From Physics to Number theory via Noncommutative Geometry

Alain Connes and Matilde Marcolli

Contents

1 Quantum Statistical Mechanics of ℚ-Lattices 11 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Quantum Statistical Mechanics . . . . . . . . . . . . . . . . . . . 14 1.3 ℚ^{ab} and KMS states . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.4 Further Developments . . . . . . . . . . . . . . . . . . . . . . . . 23 1.5 Fabulous States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.6 The subalgebra A_{ℚ} and Eisenstein Series . . . . . . . . . . . . . 27 1.7 The Determinant part of the GL_{2}-System . . . . . . . . . . . . . 35 1.8 Commensurability of ℚ-Lattices in ℂ and the full GL_{2}-System . 41 1.9 The subalgebra A_{ℚ} and the Modular Field . . . . . . . . . . . . . 57 1.10 The noncommutative boundary of modular curves . . . . . . . . 68 1.11 The BC algebra and optical coherence . . . . . . . . . . . . . . . 73

Introduction

e volta nostra poppa nel mattino, de’ remi facemmo ali al folle volo —— Dante, Inf. XXVI 124-125

Several recent results reveal a surprising connection between modular forms and noncommutative geometry. The first occurrence came from the classification of noncommutative three spheres, [C–DuboisViolette-I] [C–DuboisViolette-II]. Hard computations with the noncommutative analog of the Jacobian involving the ninth power of the Dedekind eta function were necessary in order to analyze the relation between such spheres and noncommutative nilmanifolds. Another occurrence can be seen in the computation of the explicit cyclic cohomology Chern character of a spectral triple on SU_{q}(2) [C–02]. Another surprise came recently from a remarkable action of the Hopf algebra of transverse geometry of foliations of codimension one on the space of lattices modulo Hecke correspondences, described in the framework of noncommutative geometry, using a modular Hecke algebra obtained as the cross product of modular forms by the action of Hecke correspondences [C–Moscovici-I] [C–Moscovici-II]. This action determines a differentiable structure on this noncommutative space, related to the Rankin–Cohen brackets of modular forms, and shows their compatibility with Hecke operators. Another instance where properties of modular forms can be recast in the context of noncommutative geometry can be found in the theory of modular symbols and Mellin transforms of cusp forms of weight two, which can be recovered from the geometry of the moduli space of Morita equivalence classes of noncommutative tori viewed as boundary of the modular curve [Manin–M]. In this paper we show that the theory of modular Hecke algebras, the spectral realization of zeros of L-functions, and the arithmetic properties of KMS states in quantum statistical mechanics combine into a unique general picture based on the noncommutative geometry of the space of commensurability classes of ℚ-lattices. An n-dimensional ℚ-lattice consists of an ordinary lattice Λ in ℝ^{n} and a homomorphism

ϕ : ℚ^{n}/ℤ^{n} → ℚΛ/Λ.

Two such ℚ-lattices are commensurable if and only if the corresponding lattices are commensurable and the maps agree modulo the sum of the lattices.

The description of the spaces of commensurability classes of ℚ-lattices via noncommutative geometry yields two quantum systems related by a duality. The first system is of quantum statistical mechanical nature, with the algebra of coordinates parameterizing commensurability classes of ℚ-lattices modulo scaling and with a time evolution with eigenvalues given by the index of pairs of commensurable ℚ-lattices. There is a symmetry group acting on the system, in general by endomorphisms. It is this symmetry that is spontaneously broken at low temperatures, where the system exhibits distinct phases parameterized by arithmetic data. We completely analyze the phase transition with spontaneous symmetry breaking in the two-dimensional case, where a new phenomenon appears, namely that there is a second critical temperature, beyond which no equilibrium state survives. In the “dual system”, which corresponds just to commensurability of ℚ-lattices, the scaling group is acting. In physics language, what emerges is that the zeros of zeta appear as an absorption spectrum of the scaling action in the L^{2} space of the space of commensurability classes of ℚ-lattices as in [C–99]. While the zeros of zeta and L-functions appear at the critical temperature, the analysis of the low temperature equilibrium states concentrates on the subspace

GL_{n}(ℚ)\GL_{n}(𝔸)

of invertible ℚ-lattices, which as is well known plays a central role in the theory of automorphic forms. While, at first sight, at least in the 1-dimensional case, it would seem easy to classify commensurability classes of ℚ-lattices, we shall see that ordinary geometric tools fail because of the ergodic nature of the equivalence relation. Such quotients are fundamentally of “quantum nature”, in that, even though they are sets in the ordinary sense, it is impossible to distinguish points by any finite (or countable) collection of invariants. Noncommutative geometry is specifically designed to handle such quantum spaces by encoding them by algebras of non-commuting coordinates and extending the techniques of ordinary geometry using the tools of functional analysis, noncommutative algebra, and quantum physics. Direct attempts to define function spaces for such quotients lead to invariants that are of a cohomological nature. For instance, let the fundamental group Γ of a Riemann surface act on the boundary ℙ^{1}(ℝ) of its universal cover identified with the Poincaré disk. The space

L^{∞}(Γ\ℙ^{1}(ℝ)) := L^{∞}(ℙ^{1}(ℝ))^{Γ}

is in natural correspondence with global sections of the sheaf of (real parts of) holomorphic functions on the Riemann surface, as boundary values. More generally, the cyclic cohomology of the noncommutative algebra of coordinates on such quotients is obtained by applying derived functors to these naive functorial definition of function spaces.

In the 1-dimensional case, the states at zero temperature are related to the Kronecker–Weber construction of the maximal abelian extension ℚ^{ab}. In fact, in this case the quantum statistical mechanical system is the one constructed in [Bost–C], which has underlying geometric space X_{1} parameterizing commensurability classes of 1-dimensional ℚ-lattices modulo scaling by ℝ_{+}^{∗}. The corresponding algebra of coordinates is a Hecke algebra for an almost normal pair of solvable groups. The regular representation is of type III_{1} and determines the time evolution of the system, which has the set of log(p), p a prime number, as set of basic frequencies. The system has an action of the idèles class group modulo the connected component of identity as a group of symmetries. This induces a Galois action on the ground states of the system at zero temperature. When raising the temperature the system has a phase transition, with a unique equilibrium state above the critical temperature. The Riemann zeta function appears as the partition function of the system, as in [Julia]. Each equivalence class of ℚ-lattices determines an irreducible covariant representation, where the Hamiltonian is implemented by minus the log of the covolume. For a general class, this is not bounded below. It is so, however, in the case of equivalence classes of invertible ℚ-lattices, i.e. where the labelling of torsion points is one to one. These classes then define positive energy representations and corresponding KMS states for all temperatures below critical. In the 2-dimensional case, as the temperature lowers, the system settles down on these invertible ℚ-lattices, so that the zero temperature space is commutative and is given by the Shimura variety

GL_{2}(ℚ)\GL_{2}(𝔸)/ℂ^{∗}.

The action of the symmetry group, which in this case is nonabelian and isomorphic to ℚ^{∗}\GL_{2}(𝔸_{f}), is more subtle due to the presence of inner automorphisms and the necessary use of the formalism of superselection sectors. Moreover, its effect on the zero temperature states is not obtained directly but is induced by the action at non-zero temperature, which involves the full noncommutative system. The quotient GL_{2}(ℚ)\(M_{2}(𝔸_{f}) × GL_{2}(ℝ))/ℂ^{∗} and the space of 2–dimensional ℚ-lattices modulo commensurability and scaling are the same, hence the corresponding algebras are Morita equivalent. However, it is preferable to work with the second description, since, by taking the classical quotient by the action of the subgroup SL_{2}(ℤ), it reduces the group part in the cross product to the classical Hecke algebra. The GL_{2} system has an arithmetic structure provided by a rational subalgebra, given by a natural condition on the coefficients of the q-series. We show that it is a Hecke algebra of modular functions, closely related to the modular Hecke algebra of [C–Moscovici-I], [C–Moscovici-II]. The symmetry group acts on the values of ground states on this rational subalgebra as the automorphism group of the modular field. Evaluation of a generic ground state φ of the system on the rational subalgebra generates an embedded copy of the modular field in ℂ and there exists a unique isomorphism of the symmetry group of the system with the Galois group of the embedded modular field, which intertwines the Galois action on the image with the symmetries of the system,

θ(σ) ∘ φ = φ ∘ σ.

The relation between this GL_{2} system and class field theory is being investigated in ongoing work [C–M–Ramachandran]. The arithmetic structure is inherited by the dual of the GL_{2} system and enriches the structure of the noncommutative space of commensurability classes of 2dimensional ℚ-lattices to that of a “noncommutative arithmetic variety”. The relation between this dual system and the spectral realization of zeros of Lfunctions is the central topic of Chapter 2. The dual of the GL_{1} system, under the duality obtained by taking the cross product by the time evolution, corresponds to the space of commensurability classes of 1-dimensional ℚ-lattices, not considered up to scaling. This corresponds geometrically to the total space L of a principal ℝ_{+}^{∗} bundle over the base X_{1}, and determines a natural scaling action of ℝ_{+}^{∗}. The space L is described by the quotient

L = GL_{1}(ℚ)\𝔸^{·},

where 𝔸^{·} denotes the set of adèles with nonzero archimedean component. The corresponding algebra of coordinates is Morita equivalent to C(X_{1}) ⋊_{σ_{t}} ℝ. Any approach to a spectral realization of the zeros of zeta through the quantization of a classical dynamical system faces the problem of obtaining the leading term in the Riemann counting function for the number of zeros of imaginary part less than E as a volume in phase space. The solution [C–99] of this issue is achieved in a remarkably simple way, by the scaling action of ℝ_{+}^{∗} on the phase space of the real line ℝ, and will be the point of departure for the second part of the paper. In particular, this shows that the space L requires a further compactification at the archimedean place, obtained by replacing the quotient L = GL_{1}(ℚ)\𝔸^{·} by \overline{L} = GL_{1}(ℚ)\𝔸 i.e. dropping the non vanishing of the archimedean component. This compactification has an analog for the GL_{2} case, given by the noncommutative boundary of modular curves considered in [Manin–M], which corresponds to replacing GL_{2}(ℝ) by M_{2}(ℝ) at the archimedean place, and is related to class field theory for real quadratic fields through Manin’s real multiplication program. The space \overline{L} appears as the configuration space for a quantum field theory, where the degrees of freedom are parameterized by prime numbers, including infinity. When only finitely many degrees of freedom are considered, and in particular only the place at infinity, the semiclassical approximation exhibits the main terms in the asymptotic formula for the number of zeros of the Riemann zeta function. The zeros of zeta appear as an absorption spectrum, namely as lacunae in a continuous spectrum, where the width of the absorption lines depends on the presence of a cutoff. The full idèles class group appears as symmetries of the system and L-functions with Grössencharakter replace the Riemann zeta function in nontrivial sectors. From the point of view of quantum field theory, the field configurations are given by adèles, whose space 𝔸 is then divided by the action of the gauge group GL_{1}(ℚ). As mentioned above, the quotient space is essentially the same as the space L of commensurability classes of 1-dimensional ℚ-lattices. The log(p) appear as periods of the orbits of the scaling action. The Lefschetz formula for the scaling action recovers the Riemann–Weil explicit formula as a semi-classical approximation. The exact quantum calculation for finitely many degrees of freedom confirms this result. The difficulty in extending this calculation to the global case lies in the quantum field theoretic problem of passing to infinitely many degrees of freedom. The main features of the dual systems in the GL_{1} case are summarized in the following table:

Quantum statistical mechanics  Quantum field theory

Commensurability classes  Commensurability classes
of ℚ-lattices modulo scaling  of ℚ-lattices

A = C^{∗}(ℚ/ℤ) ⋊ ℕ^{×}  A ⋊_{σ_{t}} ℝ

Time evolution σ_{t}  Energy scaling U(λ), λ ∈ ℝ_{+}^{∗}

{log p} as frequencies  {log p} as periods of orbits

Arithmetic rescaling μ_{n}  Renormalization group flow μ∂_{μ}

Symmetry group ℤ̂^{∗}  Idèles class group
as Galois action on T = 0 states  as gauge group

System at zero temperature  GL_{n}(ℚ)\GL_{n}(𝔸)

System at critical temperature  Spectral realization
(Riemann’s ζ as partition function) (Zeros of ζ as absorption spectrum)

Type III_{1}  Type II_{∞}

There is a similar duality (and table) in the GL_{2} case, where part of the picture remains to be clarified. The relation with the modular Hecke algebra of [C–Moscovici-I], [C–Moscovici-II] is more natural in the dual system where modular forms with non-zero weight are naturally present. The dual system L can be interpreted physically as a “universal scaling system”, since it exhibits the continuous renormalization group flow and its relation with the discrete scaling by powers of primes. For the primes two and three, this discrete scaling manifests itself in acoustic systems, as is well known in western classical music, where the two scalings correspond, respectively, to passing to the octave (frequency ratio of 2) and transposition (the perfect fifth is the frequency ratio 3/2), with the approximate value log(3)/ log(2) ∼ 19/12 responsible for the difference between the “circulating temperament” of the Well Tempered Clavier and the “equal temperament” of XIX century music. It is precisely the irrationality of log(3)/ log(2) which is responsible for the noncommutative nature of the quotient corresponding to the three places {2, 3, ∞}. The key role of the continuous renormalization group flow as a symmetry of the dual system L and its similarity with a Galois group at the archimedean place brings us to the last part of this paper. In Chapter 3, we analyze the quantum statistical mechanics of ℚ-lattices at critical temperature. The fact that the KMS state at critical temperature can be expressed as a noncommutative residue (Dixmier trace) shows that the system at critical temperature should be analyzed with tools from quantum field theory and renormalization. The mathematical theory of renormalization in QFT developed in [C–Kreimer-I] [C–Kreimer-II] shows in geometric terms that the procedure of perturbative renormalization can be described as the Birkhoff decomposition

g_{eff}(ε) = g_{eff_{+}}(ε) (g_{eff_{−}}(ε))^{−1}

on the projective line of complexified dimensions ε of the loop

g_{eff}(ε) ∈ G = formal diffeomorphisms of ℂ

given by the unrenormalized effective coupling constant. The g_{eff_{−}} side of the Birkhoff decomposition yields the counterterms and the g_{eff_{+}} side evaluated at the critical dimension gives the renormalized value of the effective coupling. This explicit knowledge of the counterterms suffices to determine the full renormalized theory. The principal G-bundle on ℙ^{1}(ℂ), with trivialization given by the Birkhoff decomposition, has a flat connection with regular singularities, coming from a Riemann–Hilbert problem determined by the representation datum given by the β-function of renormalization, viewed as the logarithm of the monodromy around the singular dimension. The problem of incorporating nonperturbative effects leads to a more sophisticated Riemann–Hilbert problem in terms of a representation of the wild fundamental group of Martinet–Ramis, related to applying Borel summation techniques to the unrenormalized effective coupling constant. In the perturbative theory the renormalization group appears as a natural 1parameter subgroup of the group of “diffeographisms” which governs the ambiguity in the choice of the physical solution. In fact, the renormalization group, the wild fundamental group and the connected component of identity in the idèles class group are all incarnations of a still mysterious Galois theory at the archimedean place. It was shown in [C–00] that the classification of approximately finite factors provides a nontrivial Brauer theory for central simple algebras over ℂ, and an archimedean analog of the module of central simple algebras over nonarchimedean fields. The relation of Brauer theory to the Galois group is via the construction of central simple algebras as cross products of a field by a group of automorphisms. It remained for a long time an elusive point to obtain in a natural manner factors as cross product by a group of automorphisms of a field, which is a transcendental extension of ℂ. This was achieved in [C–DuboisViolette-II] for type II_{1}, via the cross product of the field of elliptic functions by an automorphism given by translation on the elliptic curve. The results on the GL_{2} system give an analogous construction for type III_{1} factors using the modular field. The physical reason for considering such field extensions of ℂ lies in the fact that the coupling constants g of the fundamental interactions (electromagnetic, weak and strong) are not really constants but depend on the energy scale μ and are therefore functions g(μ). Thus, high energy physics implicitly extends the “field of constants”, passing from the field of scalars ℂ to a field of functions containing all the g(μ). On this field, the renormalization group provides the corresponding theory of ambiguity, and acquires an interpretation as the missing Galois group at archimedean places.

