ALAIN CONNES, MATILDE MARCOLLI, AND NIRANJAN RAMACHANDRAN

30 Pages·2011·0.29 MB·English

KMS STATES AND COMPLEX MULTIPLICATION
ALAINCONNES,MATILDEMARCOLLI,ANDNIRANJANRAMACHANDRAN
1. Introduction
Several results point to deep relations between noncommutative geometry and class (cid:12)eld theory ([2],
[9], [18], [20]). In [2] a quantum statistical mechanical system (BC) is exhibited, with partition func-
tiontheRiemannzetafunction(cid:16)((cid:12)), andwhosearithmeticpropertiesarerelatedtotheGaloistheory
of the maximal abelian extension of Q. In [9], this system is reinterpreted in terms of the geome-
try of commensurable 1-dimensional Q-lattices, and a generalizationis constructed for 2-dimensional
Q-lattices. The arithmetic properties of this GL -system and its extremal KMS states at zero tem-
2
perature are related to the Galois theory of the modular (cid:12)eld F, that is, the (cid:12)eld of elliptic modular
functions. These are functions on modular curves, i.e. on moduli spaces of elliptic curves. The low
temperatureextremalKMSstatesandthe Galoispropertiesofthe GL -systemareanalyzedin[9] for
2
the generic case of elliptic curves with transcendental j-invariant. As the results of [9] show, one of
the main new features of the GL -system is the presence of symmetries by endomorphism, through
2
which the full Galoisgroup of the modular (cid:12)eld appearsas symmetries acting on the KMS states of
(cid:12)
the system, for large inverse temperature (cid:12).
Inboth the originalBCsystemandin the GL -system,the arithmeticpropertiesof zerotemperature
2
KMS states rely on an underlying result of compatibility between ad(cid:18)elic groups of symmetries and
Galois groups. This correspondence between ad(cid:18)elic and Galois groups naturally arises within the
context of Shimura varieties. In fact, a Shimura variety is a pro-variety de(cid:12)ned over Q, with a
rich ad(cid:18)elic group of symmetries. In that context, the compatibility of the Galois action and the
automorphisms is at the heart of Langlands program. This leads us to give a reinterpretation of the
BC and the GL systems in the language of Shimura varieties, with the BC system corresponding
2
to the simplest (zero dimensional) Shimura variety Sh(GL ; 1). In the case of the GL system, we
1 2
(cid:6)
show how the data of 2-dimensional Q-lattices and commensurability can be also described in terms
of elliptic curves together with a pair of points in the total Tate module, and the system is related
to the Shimura variety Sh(GL ;H(cid:6)) of GL . This viewpoint suggests considering our systems as
2 2
noncommutative pro-varieties de(cid:12)nedoverQ,morespeci(cid:12)callyasnoncommutativeShimuravarieties.
We then present our main result, which is the construction of a new (CM) system, whose arithmetic
propertiesfullyincorporatetheexplicitclass(cid:12)eldtheoryforanimaginaryquadratic(cid:12)eldK,andwhose
partition function is the Dedekind zeta function (cid:16) ((cid:12)) of K. The underlying geometric structure is
K
given by commensurability of 1-dimensional K-lattices.
This new CM system can be regarded in two di(cid:11)erent ways. On the one hand, it is a generalization
of the BC system of [2], when changing the (cid:12)eld from Q to K, and is in fact Morita equivalent to
the one considered in [18], but with no restriction on the class number. On the other hand, it is also
a specialization of the GL -system of [9] to elliptic curves with complex multiplication by K. The
2
KMS states of the CM system can be related to the non-generic KMS states of the GL -system,
1 1 2
associated to points (cid:28) H with complex multiplication by K, and the group of symmetries is the
2
Galois group of the maximal abelian extension of K.
Herealsoweshowthatsymmetriesbyendomorphismsplayacrucialrole,astheyallowforthe action
of the class group Cl( ) of the ring of algebraic integers of K. Thus, our results hold for any K
O O
with no restriction on the classnumber. Since this complex multiplication (CM) case can be realized
as a subgroupoid of the GL -system, it has a natural choice of a rational subalgebra (an arithmetic
2
1
2 CONNES,MARCOLLI,ANDRAMACHANDRAN
structure) inherited from that of the GL -system. This is crucial, in order to obtain the intertwining
2
of Galois action on the values of extremal KMS states and action of symmetries of the system.
We summarize and compare the main properties of the three systems (BC, GL , and CM) in the
2
following table.
System GL GL CM
1 2
Partition function (cid:16)((cid:12)) (cid:16)((cid:12))(cid:16)((cid:12) 1) (cid:16) ((cid:12))
K
(cid:0)
Symmetries A(cid:3)=Q(cid:3) GL (A )=Q(cid:3) A(cid:3) =K(cid:3)
f 2 f K;f
Symmetry group Compact Locally compact Compact
Automorphisms Z^(cid:3) GL (Z^) ^(cid:3)= (cid:3)
2
O O
Endomorphisms GL+(Q) Cl( )
2 O
Galois group Gal(Qab=Q) Aut(F) Gal(Kab=K)
Extremal KMS Sh(GL ; 1) Sh(GL ;H(cid:6)) A(cid:3) =K(cid:3)
1 1 (cid:6) 2 K;f
Here we denote by Z^ the pro(cid:12)nite completion of Z and by A =Z^ Q the ring of (cid:12)nite adeles of Q.
f
(cid:10)
For any abelian group G, we denote by G the subgroup of elements of (cid:12)nite order. For any ring
tors
R, we write R(cid:3) for the group of invertible elements, while R(cid:2) denotes the set of nonzero elements of
R, which is a semigroup if R is an integral domain. We write for the ring of algebraic integers of
the imaginary quadratic (cid:12)eld K =Q(p d), where d is a positiOve integer. We set ^ :=( Z^) and
(cid:0) O O(cid:10)
write A =A K and I =A(cid:3) =GL (A ). Note that K(cid:3) embeds diagonally into I .
