Noncommutative geometry and the Hilbert 12th prob- lem

by all possible embedding of Q^cyc into C. The Galois group G acts naturally as a group of automorphisms ofA1=H1 commuting with the time evolution σ and the spontaneous symmetry breaking occurs for β >1.

As shown in [41], thatQ-algebraA1,Q can also be obtained as the algebra generated by μn, μ^∗n forn∈N^× and by homogeneous functions of weight zero on 1-dimensional Q-lattices obtained as a normalization of the functions

ξk,a(Λ, ϕ) =

y∈Λ+ϕ(a)

1

y^k by covolume.

Namely, consider the functionsc^kξk,a, wherec(Λ) is proportional to the covol- ume|Λ|and satisfies 2π√

−1c(Z) = 1.

The choice of such an arithmetic subalgebra ofA1 corresponds to endowing the noncommutative space (asA1,Q) with an arithmetic structure. The subal- gebra corresonds to the rational functions and the values of KMS_∞ states at elements 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 prob-

theory in the case of K = Q, already mentioned. In this case, the maximal abelian extension ofQcan be identified with the cyclotomic field Q^cyc. Equiv- alently, the torsion points of themultiplicative groupC^∗, as the roots of unity, generateQ^ab⊂C.

As a remark, the only other case of number fields where such formulation is carried out completely is that of imaginary quadratic fields where the construc- tion relies on the theory of elliptic curves with complex multiplication and on the Galois theory of the fields ofmodular functions (cf. [148] as a survey).

Remark. (Added). Now may recall from [116] the following fundamental theoremon theclass fieldtheory.

[TheArtinreciprocity law]. For a finite Abel extensionK over an algebraic number field K that is a finite extension over Q, there is a non-zero ideal m of the ring OK of algebric integers of K such that the reciprocity map from

⊕S_mZtoGal(K/K)mapsΓ_m to zero, where(⊕S_mZ)/Γm∼=Cl(m)the ray class group that is a finite abelian group, where there is a surjective map from⊕^SmZ to Cl(m), induced by Z ∼=K^∗_v/O^∗v for v ∈ S_m a finite prime point of K which does not divide m, identified with a fractional ideal relatively prime withm.

There is themaximal idealmsatisfying the condition above, called thecon- ductorofK, so that induced is the homomorphism fromCl(m) to Gal(K/K).

[Existence theorem]. Conversely, for a non-zero ideal m of Ok, there is a finite Abel extension K_m ofKsuch that the reciprocity map induces the isomor- phism fromCl(m)toGal(K_m/K).

Such Abel extensionK_m form is unique up to isomorphisms overK, called therayclass field of (the conductor)m. Ifm=OK, thenK_mis the absolute class field of K. A fintie Abel extension overKwithmas the conductor is a subfield of K_m. A class fieldof Kis defined to be a subfield of K_m corresponding to the image of a subgroup ofCl(m) under the reciprocitymap.

For instance, ifK=Qandm=nZ, thenK_m=Q(ζn). IfK=K_m, then Cl(m)∼= Gal(K/Q)∼=Z^∗n.

It then follows [Kronecker-Weber] that the largest Abel extensionQ^Ab ofQ isQ(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]. Amore detailed account of various results related to the BC system and generalizations is given in the section of [41] as futher developments. Because of such a close relation to the Hilbert 12th problem, it seems to be clear that obtaining generalizations of the Bost-Connes systemto 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 systemin 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 ad`eles of Kand by A<sup>∗</sup><sub>K</sub> =GL1(AK) the group ofid`eles of K.

Let C_K = A^∗_K/K^∗ be the group of id`ele classes of K and D_K the connected component of the identity inC_K. Construct a C^∗-algebra dynamical system (A_K, σ,R)and an arithmetic subalgebraA_K,Q such that

(1)The(quotient)id`ele class groupC_K/D_Kacts by symmetries on the system (A_K, σ,R)preserving the subalgebraA_K,Q.