The structure of the paper is organized as follows.

• The first part is dedicated to the quantum statistical mechanical system of ℚ-lattices, in the cases of dimension one and two, and its behavior at zero temperature.

• The second part deals with the GL_{1} system at critical temperature and its dual system, and their relation to the spectral realization of the zeros of zeta of [C–99].

• The last part is dedicated to the system at critical temperature, the theory of renormalization of [C–Kreimer-I], [C–Kreimer-II], the Riemann-Hilbert problem and the missing Galois theory at the archimedean place.

Bibliography

[Bost–C] J.B. Bost, A. Connes, Hecke algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. (New Series) Vol.1 (1995) N.3, 411–457.

[C–02] A. Connes, Cyclic cohomology, Quantum group Symmetries and the Local Index Formula for SU_{q}(2).(2002), Math QA/0209142.

[C–00] A. Connes, Noncommutative Geometry and the Riemann Zeta Function, Mathematics: Frontiers and perspectives, IMU 2000 volume.

[C–99] A. Connes, Trace formula in Noncommutative Geometry and the zeros of the Riemann zeta function. Selecta Mathematica. (New Series) Vol.5 (1999) 29–106.

[C–DuboisViolette-I] A. Connes, M. Dubois-Violette, Noncommutative finitedimensional manifolds. I. spherical manifolds and related examples, Comm. Math. Phys. Vol.230 (2002) N.3, 539–579.

[C–DuboisViolette-II] A. Connes, M. Dubois-Violette, Moduli space and structure of noncommutative 3-spheres, Lett. Math. Phys., Vol.66 (2003) N.1-2, 91–121.

[C–Kreimer-I] A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem. Comm. Math. Phys. 210 (2000), N.1, 249– 273.

[C–Kreimer-II] 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), N.1, 215–241.

[C–M–Ramachandran] A. Connes, M. Marcolli, N. Ramachandran, KMS states and complex multiplication, in preparation.

[C–Moscovici-I] A. Connes, H. Moscovici, Modular Hecke algebras and their Hopf symmetry, Moscow Math. Journal, Vol.4 (2004) N.1, 67–109.

[C–Moscovici-II] A. Connes, H. Moscovici, Rankin-Cohen Brackets and the Hopf Algebra of Transverse Geometry, Moscow Math. Journal, Vol.4 (2004) N.1, 111–130.

[Julia] B. Julia, Statistical theory of numbers, in Number Theory and Physics, J.-M. Luck, P. Moussa and M. Waldschmidt (Eds.), Springer Verlag, Berlin, 1990.

[Manin–M] Yu.I. Manin, M. Marcolli, Continued fractions, modular symbols, and noncommutative geometry, Selecta Mathematica (New Series) Vol.8 (2002) N.3, 475–520.

Chapter 1

Quantum Statistical Mechanics of ℚ-Lattices

1.1 Introduction

In this chapter we shall start by giving a geometric interpretation in terms of the space of commensurability classes of 1-dimensional ℚ-lattices of the quantum statistical dynamical system (BC [5]). This system exhibits the relation between the phenomenon of spontaneous symmetry breaking and number theory. Its dual system obtained by taking the cross product by the time evolution is basic in the spectral interpretation of zeros of zeta. Since ℚ-lattices and commensurability continue to make sense in dimension n, we shall obtain an analogous system in higher dimension and in particular we derive a complete picture of the system in dimension n = 2. This shows two distinct phase transitions with arithmetic spontaneous symmetry breaking. In the initial model of BC ([5]) the partition function is the Riemann zeta function. Equilibrium states are characterized by the KMS-condition. While at large temperature there is only one equilibrium state, when the temperature gets smaller than the critical temperature, the equilibrium states are no longer unique but fall in distinct phases parameterized by number theoretic data. The pure phases are parameterized by the various embeddings of the cyclotomic field ℚ^{ab} in ℂ. The physical observables of the BC system form a C^{∗}-algebra endowed with a natural time evolution σ_{t}. This algebra is interpreted here as the algebra of noncommuting coordinates on the space of commensurability classes of 1dimensional ℚ-lattices up to scaling by ℝ_{+}^{∗}. What is remarkable about the ground states of this system is that, when evaluated on the rational observables of the system, they only affect values that are algebraic numbers. These span the maximal abelian extension of ℚ. Moreover, the class field theory isomorphism intertwines the two actions of the idèles class group, as symmetry group of the system, and of the Galois group, as permutations of the expectation values of the rational observables. That the latter action preserves positivity is a rare property of states. We abstract this property as a definition of “fabulous^{1} states”, in the more general context of arbitrary number fields and review recent developments in the direction of extending this result to other number fields. We present a new approach, based on the construction of an analog of the BC system in the GL_{2} case. Its relation to the complex multiplication case of the Hilbert 12th problem will be discussed specifically in ongoing work of the two authors with N. Ramachandran [13]. The C^{∗}-algebra of observables in the GL_{2}-system describes the non-commutative space of commensurability classes of ℚ-lattices in ℂ up to scaling by ℂ^{∗}. A ℚ-lattice in ℂ is a pair (Λ, ϕ) where Λ ⊂ ℂ is a lattice while

ϕ : ℚ^{2}/ℤ^{2} ⟶ ℚΛ/Λ

is a homomorphism of abelian groups (not necessarily invertible). Two ℚlattices (Λ_{j}, ϕ_{j}) are commensurable iff the lattices Λ_{j} are commensurable (i.e. ℚΛ_{1} = ℚΛ_{2}) and the maps ϕ_{j} are equal modulo Λ_{1} + Λ_{2}. The time evolution corresponds to the ratio of covolumes of pairs of commensurable ℚ-lattices. The group

S = ℚ^{∗}\GL_{2}(𝔸_{f})

quotient of the finite adèlic group of GL_{2} by the multiplicative group ℚ^{∗} acts as symmetries of the system, and the action is implemented by endomorphisms, as in the theory of superselection sectors of Doplicher-Haag-Roberts ([16]). It is this symmetry which is spontaneously broken below the critical temperature T = \frac{1}{2}. The partition function of the GL_{2} system is ζ(β)ζ(β − 1), for β = 1/T, and the system exhibits three distinct phases, with two phase transitions at T = \frac{1}{2} and at T = 1. At low temperatures (T < \frac{1}{2}) the pure phases are parameterized by the set

GL_{2}(ℚ)\GL_{2}(𝔸)/ℂ^{∗}

of classes of invertible ℚ-lattices (up to scaling). The equilibrium states of the “crystalline phase” merge as T → 1/2 from below, as the system passes to a “liquid phase”, while at higher temperatures (T ≥ 1) there are no KMS states. The subalgebra of rational observables turns out to be intimately related to the modular Hecke algebra introduced in (Connes-Moscovici [10]) where its surprising relation with transverse geometry of foliations is analyzed. We show that the KMS states at zero temperature when evaluated on the rational observables generate a specialization of the modular function field F. Moreover, as in the BC system the state intertwines the two actions of the group S, as symmetry group of the system, and as permutations of the expectation values of the rational observables by the Galois group of the modular field, identified with S by Shimura’s theorem ([51]). We shall first explain the general framework of quantum statistical mechanics, in terms of C^{∗}-algebras and KMS states. Noncommutative algebras concretely represented in Hilbert space inherit a canonical time evolution, which allows for phenomena of phase transition and spontaneous symmetry breaking for KMS states at different temperatures. There are a number of important nuances between the abelian BC case and the higher dimensional non-abelian cases. For instance, in the abelian case, the subfield of ℂ generated by the image of the rational subalgebra under an extremal KMS_{∞} state does not depend on the choice of the state and the intertwining between the symmetry and the Galois actions is also independent of the state. This no longer holds in the non-abelian case, because of the presence of inner automorphisms of the symmetry group S. Moreover, in the GL_{2} case, the action of S on the extremal KMS_{∞} states is not transitive, and the corresponding invariant of the orbit of a state φ under S is the subfield F_{φ} ⊂ ℂ, which is the specialization of the modular field given by evaluation at the point in the upper half plane parameterizing the ground state φ. Another important nuance is that the algebra A is no longer unital while A_{ℚ} is a subalgebra of the algebra of unbounded multipliers of A. Just as an ordinary function need not be bounded to be integrable, so states can be evaluated on unbounded multipliers. In our case, the rational subalgebra A_{ℚ} is not selfadjoint. Finally the action of the symmetry group on the ground states is obtained via the action on states at positive temperature. Given a ground state, one warms it up below the critical temperature and acts on it by endomorphisms. When taking the limit to zero temperature of the resulting state, one obtains the corresponding transformed ground state. In our framework, the correct notion of ground states is given by a stronger form of the KMS_{∞} condition, where we also require that these are weak limits of KMS_{β} states for β → ∞. We then consider the “dual” system of the GL_{2}-system, which describes the space of commensurability classes of 2-dimensional ℚ-lattices (not up to scaling). The corresponding algebra is closely related to the modular Hecke algebra of [10]. As in the 1-dimensional case, where the corresponding space is compactified by removing the non-zero condition for the archimedean component of the adèle, the compactification of the two-dimensional system amounts to replacing the archimedean component GL_{2}(ℝ) with matrices M_{2}(ℝ). This corresponds to the noncommutative compactification of modular curves considered in [37]. In terms of ℚ-lattices this corresponds to degenerations to pseudo-lattices, as in [34]. It is desirable to have a concrete physical (experimental) system realizing the BC symmetry breaking phenomenon (as suggested in [43]). In fact, we shall show that the explicit presentation of the BC algebra not only exhibits a strong analogy with phase states, as in the theory of optical coherence, but it also involves an action on them of a discrete scaling group, acting by integral multiplication of frequencies.

1This terminology is inspired from John Conway’s talk on “fabulous” groups.

Acknowledgements. We are very grateful to Niranjan Ramachandran for many extremely useful conversations on class field theory and KMS states, that motivated the GL_{2} system described here, whose relation to the theory of complex multiplication is being investigated in [13]. We thank Marcelo Laca for giving us an extensive update on the further developments on [5]. We benefited from visits of the first author to MPI and of the second author to IHES and we thank both institutions for their hospitality. The second author is partially supported by a Sofja Kovalevskaya Award of the Humboldt Foundation and the German Government.

1.2 Quantum Statistical Mechanics

In classical statistical mechanics a state is a probability measure μ on the phase space that assigns to each observable f an expectation value, in the form of an average

∫ f dμ.  (1.1)

In particular for a Hamiltonian system, the Gibbs canonical ensemble is a measure defined in terms of the Hamiltonian and the symplectic structure on the phase space. It depends on a parameter β, which is an inverse temperature, β = 1/kT with k the Boltzmann constant. The Gibbs measure is given by

dμ_{G} = \frac{1}{Z} e^{−βH}dμ_{Liouville},  (1.2)

normalized by Z = ∫ e^{−βH}dμ_{Liouville}. When passing to infinitely many degrees of freedom, where the interesting phenomena of phase transitions and symmetry breaking happen, the definition of the Gibbs states becomes more involved (cf. [45]). In the quantum mechanical framework, the analog of the Gibbs condition is given by the KMS condition at inverse temperature β ([17]). This is simpler in formulation than its classical counterpart, as it relies only on the involutive algebra A of observables and its time evolution σ_{t} ∈ Aut(A), and does not involve any additional structure like the symplectic structure or the approximation by regions of finite volume. In fact, quantum mechanically, the observables form a C^{∗}-algebra A, the Hamiltonian is the infinitesimal generator of the (pointwise norm continuous) one parameter group of automorphisms σ_{t} ∈ Aut(A), and the analog of a probability measure, assigning to every observable a certain average, is given by a state.

Definition 1.1 A state on a C^{∗}-algebra A is a linear form on A such that

φ(1) = 1 , φ(a^{∗}a) ≧ 0  ∀ a ∈ A .  (1.3)

When the C^{∗}-algebra A is non unital the condition φ(1) = 1 is replaced by ||φ|| = 1 where

||φ|| := sup_{x∈A,||x||≤1}|φ(x)| .  (1.4)

Such states are restrictions of states on the unital C^{∗}-algebra Ã obtained by adjoining a unit. The evaluation φ(a) gives the expectation value of the observable a in the statistical state φ. The Gibbs relation between a thermal state at inverse temperature β = \frac{1}{kT} and the time evolution

σ_{t} ∈ Aut(A)  (1.5)

is encoded by the KMS condition ([17]) which reads

∀ a, b ∈ A , ∃ F bounded holomorphic in the strip {z | Im z ∈ [0, β]} (1.6)
F (t) = φ(a σ_{t}(b)) F (t + iβ) = φ(σ_{t}(b)a) ∀t ∈ ℝ.

In the case of a system with finitely many quantum degrees of freedom, the algebra of observables is the algebra of operators in a Hilbert space H and the time evolution is given by σ_{t}(a) = e^{itH} a e^{−itH}, where H is a positive self-adjoint operator such that exp(−βH) is trace class for any β > 0. For such a system, the analog of (1.2) is

φ(a) = \frac{1}{Z} Tr (a e^{−βH}) ∀a ∈ A,  (1.7)

with the normalization factor Z = Tr(exp(−βH)). It is easy to see that (1.7) satisfies the KMS condition (1.6) at inverse temperature β.

In the nonunital case, the KMS condition is defined in the same way by (1.6). Let M(A) be the multiplier algebra of A and let B ⊂ M(A) be the C^{∗}-subalgebra of elements x ∈ M(A) such that t ↦ σ_{t}(x) is norm continuous.