K;f f (cid:10)Q K K;f 1 K;f K
G. Shimura determined the automorphisms of the modular (cid:12)eld. His result
GL (A )=Q(cid:3) (cid:24) Aut(F);
2 f
(cid:0)!
is a non-commutative analogue of the class (cid:12)eld theory isomorphism which provides the canonical
identi(cid:12)cations
(1.1) (cid:18) :I =K(cid:3) (cid:24) Gal(Kab=K);
K
(cid:0)!
and A(cid:3)=Q(cid:3) (cid:24) Gal(Qab=Q).
f + (cid:0)!
The paper consists of two parts, with sections 2 and 3 centered on the relation of the BC and GL
2
system to the arithmetic of Shimura varieties, and sections 4 and 5 dedicated to the construction of
the CM system and its relation to the explicit class (cid:12)eld theory for imaginary quadratic (cid:12)elds. The
two parts are closely interrelated, but can also be read independently.
2. Quantum Statistical Mechanics and Explicit Class Field Theory
The BC quantum statistical mechanical system [1, 2] exhibits generators of the maximal abelian
extension of Q, parameterizing extremal zero temperature states. Moreover, the system has the
remarkable property that extremal KMS states take algebraicvalues, when evaluated on a rational
1
subalgebra of the C(cid:3)-algebraof observables. The action on these values of the absolute Galois group
factors through the abelianization Gal(Qab=Q) and is implemented by the action of the id(cid:18)ele class
groupassymmetriesofthesystem,viatheclass(cid:12)eldtheoryisomorphism. Thissuggeststheintriguing
possibility of using the setting of quantum statistical mechanics to address the problem of explicit
class (cid:12)eld theory for other number (cid:12)elds.
KMS AND CM 3
In this sectionwerecallsomebasicnotionsof quantumstatistical mechanicsandof class(cid:12)eld theory,
which will be used throughout the paper. We also formulate a general conjectural relation between
quantum statistical mechanics and the explicit class (cid:12)eld theory problem for number (cid:12)elds.
Quantum Statistical Mechanics.
A quantum statistical mechanical system consists of an algebra of observables, given by a unital C(cid:3)-
algebra , together with a time evolution, consisting of a 1-parameter group of automorphisms (cid:27) ,
t
A
(t R), whose in(cid:12)nitesimal generator is the Hamiltonian of the system. The analog of a probability
2
measure, assigning to every observable a certain average, is given by a state, namely a continuous
linear functional ’ : C satisfying positivity, ’(x(cid:3)x) 0, for all x , and normalization,
A ! (cid:21) 2 A
’(1) = 1. In the quantum mechanical framework, the analog of the classical Gibbs measure is given
by states satisfying the KMS condition (cf. [13]).
De(cid:12)nition 2.1. A triple ( ;(cid:27) ;’) satis(cid:12)es the Kubo-Martin-Schwinger (KMS) condition at inverse
t
A
temperature 0 (cid:12) < , if, for all x;y , there exists a bounded holomorphic function F (z) on
x;y
(cid:20) 1 2 A
the strip 0<Im(z)<(cid:12), continuous on the boundary of the strip, such that
(2.1) F (t)=’(x(cid:27) (y)) and F (t+i(cid:12))=’((cid:27) (y)x); t R:
x;y t x;y t
8 2
We also say that ’ is a KMS state for ( ;(cid:27) ). The set of KMS states is a compact convex
(cid:12) t (cid:12) (cid:12)
A K
Choquet simplex [3, II 5] whose set of extreme points consists of the factor states.
(cid:12)
x E
At 0 temperature ((cid:12) = ) the KMS condition (2.1) says that, for all x;y , the function
1 2A
(2.2) F (t)=’(x(cid:27) (y))
x;y t
extends to a bounded holomorphic function in the upper half plane H. This implies that, in the
Hilbert space of the GNS representation of ’ (i.e. the completion of in the inner product ’(x(cid:3)y)),
A
the generator H of the one-parameter group (cid:27) is a positive operator (positive energy condition).
t
However,this notion of 0-temperatureKMSstatesis in generaltoo weak, hencethe notion ofKMS
1
states that we shall consider is the following.
De(cid:12)nition 2.2. A state ’ is a KMS state for ( ;(cid:27) ) if it is a weak limit of (cid:12)-KMS states for
1 t
A
(cid:12) .
!1
One can easily see the di(cid:11)erence between these two notions in the case of the trivial evolution (cid:27) =
t
id; t R, where any state has the property that (2.2) extends to the upper half plane (as a
8 2
constant), whileweaklimits of (cid:12)-KMSstatesareautomaticallytracialstates. With De(cid:12)nition 2.2we
still obtain a weakly compact convex set (cid:6) and we can consider the set of its extremal points.
