KMS STATES AND COMPLEX MULTIPLICATION (PART II) ALAINCONNES,MATILDEMARCOLLI,ANDNIRANJANRAMACHANDRAN Contents 1. Introduction 1 1.1. Notation 3 2. Quantum Statistical Mechanics and Explicit Class Field Theory 4 2.1. Quantum Statistical Mechanics 4 2.2. Hilbert’s 12th problem 6 2.3. Fabulous states for number (cid:12)elds 6 2.4. Noncommutative pro-varieties 7 3. Q-lattices and noncommutative Shimura varieties 8 3.1. Tower Power 9 3.2. The cyclotomic tower and the BC system 9 3.3. Arithmetic structure of the BC system 11 3.4. The modular tower and the GL -system 14 2 3.5. Tate modules and Shimura varieties 16 3.6. Arithmetic properties of the GL -system 18 2 3.7. Crossed products and functoriality 20 4. Quantum statistical mechanics for imaginary quadratic (cid:12)elds 22 4.1. K-lattices and commensurability 23 4.2. Algebras of coordinates 25 4.3. Symmetries 27 4.4. Comparison with other systems 28 5. KMS states and complex multiplication 28 5.1. Low temperature KMS states and Galois action 29 5.2. Open Questions 31 References 31 1. Introduction Several results point to deep relations between noncommutative geometry and class (cid:12)eld theory ([3], [10], [20], [22]). In[3]aquantumstatisticalmechanicalsystem(BC) isexhibited, with partitionfunc- tiontheRiemannzetafunction(cid:16)((cid:12)), andwhosearithmeticpropertiesarerelatedtotheGaloistheory of the maximal abelian extension of Q. In [10], 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 temperature extremal KMS states and the Galois properties of the GL -system are analyzed in [10] 2 for the generic case of elliptic curves with transcendental j-invariant. As the results of [10] show, one of the main new features of the GL -system is the presence of symmetries by endomorphism, as in 2 (2.6) below. The full Galoisgroupof the modular(cid:12)eld appearsthen assymmetries, acting onthe set of extremal KMS states of the system, for large inverse temperature (cid:12). (cid:12) 1 2 CONNES,MARCOLLI,ANDRAMACHANDRAN 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. This point of view, which we discuss here only in the simplest case of G = GL and G = GL was 1 2 extendedtoawideclassofShimuravarietiesbyE.HaandF.Paugam,[13]. Theyconstructedgeneral- izationsof the BC andGL system that providenoncommutativeShimuravarieties, and investigated 2 the arithmetic properties of their partition functions and KMS states. In our paper [11], we construct a quantum statistical mechanical system associated to an imaginary quadratic (cid:12)eld K = Q(p d), d > 0 a positive integer. Just like the BC and the GL systems 2 (cid:0) are based on the geometric notion of Q-lattices and commensurability, this \complex multiplication system" (CM) is based on an analogous geometric notion of commensurability of 1-dimensional K- lattices. The arithmetic properties of the CM system fully incorporate the explicit class (cid:12)eld theory for the imaginary quadratic (cid:12)eld K, and its partition function is the Dedekind zeta function (cid:16) ((cid:12)) of K. K Thus, the main result of [11], which we recallhere in Theorem5.2 below, gives a complete answer,in the case of an imaginary quadratic (cid:12)eld K, to the following question, which has been open since the work of Bost and Connes [3]. Problem 1.1. For some number (cid:12)eld K (other than Q) exhibit an explicit quantum statistical me- chanical system ( ;(cid:27) ) with the following properties: t A (1) The partition function Z((cid:12)) is the Dedekind zeta function of K. (2) The system has a phase transition with spontaneous symmetry breaking at the pole (cid:12) = 1 of the zeta function. (3) There is a unique equilibrium state above critical temperature. (4) ThequotientC =D of theid(cid:18)eles classgroupof K bytheconnectedcomponentofthe identity K K acts as symmetries of the system ( ;(cid:27) ). t A (5) There is a subalgebra of with the property that the values