Proposition 1.2 Any state φ on A admits a canonical extension to a state still noted φ on the multiplier algebra M(A) of A. The canonical extension of a KMS state still satisfies the KMS condition on B.

Proof. For the first statement we refer to [42]. The proof of the second statement illustrates a general density argument, where the continuity of t ↦ σ_{t}(x) is used to control the uniform continuity in the closed strip, in order to apply the Montel theorem of normal families. Indeed, by weak density of A in M(A), one obtains a sequence of holomorphic functions, but one only controls their uniform continuity on smooth elements of B. □

Figure 1.1: The KMS condition.

As we shall see, it will also be useful to extend whenever possible the integration provided by a state to unbounded multipliers of A. In the unital case, for any given value of β, the set Σ_{β} of KMS_{β} states on A forms a convex compact Choquet simplex (possibly empty and in general infinite dimensional). In the nonunital case, given a σ_{t}-invariant subalgebra C of B, the set Σ_{β}(C) of KMS_{β} states on C should be viewed as a compactification of the set of KMS_{β} states on A. The restriction from C to A maps Σ_{β}(C) to KMS_{β} positive linear forms on A of norm less than or equal to one (quasi-states). The typical pattern for a system with a single phase transition is that this simplex consists of a single point for β ≦ β_{c} i.e. when the temperature is larger than the critical temperature T_{c}, and is non-trivial (of some higher dimension in general) when the temperature lowers. Systems can exhibit a more complex pattern of multiple phase transitions, where no KMS state exists above a certain temperature. The GL_{2} system, which is the main object of study in this paper, will actually exhibit this more elaborate behavior. We refer to the books ([6], [16]) for the general discussion of KMS states and phase transitions. The main technical point is that for finite β a β-KMS state is extremal iff the corresponding GNS representation is factorial. The decomposition into extremal β-KMS states is then the primary decomposition for a given β-KMS state. At 0 temperature (β = ∞) the interesting notion is that of weak limit of β-KMS states for β → ∞. It is true that such states satisfy a weak form of the KMS condition. This can be formulated by saying that, for all a, b ∈ A, the function

F (t) = φ(a σ_{t}(b))

extends to a bounded holomorphic function in the upper half plane ℍ. This implies that, in the Hilbert space of the GNS representation of φ (i.e. the completion of A in the inner product φ(a^{∗}b)), the generator H of the oneparameter group σ_{t} is a positive operator (positive energy condition). However, this condition is too weak in general to be interesting, as one sees by taking the trivial evolution (σ_{t} = id , ∀t ∈ ℝ). In this case any state fulfills it, while weak limits of β-KMS states are automatically tracial states. Thus, we shall define Σ_{β=∞} as the set of weak limit points of the sets Σ_{β} of β-KMS states for β → ∞. The framework for spontaneous symmetry breaking ([16]) involves a (compact) group of automorphisms G ⊂ Aut(A) of A commuting with the time evolution,

σ_{t} α_{g} = α_{g} σ_{t}  ∀ g ∈ G , t ∈ ℝ .  (1.8)

The group G is the symmetry group of the system, and the choice of an equilibrium state φ may break it to a smaller subgroup given by the isotropy group of φ

G_{φ} = {g ∈ G , gφ = φ}.  (1.9)

The group G acts on Σ_{β} for any β, hence on its extreme points E(Σ_{β}) = E_{β}. The unitary group U of the fixed point algebra of σ_{t} acts by inner automorphisms of the dynamical system (A, σ_{t}): for u ∈ U,

(Adu) (x) := u x u^{∗} , ∀x ∈ A .

These inner automorphisms of (A, σ_{t}) act trivially on KMS_{β} states, as one checks using the KMS condition. This gives us the freedom to wipe out the group Int(A, σ_{t}) of inner symmetries and to define an action modulo inner of a group G on the system (A, σ_{t}) as a map

α : G → Aut(A, σ_{t})

fulfilling the condition

α(g_{1}g_{2}) α(g_{2})^{−1} α(g_{1})^{−1} ∈ Int(A, σ_{t}) , ∀g_{j} ∈ G .

Such an action gives an action of the group G on the set Σ_{β} of KMS_{β} states since the ambiguity coming from Int(A, σ_{t}) disappears in the action on Σ_{β}. In fact there is one more generalization of the above obvious notion of symmetries that we shall crucially need later – it involves actions by endomorphisms. This type of symmetry plays a key role in the theory of superselection sectors developed by Doplicher-Haag-Roberts (cf.[16], Chapter IV).

Definition 1.3 An endomorphism ρ of the dynamical system (A, σ_{t}) is a ∗- homomorphism ρ : A → A commuting with σ_{t}.

It follows then that ρ(1) = e is an idempotent fixed by σ_{t}. Given a KMS_{β} state φ the equality

ρ^{∗}(φ) := Z^{−1} φ ∘ ρ , Z = ϕ(e)

gives a KMS_{β} state, provided that φ(e) ≠ 0. Exactly as above for unitaries, consider an isometry

u ∈ A , u^{∗} u = 1

which is an eigenvector for σ_{t}, i.e. that fulfills, for some λ ∈ ℝ_{+}^{∗} (λ ≥ 1), the condition

σ_{t}(u) = λ^{it} u , ∀t ∈ ℝ .

Then u defines an inner endomorphism Adu of the dynamical system (A, σ_{t}) by the equality

(Adu) (x) := u x u^{∗} , ∀x ∈ A ,

and one obtains the following.

Proposition 1.4 The inner endomorphisms of the dynamical system (A, σ_{t}) act trivially on the set of KMS_{β} states,

(Adu)^{∗}(φ) = φ , ∀φ ∈ Σ_{β} .

Proof. The KMS_{β} condition shows that φ(u u^{∗}) = λ^{−β} > 0 so that (Adu)^{∗}(φ) is well defined. The same KMS_{β} condition applied to the pair (x u^{∗}, u) shows that (Adu)^{∗}(φ) = φ. □ At 0 temperature (β = ∞) it is no longer true that the endomorphisms act directly on the set Σ_{∞} of KMS_{∞} states, but one can use their action on KMS_{β}- states together with the “warming up” operation. This is defined as the map

W_{β}(φ)(x) = Z^{−1} Trace(π(x) e^{−β H}) , ∀x ∈ A ,  (1.10)

where H is the positive energy Hamiltonian, implementing the time evolution in the representation π associated to the KMS_{∞} state φ and Z = Trace( e^{−β H}). Typically, W_{β} gives a bijection

W_{β} : Σ_{∞} → Σ_{β},

for β larger than critical. Using the bijection W_{β}, one can transfer the action back to zero temperature states. Another property of KMS states that we shall need later is the following functoriality. Namely, besides the obvious functoriality under pullback, discussed above, KMS states push forward under equivariant surjections, modulo normalization.

Proposition 1.5 Let (A, σ_{t}) be a C^{∗}-dynamical system (A separable) and J a norm closed two sided ideal of A globally invariant under σ_{t}. Let u_{n} be a quasi central approximate unit for J. For any KMS_{β}-state φ on (A, σ_{t}) the following sequence converges and defines a KMS_{β} positive linear form on (A/J, σ_{t}),

ψ(x) = lim_{n→∞} φ((1 − u_{n}) x) , ∀x ∈ A .

Proof. Let A′′ be the double dual of A and p ∈ A′′ the central open projection corresponding to the ideal J (cf. [42]). By construction the u_{n} converge weakly to p (cf. [42] 3.12.14) so the convergence follows as well as the positivity of ψ. By construction ψ vanishes on J. To get the KMS_{β} condition one applies (1.6) with a = (1 − u_{n}) x, b = y where y is a smooth element in A. Then one gets a bounded uniformly continuous sequence F_{n}(z) of holomorphic functions in the strip {z | Im z ∈ [0, β]} with

F_{n}(t) = φ((1 − u_{n}) x σ_{t}(y)) F_{n}(t + iβ) = φ(σ_{t}(y)(1 − u_{n}) x) ∀t ∈ ℝ.

Using the Montel theorem on normal families and the quasi-central property of u_{n} one gets the KMS_{β} condition for ψ. □

1.3 ℚ^{ab} and KMS states

We shall now describe an explicit system (cf. [4], [5]) that will make contact between the general framework above and arithmetic. The algebra A of this system is defined over the rationals,

A = A_{ℚ} ⊗_{ℚ} ℂ ,  (1.11)

where A_{ℚ} is a ℚ-algebra and is of countable (infinite) dimension as a vector space over ℚ. The algebra A has a C^{∗}-completion A and a natural time evolution σ_{t}. To any vacuum state φ ∈ E_{∞} for (A, σ_{t}) we attach the ℚ-vector space of complex numbers,

V_{φ} := {φ(a) ; a ∈ A_{ℚ}}  (1.12)

that is of countable dimension over ℚ. It turns out that V_{φ} is included in algebraic numbers, so that one can act on these numbers by the Galois group

Gal (\overline{ℚ}/ℚ) .  (1.13)

The symmetry group G is the inverse limit with the profinite topology

G = ℤ̂^{∗} = \underleftarrow{lim}_{n} GL_{1}(ℤ/nℤ).  (1.14)

This can also be described as the quotient of the idèle class group of ℚ by the connected component of the identity,

G = GL_{1}(ℚ)\GL_{1}(𝔸)/ℝ_{+}^{∗} = C_{ℚ}/D_{ℚ} .  (1.15)

Here 𝔸 = 𝔸_{ℚ} denotes the adèles of ℚ, namely 𝔸 = 𝔸_{f} × ℝ, with 𝔸_{f} = ℤ̂ ⊗ ℚ. The following amazing fact holds:

For any φ ∈ E_{∞} and any γ ∈ Gal(\overline{ℚ}/ℚ), the composition  (1.16)

γ ∘ φ defined on A_{ℚ} does extend to a state on A.

What is “unreasonable” in this property defining “fabulous” states is that, though elements

γ ∈ Gal(\overline{ℚ}/ℚ)  (1.17)

extend to automorphisms of ℂ, these are extremely discontinuous and not even Lebesgue measurable (except for z ↦ z̄), and certainly do not preserve positivity. It follows from (1.16) that the composition φ ↦ γ ∘φ defines uniquely an action of Gal(\overline{ℚ}/ℚ) on E_{∞} and the equation

γ ∘ φ = φ ∘ g  (1.18)

gives a relation between Galois automorphisms and elements of G, i.e. idèle classes (1.15), which is in fact the class field theory isomorphism C_{ℚ}/D_{ℚ} ≅ Gal(ℚ^{ab}/ℚ).

Let us now concretely describe our system, consisting of the algebra A (defined over ℚ) and of the time evolution σ_{t}. The main conceptual steps involved in the construction of this algebra are:

• The construction, due to Hecke, of convolution algebras associated to double cosets on algebraic groups over the rational numbers;

• The existence of a canonical time evolution on a von Neumann algebra.

More concretely, while Hecke was considering the case of GL_{2}, where Hecke operators appear in the convolution algebra associated to the almost normal subgroup GL_{2}(ℤ) ⊂ GL_{2}(ℚ), the BC system arises from the Hecke algebra associated to the corresponding pair of parabolic subgroups. Indeed, let P be the algebraic group “ax + b”, i.e. the functor which to any abelian ring R assigns the group P_{R} of 2 by 2 matrices over R of the form

P_{R} = \{[\begin{matrix}1 & b \\ 0 & a \end{matrix}] ; a, b ∈ R , a invertible\} .  (1.19)

By construction P_{ℤ}^{+} ⊂ P_{ℚ}^{+} is an inclusion Γ_{0} ⊂ Γ of countable groups, where P_{R}^{+} denotes the restriction to a > 0. This inclusion fulfills the following commensurability condition:

The orbits of the left action of Γ_{0} on Γ/Γ_{0} are all finite.  (1.20)

For obvious reasons the same holds for orbits of Γ_{0} acting on the right on Γ_{0}\Γ. The Hecke algebra A_{ℚ} = H_{ℚ}(Γ, Γ_{0}) is by definition the convolution algebra of functions of finite support

f : Γ_{0}\Γ → ℚ ,  (1.21)

which fulfill the Γ_{0}-invariance condition

f(γγ_{0}) = f(γ)  ∀ γ ∈ Γ , γ_{0} ∈ Γ_{0}  (1.22)

so that f is really defined on Γ_{0}\Γ/Γ_{0}. The convolution product is then given by

(f_{1} ∗ f_{2})(γ) = ∑_{Γ_{0}\Γ} f_{1}(γγ_{1}^{−1})f_{2}(γ_{1}) .  (1.23)

The time evolution appears from the analysis of the regular representation of the pair (Γ, Γ_{0}). It is trivial when Γ_{0} is normal, or in the original case of Hecke, but it becomes interesting in the parabolic case, due to the lack of unimodularity of the parabolic group, as will become clear in the following. The regular representation

(π(f)ξ)(γ) = ∑_{Γ_{0}\Γ} f(γγ_{1}^{−1})ξ(γ_{1})  (1.24)

in the Hilbert space

H = ℓ^{2}(Γ_{0}\Γ)  (1.25)

extends to the complexification

A_{ℂ} = A_{ℚ} ⊗_{ℚ} ℂ  (1.26)

of the above algebra, which inherits from this representation the involution a ↦ a^{∗}, uniquely defined so that π(a^{∗}) = π(a)^{∗} (the Hilbert space adjoint), namely

f^{∗}(γ) := \overline{f(γ^{−1})}  ∀ γ ∈ Γ_{0}\Γ/Γ_{0}.  (1.27)

It happens that the time evolution (cf. [54]) of the von Neumann algebra generated by A in the regular representation restricts to the dense subalgebra A. This implies that there is a uniquely determined time evolution σ_{t} ∈ Aut(A), such that the state φ_{1} given by

φ_{1}(f) = ⟨π(f)ε_{e}, ε_{e}⟩  (1.28)

is a KMS_{1} state i.e. a KMS state at inverse temperature β = 1. Here ε_{e} is the cyclic and separating vector for the regular representation given by the left coset {Γ_{0}} ∈ Γ_{0}\Γ. Explicitly, one gets the following formula for the time evolution:

σ_{t}(f)(γ) = (\frac{L(γ)}{R(γ)})^{−it} f(γ)  ∀γ ∈ Γ_{0}\Γ/Γ_{0} ,  (1.29)

where the integer valued functions L and R on the double coset space are given respectively by

L(γ) = Cardinality of left Γ_{0} orbit of γ in Γ/Γ_{0} , R(γ) = L(γ^{−1}) . (1.30)

Besides the conceptual description given above, the algebra A_{ℚ} also has a useful explicit presentation in terms of generators and relations (cf. [5] §4, Prop.18). We recall it here, in the slightly simplified version of [21], Prop.24.