1 1
E
Thetypicalframeworkforspontaneoussymmetrybreakinginasystemwithauniquephasetransition
(cf. [12]) is that the simplex (cid:6) consists of a single point for (cid:12) 5 (cid:12) i.e. when the temperature is
(cid:12) c
largerthanthe criticaltemperatureT , andisnon-trivial(of somehigherdimension ingeneral)when
c
the temperaturelowers. A(compact)groupofautomorphismsG Aut( )commutingwiththe time
(cid:26) A
evolution,
(2.3) (cid:27) (cid:11) =(cid:11) (cid:27) g G; t R;
t g g t
8 2 2
is a symmetry group of the system. Such G acts on (cid:6) for any (cid:12), hence on the extreme points
(cid:12)
((cid:6) ) = . The choice of an equilibrium state ’ may break this symmetry to a smaller
(cid:12) (cid:12) (cid:12)
E E 2 E
subgroup given by the isotropy group G = g G; g’=’ .
’
f 2 g
The unitary group of the (cid:12)xed point algebra of (cid:27) acts by inner automorphisms of the dynamical
t
U
system ( ;(cid:27) ), by
t
A
(2.4) (Adu)(a):= uau(cid:3); a ;
8 2A
for all u . One can de(cid:12)ne an action modulo inner of a group G on the system ( ;(cid:27) ) as a map
t
2 U A
(cid:11):G Aut( ;(cid:27) ) ful(cid:12)lling the condition
t
! A
(2.5) (cid:11)(gh)(cid:11)(h)(cid:0)1(cid:11)(g)(cid:0)1 Inn( ;(cid:27) ); g;h G;
t
2 A 8 2
4 CONNES,MARCOLLI,ANDRAMACHANDRAN
i.e. , asahomomorphismof G to Aut(A;(cid:27) )= . The KMS condition showsthat the inner automor-
t (cid:12)
U
phisms Inn( ;(cid:27) ) act trivially on KMS states, hence (2.5) induces an action of the group G on the
t (cid:12)
A
set (cid:6) of KMS states, for 0<(cid:12) .
(cid:12) (cid:12)
(cid:20)1
Moregenerally,onecanconsideractionsby endomorphisms (cf. [9]), whereanendomorphism(cid:26)ofthe
dynamical system ( ;(cid:27) ) is a -homomorphism (cid:26) : commuting with the evolution (cid:27) . There
t t
A (cid:3) A ! A
is an induced action of (cid:26) on KMS states, for 0<(cid:12) < , given by
(cid:12)
1
(2.6) (cid:26)(cid:3)(’):=Z(cid:0)1’ (cid:26); Z =’(e);
(cid:14)
provided that ’(e)=0, where e=(cid:26)(1) is an idempotent (cid:12)xed by (cid:27) .
t
6
An isometry u , u(cid:3)u = 1, satisfying (cid:27) (u) = (cid:21)itu for all t R and for some (cid:21) R(cid:3), de(cid:12)nes
2 A t 2 2 +
an inner endomorphism Adu of the dynamical system ( ;(cid:27) ), again of the form (2.4). The KMS
t (cid:12)
A
condition shows that the induced action of Adu on (cid:6) is trivial, cf. [9]. The induced action (modulo
(cid:12)
inner) of a semigroup of endomorphisms of ( ;(cid:27) ) on the KMS states in general may not extend
t (cid:12)
A
directly to KMS states (in a nontrivial way), but it may be de(cid:12)ned on by \warming up and
1 1
E
coolingdown"(cf. [9]), providedthe \warmingup"mapW : isabijectionbetweenKMS
(cid:12) 1 (cid:12) 1
E !E
states (in the sense of De(cid:12)nition 2.2) and KMS states, for su(cid:14)ciently large (cid:12). The map is given by
(cid:12)
Tr((cid:25) (a)e(cid:0)(cid:12)H)
’
(2.7) W (’)(a)= ; a ;
(cid:12) Tr(e(cid:0)(cid:12)H) 8 2A
with H the positive energy Hamiltonian, implementing the time evolution in the representation (cid:25)
’
associated to the extremal KMS state ’.
1
This type of symmetries, implemented by endomorphisms instead of automorphisms, plays a crucial
role in the theory of superselection sectors in quantum (cid:12)eld theory, developed by Doplicher{Haag{
Roberts (cf.[12], Chapter IV).
StatesonaC(cid:3)-algebraextendthenotionofintegrationwith respecttoameasureinthecommutative
case. In the case of a non-unital algebra, the multipliers algebra provides a compacti(cid:12)cation, which
correspondstothe Stone{C(cid:20)echcompacti(cid:12)cationin the commutativecase. A stateadmits acanonical
extension to the multiplier algebra. Moreover, just as in the commutative case one can extend inte-
gration to certain classes of unbounded functions, it is preferable to extend, whenever possible, the
integration provided by a state to certain classes of unbounded multipliers.
Hilbert’s 12th problem.
The main theorem of class (cid:12)eld theory provides a classi(cid:12)cation of (cid:12)nite abelian extensions of a local
or global (cid:12)eld K in terms of subgroups of a locally compact abelian group canonically associated to
the (cid:12)eld. This is the multiplicative group K(cid:3) = GL (K) in the local non-archimedean case, while
1
in the global case it is the quotient of the id(cid:18)ele class group C by the connected component of the
K
identity. The construction of the group C is at the origin of the theory of id(cid:18)eles and ad(cid:18)eles.