of extremal ground states on 0 A A elements of are algebraic numbers and generate the maximal abelian extension Kab. 0 A (6) The Galois action on these values is realized by the induced action of C =D on the ground K K states, via the class (cid:12)eld theory isomorphism (cid:18) :C =D Gal(Kab=K). K K ! The BC system satis(cid:12)es all the properties listed in Problem 1.1, in the case of K =Q. It is natural, therefore,toposethe analogousquestion inthe caseofother number(cid:12)elds. Someimportantprogress inthedirectionofgeneralizingtheBCsystemtoothernumber(cid:12)eldswasdonebyHarariandLeichtnam [16], Cohen [5], Arledge, Laca and Raeburn [1], Laca and van Frankenhuijsen [20]. However, to our knowledge, the (cid:12)rst construction of a system satisfying all the conditions posed in Problem 1.1, with no restriction on the class number of K, was the one obtained in [11]. As we shall discuss at length in the present paper, one reason why progress in the solution of Problem 1.1 is di(cid:14)cult is that the requirement on the Galois action on ground states (zero temperature KMS states) of the system is verycloselyrelatedtoadi(cid:14)cultprobleminnumbertheory,namelyHilbert’s12thproblemonexplicit class(cid:12)eld theory. We will arguein this paper that Problem1.1 mayprovidea new possible approach to Hilbert’s 12th problem. KMS AND CM, II 3 In this perspective, the BC system and the CM system of [11] cover the two known cases (K = Q and K =Q(p d)) of Hilbert’s 12th problem. We analyze closely the properties of both the BC and (cid:0) the CM system, in order to understand what new insight they maygive on this problem and in what part instead they depend on the number theoretic solution. The new CM system we constructed in [11] can be regarded in two di(cid:11)erent ways. On the one hand, it is a generalization of the BC system of [3], when changing the (cid:12)eld from Q to K = Q(p d), and (cid:0) is infact Moritaequivalenttothe oneconsideredin [20], but with norestrictiononthe class number. On the other hand, it isalsoaspecializationof the GL -systemof [10]toelliptic curveswith complex 2 multiplication by K. The KMS states of the CM system can be related to the non-generic KMS 1 1 statesoftheGL -system,associatedtopoints(cid:28) HwithcomplexmultiplicationbyK,andthegroup 2 2 of symmetries is the Galois group of the maximal abelian extension of K. Here also symmetries by endomorphisms play a crucial role, as they allow for the action of the class groupCl( )ofthering ofalgebraicintegersofK,sothatthepropertiesofProblem1.1aresatis(cid:12)ed O O in all cases, with no restriction on the class number of K. Asweshowedin[11],theCMsystemcanberealizedasasubgroupoidoftheGL -system. Ithasthen 2 a natural choice of an arithmetic subalgebra inherited from that of the GL -system. This is crucial, 2 in order to obtain the intertwining of Galois action on the values of extremal KMS states and action of symmetries of the system. The paper is structured as follows. In Sections 2 and 3 we discuss the relation between Problem 1.1 and Hilbert’s 12th problem and the geometry of the BC and GL system from the point of view of 2 Shimura varieties. In Section 4 we recall the construction and main properties of the CM system, especiallyitsrelationtotheexplicitclass(cid:12)eldtheoryforimaginaryquadratic(cid:12)elds. Wealsocompare it with the BC and GL systems and with previous systems introduced as generalizations of the BC 2 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 1.1. Notation. In the table above and in the rest of the paper, we denote by Z^ the pro(cid:12)nite completion of Z and by A =Z^ Qthe ringof (cid:12)nite adeles of Q. Forany abeliangroupG, we denote by G the subgroup f tors (cid:10) of elements of (cid:12)nite order. For any ring 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 4 CONNES,MARCOLLI,ANDRAMACHANDRAN write for the ring of algebraic integers