Proposition 1.6 The algebra A_{ℚ} is generated by elements μ_{n}, n ∈ ℕ^{×} and e(r), for r ∈ ℚ/ℤ, satisfying the relations

• μ_{n}^{∗}μ_{n} = 1, for all n ∈ ℕ^{×},

• μ_{k}μ_{n} = μ_{kn}, for all k, n ∈ ℕ^{×},

• e(0) = 1, e(r)^{∗} = e(−r), and e(r)e(s) = e(r + s), for all r, s ∈ ℚ/ℤ,

• For all n ∈ ℕ^{×} and all r ∈ ℚ/ℤ,

μ_{n} e(r) μ_{n}^{∗} = \frac{1}{n} ∑_{ns=r} e(s).  (1.31)

In this form the time evolution preserves pointwise the subalgebra R_{ℚ} = ℚ[ℚ/ℤ] generated by the e(r) and acts on the μ_{n}’s as

σ_{t}(μ_{n}) = n^{it} μ_{n}.

The Hecke algebra considered above admits an automorphism α, α^{2} = 1 whose fixed point algebra is the Hecke algebra of the pair P_{ℤ} ⊂ P_{ℚ}. The latter admits an equivalent description^{2}, from the pair

(P_{R}, P_{𝔸_{f}}),

where R is the maximal compact subring of the ring of finite adèles

𝔸_{f} = ∏ ; res ℚ_{p} .  (1.32)

This adèlic description displays, as a natural symmetry group, the quotient G of the idèle class group of ℚ by the connected component of identity (1.15).

Let \overline{ℚ} be an algebraic closure of ℚ and ℚ^{ab} ⊂ \overline{ℚ} be the maximal abelian extension of ℚ. Let r ↦ ζ_{r} be a (non-canonical) isomorphism of ℚ/ℤ with the multiplicative group of roots of unity inside ℚ^{ab}. We can now state the basic result that gives content to the relation between phase transition and arithmetic (BC [5]):

Theorem 1.7 1. For 0 < β ≦ 1 there exists a unique KMS_{β} state φ_{β} for the above system. Its restriction to R_{ℚ} = ℚ[ℚ/ℤ] ⊂ A is given by

φ_{β} (e(a/b)) = b^{−β} ∏_{p prime, p|b} (\frac{1 − p^{β−1}}{1 − p^{−1}} ) .  (1.33)

2This procedure holds more generally (cf. [47] [48]) for arbitrary almost normal inclusions (Γ_{0}, Γ).

2. For β > 1 the extreme KMS_{β} states are parameterized by embeddings ρ : ℚ^{ab} → ℂ and

φ_{β,ρ} (e(a/b)) = Z(β)^{−1} ∑_{n=1}^{∞} n^{−β}ρ (ζ_{a/b}^{n}) ,  (1.34)

where the partition function Z(β) = ζ(β) is the Riemann zeta function.

3. For β = ∞, the Galois group Gal(\overline{ℚ}/ℚ) acts by composition on E_{∞}. The action factors through the abelianization Gal(ℚ^{ab}/ℚ), and the class field theory isomorphism θ : G → Gal(ℚ^{ab}/ℚ) intertwines the actions,

α ∘ φ = φ ∘ θ^{−1}(α) , α ∈ Gal(ℚ^{ab}/ℚ) .

1.4 Further Developments

The main theorem of class field theory provides a classification of 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^{∗} = GL_{1}(K) in the local nonarchimedean case, while in the global case it is the quotient C_{K}/D_{K} of the idèle class group C_{K} by the connected component of the identity. The construction of the group C_{K} is at the origin of the theory of idèles and adèles. Hilbert’s 12th problem can be formulated as the question of providing an explicit set of generators of the maximal abelian extension K^{ab} of a number field K, inside an algebraic closure K̄, and of the action of the Galois group Gal(K^{ab}/K). The typical example where this is achieved, which motivated Hilbert’s formulation of the explicit class field theory problem, is the Kronecker–Weber case: the construction of the maximal abelian extension of ℚ. In this case the torsion points of ℂ^{∗} (roots of unity) generate ℚ^{ab} ⊂ ℂ. Remarkably, the only other case for number fields where this program has been carried out completely is that of imaginary quadratic fields, where the construction relies on the theory of elliptic curves with complex multiplication (cf. e.g. [53]). Generalizations to other number fields involve other remarkable problems in number theory like the Stark conjectures. Recent work of Manin [34] [35] suggests a close relation between the real quadratic case and noncommutative geometry. To better appreciate the technical difficulties underlying any attempt to address the Hilbert 12th problem of explicit class field theory via the BC approach, in view of the problem of fabulous states that we shall formulate in §1.5, we first summarize briefly the state of the art (to this moment and to our knowledge) in the study of C^{∗}-dynamical systems with phase transitions associated to number fields.

Some progress from the original BC paper followed in various directions, and some extensions of the BC construction to other global fields (number fields and function fields) were obtained. Harari and Leichtnam [19] produced a C^{∗}- dynamical system with phase transition for function fields and algebraic number fields. In the number fields case a localization is used in order to deal with lack of unique factorization into primes, and this changes finitely many Euler factors in the zeta function. The construction is based on the inclusion O⋊1 ⊂ K ⋊K^{∗}, where O is the ring of integers of K. The symmetry group G of (1.15) is replaced by the group

G = Ô^{∗} = GL_{1}(Ô),

with Ô the profinite completion of the ring of integers O. Their result on fixed point algebras of compact group actions shows that the contribution of the group of units O^{∗} can be factored out in the cases when this group is finite. There is a group homomorphism s : G → C_{K}/D_{K}, but it is in general neither injective nor surjective, hence, even in the case of imaginary quadratic fields, the construction does not capture the action of the Galois group Gal(K^{ab}/K), except in the very special class number one case. P. Cohen gave in [8] a construction of a C^{∗}-dynamical system associated to a number field K, which has spontaneous symmetry breaking and recovers the full Dedekind zeta function as partition function. The main point of her approach is to involve the semigroup of all ideals rather than just the principal ideals used in other approaches as the replacement of the semi-group of positive integers involved in BC. Still, the group of symmetries is G = Ô^{∗} and not the desired C_{K}/D_{K}. Typically, the extensions of the number field K obtained via these constructions are given by roots of unity, hence they do not recover the maximal abelian extension. The Hecke algebra of the inclusion O ⋊ 1 ⊂ K ⋊ K^{∗} for an arbitrary algebraic number field K was also considered by Arledge, Laca, and Raeburn in [1], where they discuss its structure and representations, but not the problem of KMS states. Further results on this Hecke algebra have been announced by Laca and van Frankenhuysen [27]: they obtain some general results on the structure and representations for all number fields, while they analyze the structure of KMS states only for the class number one case. In this case, their announced result is that there are enough ground states to support a transitive free action of Gal(K^{ab}/K) (up to a copy of {±1} for each real embedding). However, it appears that the construction does not give embeddings of K^{ab} as actual values of the ground states on the Hecke algebra over K, hence it does not seem suitable to treat the class field theory problem of providing explicit generators of K^{ab}. The structure of the Hecke algebra of the inclusion O ⋊ O^{∗} ⊂ K ⋊ K^{∗} was further clarified by Laca and Larsen in [23], using a decomposition of the Hecke algebra of a semidirect product as the cross product of the Hecke algebra of an intermediate (smaller) inclusion by an action of a semigroup.

The original BC algebra was also studied in much greater details in several following papers. It was proved by Brenken in [7] and by Laca and Raeburn in [25] that the BC algebra can be written as a semigroup cross product. Brenken also discusses the case of Hecke algebras from number fields of the type considered in [25, 1]. Laca then re-derived the original BC result from the point of view of semigroup cross products in [21]. This allows for significant simplifications of the argument in the case of β > 1, by looking at the conditional expectations and the KMS condition at the level of the “predual” (semigroup) dynamical system. A further simplification of the original phase transition theorem of BC was given by Neshveyev in [40], via a direct argument for ergodicity, which implies uniqueness of the KMS states for 0 ≤ β ≤ 1. The BC algebra can also be realized as a full corner in the cross product of the finite adèles by the multiplicative rationals, as was shown by Laca in [22], by dilating the semigroup action to a minimal full group action. Laca and Raeburn used the dilation results of [22] to calculate explicitly the primitive (and maximal) ideal spaces of the BC algebra as well as of the cross product of the full adèles by the action of the multiplicative rationals. Using the cross product description of the BC algebra, Leichtnam and Nistor computed Hochschild, cyclic, and periodic cyclic homology groups of the BC algebra, by computing the corresponding groups for the C^{∗}-dynamical system algebras arising from the action of ℚ^{∗} on the adèles of ℚ. The calculation for the BC algebra then follows by taking an increasing sequence of smooth subalgebras and an inductive limit over certain Morita equivalent subalgebras. Further results related to aspects of the BC construction and generalizations can be found in [3], [15], [24], [29], [30], [55].

1.5 Fabulous States

Given a number field K, we let 𝔸_{K} denote the adèles of K and J_{K} = GL_{1}(𝔸_{K}) be the group of idèles of K. We write C_{K} for the group of idèles classes C_{K} = J_{K}/K^{∗} and D_{K} for the connected component of the identity in C_{K}.

If we remain close to the spirit of the Hilbert 12th problem, we can formulate a general question, aimed at extending the results of [5] to other number fields K. Given a number field K, with a choice of an embedding K ⊂ ℂ, the “problem of fabulous states” consists in constructing a C^{∗}-dynamical system (A, σ_{t}) and an “arithmetic” subalgebra A, which satisfy the following properties:

1. The idèles class group G = C_{K}/D_{K} acts by symmetries on (A, σ_{t}) preserving the subalgebra A.

2. The states φ ∈ E_{∞}, evaluated on elements of A, satisfy:

• φ(a) ∈ K̄, the algebraic closure of K in ℂ;

• the elements of {φ(a) : a ∈ A}, for φ ∈ E_{∞} generate K^{ab}.

3. The class field theory isomorphism

θ : C_{K}/D_{K} ⟶^{≃} Gal(K^{ab}/K)

intertwines the actions,

α ∘ φ = φ ∘ θ^{−1}(α),  (1.35)

for all α ∈ Gal(K^{ab}/K) and for all φ ∈ E_{∞}.

Notice that, with this formulation, the problem of the construction of fabulous states is intimately related to Hilbert’s 12th problem. This question will be pursued in [13]. We shall construct here a system which is the analog of the BC system for GL_{2}(ℚ) instead of GL_{1}(ℚ). This will extend the results of [5] to this nonabelian GL_{2} case and will exhibit many new features which have no counterpart in the abelian case. Our construction involves the explicit description of the automorphism group of the modular field, [51]. The construction of fabulous states for imaginary quadratic fields, which will be investigated with N. Ramachandran in [13], involves specializing the GL_{2} system to a subsystem compatible with complex multiplication in a given imaginary quadratic field.

The construction of the GL_{2} system gives a C^{∗}-dynamical system (A, σ_{t}) and an involutive subalgebra A_{ℚ} defined over ℚ, satisfying the following properties:

• The quotient group S := ℚ^{∗}\GL_{2}(𝔸_{f}) of the finite adèlic group of GL_{2} acts as symmetries of the dynamical system (A, σ_{t}) preserving the subalgebra A_{ℚ}.

• For generic φ ∈ E_{∞}, the values {φ(a) ∈ ℂ : a ∈ A_{ℚ}} generate a subfield F_{φ} ⊂ ℂ which is an extension of ℚ of transcendence degree 1.

• For generic φ ∈ E_{∞}, there exists an isomorphism

θ : S ⟶^{≃} Gal(F_{φ}/ℚ)

which intertwines the actions

α ∘ φ = φ ∘ θ^{−1}(α),  (1.36)

for all α ∈ Gal(F_{φ}/ℚ).

There are a number of important nuances between the abelian case above and the non-abelian one. For instance, in the abelian case the field generated by φ(A) does not depend on the choice of φ ∈ E_{∞} and the isomorphism θ is also independent of φ. This no longer holds in the non-abelian case, as is clear from the presence of inner automorphisms of the symmetry group S. Also, in the latter case, the action of S on E_{∞} is not transitive and the corresponding invariant of the orbit of φ under S is the subfield F_{φ} ⊂ ℂ. Another important nuance is that the algebra A is no longer unital while A_{ℚ} is an algebra of unbounded multipliers of A. Finally, the symmetries require the full framework of endomorphisms as explained above in §1.2.

1.6 The subalgebra A_{ℚ} and Eisenstein Series

In this section we shall recast the BC algebra in terms of the trigonometric analog of the Eisenstein series, following the analogy developed by Eisenstein and Kronecker between trigonometric and elliptic functions, as outlined by A.Weil in [56]. This will be done by first giving a geometric interpretation in terms of ℚ-lattices of the noncommutative space X whose algebra of continuous functions C(X) is the BC C^{∗}-algebra. The space X is by construction the quotient of the Pontrjagin dual of the abelian group ℚ/ℤ by the equivalence relation generated by the action by multiplication of the semi-group ℕ^{×}. Let

R = ∏_{p} ℤ_{p}

be the compact ring product of the rings ℤ_{p} of p-adic integers. It is the maximal compact subring of the locally compact ring of finite adèles

𝔸_{f} = ∏_{res} ℚ_{p}

We recall the following standard fact

Proposition 1.8 • The inclusion ℚ ⊂ 𝔸_{f} gives an isomorphism of abelian

groups

ℚ/ℤ = 𝔸_{f}/R .

• The following map is an isomorphism of compact rings

j : R → Hom(ℚ/ℤ, ℚ/ℤ) , j(a)(x) = a x , ∀x ∈ 𝔸_{f}/R , ∀a ∈ R .

We shall use j from now on to identify R with Hom(ℚ/ℤ, ℚ/ℤ). Note that by construction Hom(ℚ/ℤ, ℚ/ℤ) is endowed with the topology of pointwise convergence. It is identified with \underleftarrow{lim} ℤ/Nℤ using the restriction to N-torsion elements. For every r ∈ ℚ/ℤ one gets a function e(r) ∈ C(R) by,

e(r)(ρ) := exp2πiρ(r) ∀ρ ∈ Hom(ℚ/ℤ, ℚ/ℤ)

and this gives the identification of R with the Pontrjagin dual of ℚ/ℤ and of C(R) with the group C^{∗}-algebra C^{∗}(ℚ/ℤ).