K
Hilbert’s 12th problem can be formulated as the question of providing an explicit description of a set
ofgeneratorsofthemaximalabelianextensionKab ofanumber(cid:12)eldK andoftheactionoftheGalois
group Gal(Kab=K). This is the maximal abelian quotient of the absolute Galois group Gal(K(cid:22)=K) of
K, where K(cid:22) denotes an algebraic closure of K.
Remarkably,theonlycasesofnumber(cid:12)eldsforwhichthereisacompleteanswertoHilbert’s12thprob-
lemaretheconstructionofthemaximalabelianextensionofQusingtorsionpointsofC(cid:3) (Kronecker{
Weber)andthecaseofimaginaryquadratic(cid:12)elds,wheretheconstructionreliesonthetheoryofelliptic
curves with complex multiplication (cf. e.g. the survey [30]).
IfA denotesthead(cid:18)elesofanumber(cid:12)eldK andJ =GL (A )isthe groupofid(cid:18)elesofK,wewrite
K K 1 K
C for the group of id(cid:18)ele classes C =J =K(cid:3) and D for the connected component of the identity
K K K K
in C .
K
KMS AND CM 5
Fabulous states for number (cid:12)elds.
Theconnectionbetweenclass(cid:12)eldtheoryandquantumstatisticalmechanicscanbeformulatedasthe
problem of constructing a class of quantum statistical mechanical systems, whose set of extremal
1
E
zero temperature KMS states has special arithmetic properties, because of which we refer to such
states as \fabulous states".
Given a number (cid:12)eld K, with a choice of an embedding K C, the \problem of fabulous states"
(cid:26)
consists of constructing a C(cid:3)-dynamical system ( ;(cid:27) ), with an arithmetic subalgebra of , with
t Q
A A A
the following properties:
(1) The quotient group G=C =D acts on as symmetries compatible with (cid:27) .
K K t
A
(2) The states ’ , evaluated on elements of the arithmetic subalgebra , satisfy:
1 Q
2E A
’(a) K, the algebraicclosure of K in C;
(cid:15) 2
the elements of ’(a): a ; ’ generate Kab.
K 1
(cid:15) f 2A 2E g
(3) The class (cid:12)eld theory isomorphism
(2.8) (cid:18):C =D ’ Gal(Kab=K)
K K
(cid:0)!
intertwines the actions,
(cid:11) ’=’ (cid:18)(cid:0)1((cid:11));
(cid:14) (cid:14)
for all (cid:11) Gal(Kab=K) and for all ’ .
1
2 2E
InthesettingdescribedabovetheC(cid:3)-dynamicalsystem( ;(cid:27) )togetherwithaQ-structurecompatible
t
A
with the (cid:13)ow (cid:27) (i.e. a rational subalgebra such that (cid:27) ( C)= C) de(cid:12)nes a non-
t Q t Q Q
A (cid:26)A A (cid:10) A (cid:10)
commutativealgebraic (pro-)variety X overQ. Thering (or C),whichneednotbeinvolutive,
Q Q
A A (cid:10)
istheanalogoftheringofalgebraicfunctionsonX andthesetofextremalKMS -statesistheanalog
1
of the set of points of X. The action of the subgroup of Aut( ;(cid:27) ) which takes C into itself is
t Q
A A (cid:10)
analogous to the action of the Galois group on the (algebraic) values of algebraic functions at points
of X.
The analogy illustrated above leads us to speak somewhat loosely of \classical points" of such a
noncommutative algebraic pro-variety. We do not attempt to give a general de(cid:12)nition of classical
points, while we simply remark that, for the speci(cid:12)c construction considered here, such a notion is
provided by the zero temperature extremal states.
Abroadertypeofquestion,inasimilarspirit,canbeformulatedregardingtheconstructionofquantum
statistical mechanical systems with ad(cid:18)elic groups of symmetries and the arithmetic properties of its
action on zero temperature extremal KMS states. The case of the GL -system of [9] (cid:12)ts into this
2
general program.
3. Q-lattices and noncommutative Shimura varieties
In this section we recall the main properties of the BC and the GL system, which will be useful for
2
our main result.
Both cases can be formulated starting with the same geometric notion, that of commensurability
classes of Q-lattices, in dimension one and two, respectively.
De(cid:12)nition 3.1. A Q-lattice in Rn is a pair ((cid:3);(cid:30)), with (cid:3) a lattice in Rn, and
(3.1) (cid:30):Qn=Zn Q(cid:3)=(cid:3)
(cid:0)!
a homomorphism of abelian groups. A Q-lattice is invertible if the map (3.1) is an isomorphism. Two
Q-lattices((cid:3) ;(cid:30) )and((cid:3) ;(cid:30) )arecommensurableifthelatticesarecommensurable(i.e.Q(cid:3) =Q(cid:3) )
1 1 2 2 1 2
and the maps (cid:30) and (cid:30) agree modulo the sum of the lattices.
1 2
It is essential here that one does not require the homomorphism (cid:30) to be invertible in general.