of the imaginary quadratic (cid:12)eld K =Q(p d), where d is a O (cid:0) positive integer. We also use the notation (1.1) ^ :=( Z^) A =A K and I =A(cid:3) =GL (A ): O O(cid:10) K;f f (cid:10)Q K K;f 1 K;f Notice that K(cid:3) embeds diagonallyinto I . K The modular (cid:12)eld F is the (cid:12)eld of modular functions overQab (cf. e.g. [21]). This is the union of the (cid:12)elds F of modular functions of level N rational over the cyclotomic (cid:12)eld Q((cid:16) ), that is, such that N N the q-expansion at a cusp has coe(cid:14)cients in the cyclotomic (cid:12)eld Q((cid:16) ). N G. Shimura determined the automorphisms of F (cf. [32]). 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 (1.2) (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)! 2. Quantum Statistical Mechanics and Explicit Class Field Theory The BC quantum statistical mechanical system [2, 3] 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. 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. 2.1. 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, (cid:27) (x) = eitHxe(cid:0)itH. The t 2 analog of a probability 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 A! (cid:21) 2A normalization, ’(1) = 1. In the quantum mechanical framework, the analog of the classical Gibbs measure is given by states satisfying the KMS condition (cf. [15]). 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 the following holds. For all x;y , there exists a holomorphic function (cid:20) 1 2 A F (z) on the strip 0 < Im(z) < (cid:12), which extends as a continuous function on the boundary of the x;y strip, with the property 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 Choquetsimplex[4,II 5]whosesetofextremepoints consistsofthefactorstates. Onecanexpress (cid:12) x E any KMS state uniquely in terms of extremal states, because of the uniqueness of the barycentric (cid:12) decomposition of a Choquet simplex. KMS AND CM, II 5 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 KMS states, that is, 1 t (cid:12) A ’ (a)=lim ’ (a) for all a . 1 (cid:12)!1 (cid:12) 2A One can easily see the di(cid:11)erence between the KMS condition at zero tempretature and the notion of KMS statesgiveninDe(cid:12)nition2.2,inthesimplecaseofthetrivialtimeevolution(cid:27) =id; t R. 1 t 8 2 In this case, any state has the property that (2.2) extends to the upper half plane (as a constant). On the other hand, only tracial states can be weak limits of (cid:12)-KMS states, hence the notion given in De(cid:12)nition 2.2 is more restrictive. With De(cid:12)nition 2.2 we still obtain a weakly compact convex set (cid:6) and we can consider the set of its extremal points. 1 1 E Thetypicalframeworkforspontaneoussymmetrybreakinginasystemwithauniquephasetransition (cf. [14]) 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 set of extreme (cid:12) points ((cid:6) )= . The choice of an equilibrium state ’ may break this symmetry to a smaller (cid:12) (cid:12) (cid:12) E E 2E 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 i.e. , as a homomorphism (cid:11) : G Aut(A;(cid:27) )= . The KMS condition shows that the inner auto- t (cid:12) ! U morphisms Inn( ;(cid:27) ) act trivially on KMS states, hence (2.5) induces an action of the group G on t (cid:12) A the set (cid:6) of KMS states, for 0<(cid:12) . (cid:12) (cid:12) (cid:20)1 More generally, one can consider actions by endomorphisms (cf. [10]), where an endomorphism (cid:26) of the dynamical system ( ;(cid:27) ) is a -homomorphism (cid:26) : commuting with the evolution (cid:27) . t t A (cid:3) A ! A There is an induced action of (cid:26) on KMS states, for 0<(cid:12) < , given by (cid:12) 1 ’ (cid:26) (2.6) (cid:26)(cid:3)(’):= (cid:14) ; ’((cid:26)(1)) provided that ’((cid:26)(1))=0, where (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. [10]. (cid:12) In general, the induced action (modulo inner) of a semigroup of endomorphisms of ( ;(cid:27) ) on the t A KMS states need not extend directly (in a nontrivial way) to KMS states. In fact, even though (cid:12) 1 (2.6) is de(cid:12)ned for states ’ , it can happen that, when passing to the weak limit ’ = lim ’ (cid:12) (cid:12) (cid:12) (cid:12) 2 E one has ’((cid:26)(1))=0 and can no longer apply (2.6). 