One can then describe the BC C^{∗}-algebra as the cross product of C(R) by the semigroup action of ℕ^{×} as follows. For each integer n ∈ ℕ^{×} we let nR ⊂ R be the range of multiplication by n. It is an open and closed subset of R whose characteristic function π_{n} is a projection π_{n} ∈ C(R). One has by construction

π_{n} π_{m} = π_{n∨m} , ∀n , m ∈ ℕ^{×}

where n ∨ m denotes the lowest common multiple of n and m. The semigroup action of ℕ^{×} on C(R) corresponds to the isomorphism

α_{n}(f)(ρ) := f(n^{−1} ρ) , ∀ρ ∈ nR .  (1.37)

of C(R) with the reduced algebra C(R)_{π_{n}} of C(R) by the projection π_{n}. In the BC algebra one has

μ_{n} f μ_{n}^{∗} = α_{n}(f) , ∀f ∈ C(R).  (1.38)

There is an equivalent description of the BC algebra in terms of the étale groupoid G of pairs (r, ρ), where r ∈ ℚ_{+}^{∗}, ρ ∈ R and r ρ ∈ R. The composition in G is given by

(r_{1}, ρ_{1}) ∘ (r_{2}, ρ_{2}) = (r_{1} r_{2}, ρ_{2}) , if r_{2} ρ_{2} = ρ_{1} ,  (1.39)

and the convolution of functions by

f_{1} ∗ f_{2}(r, ρ) := ∑ f_{1}(rs^{−1}, s ρ) f_{2}(s, ρ) ,  (1.40)

while the adjoint of f is

f^{∗}(r, ρ) := \overline{f(r^{−1}, r ρ)} .  (1.41)

All of this is implicit in ([5]) and has been amply described in the subsequent papers mentioned in §1.4. In the description above, μ_{n} is given by the function μ_{n}(r, ρ) which vanishes unless r = n and is equal to 1 for r = n. The time evolution is given by

σ_{t}(f)(r, ρ) := r^{it} f(r, ρ) , ∀f ∈ C^{∗}(G) .  (1.42)

We shall now describe a geometric interpretation of this groupoid G in terms of commensurability of ℚ-lattices. In particular, it will pave the way to the generalization of the BC system to higher dimensions. The basic simple geometric objects are ℚ-lattices in ℝ^{n}, defined as follows.

Definition 1.9 A ℚ-lattice in ℝ^{n} is a pair (Λ, ϕ) , with Λ a lattice in ℝ^{n}, and ϕ : ℚ^{n}/ℤ^{n} ⟶ ℚΛ/Λ an homomorphism of abelian groups.

Two lattices Λ_{j} in ℝ^{n} are commensurable iff their intersection Λ_{1} ∩ Λ_{2} is of finite index in Λ_{j}. Their sum Λ = Λ_{1} + Λ_{2} is then a lattice and, given two homomorphisms of abelian groups ϕ_{j} : ℚ^{n}/ℤ^{n} ⟶ ℚ Λ_{j}/Λ_{j}, the difference ϕ_{1} − ϕ_{2} is well defined modulo Λ = Λ_{1} + Λ_{2}. Notice that in Definition 1.9 the homomorphism ϕ, in general, is not an isomorphism.

Definition 1.10 A ℚ-lattice (Λ, ϕ) is invertible if the map ϕ is an isomorphism of abelian groups.

We consider a natural equivalence relation on the set of ℚ-lattices defined as follows.

Proposition 1.11 The following defines an equivalence relation called commensurability between ℚ-lattices: (Λ_{1}, ϕ_{1}) , (Λ_{2}, ϕ_{2}) are commensurable iff Λ_{j} are commensurable and ϕ_{1} − ϕ_{2} = 0 modulo Λ = Λ_{1} + Λ_{2}.

Proof. Indeed, let (Λ_{j}, ϕ_{j}) be three ℚ-lattices and assume commensurability between the pairs (1, 2) and (2, 3). Then the lattices Λ_{j} are commensurable and are of finite index in Λ = Λ_{1} + Λ_{2} + Λ_{3}. One has ϕ_{1} − ϕ_{2} = 0 modulo Λ, ϕ_{2} − ϕ_{3} = 0 modulo Λ and thus ϕ_{1} − ϕ_{3} = 0 modulo Λ. But Λ′ = Λ_{1} + Λ_{3} is of finite index in Λ and thus ϕ_{1} − ϕ_{3} gives a group homomorphism

ℚ^{n}/ℤ^{n} ⟶ Λ/Λ′

which is zero since ℚ^{n}/ℤ^{n} is infinitely divisible and Λ/Λ′ is finite. This shows that ϕ_{1} − ϕ_{3} = 0 modulo Λ′ = Λ_{1} + Λ_{3} and hence that the pair (1, 3) is commensurable. □ Notice that every ℚ-lattice in ℝ is uniquely of the form

(Λ, ϕ) = (λ ℤ, λ ρ) , λ > 0,  (1.43)

with ρ ∈ Hom(ℚ/ℤ, ℚ/ℤ) = R.

Proposition 1.12 The map

γ(r, ρ) = ((r^{−1} ℤ, ρ) , (ℤ, ρ)) , ∀(r, ρ) ∈ G ,

defines an isomorphism of locally compact étale groupoids between G and the quotient R / ℝ_{+}^{∗} of the equivalence relation R of commensurability on the space of ℚ-lattices in ℝ by the natural scaling action of ℝ_{+}^{∗}.

Proof. First since r ρ ∈ R the pair (r^{−1} ℤ, ρ) = r^{−1}(ℤ, r ρ) is a ℚ-lattice and is commensurable to (ℤ, ρ). Thus, the map γ is well defined. Using the identification (1.43), we see that the restriction of γ to the objects G^{(0)} of G is an isomorphism of R with the quotient of the space of ℚ-lattices in ℝ by the natural scaling action of ℝ_{+}^{∗}. The freeness of this action shows that the quotient R / ℝ_{+}^{∗} is still a groupoid, and one has

γ(r_{1}, ρ_{1}) ∘ γ(r_{2}, ρ_{2}) = γ(r_{1} r_{2}, ρ_{2}) if r_{2} ρ_{2} = ρ_{1} .

Finally, up to scaling, every element of R is of the form

((r^{−1} ℤ, r^{−1} ρ′) , (ℤ, ρ))

where both ρ′ and ρ are in R and r = \frac{a}{b} ∈ ℚ_{+}^{∗}. Moreover since r^{−1} ρ′ = ρ modulo \frac{1}{a}ℤ one gets aρ − bρ′ = 0 and r^{−1} ρ′ = ρ. Thus γ is surjective and is an isomorphism. □

This geometric description of the BC algebra allows us to generate in a natural manner a rational subalgebra which will generalize to the two dimensional case. In particular the algebra C(R) can be viewed as the algebra of homogeneous functions of “weight 0” on the space of ℚ-lattices for the natural scaling action of the multiplicative group ℝ_{+}^{∗} where weight k means

f(λ Λ, λ ϕ) = λ^{−k} f(Λ, ϕ) , ∀λ ∈ ℝ_{+}^{∗} .

We let the function c(Λ) be the multiple of the covolume |Λ| of the lattice, specified by

2 π i c(ℤ) = 1  (1.44)

The function c is homogeneous of weight −1 on the space of ℚ-lattices. For a ∈ ℚ/ℤ, we then define a function e_{1,a} of weight 0 by

e_{1,a}(Λ, ϕ) = c(Λ) ∑_{y∈Λ+ϕ(a)} y^{−1} ,  (1.45)

where one uses Eisenstein summation i.e. lim_{N→∞} ∑_{−N}^{N} when ϕ(a) ≠ 0 and one lets e_{1,a}(Λ, ϕ) = 0 when ϕ(a) = 0. The main result of this section is the following

Theorem 1.13 • The e_{1,a} , a ∈ ℚ/ℤ generate ℚ[ℚ/ℤ].

• The rational algebra A_{ℚ} is the subalgebra of A = C^{∗}(G) generated by the e_{1,a} , a ∈ ℚ/ℤ and the μ_{n}, μ_{n}^{∗}.

We define more generally for each weight k ∈ ℕ and each a ∈ ℚ/ℤ a function ϵ_{k,a} on the space of ℚ-lattices in ℝ by

ϵ_{k,a}(Λ, ϕ) = ∑_{y∈Λ+ϕ(a)} y^{−k} .  (1.46)

This is well defined provided ϕ(a) ≠ 0. For ϕ(a) = 0 we let

ϵ_{k,a}(Λ, ϕ) = λ_{k} c(Λ)^{−k},  (1.47)

where we shall fix the constants λ_{k} below in (1.50). The function ϵ_{k,a} has weight k i.e. it satisfies the homogeneity condition

ϵ_{k,a}(λ Λ, λ ϕ) = λ^{−k}ϵ_{k,a}(Λ, ϕ) , ∀λ ∈ ℝ_{+}^{∗} .

When a = \frac{b}{N} the function ϵ_{k,a} has level N in that it only uses the restriction ϕ_{N} of ϕ to N-torsion points of ℚ/ℤ,

ϕ_{N} : \frac{1}{N} ℤ/ℤ ⟶ \frac{1}{N} Λ/Λ.

The products

e_{k,a} := c^{k} ϵ_{k,a}  (1.48)

are of weight 0 and satisfy two types of relations. The first relations are multiplicative and express e_{k,a} as a polynomial in e_{1,a},

e_{k,a} = P_{k}(e_{1,a})  (1.49)

where the P_{k} are the polynomials with rational coefficients uniquely determined by the equalities

P_{1}(u) = u , P_{k+1}(u) = \frac{1}{k} (u^{2} − \frac{1}{4}) ∂_{u}P_{k}(u) .

This follows for ϕ(a) ∉ Λ from the elementary formulas for the trigonometric analog of the Eisenstein series ([56] Chapter II). Since e_{1,a}(Λ, ϕ) = 0 is the natural choice for ϕ(a) ∈ Λ, the validity of (1.49) uniquely dictates the choice of the normalization constants λ_{k} of (1.47). One gets

λ_{k} = P_{k}(0) = (2^{k} − 1) γ_{k} ,  (1.50)

where γ_{k} = 0 for odd k and γ_{2j} = (−1)^{j} \frac{B_{j}}{(2j)!} with B_{j} ∈ ℚ the Bernoulli

numbers. Equivalently,

\frac{x}{e^{x} − 1} = 1 − \frac{x}{2} − ∑_{1}^{∞} γ_{2j} x^{2j}.

One can express the e_{k,a} as ℚ-linear combinations of the generators e(r). We view e(r) as the function on ℚ-lattices which assigns to (Λ, ϕ) = (λ ℤ, λ ρ), λ > 0, the value

e(r)(Λ, ϕ) := exp2πiρ(r).

One then has

Lemma 1.14 Let a ∈ ℚ/ℤ, and n > 0 with n a = 0. Then

e_{1,a} = ∑_{k=1}^{n−1}(\frac{k}{n} − \frac{1}{2}) e(k a).  (1.51)

Proof. We evaluate both sides on (Λ, ϕ) = (λ ℤ, λ ρ), λ > 0. Both sides only depend on the restriction x ↦ c x of ρ to n-torsion elements of ℚ/ℤ which we write as multiplication by c ∈ ℤ/nℤ. Let a = \frac{b}{n}. If bc = 0(n) then ϕ(a) = 0 and both sides vanish since e(k a)(Λ, ϕ) = exp2πi(\frac{kbc}{n}) = 1 for all k. If bc ≠ 0(n) then ϕ(a) ≠ 0 and the left side is \frac{1}{2}(U +1)/(U −1) where U = exp2πi\frac{bc}{n}, U^{n} = 1, U ≠ 1. The right hand side is

∑_{k=1}^{n−1}(\frac{k}{n} − \frac{1}{2}) U^{k} ,

which gives \frac{1}{2}(U + 1) after multiplication by U − 1. □

This last equality shows that e_{1,a} is (one half of) the Cayley transform of e(a) with care taken where e(a) − 1 fails to be invertible. In particular while e(a) is unitary, e_{1,a} is skew-adjoint,

e_{1,a}^{∗} = − e_{1,a} .

We say that a ℚ-lattice (Λ, ϕ) is divisible by an integer n ∈ ℕ when ϕ_{n} = 0. We let π_{n} be the characteristic function of the set of ℚ-lattices divisible by n. It corresponds to the characteristic function of n R ⊂ R. Let N > 0 and (Λ, ϕ) a ℚ-lattice with ϕ_{N}(a) = c a for c ∈ ℤ/Nℤ. The order of the kernel of ϕ_{N} is m = gcd(N, c). Also a divisor b|N divides (Λ, ϕ) iff it divides c. Thus for any function f on ℕ^{∗} one has

∑_{b|N} f(b) π_{b}(Λ, ϕ) = ∑_{b|gcd(N,c)} f(b) ,

which allows one to express any function of the order m = gcd(N, c) of the kernel of ϕ_{N} in terms of the projections π_{b} , b|N. In order to obtain the function m ↦ m^{j} we let

f_{j}(n) := ∑_{d|n} μ(d)(n/d)^{j} ,

where μ is the Möbius function so that

f_{j}(n) = n^{j} ∏_{p prime, p|n} (1 − p^{−j}) .

Notice that f_{1} is the Euler totient function and that the ratio f_{−β+1}/f_{1} gives the r.h.s. of (1.33) in Theorem 1.7. The Möbius inversion formula gives

∑_{b|N} f_{j}(b) π_{b}(Λ, ϕ) = m^{j} , m = gcd(N, c) .  (1.52)

We can now write division relations fulfilled by the functions (1.48).

Lemma 1.15 Let N > 0 then

∑_{N a=0} e_{k,a} = γ_{k} ∑_{d|N}((2^{k} − 2) f_{1}(d) + N^{k}f_{−k+1}(d)) π_{d}.  (1.53)

Proof. For a given ℚ-lattice (Λ, ϕ) with Ker ϕ_{N} of order m|N, N = m d, the result follows from

∑_{N a=0} ϵ_{k,a}(Λ, ϕ) = m ∑_{y∈ \frac{1}{d} Λ\Λ} y^{−k} +m (2^{k} −1) γ_{k} c^{−k}(Λ) = m (d^{k} +2^{k} −2)γ_{k} c^{−k}

together with (1.52) applied for j = 1 and j = 1 − k. □ The semigroup action of ℕ^{×} is given on functions of ℚ-lattices by the endomorphisms

α_{n}(f)(Λ, ϕ) := f(n Λ, ϕ) , ∀(Λ, ϕ) ∈ π_{n} ,  (1.54)

while α_{n}(f)(Λ, ϕ) = 0 outside π_{n}. This semigroup action preserves the rational subalgebra B_{ℚ} generated by the e_{1,a} , a ∈ ℚ/ℤ, since one has

α_{n}(e_{k,a}) = π_{n} e_{k,a/n} ,  (1.55)

(independently of the choice of the solution b = a/n of n b = a) and we shall now show that the projections π_{n} belong to B_{ℚ}.

Proof of Theorem 1.13 Using (1.53) one can express π_{n} as a rational linear combination of the e_{k,a}, with k even, but special care is needed when n is a power of two. The coefficient of γ_{k} π_{N} in (1.53), when N = p^{b} is a prime power, is given by (2^{k} −2)(p−1)p^{b−1} − p^{b}(p^{k−1} − 1), which does not vanish unless p = 2, and is −p^{b−1}(2 − 3p + p^{2}) for k = 2. Thus, one can express π_{N} as a linear combination of the e_{2,a} by induction on b. For p = 2, N = 2^{b}, b > 1 the coefficient of γ_{k} π_{N} in (1.53) is zero but the coefficient of γ_{k} π_{N/2} is −2^{b−2}(2^{k} − 1)(2^{k} − 2) ≠ 0 for k even. This allows one to express π_{N} as a linear combination of the e_{2,a} by induction on b. Thus, for instance, π_{2} is given by

π_{2} = 3 + 2 ∑_{4 a=0} e_{2,a}.