ThesetofQ-latticesmodulotheequivalencerelationofcommensurabilityandconsidereduptoscaling
isbestdescribedwiththetoolsofnoncommutativegeometry,asexplainedin[9]. Infact,onecan(cid:12)rst
considerthe groupoid of the equivalencerelationof commensurabilityonthe setof Q-lattices(not up
6 CONNES,MARCOLLI,ANDRAMACHANDRAN
toscaling). Thisisalocallycompact(cid:19)etalegroupoid . When consideringthequotientbythe scaling
R
action (by S =R(cid:3) in the 1-dimensionalcase, orby S =C(cid:3) in the 2-dimensional case), the algebraof
+
coordinates associated to the quotient =S is obtained by restricting the convolution product of the
R
algebraof toweight zerofunctions with S-compact support. The algebraobtainedthis way, which
R
is unital in the 1-dimensional case, but not in the 2-dimensional case, has a natural time evolution
given by the ratio of the covolumes of a pair of commensurable lattices. Every unit y (0) of
2 R R
de(cid:12)nes a representation(cid:25) by left convolution of the algebraof on the Hilbert space =‘2( ),
y y y
R H R
where isthe setof elementswith sourcey. Thisconstructionpassestothe quotientbythe scaling
y
R
action of S. Representations corresponding to points that acquire a nontrivial automorphism group
will no longer be irreducible. If the unit y (0) corresponds to an invertible Q-lattice, then (cid:25) is a
y
2R
positive energy representation.
In both the 1-dimensional and the 2-dimensional case, the set of extremal KMS states at low tem-
perature is given by a classical ad(cid:18)elic quotient, namely, by the Shimura varieties for GL and GL ,
1 2
respectively,henceweargueherethatthenoncommutativespacedescribingcommensurabilityclasses
ofQ-latticesuptoscalecanbethoughtofasanoncommutativeShimuravariety,whosesetofclassical
points is the corresponding classical Shimura variety.
In both cases, a crucial step for the arithmetic properties of the action of symmetries on extremal
KMS states at zero temperature is the choice of an arithmetic subalgebra of the system, on which
the extremal KMS states are evaluated. Such choice gives the underlying noncommutative space a
1
more rigid structure, of \noncommutative arithmetic variety".
Tower Power.
IfV isanalgebraicvariety{oraschemeorastack{overa(cid:12)eldk,a\tower" overV isafamilyV
i
T
(i ) of (cid:12)nite (possibly branched) covers of V such that for any i;j , there is a l with V a
l
2 I 2 I 2I
coverof V and V . Thus, is apartially orderedset. In caseof a toweroverapointed variety(V;v),
i j
I
one(cid:12)xesapointv overv ineachV . EventhoughV maynotbe irreducible,weshallallowourselves
i i i
to loosely refer to V as a variety. It is convenient to view a \tower" as a category with objects
i
T C
(V V) and morphisms Hom(V ;V ) being maps of covers of V. One has the group Aut (V ) of
i i j T i
!
invertibleself-mapsofV overV (thegroupofdecktransformations);thedecktransformationsarenot
i
required to preserve the points v . There is a (pro(cid:12)nite) group of symmetries associated to a tower,
i
namely
(3.2) :=lim Aut (V ):
T i
G i
(cid:0)
The simplest example of a tower is the \fundamental group" tower associated with a (smooth con-
nected) complexalgebraicvariety(V;v) anditsuniversalcovering(V~;v~). Let be the categoryof all
C
(cid:12)nite(cid:19)etale(unbranched)intermediatecoversV~ W V of V . Inthiscase,the symmetrygroup
! ! G
of (3.2)isthealgebraicfundamentalgroupofV;itisalsothepro(cid:12)nitecompletionofthe(topological)
fundamental group(cid:25) (V;v). Simple variantsof this exampleinclude allowingcontrolled rami(cid:12)cation.
1
Other examples of towers are those de(cid:12)ned by iteration of self maps of algebraic varieties.
Forus,themostimportantexamplesof\towers"willbethecyclotomictowerandthe modulartower.
AnotherveryinterestingcaseoftowersisthatofmoregeneralShimuravarieties. These,however,will
not be treated in this paper. (For another example of noncommutative Shimura varieties see [11].)
The cyclotomic tower and the BC system.
In the case of Q, an explicit description of Qab is provided by the Kronecker{Weber theorem. This
shows that the (cid:12)eld Qab is equal to Qcyc, the (cid:12)eld obtained by attaching all roots of unity to Q.
Namely, Qab is obtained by attaching the values of the exponential function exp(2(cid:25)iz) at the torsion
points of the circle group R=Z. Using the isomorphism of abelian groups Q(cid:22)(cid:3) = Q=Z and the
tors (cid:24)
identi(cid:12)cation Aut(Q=Z) = GL (Z^) = Z^(cid:3), the restriction to Q(cid:22)(cid:3) of the natural action of Gal(Q(cid:22)=Q)
1 tors
on Q(cid:22)(cid:3) factors as
Gal(Q(cid:22)=Q) Gal(Q(cid:22)=Q)ab =Gal(Qab=Q) (cid:24) Z^(cid:3):
! (cid:0)!
KMS AND CM 7
Geometrically, the above setting can be understood in terms of the cyclotomic tower. This has base
Spec Z = V . The family is Spec Z[(cid:16) ] = V where (cid:16) is a primitive n-th root of unity (n N(cid:3)).
1 n n n
2
The set Hom (V V ), non-trivial for nm, corresponds to the map Z[(cid:16) ] , Z[(cid:16) ] of rings. The
m n n m
! j !
group Aut(V ) = GL (Z=nZ) is the Galois group Gal(Q((cid:16) )=Q). The group of symmetries (3.2) of
n 1 n
the tower is then
(3.3) =limGL (Z=nZ)=GL (Z^);
1 1
G
n(cid:0)
which is isomorphic to the Galois group Gal(Qab=Q) of the maximal abelian extension of Q.