6 CONNES,MARCOLLI,ANDRAMACHANDRAN In suchcases,it isoften still possibletoobtainan inducednontrivialactionon . This canbe done 1 E viathe followingprocedure,whichwasnamed\warmingupandcoolingdown"in[10]. Oneconsiders (cid:12)rst a map W : (called the \warmingup" map) given by (cid:12) 1 (cid:12) E !E 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 HereH isthepositiveenergyHamiltonian,implementingthetimeevolutionintheGNSrepresentation (cid:25) associated to the extremal KMS state ’. Assume that, for su(cid:14)ciently large (cid:12), the map (2.7) ’ 1 gives a bijection between KMS states (in the sense of De(cid:12)nition 2.2) and KMS states. The action 1 (cid:12) by endomorphisms on is then de(cid:12)ned as 1 E (2.8) (cid:26)(cid:3)(’)(a):= lim (cid:26)(cid:3)(W (’))(a) a : (cid:12) (cid:12)!1 8 2A 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.[14], 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. 2.2. 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 of generators of the maximal abelian extension Kab of a number (cid:12)eld K and an explicit description of the action of the Galois group Gal(Kab=K) on them. This Galois group 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 [33]). 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 2.3. Fabulous states for number (cid:12)elds. Wediscussherebrie(cid:13)ythe relationof Problem1.1toHilbert’s12thproblem,byconcentratingonthe arithmetic properties of the action of symmetries on the set of extremal zero temperature KMS 1 E states. We abstract these properties in the notion of \fabulous states" discussed below. Given a number (cid:12)eld K, with a choice of an embedding K C, the \problem of fabulous states" (cid:26) consists of the following question. Problem 2.3. Construct a C(cid:3)-dynamical system ( ;(cid:27) ), with an arithmetic subalgebra of , t K A A A with the following properties: (1) The quotient group G=C =D acts on as symmetries compatible with (cid:27) and preserving K K t A . K A KMS AND CM, II 7 (2) The states ’ , evaluated on elements of the arithmetic subalgebra , satisfy: 1 K 2E A ’(a) K, the algebraic closure 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 (2.9) (cid:18):C =D ’ Gal(Kab=K) K K (cid:0)! intertwines the actions, (2.10) (cid:11)(’(a))=(’ (cid:18)(cid:0)1((cid:11)))(a); (cid:14) for all (cid:11) Gal(Kab=K), for all ’ , and for all a . 1 K 2 2E 2A (4) The algebra has an explicit presentation by generators and relations. K A ItisimportanttomakeageneralremarkaboutProblem2.3versustheoriginalHilbert’s12thproblem. Itmayseemat(cid:12)rstthattheformulationofProblem2.3willnotleadtoanynewinformationaboutthe Hilbert12thproblem. Infact,onecanalwaysconsiderherea\trivialsystem"where =C(C =D ) K K A and(cid:27) =id. AllstatesareKMSstatesandthe extremalonesarejust valuationsatpoints. Thenone t can alwayschoose as the \rationalsubalgebra"the (cid:12)eld Kab itself, embedded in the space of smooth functions on C =D as x f (g) = (cid:18)(g)(x), for x Kab. Then this trivial system tautologically K K x 7! 