In general, π_{2^{n}} involves ∑_{2^{n+1} a=0} e_{2,a}.

Since for relatively prime integers n, m one has π_{nm} = π_{n} π_{m}, we see that the algebra B_{ℚ} generated over ℚ by the e_{1,a} contains all the projections π_{n}. In order to show that B_{ℚ} contains the e(r) it is enough to show that for any prime power N = p^{b} it contains e(\frac{1}{N}). This is proved by induction on b. Multiplying (1.53) by 1 − π_{p} and using (1 − π_{p}) π_{p^{l}} = 0 for l > 0 we get the equalities

(1 − π_{p}) ∑_{N a=0} e_{k,a} = (N^{k} + 2^{k} − 2) γ_{k} (1 − π_{p}) .

Let then z(j) = (1 − π_{p}) e_{1, \frac{j}{N}}. The above relations together with(1.49) show that in the reduced algebra (B_{ℚ})_{1−π_{p}} one has, for all k,

∑_{j=1}^{N−1} P_{k}(z(j)) = (N^{k} − 1) γ_{k} .

Thus, for j ∈ {1, ..., N −1}, the symmetric functions of the z(j) are fixed rational numbers σ_{h}. In particular z = z(1) fulfills

Q(z) = z^{N−1} + ∑_{1}^{N−1}(−1)^{h} σ_{h} z^{N−1−h} = 0

and ± \frac{1}{2} is not a root of this equation, whose roots are the \frac{1}{2i} cot(\frac{π j}{N}). This allows us, using the companion matrix of Q, to express the Cayley transform of 2z as a polynomial with rational coefficients,

\frac{2z + 1}{2z − 1} = ∑_{0}^{N−2} α_{n} z^{n}.

One then has

∑_{0}^{N−2} α_{n} z^{n} = (1 − π_{p}) e(\frac{1}{N}),

where the left-hand side belongs to B_{ℚ} by construction. Now π_{p} e(\frac{1}{N}) is equal to α_{p}(e(\frac{p}{N})). It follows from the induction hypothesis on b, (N = p^{b}), that e(\frac{p}{N}) ∈ B_{ℚ} and therefore using (1.55) that α_{p}(e(\frac{p}{N})) ∈ B_{ℚ}. Thus, we get e(\frac{1}{N}) ∈ B_{ℚ} as required. This proves the first part. To get the second notice that the cross product by ℕ^{×} is obtained by adjoining to the rational group ring of ℚ/ℤ the isometries μ_{n} and their adjoints μ_{n}^{∗} with the relation

μ_{n} f μ_{n}^{∗} = α_{n}(f) , ∀f ∈ ℚ[ℚ/ℤ] ,

which gives the rational algebra A_{ℚ}. □

It is not true, however, that the division relations (1.53) combined with the multiplicative relations (1.49) suffice to present the algebra. In particular there are more elaborate division relations which we did not need in the above proof. In order to formulate them, we let for d|N, π(N, d) be the projection belonging to the algebra generated by the π_{b}, b|N, and corresponding to the subset

gcd(N, (Λ, ϕ)) = N/d

so that

π(N, d) = π_{N/d} ∏_{k|d}(1 − π_{k N/d}),

where the product is over non trivial divisors k ≠ 1 of d.

Proposition 1.16 The e_{k,a} , a ∈ ℚ/ℤ , k odd, fulfill for any x ∈ ℚ/ℤ and any integer N the relation

\frac{1}{N} ∑_{N a=0} e_{k,x+a} = ∑_{d|N} π(N, d) d^{k−1} e_{k,d x}.

Proof. To prove this, let (Λ, ϕ) be such that gcd(N, (Λ, ϕ)) = N/d = m and assume by homogeneity that Λ = ℤ. Then when a ranges through the \frac{j}{N} , j ∈ \{0, ..., N − 1\}, the ϕ(a) range m-times through the \frac{j}{d}, j ∈ \{0, ..., d − 1\}. Thus the left-hand side of (1.16) gives m-times

c(ℤ)^{k} ∑_{j=0}^{d−1} ∑_{y∈ℤ+ϕ(x)+ \frac{j}{d}} y^{−k} = c(ℤ)^{k} d^{k} ∑_{y∈ℤ+ϕ(d x)} y^{−k}.

This is clear when y = 0 does not appear in the sums involved. When it does one has, for ϵ ∉ \frac{ℤ}{d},

∑_{j=0}^{d−1} ∑_{y∈ℤ+ϕ(x)+ \frac{j}{d}}(y + ϵ)^{−k} = d^{k} ∑_{y∈ℤ+ϕ(d x)}(y + d ϵ)^{−k}.

Subtracting the pole part on both sides and equating the finite values gives the desired equality, since for odd k the value of ϵ_{k,a}(Λ, ϕ) for ϕ(a) = 0 can be written as the finite value of

∑_{y∈Λ+ϕ(a)}(y + ϵ)^{−k}.

For even k this no longer holds and the finite value γ_{k}c(Λ)^{k} is replaced by (2^{k} − 1) γ_{k}c(Λ)^{k}. Thus when ϕ(d x) ∈ ℤ one gets an additional term which is best taken care of by multiplying the right hand side in Proposition 1.16 by (1 − π_{δ(d x)}), with δ(y) the order of y in ℚ/ℤ, and adding corresponding terms to the formula, which becomes

\frac{1}{N} ∑_{N a=0} e_{k,x+a} = ∑_{d|N} π(N, d)(1 − π_{δ(d x)}) d^{k−1} e_{k,d x}  (1.56)
+ γ_{k} ∑_{d|N} (d^{k−1} + d^{−1}(2^{k} − 2)) π(N, d) π_{δ(d x)}

These relations are more elaborate than the division relations for trigonometric functions. They restrict to the latter on the subset of invertible ℚ-lattices, for which all π_{n}, n ≠ 1 are zero and the only non-zero term in the r.h.s. is the term in d = N. The standard discussion of Eisenstein series in higher dimension is restricted to invertible ℚ-lattices, but in our case the construction of the endomorphisms implemented by the μ_{n} requires the above extension to non-invertible ℚ-lattices. We shall now proceed to do it in dimension 2.

1.7 The Determinant part of the GL_{2}-System

As we recalled in the previous sections, the algebra of the 1-dimensional system can be described as the semigroup cross product

C(R) ⋊ ℕ^{×}.

Thus, one may wish to follow a similar approach for the 2-dimensional case, by replacing C(R) by C(M_{2}(R)) and the semigroup action of ℕ^{×} by the semigroup action of M_{2}(ℤ)^{+}. Such construction can be carried out, as we discuss in this section, and it corresponds to the “determinant part” of the GL_{2} system. It is useful to analyze what happens in this case first, before we discuss the full GL_{2}-system in the next section. In fact, this will show quite clearly where some important technical issues arise. For instance, just as in the case of the BC algebra, where the time evolution acts on the isometries μ_{n} by n^{it} and leaves the elements of C(R) fixed, the time evolution here is given by Det (m)^{it} on the isometries implementing the semigroup action of m ∈ M_{2}(ℤ)^{+}, while leaving C(M_{2}(R)) pointwise fixed. In this case, however, the vacuum state of the corresponding Hamiltonian is highly degenerate, because of the presence of the SL_{2}(ℤ) symmetry. This implies that the partition function and the KMS states below critical temperature can only be defined via the type II_{1} trace Trace_{Γ}. This issue will be taken care more naturally in the full GL_{2}-system, by first taking the classical quotient by Γ = SL_{2}(ℤ) on the space M_{2}(R) × ℍ. This will resolve the degeneracy of the vacuum state and the counting of modes of the Hamiltonian will be on the coset classes Γ\M_{2}(ℤ)^{+}. The whole discussion of this section extends to GL(n) for arbitrary n and we shall briefly indicate how this is done, but we stick to n = 2 for definiteness. We start with the action of the semigroup

M_{2}(ℤ)^{+} = {m ∈ M_{2}(ℤ), Det (m) > 0} = GL_{2}^{+}(ℚ) ∩ M_{2}(R)  (1.57)

on the compact space M_{2}(R), given by left multiplication

ρ ↦ m ρ,  (1.58)

where the product mρ takes place in M_{2}(R) using the natural homomorphism

M_{2}(ℤ)^{+} → M_{2}(R),  (1.59)

which is the extension to two by two matrices of the inclusion of the ring ℤ in ℤ̂ = R. The relevant C^{∗}-algebra is the semi-group cross product

A = C(M_{2}(R)) ⋊ M_{2}(ℤ)^{+}.  (1.60)

It can be viewed as the C^{∗}-algebra C^{∗}(G_{2}) of the étale groupoid G_{2} of pairs (r, ρ), with r ∈ GL_{2}^{+}(ℚ), ρ ∈ M_{2}(R) and r ρ ∈ M_{2}(R), where the product takes place in M_{2}(𝔸_{f}). The composition in G_{2} is given by

(r_{1}, ρ_{1}) ∘ (r_{2}, ρ_{2}) = (r_{1} r_{2}, ρ_{2}) , if r_{2} ρ_{2} = ρ_{1}

and the convolution of functions by

f_{1} ∗ f_{2}(r, ρ) := ∑ f_{1}(rs^{−1}, s ρ) f_{2}(s, ρ) ,  (1.61)

while the adjoint of f is

f^{∗}(r, ρ) := \overline{f(r^{−1}, r ρ)}  (1.62)

(cf. the analogous expressions (1.39), (1.40), (1.41) in the 1-dimensional case). A homomorphism G_{2} → H of the groupoid G_{2} to an abelian group H determines a dual action of the Pontrjagin dual of H on the algebra of G_{2}, as in the case of the time evolution σ_{t}, with H = ℝ_{+}^{∗} and its dual identified with ℝ. We shall use the same term “dual action” for nonabelian H. The main structure is given by the dual action of GL_{2}^{+}(ℝ) corresponding to the groupoid homomorphism j

j : G_{2} → GL_{2}^{+}(ℝ) , j(r, ρ) = r  (1.63)

obtained from the inclusion GL_{2}^{+}(ℚ) ⊂ GL_{2}^{+}(ℝ). As a derived piece of structure one gets the one parameter group of automorphisms σ_{t} ∈ Aut(A) which is dual to the determinant of the homomorphism j,

σ_{t}(f)(r, ρ) := Det(r)^{it} f(r, ρ) , ∀f ∈ A .  (1.64)

The obtained C^{∗}-dynamical system (A, σ_{t}) only involves Det ∘j and it does not fully correspond to the BC system. We shall make use of the full dual action of GL_{2}^{+}(ℝ) later in the construction of the full GL_{2} system. The algebra C(M_{2}(R)) embeds as a ∗-subalgebra of A. The analogs of the isometries μ_{n}, n ∈ ℕ^{×} are the isometries μ_{m}, m ∈ M_{2}(ℤ)^{+} given by

μ_{m}(m, ρ) = 1 , μ_{m}(r, ρ) = 0 , ∀r ≠ m .

The range μ_{m} μ_{m}^{∗} of μ_{m} is the projection given by the characteristic function of the subset P_{m} = m M_{2}(R) ⊂ M_{2}(R). It depends only on the lattice L = m(ℤ^{2}) ⊂ ℤ^{2}. Indeed, if m, m′ ∈ M_{2}(ℤ)^{+} fulfill m(ℤ^{2}) = m′(ℤ^{2}), then m′ = mγ for some γ ∈ Γ, hence m M_{2}(R) = m′ M_{2}(R). Thus, we shall label this analog of the π_{n} by lattices

L ⊂ ℤ^{2} ↦ π_{L} ∈ C(M_{2}(R) ,  (1.65)

where π_{L} is the characteristic function of P_{m}, for any m such that m(ℤ^{2}) = L. The algebra generated by the π_{L} is then governed by

π_{L} π_{L′} = π_{L∩L′} ,  π_{ℤ^{2}} = 1 .  (1.66)

In fact, the complete rules are better expressed in terms of partial isometries μ(g, L), with g ∈ GL_{2}^{+}(ℚ), L ⊂ ℤ^{2} a lattice, and g(L) ⊂ ℤ^{2}, satisfying

μ(g, L)(g, ρ) = π_{L}(ρ) , μ(g, L)(r, ρ) = 0 , ∀r ≠ g .

One has

μ(g_{1}, L_{1}) μ(g_{2}, L_{2}) = μ(g_{1}g_{2}, g_{2}^{−1}(L_{1}) ∩ L_{2}) ,  (1.67)

and

μ(g, L)^{∗} = μ(g^{−1}, g(L)) .  (1.68)

The μ(g, L) generate the semi-group C^{∗}-subalgebra C^{∗}(M_{2}(ℤ)^{+}) ⊂ A and together with C(M_{2}(R)) they generate A. The additional relations are

f μ(g, L) = μ(g, L) f^{g} , ∀f ∈ C(M_{2}(R)) , g ∈ GL_{2}^{+}(ℚ),  (1.69)

where f^{g}(y) := f(gy) whenever gy makes sense. The action of GL_{2}(R) on M_{2}(R) by right multiplication commutes with the semi-group action (1.58) of M_{2}(ℤ)^{+} and with the time evolution σ_{t}. They define symmetries

α_{θ} ∈ Aut(A, σ) .

Thus, we have a C^{∗}-dynamical system with a compact group of symmetries. The following results show how to construct KMS_{β}-states for β > 2. We first describe a specific positive energy representation of the C^{∗}-dynamical subsystem (C^{∗}(M_{2}(ℤ)^{+}), σ_{t}). We let H = ℓ^{2}(M_{2}(ℤ)^{+}) with canonical basis ε_{m}, m ∈ M_{2}(ℤ)^{+}. We define π(μ(g, L)) as the partial isometry in H with initial domain given by the span of

ε_{m} , m = [\begin{matrix}m_{11} & m_{12} \\ m_{21} & m_{22} \end{matrix}] , (m_{11}, m_{21}) ∈ L , (m_{12}, m_{22}) ∈ L ,  (1.70)

i.e. matrices m whose columns belong to the lattice L ⊂ ℤ^{2}. On this domain we define the action of π(μ(g, L)) by

π(μ(g, L)) ε_{m} = ε_{gm} .  (1.71)

Notice that the columns of gm belong to gL.

Proposition 1.17 1) π is an involutive representation of C^{∗}(M_{2}(ℤ)^{+}) in H.

2) The Hamiltonian H given by Hε_{m} = log Det(m) ε_{m} is positive and implements the time evolution σ_{t}:

π(σ_{t}(x)) = e^{itH} π(x) e^{−itH} ∀ x ∈ C^{∗}(M_{2}(ℤ)^{+}) .

