/0410254.
[133] F. Paugam, Quelques bords irrationnels de varietes de Shimura, arXiv
math.AG/0410269.
[134] M. Pimsner, KK-groups of crossed products by groups acting on trees,
Invent. Math. 86 (1986), 603-634.
[135] M. Pimsner and D. Voiculescu, Exact sequences for K-groups andExt-group of certain crossed product C ∗ -algebras, J. Operator Th. 4(1980) 93-118.
[136] A. Polishchuk, Noncommutative two-tori with real multiplication as
noncommutative projective varieties, J. Geom. Phys. 50 (2004) no. 1-4,
162-187.
[137] M. Pollicott and H. Weiss, Multifractal analysis of Lyapunov exponent for continued fraction and Manneville-Pomeau transformations
and applications to Diophantine approximation, Comm. Math. Phys.
207 (1999) no. 1, 145-171.
[138] D. B. Ray and I. M. Singer, Analytic torsion for complex manifolds,
Ann. Math. (2) 98 (1973), 154-177.
[139] Luis Ribes and Pavel Zalesskii, Proﬁnite groups, Second Edition,
Vol. 40, Springer (2000, 2010).
[140] Marc A. Rieffel, C ∗ -algebras associated with irrational rotations,Pac. J. Math. 93 (1981) 415-429.
[141] M. A. Rieffel, Morita equivalence for operator algebras, Proceedingsof Symposia in Pure Math., 38 (1982), Part I, 285-298.
[142] M. A. Rieffel, Applications of strong Morita equivalence to transformation group C ∗ -algebras, Proceedings of Symposia in Pure Math., 38(1982), Part I, 299-310.

－ 162 －

[143] A. L. Rosenberg, Noncommutative schemes, Compositio Math. 112(1998) no. 1, 93-125.
[144] J. P. Serre, Facteurs locaux des fonctions zêta des variétés algébriques(déﬁnitions et conjectures)., Sém. Delange-Pisot-Poitou, exp. 19,
1969/70, Ouvres Vol. II, Springer (1986) 581-592.
[145] H. Shiga, Automorphic functions, Classical theory and its modern
application, Kyoritsu Publ. Co. (2017).
[146] G. Shimura, Arithmetic Theory of Automorphic Functions, Iwanami
and Princeton (1971).
[147] H. M. Stark, L-functions at s = 1. IV. First derivatives at s = 0,
Adv. Math. 35 (1980) 197-235.
[148] P. Stevenhagen, Hilbert’s 12th problem, complex multiplication andShimura reciprocity, “Class Field Theory – its centenary and prospect”
Advaned Studies in Pure Math. 30 (2001) 161-176.
[149] T. Sudo, The K-theory and the E-theory for C ∗ -algebras and the
Baum-Connes conjecture for discrete groups — a commentative local
study, Ryukyu Math. J. 29 (2016). 33-215.
[150] T. Sudo, A locally commentative and transcriptive exercising in thequantum algebraic, analytic, and geometric garden stadium, Ryukyu
Math. J., 30 (2017), 9-192.
[151] Mitsuo Sugiura, Introduction to Analysis II, Tokyo Univ. Press(1990) (in Japanese).
[152] Yoichiro Takahashi, Introduction to Diﬀerential Equations, Tokyo
Univ. Press (1988) (in Japanese).
[153] H. Umegaki, M. Ohya, and F. Hiai, Introduction to Operator Algebras, Kyoritsu Publ. Co. (1985) (in Japanese).
[154] N. E. Wegge-Olsen, K-theory and C ∗ -algebras, Oxford Univ. Press(1993).
[155] André Weil, Basic Number Theory, Third Edition, 144, Springer(1974).
[156] R. O. Wells, Diﬀerential analysis on complex manifolds, Springer(1980).
[157] A. Werner, Local heights on Mumford curves, Math. Ann. 306 (1996),
no. 4, 819-831.
[158] Ichiro Yokota, Groups and Topology, Sho-Ka-Bo, (1971) (inJapanese).

－ 163 －

[159] Don Zagier, Modular forms and diﬀerential operators, Proc. Indian
Acad. Sci. (Math. Sci.), 104 (1), (1994), 57-75.
Department of Mathematical Sciences, Faculty of Science, University of the
Ryukyus, Senbaru 1, Nishihara, Okinawa 903-0213, Japan.
Email: sudo@math.u-ryukyu.ac.jp(Visit: www.math.u-ryukyu.ac.jp)－ 164 －