2 satis(cid:12)es all the properties of Problem 2.3, except the last one. This last property, for this system, is then exactly as di(cid:14)cult as the original Hilbert 12th problem. This appears to show that the constructionof \fabulousstates"isjust areformulationof the originalproblem andneednotsimplify the task of obtaining explicit information about the generatorsof Kab and the Galois action. Thepointispreciselythatsuch\trivialexample"isonlyareformulationoftheHilbert12thproblem, whileProblem2.3allowsfornontrivialconstructionsofquantumstatisticalmechanicalsystems,where onecanuseessentiallythefactofworkingwithalgebrasinsteadof(cid:12)elds. Thehopeisthat,forsuitable systems, giving a presentation of the algebra will prove to be an easier problem than giving K A generators of the (cid:12)eld Kab and similarly for the action by symmetries on the algebra as opposed to the Galois action on the (cid:12)eld. SolutionstoProblem1.1givenontrivialquantumstatisticalmechanicalsystemswiththe rightGalois symmetriesandarithmeticpropertiesofzerotemperatureKMSstates. (Noticethat thetrivialexam- ple abovecertainly will not satisfy the other properties soughtfor in Problem 1.1.) The hope is that, for such systems, the problem of giving a presentation of the algebra may turn out to be easier K A than the original Hilbert 12th problem. We show in Proposition 3.3 in Section 3.3 below how, in the case of K =Q, using the algebra of the BC system of [3], one can obtain a verysimple description of Qab that uses only the explicit presentation of the algebra, without any reference to (cid:12)eld extensions. This is possible precisely because the algebra is larger than the part that corresponds to the trivial system. 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 [10] (cid:12)ts into this 2 general program. 2.4. Noncommutative pro-varieties. Inthesettingabove,theC(cid:3)-dynamicalsystem( ;(cid:27) )togetherwithaQ-structurecompatiblewiththe t A (cid:13)ow(cid:27) (i.e.arationalsubalgebra suchthat(cid:27) ( C)= C)de(cid:12)nesanon-commutative t Q t Q Q A (cid:26)A A (cid:10) A (cid:10) algebraic (pro-)variety X overQ. Thering (or C),whichneednotbeinvolutive,istheanalog Q Q A A (cid:10) of the ring of algebraic functions on X and the set of extremal KMS -states is the analog of the set 1 of points of X. As we will see for example in (3.3) and (3.31) below, the action of the subgroup of Aut( ;(cid:27) ) which takes C into itself is analogous to the action of the Galois group on the t Q A A (cid:10) (algebraic) values of algebraic functions at points of X. 8 CONNES,MARCOLLI,ANDRAMACHANDRAN 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. 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 agree modulo the sum of the lattices, (cid:30) (cid:30) mod (cid:3) +(cid:3) : 1 2 1 2 (cid:17) It is essential here that one does not require the homomorphism (cid:30) to be invertible in general. The set of Q-lattices modulo the equivalence relation of commensurability is best described with the tools of noncommutative geometry, as explained in [10]. We will be concerned here only with the case of n = 1 or n = 2 and we will consider also the set of commensurability classes of Q-lattices up to scaling, where the scaling action is given by the group S =R(cid:3) in the 1-dimensional case and S =C(cid:3) in the 2-dimensional case. + Inthesecases,onecan(cid:12)rstconsiderthegroupoid oftheequivalencerelationofcommensurabilityon R thesetofQ-lattices(notuptoscaling). Thisisalocallycompact(cid:19)etalegroupoid. Whenconsideringthe quotient by the scaling action, the algebra of coordinates associated to the quotient =S is obtained R by restricting the convolution product of the algebra of to weight zero functions with S-compact R support. The algebra obtained this way, which 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 de(cid:12)nes a representation (cid:25) by left convolution of y 2 R R the algebra of on the Hilbert space = ‘2( ), where is the set of elements with source y. y y y R H R R This constructionpassestothe quotientbythe scalingactionof S. Representationscorrespondingto pointsthatacquireanontrivialautomorphismgroupwillnolongerbeirreducible. Iftheunity (0) 2R corresponds to an invertible Q-lattice, then (cid:25) is a positive energy representation. y 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". KMS AND CM, II 9 3.1. 