3) Γ = SL_{2}(ℤ) acts on the right in H by

ρ(γ) ϵ_{m} := ϵ_{m γ^{−1}} , ∀γ ∈ Γ , m ∈ M_{2}(ℤ)^{+} .

and this action commutes with π(C^{∗}(M_{2}(ℤ)^{+}).

Proof. The map m ↦ gm is injective so that π(μ(g, L)) is a partial isometry. Its range is the set of h ∈ M_{2}(ℤ)^{+} of the form g m where Det(m) > 0 and the columns of m are in L. This means that Det(h) > 0 and the columns of h are in gL ⊂ ℤ^{2}. This shows that

π(μ(g, L))^{∗} = π(μ(g^{−1}, gL)),  (1.72)

so that π is involutive on these elements. Then the support of π(μ(g_{1}, L_{1})) π(μ(g_{2}, L_{2})) is formed by the ε_{m} with columns of m in L_{2}, such that the columns of g_{2}m are in L_{1}. This is the same as the support of π(μ(g_{1}g_{2}, g_{2}^{−1}L_{1} ∩ L_{2})) and the two partial isometries agree there. Thus, we get

π(μ(g_{1}, L_{1}) μ(g_{2}, L_{2})) = π(μ(g_{1}, L_{1})) π(μ(g_{2}, L_{2})) .  (1.73)

Next, using (1.71) we see that

H π(μ(g, L)) − π(μ(g, L)) H = log(Det g) π(μ(g, L)),  (1.74)

since both sides vanish on the kernel while on the support one can use the multiplicativity of Det. Now Γ = SL_{2}(ℤ) acts on the right in H by

ρ(γ) ϵ_{m} := ϵ_{m γ^{−1}} , ∀γ ∈ Γ , m ∈ M_{2}(ℤ)^{+}  (1.75)

and this action commutes by construction with the algebra π(C^{∗}(M_{2}(ℤ)^{+}). □ The image ρ(C^{∗}(Γ)) generates a type II_{1} factor in H, hence one can evaluate the corresponding trace Trace_{Γ} on any element of its commutant. We let

φ_{β}(x) := Trace_{Γ}(π(x) e^{−βH}) , ∀x ∈ C^{∗}(M_{2}(ℤ)^{+})  (1.76)

and we define the normalization factor by

Z(β) = Trace_{Γ}(e^{−βH}) .  (1.77)

We then have the following:

Lemma 1.18 1) The normalization factor Z(β) is given by

Z(β) = ζ(β) ζ(β − 1),

where ζ is the Riemann ζ-function.

2) For all β > 2, Z^{−1} φ_{β} is a KMS_{β} state on C^{∗}(M_{2}(ℤ)^{+}).

Proof. Any sublattice L ⊂ ℤ^{2} is uniquely of the form L = [\begin{matrix}a & b \\ 0 & d \end{matrix}] ℤ^{2}, where

a, d ≥ 1, 0 ≤ b < d (cf. [50] p. 161). Thus, the type II_{1} dimension of the action of Γ in the subspace of H spanned by the ε_{m} with Det m = N is the same as the cardinality of the quotient of {m ∈ M_{2}(ℤ)^{+}, Det m = N} by Γ acting on

the right. This is equal to the cardinality of the set of matrices [\begin{matrix}a & b \\ 0 & d \end{matrix}] as above

with determinant = N. This gives σ_{1}(N) = ∑_{d|N} d. Thus, Z(β) is given by

∑_{N=1}^{∞} \frac{σ_{1}(N)}{N^{β}} = ζ(β) ζ(β − 1) .  (1.78)

One checks the KMS_{β}-property of φ_{β} using the trace property of Trace_{Γ} together with the second equality in Proposition 1.17. □

Proposition 1.19 1) For any θ ∈ GL_{2}(R) the formula

π_{θ}(f) ϵ_{m} := f(m θ) ϵ_{m} , ∀m ∈ M_{2}(ℤ)^{+}

extends the representation π to an involutive representation π_{θ} of the cross product A = C(M_{2}(R)) ⋊ M_{2}(ℤ)^{+} in H.

2) Let f ∈ C(M_{2}(ℤ/Nℤ)) ⊂ C(M_{2}(R)). Then π_{θ}(f) ∈ ρ(Γ_{N})′ where Γ_{N} is the congruence subgroup of level N.

3) For each β > 2 the formula

ψ_{β}(x) := Lim_{N→∞} Z_{N}^{−1} Trace_{Γ_{N}} (π_{θ}(x) e^{−βH}) , ∀x ∈ A

defines a KMS_{β} state on A, where Z_{N} := Trace_{Γ_{N}} (e^{−βH}).

Proof. 1) The invertibility of θ shows that letting f_{L} be the characteristic function of P_{L} one has π_{θ}(f_{L}) = π_{L} independently of θ. Indeed f_{L}(m θ) = 1 iff m θ ∈ P_{L} and this holds iff m(ℤ^{2}) ⊂ L. To check (1.69) one uses

f(gm θ) = f^{g}(m θ)) , ∀g ∈ GL_{2}^{+}(ℚ) .

2) Let p_{N} : M_{2}(R) → M_{2}(ℤ/Nℤ) be the canonical projection. It is a ring homomorphism. Let then f = h ∘ p_{N} where h is a function on M_{2}(ℤ/Nℤ). One has, for any γ ∈ Γ_{N},

π_{θ}(f) ρ(γ) ϵ_{m} = π_{θ}(f) ϵ_{m γ^{−1}} =
f(m γ^{−1}θ) ϵ_{m γ^{−1}} = h(p_{N}(m γ^{−1}θ)) ϵ_{m γ^{−1}} .

The equality p_{N}(γ) = 1 shows that

p_{N}(m γ^{−1}θ) = p_{N}(m) p_{N}(γ^{−1}) p_{N}(θ) = p_{N}(m) p_{N}(θ) = p_{N}(m θ),

hence

π_{θ}(f) ρ(γ) ϵ_{m} = ρ(γ) π_{θ}(f) ϵ_{m} .

3) One uses 2) to show that, for all N and f ∈ C(M_{2}(ℤ/Nℤ)) ⊂ C(M_{2}(R)), the products f μ(g, L) belong to the commutant of ρ(Γ_{N}). Since Γ_{N} has finite index in Γ it follows that for β > 2 one has Z_{N} := Trace_{Γ_{N}} (e^{−βH}) < ∞. Thus, the limit defining ψ_{β}(x) makes sense on a norm dense subalgebra of A and extends to a state on A by uniform continuity. One checks the KMS_{β} condition on the dense subalgebra in the same way as above. □

It is not difficult to extend the above discussion to arbitrary n using ([51]). The normalization factor is then given by

Z(β) = ∏_{0}^{n−1} ζ(β − k) .

What happens, however, is that the states ψ_{β} only depend on the determinant of θ. This shows that the above construction should be extended to involve not only the one-parameter group σ_{t} but in fact the whole dual action given by the groupoid homomorphism (1.63).

Definition 1.20 Given a groupoid G and a homomorphism j : G → H to a group H, the “cross product” groupoid G ×_{j} H is defined as the product G × H with units G^{(0)} × H, range and source maps

r(γ, α) := (r(γ), j(γ) α) , s(γ, α) := (s(γ), α)

and composition

(γ_{1}, α_{1}) ∘ (γ_{2}, α_{2}) := (γ_{1} ∘ γ_{2}, α_{2}) .

In our case this cross product G̃_{2} = G_{2} ×_{j} GL_{2}^{+}(ℝ) corresponds to the groupoid of the partially defined action of GL_{2}^{+}(ℚ) on the locally compact space Z_{0} of pairs (ρ, α) ∈ M_{2}(R) × GL_{2}^{+}(ℝ) given by

g(ρ, α) := (g ρ, g α) , ∀g ∈ GL_{2}^{+}(ℚ) , g ρ ∈ M_{2}(R) .

Since the subgroup Γ ⊂ GL_{2}^{+}(ℚ) acts freely and properly by translation on GL_{2}^{+}(ℝ), one obtains a Morita equivalent groupoid S_{2} by dividing G̃_{2} by the following action of Γ × Γ:

(γ_{1}, γ_{2}).(g, ρ, α) := (γ_{1} gγ_{2}^{−1}, γ_{2}ρ, γ_{2}α).  (1.79)

The space of units S_{2}^{(0)} is the quotient Γ\Z_{0}. We let p : Z_{0} → Γ\Z_{0} be the quotient map. The range and source maps are given by

r(g, ρ, α) := p(gρ, gα) , s(g, ρ, α) := p(ρ, α)

and the composition is given by

(g_{1}, ρ_{1}, α_{1}) ∘ (g_{2}, ρ_{2}, α_{2}) = (g_{1}g_{2}, ρ_{2}, α_{2}),

which passes to Γ × Γ-orbits.

1.8 Commensurability of ℚ-Lattices in ℂ and the full GL_{2}-System

We shall now describe the full GL_{2} C^{∗}-dynamical system (A, σ_{t}). It is obtained from the system of the previous section by taking a cross product with the dual action of GL_{2}^{+}(ℝ) i.e. from the groupoid S_{2} that we just described. It admits an equivalent and more geometric description in terms of the notion of commensurability between ℚ-lattices developed in section 1.6 above and we shall follow both points of view. The C^{∗}-algebra A is a Hecke algebra, which is a variant of the modular Hecke algebra defined in ([10]). Recall from section 1.6:

Figure 1.2: ℚ-Lattices in ℂ.

Definition 1.21 1) A ℚ-lattice in ℂ is a pair (Λ, ϕ) , with Λ a lattice in ℂ, and ϕ : ℚ^{2}/ℤ^{2} ⟶ ℚ Λ/Λ an homomorphism of abelian groups.

2) Two ℚ-lattices (Λ_{j}, ϕ_{j}) are commensurable iff Λ_{j} are commensurable and ϕ_{1} − ϕ_{2} = 0 modulo Λ = Λ_{1} + Λ_{2}.

This is an equivalence relation R between ℚ-lattices (Proposition 1.11). We use the basis {e_{1} = 1, e_{2} = −i} of the ℝ-vector space ℂ to let GL_{2}^{+}(ℝ) act on ℂ as ℝ-linear transformations. We let

Λ_{0} := ℤ e_{1} + ℤ e_{2} = ℤ + iℤ

Also we view ρ ∈ M_{2}(R) as the homomorphism

ρ : ℚ^{2}/ℤ^{2} ⟶ ℚ.Λ_{0}/Λ_{0} , ρ(a) = ρ_{1}(a)e_{1} + ρ_{2}(a)e_{2} .

Proposition 1.22 The map

γ(g, ρ, α) = ((α^{−1} g^{−1}Λ_{0}, α^{−1} ρ) , (α^{−1} Λ_{0}, α^{−1} ρ)) , ∀(g, ρ, α) ∈ S_{2}

defines an isomorphism of locally compact étale groupoids between S_{2} and the equivalence relation R of commensurability on the space of ℚ-lattices in ℂ.

Proof. The proof is the same as for Proposition 1.12. □ We shall now describe the quotient of S_{2} ∼ R by the natural scaling action of ℂ^{∗}. We view ℂ^{∗} as a subgroup of GL_{2}^{+}(ℝ) by the map

λ = a + i b ∈ ℂ^{∗} ↦ [\begin{matrix}a & b \\ −b & a \end{matrix}] ∈ GL_{2}^{+}(ℝ)  (1.80)

and identify the quotient GL_{2}^{+}(ℝ)/ℂ^{∗} with ℍ by the map

α ∈ GL_{2}^{+}(ℝ) ↦ τ = α(i) ∈ ℍ.  (1.81)

Given a pair (Λ_{j}, ϕ_{j}) of commensurable ℚ-lattices and a non zero complex number λ ∈ ℂ^{∗} the pair (λΛ_{j}, λϕ_{j}) is still a pair of commensurable ℚ-lattices. Moreover, one has

γ(g, ρ, α λ^{−1}) = λ γ(g, ρ, α) , ∀λ ∈ ℂ^{∗} .  (1.82)

The scaling action of ℂ^{∗} on ℚ-lattices in ℂ is not free, since the lattice Λ_{0} for instance is invariant under multiplication by i. It follows that the quotient S_{2}/ℂ^{∗} ∼ R/ℂ^{∗} is not a groupoid. One can nevertheless define its convolution algebra in a straightforward manner by restricting the convolution product on S_{2} ∼ R to functions which are homogeneous of weight 0, where weight k means

f(g, ρ, α λ) = λ^{k} f(g, ρ, α) , ∀λ ∈ ℂ^{∗} .  (1.83)

Let

Y = M_{2}(R) × ℍ,  (1.84)

endowed with the natural action of GL_{2}^{+}(ℚ) by

γ · (ρ, τ) = (γ ρ, \frac{aτ + b}{cτ + d}) ,  (1.85)

for γ = [\begin{matrix}a & b \\ c & d \end{matrix}] ∈ GL_{2}^{+}(ℚ) and (ρ, τ) ∈ Y. Let then

Z ⊂ Γ\GL_{2}^{+}(ℚ) ×_{Γ} Y  (1.86)

be the locally compact space quotient of {(g, y) ∈ GL_{2}^{+}(ℚ) × Y , g y ∈ Y } by the following action of Γ × Γ :

(g, y) ↦ (γ_{1} g γ_{2}^{−1}, γ_{2} y) , ∀γ_{j} ∈ Γ .

The natural lift of the quotient map (1.81), together with proposition 1.22, first gives the identification of the quotient of Y by Γ with the space of ℚ-lattices in ℂ up to scaling, realized by the map

θ : Γ\Y → (Space of ℚ-lattices in ℂ)/ℂ^{∗} = X ,  (1.87)
θ(ρ, τ) = (Λ, ϕ) , Λ = ℤ + ℤτ , ϕ(x) = ρ_{1}(x) − τρ_{2}(x) .

It also gives the isomorphism θ : S_{2}/ℂ^{∗} = Z → R/ℂ^{∗},

θ(g, y) = (λθ(gy), θ(y)),  (1.88)

where λ = Det(g)^{−1}(cτ + d) for γ = [\begin{matrix}a & b \\ c & d \end{matrix}] ∈ GL_{2}^{+}(ℚ) and y = (ρ, τ) ∈ Y.

We let A = C_{c}(Z) be the space of continuous functions with compact support on Z. We view elements f ∈ A as functions on GL_{2}^{+}(ℚ) × Y such that

f(γ g, y) = f(g, y) f(g γ, y) = f(g, γ y) , ∀γ ∈ Γ , g ∈ GL_{2}^{+}(ℚ) , y ∈ Y .