Theclassicalobjectthatweconsider,associatedtothecyclotomictower,isthe Shimuravarietygiven
by the ad(cid:18)elic quotient
(3.4) Sh(GL ; 1 )=GL (Q) (GL (A ) 1 )=A(cid:3)=Q(cid:3):
1 f(cid:6) g 1 n 1 f (cid:2)f(cid:6) g f +
Now we consider the space of 1-dimensional Q-lattices up to scaling modulo commensurability. This
can be described as follows ([9]).
In one dimension, every Q-lattice is of the form
(3.5) ((cid:3);(cid:30)) =((cid:21)Z;(cid:21)(cid:26));
for some (cid:21)>0 and some (cid:26) Hom(Q=Z;Q=Z). Since we can identify Hom(Q=Z;Q=Z) endowed with
2
the topology of pointwise convergencewith
(3.6) Hom(Q=Z;Q=Z)=limZ=nZ=Z^;
n(cid:0)
weobtain that the algebraC(Z^) is the algebraof coordinatesof the spaceof 1-dimensionalQ-lattices
up to scaling. The group Z^ is the Pontrjagin dual of Q=Z, hence we also have an identi(cid:12)cation
C(Z^)=C(cid:3)(Q=Z).
The group of deck transformations = Z^(cid:3) of the cyclotomic tower acts by automorphisms on the
algebra of coordinates C(Z^). In addGition to this action, there is a semigroup action of N(cid:2) = Z
>0
implementing the commensurability relation. This is given by endomorphisms that move vertically
across the levels of the cyclotomic tower. They are given by
(3.7) (cid:11) (f)((cid:26))=f(n(cid:0)1(cid:26)); (cid:26) nZ^:
n
8 2
Namely, (cid:11) is the isomorphism of C(Z^) with the reduced algebra C(Z^) by the projection (cid:25) given
n (cid:25)n n
by the characteristic function of nZ^ Z^. Notice that the action (3.7) cannot be restricted to the set
(cid:26)
of invertible Q-lattices, since the range of (cid:25) is disjoint from them.
n
The algebra of coordinates on the noncommutative space of equivalence classes of 1-dimensional
1
A
Q-lattices modulo scaling, with respect to the equivalence relation of commensurability, is given then
by the semigroup crossed product
(3.8) =C(Z^)o N(cid:2):
(cid:11)
A
Equivalently,weareconsideringtheconvolutionalgebraofthegroupoid =R(cid:3) givenbythequotient
R1 +
byscalingofthegroupoidoftheequivalencerelationofcommensurabilityon1-dimensionalQ-lattices,
namely, =R(cid:3) has as algebra of coordinates the functions f(r;(cid:26)), for (cid:26) Z^ and r Q(cid:3) such that
R1 + 2 2
r(cid:26) Z^, with the convolution product
2
(3.9) f f (r;(cid:26))= f (rs(cid:0)1;s(cid:26))f (s;(cid:26));
1 2 1 2
(cid:3)
and the adjoint f(cid:3)(r;(cid:26))=f(r(cid:0)1;r(cid:26)). X
This is the C(cid:3)-algebra of the Bost{Connes (BC) system [2]. It was originally de(cid:12)ned as a Hecke
algebraforthealmostnormalpairofsolvablegroupsP+ P+,whereP isthealgebraicax+bgroup
Z (cid:26) Q
and P+ is the restriction to a > 0 (cf. [2]). It has a natural time evolution (cid:27) determined by the
t
regular representation of this Hecke algebra, which is of type III . The time evolution depends upon
1
the ratio of the lengths of P+ orbits on the left and right cosets.
Z
8 CONNES,MARCOLLI,ANDRAMACHANDRAN
As a set, the space of commensurability classes of 1-dimensional Q-lattices up to scaling can also be
described by the quotient
(3.10) GL (Q) A(cid:1)=R(cid:3) =GL (Q) (A 1 );
1 n + 1 n f (cid:2)f(cid:6) g
whereA(cid:1) :=A R(cid:3)isthesetofad(cid:18)eleswithnonzeroarchimedeancomponent. Ratherthanconsidering
f
(cid:2)
this quotient set theoretically, we regard it as a noncommutative space, so as to be able to extend to
it the ordinary tools of geometry that can be applied to the \good" quotient (3.4).
The noncommutative algebra of coordinates of (3.10) is the crossed product
(3.11) C (A )oQ(cid:3):
0 f +
This is Morita equivalent to the algebra (3.8). In fact, (3.8) is obtained as a full corner of (3.11),
C(Z^)oN(cid:2) = C (A )oQ(cid:3) ;
0 f + (cid:25)
by compression with the projection (cid:25) given by th(cid:0)e characteristi(cid:1)c function of Z^ A (cf. [17]).
f
(cid:26)
The quotient (3.10) with its noncommutative algebra of coordinates(3.11) can then be thought of as
the noncommutative Shimura variety
(3.12) Sh(nc)(GL ; 1 ):=GL (Q) (A 1 )=GL (Q) A(cid:1)=R(cid:3);
1 f(cid:6) g 1 n f (cid:2)f(cid:6) g 1 n +
whose set of classical points is the well behaved quotient (3.4).