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 cover of V and V . Thus, is a partially ordered set. This gives a corresponding compatible system i j I of covering maps V V. In case of a tower over a pointed variety (V;v), one (cid:12)xes a point v over v i i ! in eachV . Eventhough V may not be irreducible, we shall allowourselvesto loosely refer to V as a i i i variety. It is convenient to view a \tower" as a category with objects (V V) and morphisms i T C ! Hom(V ;V ) being maps of covers of V. One has the group Aut (V ) of invertible self-maps of V i j T i i overV (thegroupofdecktransformations);the decktransformationsarenotrequiredtopreservethe points v . There is a (pro(cid:12)nite) group of symmetries associated to a tower, namely i (3.2) :=lim Aut (V ): G i T i (cid:0) The simplest example of a tower is the \fundamental group" tower associated with a (smooth con- nected) complex algebraic variety (V;v) and its universal covering (V~;v~). Let be the category of C all (cid:12)nite (cid:19)etale (unbranched) intermediate covers V~ W V of V . In this case, the symmetry ! ! group of (3.2)isthe algebraicfundamentalgroupof V,whichisalsothepro(cid:12)nitecompletionofthe G (topological) fundamental group (cid:25) (V;v). (For the theory of (cid:19)etale covers and fundamental groups, 1 we refer the interested reader to SGA1.) Simple variants of this example include allowing controlled rami(cid:12)cation. Otherexamplesoftowersarethosede(cid:12)nedbyiterationofselfmapsofalgebraicvarieties. Forus,themostimportantexamplesof\towers"willbethecyclotomictowerandthemodulartower1. AnotherveryinterestingcaseoftowersisthatofmoregeneralShimuravarieties. These,however,will not be treated in this paper. (For a systematic treatment of quantum statistical mechanical systems associated to general Shimura varieties see [13] and upcoming work by the same authors.) 3.2. 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)! 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 ([10]). 1Inthispaper,wereservetheterminology\modulartower"forthetowerofmodularcurves. Itwouldbeinteresting toinvestigate inasimilarperspectivethemoregeneraltheoryofmodulartowersinthesenseofM.Fried. 10 CONNES,MARCOLLI,ANDRAMACHANDRAN 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. In fact, using the identi(cid:12)cation Q=Z(cid:24)=Af=Z^ one gets a natural character e of A =Z^ such that f e(r)= e2(cid:25)ir r Q=Z 8 2 and a pairing of Q=Z with Z^ such that r; x = e(rx) r Q=Z; x Z^: h i 8 2 2 Thus, we identify the group Z^ with the Pontrjagin dual of Q=Z and we obtain a corresponding identi(cid:12)cation (3.7) C(cid:3)(Q=Z)=C(Z^): (cid:24) The group of deck transformations = Z^(cid:3) of the cyclotomic tower acts by automorphisms on the algebraofcoordinatesC(Z^)intheobGviousway. Inadditiontothisaction,thereisasemigroupaction ofN(cid:2) =Z implementingthecommensurabilityrelation. Thisisgivenbyendomorphismsthatmove >0 vertically across the levels of the cyclotomic tower. They are given by f(n(cid:0)1(cid:26)); (cid:26) nZ^ (3.8) (cid:11) (f)((cid:26))= 2 n ( 0 otherwise. 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 (3.8) 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 (3.9) =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^, where r(cid:26) is the product in A . The product in the algebra is given by the associative f 2 convolution product (3.10) f f (r;(cid:26))= f (rs(cid:0)1;s(cid:26))f (s;(cid:26)); 1 2 1 2 (cid:3) s:Xs(cid:26)2Z^ and the adjoint is given by f(cid:3)(r;(cid:26))=f(r(cid:0)1;r(cid:26)). This is the C(cid:3)-algebra of the Bost{Connes (BC) system [3]. It has a natural time evolution (cid:27) t determined by the ratio of the covolumes of two commensurable Q-lattices, (3.11) (cid:27) (f)(r;(cid:26))=ritf(r;(cid:26)): t Thealgebra wasoriginallyde(cid:12)nedasaHeckealgebraforthealmostnormalpairofsolvablegroups 1 P+ P+, wAhere P is the algebraic ax+b group and P+ is the restriction to a>0 (cf. [3]). Z (cid:26) Q