(2) The extremal KMS_∞ states ϕ ∈ E_∞ evaluated at elements a of A_K,Q

satisfy that ϕ(a)∈Kthe algebraic closure of Kin C, and the values ϕ(a)∈C fora∈A_K,Q andϕ∈E_∞ generateK^Ab.

(3)The class field theory isomorphism θ: Gal(K^Ab/K)−→^∼= C_K/D_K inter- twines with the actions, so that forα∈Gal(K^Ab/K)andϕ∈E_∞,

A_K,Q⊂A_K −−−−→^ϕ K^Ab⊂C

θ(α)

⏐⏐

⏐⏐^α A_K,Q⊂A_K −−−−→^ϕ K^Ab⊂C.

In general, the arithmetic subalgebraA_K,Qneed not be an involutive algebra.

The setup described abovemay provide a possible new approach to the explicit class field theory problem via noncommutative geometry.

Given a number fieldKwith [K,Q] =n, there is an embedding fromitsmul- tiplicative group K^∗ into GLn(Q). Such an embedding induces an embedding fromGL1(A_K.f) intoGLn(Af), whereA_K,f =Af⊗Kare the finite ad`eles ofK.

That suggests that a possible strategy to approach the problemstated above may be to first study quantumstatisticalmechanical systems corresponding to GLn-analogues of the Bost-Connes system.

Themain result of [41] is the construction of such a system in the case of GL2and the analysis of the arithmetic properties of its KMS states. In the case ofGL1, it is observed that the geometry ofmodular curves and the algebra of modular forms appear naturally.

As the work of [48], a quantumstatisticalmechanical systemis constructed, and satisfies all the properties listed above for the case whereKis an imaginary quadratic field. The properties of this system are intermediate between those of the original BC system and those of the GL2 system of [41]. In fact, the construction is geometrically based on replacing the 1-dimensional Q-lattices of the Bost-Connes systemwith 1-dimensionalK-lattices. The groupoid of the commensurability 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 theGL2 system, it is possible to show that the complex multiplication (CM) system defined in that way recovers the full 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 integerd. It is natural to ask whether the approach outlined above, based on noncommutative geometry, may provide any new information

on this case. There is a close relation between the real quadratic case and noncommutative geometty, obtained by Manin ([109], [110]). Discussed below is the possible relation between the approach and theGL2 system.

Remark. Now recall from [116] the following. Suppose that K is a number field. LetP=Pf#P_∞ denote the set of all (finite or infinite) primes (divisors) of K(as equivalence classes of valuations ofK). For anyp ∈P, let Kp be the completion ofKwith respect to p (such as Qp ofQwithp =pZforpprime), and let K^∗_p be the multiplicative group of K_p. Moreover, let O_p the valuation ring for a finite primep, and letU_p the group of invertibles ofO_p. SinceO_p as an additive group is a compact open subgroup ofK_p, thead`eleringA_K of K is defined to be the restricted direct product of{K_p}with respect to {O_p}. It looks like

Π_p∈PK_p⊃A_K⊃O≡(Π_p∈P_fO_p)#(Π_p∈P_∞K_p),

with the quotient AK/O with discrete topology, to become a locally compact group and ring, where any element {x_p}, calledad`ele, ofAKhas anypcompo- nentx_p in O_p except finitelymanypcomponentx_p in Kp.

As well, sinceU_p as amultiplicative group is a compact open subgroup of K^∗_p, theid`ealgroup JK of Kis defined to be the restricted direct sum of{K^∗_p} with respect to {U_p}. It looks like

Π_p∈PK^∗_p⊃JK⊃U ≡(Π_p∈P_fU_p)#(Π_p∈P_∞K^∗_p),

with the similar condition as above, with any element ofJK, called id`ele. As a group, JK = A^∗_K, but with the topology stronger than the relative topology fromthe topology onA_K.