This has a \compacti(cid:12)cation", obtained by replacing A(cid:1) by A, as in [7],
(3.13) Sh(nc)(GL ; 1 )=GL (Q) A=R(cid:3):
1 f(cid:6) g 1 n +
The compacti(cid:12)cation consists of adding the trivial lattice (with a possibly nontrivial Q-structure).
Notice that, in the context of noncommutative spaces, Morita equivalence with a unital C(cid:3)-algebra
ensures compactness.
One can also consider the noncommutative space dual to (3.13), under the duality given by taking
the crossed product by the time evolution. This is the noncommutative space that gives the spectral
realization of the zeros of the Riemann zeta function in [7]. It is a principal R(cid:3)-bundle over the
+
noncommutative space
(3.14) GL (Q) A=R(cid:3):
1 n +
Arithmetic structure of the BC system.
The results of [2] show that the Galois theory of the cyclotomic (cid:12)eld Qcycl appears naturally in the
BC system when considering the action of the group of symmetries of the system on the extremal
KMS states at zero temperature.
Inthe caseof 1-dimensionalQ-latticesup toscaling,the algebraof coordinatesC(Z^)canberegarded
as the algebra of homogeneous functions of weight zero on the space of 1-dimensional Q-lattices. As
such, there is a natural choice of an arithmetic subalgebra.
This is obtained in [9] by considering functions on the space of 1-dimensional Q-lattices of the form
(3.15) (cid:15) ((cid:3);(cid:30))= y(cid:0)1;
1;a
y2(cid:3)X+(cid:30)(a)
for any a Q=Z. This is well de(cid:12)ned, for (cid:30)(a) = 0, using the summation lim N . numbers.
2 6 N!1 (cid:0)N
One can then form the weight zero functions
P
(3.16) e :=c(cid:15) ;
1;a 1;a
wherec((cid:3))isproportionaltothecovolume (cid:3) andnormalizedsothat(2(cid:25)p 1)c(Z)=1. Therational
j j (cid:0)
subalgebra of (3.8) is the Q-algebra generated by the functions e (r;(cid:26)) := e ((cid:26)) and by the
1;Q 1;a 1;a
A
functions (cid:22) (r;(cid:26)) = 1 for r = n and zero otherwise. The latter implement the semigroup action of
n
N(cid:2) in (3.8).
KMS AND CM 9
As proved in [9], the algebra is the same as the rational subalgebra considered in [2], generated
1;Q
A
over Q by the (cid:22) and the exponential functions
n
(3.17) e(r)((cid:26)):=exp(2(cid:25)i(cid:26)(r)); for (cid:26) Hom(Q=Z;Q=Z); and r Q=Z;
2 2
with relations e(r+s)=e(r)e(s), e(0)=1, e(r)(cid:3) =e( r), (cid:22)(cid:3)(cid:22) =1, (cid:22) (cid:22) =(cid:22) , and
(cid:0) n n k n kn
1
(3.18) (cid:22) e(r)(cid:22)(cid:3) = e(s):
n n n
ns=r
X
The C(cid:3)-completion of C gives (3.8).
1;Q
A (cid:10)
The algebra (3.8) has irreducible representations on the Hilbert space = ‘2(N(cid:2)), parameterized
H
by elements (cid:11) Z^(cid:3) = GL (Z^). Any such element de(cid:12)nes an embedding (cid:11) : Qcycl , C and the
1
2 !
corresponding representation is of the form
(3.19) (cid:25) (e(r))(cid:15) =(cid:11)((cid:16)k)(cid:15) (cid:25) ((cid:22) )(cid:15) =(cid:15) :
(cid:11) k r k (cid:11) n k nk
The Hamiltonian implementing the time evolution (cid:27) on is of the form H(cid:15) = logk (cid:15) and the
t k k
H
partition function of the BC system is then the Riemann zeta function
1
Z((cid:12))=Tr e(cid:0)(cid:12)H = k(cid:0)(cid:12) =(cid:16)((cid:12):):
k=1
(cid:0) (cid:1) X
The set of extremal KMS-states of the BC system enjoys the following properties (cf. [2]):
(cid:12)
E
= is a singleton for all 0<(cid:12) 1. This unique KMS state takes values
(cid:12) (cid:12)
(cid:15) E K (cid:20)
’ (e(m=n))=f (n)=f (n);
(cid:12) (cid:0)(cid:12)+1 1
where
f (n)= (cid:22)(d)(n=d)k;
k
Xdjn
with (cid:22) the Mo(cid:127)bius function, and f is the Euler totient function.
1
For 1 < (cid:12) , elements of are indexed by the classes of invertible Q-lattices (cid:26) Z^(cid:3) =
(cid:12)
(cid:15) (cid:20) 1 E 2
GL (Z^), hence by the classical points (3.4) of the noncommutative Shimura variety (3.12),
1
(3.20) E(cid:12) (cid:24)=GL1(Q)nGL1(A)=R(cid:3)+ (cid:24)=CQ=DQ (cid:24)=IQ=Q(cid:3)+:
In this range of temperatures, the values of states ’ on the elements e(r)
(cid:12);(cid:26) (cid:12) 1;Q
2 E 2 A
is given, for 1 < (cid:12) < by polylogarithms evaluated at roots of unity, normalized by the
1
Riemann zeta function,
1
1
’ (e(r))= n(cid:0)(cid:12)(cid:26)((cid:16)k):
(cid:12);(cid:26) (cid:16)((cid:12)) r
n=1
X
The group GL (Z^) acts by automorphisms of the system. The induced action of GL (Z^) on
1 1
(cid:15)
the set of extreme KMS states below critical temperature is free and transitive.
TheextremeKMSstatesat((cid:12) = )arefabulousstatesforthe(cid:12)eldK =Q,namely’( )
1;Q
(cid:15) Qcycl and the class (cid:12)eld theory i1somorphism (cid:18) : Gal(Qcycl=Q) (cid:24)= Z^(cid:3) intertwines theAGalo(cid:26)is
!
action on values with the action of Z^(cid:3) by symmetries,
(3.21) (cid:13)’(x)=’((cid:18)((cid:13))x);
for all ’ , for all (cid:13) Gal(Qcycl=Q) and for all x .
1 1;Q
2E 2 2A
10 CONNES,MARCOLLI,ANDRAMACHANDRAN
The modular tower and the GL -system.
2
Modularcurvesariseasmodulispacesofellipticcurvesendowedwithadditionallevelstructure. Every
congruence subgroup (cid:0)0 of (cid:0) = SL (Z) de(cid:12)nes a modular curve Y ; we denote by X the smooth
2 (cid:0)0 (cid:0)0
compacti(cid:12)cation of the a(cid:14)ne curve Y obtained by adding cusp points. Especiallyimportant among
(cid:0)0
these are the modular curves Y(n) and X(n) corresponding to the principal congruence subgroups
(cid:0)(n) for n N(cid:3). Any X is dominated by an X(n). We refer to [15, 29] for more details. We have
(cid:0)0
2
the following descriptions of the modular tower.
Compactversion: ThebaseisV =P1 overQ. ThefamilyisgivenbythemodularcurvesX(n),consid-
eredoverthecyclotomic(cid:12)eldQ((cid:16) )[23]. WenotethatGL (Z=nZ)= 1isthegroupofautomorphisms
n 2
(cid:6)
of the projection V =X(n) X(1)=V =V. Thus, we have
n 1
!
(3.22) =GL (Z^)= 1=limGL (Z=nZ)= 1 :
2 2
G (cid:6) f(cid:6) g
n(cid:0)
Non-compact version: The open modular curves Y(n) form a tower with base the j-line Spec Q[j]=
A1 =V . Theringof modularfunctionsisthe unionof theringsoffunctionsofthe Y(n),with
1
(cid:0)f1g
coe(cid:14)cients in Q((cid:16) ) [15].
n
This shows how the modular tower is a natural geometric way of passing from GL (Z^) to GL (Z^).
1 2
The formulation that is most convenientin our setting is the one givenin terms of Shimura varieties.
In fact, rather than the modular tower de(cid:12)ned by the projective limit
(3.23) Y =limY(n)
n(cid:0)
of the modular curves Y(n), it is better for our purposes to consider the Shimura variety
(3.24) Sh(H(cid:6);GL )=GL (Q) (GL (A ) H(cid:6))=GL (Q) GL (A)=C(cid:3);
2 2 2 f 2 2
n (cid:2) n
of which(3.23)isa connectedcomponent. In fact, it iswell knownthat, forarithmeticpurposes, it is
alwaysbetterto workwith nonconnected ratherthan with connectedShimuravarieties(cf. e.g. [23]).
Thesimple reasonwhyit isnecessarytopasstothe nonconnected caseisthe following. Thevarieties
in the tower are arithmetic varieties de(cid:12)ned over number (cid:12)elds. However, the number (cid:12)eld typically
changes along the levels of the tower (Y(n) is de(cid:12)ned over the cyclotomic (cid:12)eld Q((cid:16) )). Passing to
n
nonconnected Shimura varieties allows precisely for the de(cid:12)nition of a canonical model where the
whole tower is de(cid:12)ned over the same number (cid:12)eld.
This distinction is important to our viewpoint, since we want to work with noncommutative spaces
endowed with an arithmetic structure, speci(cid:12)ed by the choice of an arithmetic subalgebra.
Every 2-dimensional Q-lattice can be described by data
(3.25) ((cid:3);(cid:30))=((cid:21)(Z+Zz);(cid:21)(cid:11));
for some (cid:21) C(cid:3), somez H, and (cid:11) M (Z^) (using the basis(1; z) of Z+Zz as in (87) [9] to view
2
2 2 2 (cid:0)
(cid:11) as a map (cid:30)). The diagonalaction of (cid:0)=SL (Z) yields isomorphic Q-lattices, and (cf. (87) [9]) the
2
space of 2-dimensional Q-lattice up to scaling can be identi(cid:12)ed with the quotient
(3.26) (cid:0) (M (Z^) H):
2
n (cid:2)
Therelationofcommensurabilityisimplemented bythepartiallyde(cid:12)nedactionofGL+(Q)on(3.26).
2
The groupoid of the commensurability relation on 2-dimensionalQ-lattices not up to scaling (i.e.
2
R
the dual space) has as algebra of coordinates the convolution algebra of (cid:0) (cid:0)-invariant functions on
(cid:2)
(3.27) ~ = (g;(cid:11);u) GL+(Q) M (Z^) GL+(R) g(cid:11) M (Z^) :
U f 2 2 (cid:2) 2 (cid:2) 2 j 2 2 g
Up to Morita equivalence, this can also be described as the crossed product
(3.28) C (M (A ) GL (R))oGL (Q):
0 2 f 2 2
(cid:2)
When we pass to Q-lattices up to scaling, we take the quotient =C(cid:3).
2
R