This does not imply that f(g, y) only depends on the orbit Γ.y but that it only depends on the orbit of y under the congruence subgroup Γ ∩ g^{−1}Γ g. We define the convolution product of two such functions by

(f_{1} ∗ f_{2})(g, y) := ∑_{h∈Γ\GL_{2}^{+}(ℚ), hy∈Y} f_{1}(gh^{−1}, hy) f_{2}(h, y)  (1.89)

and the adjoint by

f^{∗}(g, y) := \overline{f(g^{−1}, g y)}.  (1.90)

Notice that these rules combine (1.23) and (1.61). For any x ∈ X we let c(x) be the commensurability class of x. It is a countable subset of X and we want to define a natural representation in l^{2}(c(x)). We let p be the quotient map from Y to X. Let y ∈ Y with p(y) = x be an element in the preimage of x. Let

G_{y} = {g ∈ GL_{2}^{+}(ℚ) | gy ∈ Y } .

The natural map g ∈ G_{y} ↦ p(gy) ∈ X is a surjection from Γ\G_{y} to c(x) but it fails to be injective in degenerate cases such as y = (0, τ) with τ ∈ ℍ a complex multiplication point (cf. Lemma 1.28). This corresponds to the phenomenon of holonomy in the context of foliations ([11]). To handle it one defines the representation π_{y} directly in the Hilbert space H_{y} = l^{2}(Γ\G_{y}) of left Γ-invariant functions on G_{y} by

(π_{y}(f) ξ)(g) := ∑_{h∈Γ\G_{y}} f(gh^{−1}, hy) ξ(h) , ∀g ∈ G_{y} ,  (1.91)

for f ∈ A and ξ ∈ H_{y}.

Proposition 1.23 1) The vector space A endowed with the product ∗ and the adjoint f ↦ f^{∗} is an involutive algebra.

2) For any y ∈ Y, π_{y} defines a unitary representation of A in H_{y} whose unitary equivalence class only depends on x = p(y).

3) The completion of A for the norm given by

||f|| := Sup_{y∈Y} ||π_{y}(f)||

is a C^{∗}-algebra.

The proof of (1) and (2) is similar to ([10], Proposition 2). Using the compactness of the support of f, one shows that the supremum is finite for any f ∈ A (cf. [11]). □

Remark 1.24 The locally compact space Z of (1.86) is not a groupoid, due to the torsion elements in Γ, which give nontrivial isotropy under scaling, for the square and equilateral lattices. Nonetheless, Proposition 1.23 yields a well defined C^{∗}-algebra. This can be viewed as a subalgebra of the C^{∗}-algebra of the groupoid obtained by replacing Γ by its commutator subgroup in the definition of S_{2} as in (1.79).

We let σ_{t} be the one parameter group of automorphisms of A given by

σ_{t}(f)(g, y) = (Det g)^{it} f(g, y) .  (1.92)

Notice that since X is not compact (but still locally compact) the C^{∗}-algebra A does not have a unit, hence the discussion of Proposition 1.2 applies. The one parameter group σ_{2t} (1.92) is the modular automorphism group associated to the regular representation of A. To obtain the latter we endow X = Γ\Y with the measure

dy = dρ × dμ(τ),

where dρ = ∏ dρ_{ij} is the normalized Haar measure of the additive compact group M_{2}(R) and dμ(τ) is the Riemannian volume form in ℍ for the Poincaré metric, normalized so that μ(Γ\ℍ) = 1. We then get the following result.

Proposition 1.25 The expression

φ(f) = ∫_{X} f(1, y) dy .  (1.93)

defines a state on A, which is a KMS_{2} state for the one parameter group σ_{t}.

Proof. At the measure theory level, the quotient X = Γ\Y is the total space over Γ\ℍ of a bundle with fiber the probability space M_{2}(R)/{±1}, thus the total mass ∫_{X} dy = 1. One gets

φ(f^{∗} ∗ f) = ∫_{X} ∑_{h∈Γ\GL_{2}^{+}(ℚ), hy∈Y} \overline{f(h, y)}f(h, y) dy , ∀f ∈ A ,

which suffices to get the Hilbert space H of the regular representation and the cyclic vector ξ implementing the state φ, which corresponds to

ξ(g, y) = 0 , ∀g ∉ Γ , ξ(1, y) = 1 , ∀y ∈ X .

The measure dρ is the product of the additive Haar measures on column vectors, hence one gets

d(g ρ) = (Det g)^{−2} dρ , ∀g ∈ GL_{2}^{+}(ℚ) .

Let us prove that φ is a KMS_{2} state. The above equality shows that, for any compactly supported continuous function α on Γ\GL_{2}^{+}(ℚ) ×_{Γ} Y, one has

∫_{X} ∑_{h∈Γ\GL_{2}^{+}(ℚ)} α(h, y) dy = ∫_{X} ∑_{k∈Γ\GL_{2}^{+}(ℚ)} α(k^{−1}, k y) (Det k)^{−2} dy . (1.94)

Let then f_{j} ∈ A and define α(h, y) = 0 unless hy ∈ Y while otherwise

α(h, y) = f_{1}(h^{−1}, h y) f_{2}(h, y)(Det h)^{it−2}.

The l.h.s. of (1.94) is then equal to φ(f_{1} σ_{z}(f_{2})) for z = t + 2i. The r.h.s. of (1.94) gives φ(σ_{t}(f_{2}) f_{1}) and (1.94) gives the desired equality φ(f_{1} σ_{t+2i}(f_{2})) = φ(σ_{t}(f_{2}) f_{1}). □

We can now state the main result on the analysis of KMS states on the C^{∗}- dynamical system (A, σ_{t}). Recall that a ℚ-lattice l = (Λ, ϕ) is invertible if ϕ is an isomorphism (Definition 1.10). We have the following result.

Theorem 1.26 1) For each invertible ℚ-lattice l = (Λ, ϕ), the representation π_{l} is a positive energy representation of the C^{∗}-dynamical system (A, σ_{t}).

2) For β > 2 and l = (Λ, ϕ) an invertible ℚ-lattice, the formula

φ_{β,l}(f) = Z^{−1} ∑_{Γ\M_{2}(ℤ)^{+}} f(1, m ρ, m(τ)) Det(m)^{−β} ,

defines an extremal KMS_{β} state φ_{β,l} on (A, σ_{t}), where Z = ζ(β) ζ(β − 1) is the partition function.

3) For β > 2 the map l ↦ φ_{β,l} is a bijection from the space of invertible ℚlattices (up to scaling) to the space E_{β} of extremal KMS_{β} states on (A, σ_{t}).

The proof of 1) reflects the following fact, which in essence shows that the invertible ℚ-lattices are ground states for our system.

Lemma 1.27 1) Let s : ℚ^{2}/ℤ^{2} → ℚ^{2} be a section of the projection π : ℚ^{2} → ℚ^{2}/ℤ^{2}. Then the set of s(a + b) − s(a) − s(b) , a, b ∈ ℚ^{2}/ℤ^{2}, generates ℤ^{2}.

2) Let l = (Λ, ϕ) be an invertible ℚ-lattice and l′ = (Λ′, ϕ′) be commensurable with l. Then Λ ⊂ Λ′.

Proof. 1) Let L ⊂ ℤ^{2} be the subgroup generated by the s(a + b) − s(a) − s(b) , a, b ∈ ℚ^{2}/ℤ^{2}. If L ≠ ℤ^{2} we can assume, after a change of basis, that for some prime number p one has L ⊂ p ℤ ⊕ ℤ. Restricting s to the p-torsion elements of ℚ^{2}/ℤ^{2} and multiplying it by p, we get a morphism of groups

ℤ/p ℤ ⊕ ℤ/p ℤ → ℤ/p^{2}ℤ ⊕ ℤ/p ℤ,

which is a section of the projection

ℤ/p^{2}ℤ ⊕ ℤ/p ℤ → ℤ/p ℤ ⊕ ℤ/p ℤ.

This gives a contradiction, since the group ℤ/p^{2}ℤ ⊕ ℤ/p ℤ contains elements of order p^{2}.

2) Let s (resp. s′) be a lift of ϕ modulo Λ (resp. of ϕ′ modulo Λ′). Since ϕ − ϕ′ = 0 modulo Λ′′ = Λ + Λ′ one has s(a) − s′(a) ∈ Λ + Λ′ for all a ∈ ℚ^{2}/ℤ^{2}.

This allows one to correct s modulo Λ and s′ modulo Λ′ so that s = s′. Then for any a, b ∈ ℚ^{2}/ℤ^{2} one has s(a + b) − s(a) − s(b) ∈ Λ ∩ Λ′ and the first part of the lemma together with the invertibility of ϕ show that Λ ∩ Λ′ = Λ. □

Given y ∈ Y we let H_{y} be the diagonal operator in H_{y} given by

(H_{y} ξ)(h) := log(Det(h)) ξ(h) , ∀h ∈ G_{y}  (1.95)

It implements the one parameter group σ_{t} i.e.

π_{y}(σ_{t}(x)) = e^{itH_{y}} π_{y}(x) e^{−itH_{y}} , ∀x ∈ A .  (1.96)

In general the operator H_{y} is not positive but when the lattice l = (Λ, ϕ) = θ(p(y)) is invertible one has

Det(h) ∈ ℕ^{∗} , ∀h ∈ G_{y},

hence H_{y} ≥ 0. This proves the first part of the theorem. The basis of the Hilbert space H_{y} is then labeled by the lattices Λ′ containing Λ and the operator H_{y} is diagonal with eigenvalues the logarithms of the orders Λ′ : Λ. Equivalently, one can label the orthonormal basis ϵ_{m} by the coset space Γ\M_{2}(ℤ)^{+}. Thus, the same counting as in the previous section (cf.[51]) shows that

Z = Trace(e^{−βH_{y}}) = ζ(β) ζ(β − 1)

and in particular that it is finite for β > 2. The KMS_{β} property of the functional

φ_{β,l}(f) = Z^{−1} Trace(π_{y}(f) e^{−βH_{y}})

then follows from (1.96). One has, using (1.91) for y = (ρ, τ) ∈ Y,

⟨π_{y}(f)(ϵ_{m}), ϵ_{m}⟩ = f(1, m ρ, m(τ)) ,

hence we get the following formula for φ_{β,l}:

φ_{β,l}(f) = Z^{−1} ∑_{Γ\M_{2}(ℤ)^{+}} f(1, m ρ, m(τ)) Det(m)^{−β} .  (1.97)

Finally, the irreducibility of the representation π_{y} follows as in [11] p.562 using the absence of holonomy for invertible ℚ-lattices. This completes the proof of 2) of Theorem 1.26. In order to prove 3) of Theorem 1.26 we shall proceed in two steps. The first shows (Proposition 1.30 below) that KMS_{β} states are given by measures on the space X of ℚ-lattices (up to scaling). The second shows that when β > 2 this measure is carried by the commensurability classes of invertible ℚ-lattices. We first describe the stabilizers of the action of GL_{2}^{+}(ℚ) on the space of ℚlattices in ℂ.

Lemma 1.28 Let g ∈ GL_{2}^{+}(ℚ), g ≠ 1 and y ∈ Y, y = (ρ, τ) such that gy = y. Then ρ = 0. Moreover g ∈ ℚ^{∗} ⊂ GL_{2}^{+}(ℚ) unless τ is an imaginary quadratic number in which case g ∈ K^{∗} ⊂ GL_{2}^{+}(ℚ) where K = ℚ(τ) is the corresponding quadratic field.

Proof. Let g = [\begin{matrix}a & b \\ c & d \end{matrix}], then g(τ) = τ means aτ + b = τ (cτ + d). If c ≠ 0

this shows that τ is an imaginary quadratic number. Let K = ℚ(τ) be the corresponding field and let {τ, 1} be the natural basis of K over ℚ. Then the multiplication by (cτ + d) is given by the transpose of the matrix g. Since g ≠ 1 and K is a field we get

g − 1 = [\begin{matrix}a − 1 & b \\ c & d − 1 \end{matrix}] ∈ GL_{2}(ℚ)

and thus (g − 1)ρ = 0 implies ρ = 0. If c = 0 then aτ + b = τ (cτ + d) implies a = d, b = 0 so that g ∈ ℚ^{∗} ⊂ GL_{2}^{+}(ℚ). Since g ≠ 1 one gets ρ = 0. □

We let X = Γ\Y be the quotient and p : Y → X the quotient map. We let F be the closed subset of X, F = p({(0, τ); τ ∈ ℍ}).

Lemma 1.29 Let g ∈ GL_{2}^{+}(ℚ), g ∉ Γ and x ∈ X, x ∉ F. There exists a neighborhood V of x, such that

p(g p^{−1}(V)) ∩ V = ∅

Proof. Let Γ_{0} = Γ ∩ g^{−1}Γg and X_{0} = Γ_{0}\Y. For x_{0} ∈ X_{0} the projections p_{1}(x_{0}) = p(y) and p_{2}(x_{0}) = p(gy) are independent of the representative y ∈ Y. Moreover if p_{1}(x_{0}) ∉ F then p_{1}(x_{0}) ≠ p_{2}(x_{0}) by Lemma 1.28. By construction Γ_{0} is of finite index n in Γ and the fiber p_{1}^{−1}(x) has at most n elements. Let then W ⊂ X_{0} be a compact neighborhood of p_{1}^{−1}(x) in X_{0} such that x ∉ p_{2}(W). For z ∈ V ⊂ X sufficiently close to x one has p_{1}^{−1}(z) ⊂ W and thus p_{2}(p_{1}^{−1}(z)) ⊂ V^{c} which gives the result. □ We can now prove the following.

Proposition 1.30 Let β > 0 and φ a KMS_{β} state on (A, σ_{t}). Then there exists a probability measure μ on X = Γ\Y such that

φ(f) = ∫_{X} f(1, x) dμ(x) , ∀f ∈ A .

Proof. Let g_{0} ∈ GL_{2}^{+}(ℚ) and f ∈ C_{c}(Z) such that

f(g, y) = 0 , ∀g ∉ Γ g_{0} Γ .

Since any element of C_{c}(Z) is a finite linear combination of such functions, it is enough to show that φ(f) = 0 provided g_{0} ∉ Γ. Let h_{n} ∈ C_{c}(X), 0 ≤ h_{n} ≤ 1 with support disjoint from F and converging pointwise to 1 in the complement of F. Let u_{n} ∈ A be given by

u_{n}(1, y) := h_{n}(y) , u_{n}(g, y) = 0 , ∀g ∉ Γ .

The formula

Φ(f)(g, τ) := f(g, 0, τ) ∀f ∈ A  (1.98)

defines a homomorphism of (A, σ_{t}) to the C^{∗} dynamical system (B, σ_{t}) obtained by specialization to ρ = 0, with convolution product

f_{1} ∗ f_{2}(ρ, τ) = ∑_{h} f_{1}(gh^{−1}, h(τ))f_{2}(h, τ),

where now we have no restriction on the summation, as in [10]. For each n ∈ ℕ^{∗} we let

μ_{[n]}(g, y) = 1 if g ∈ Γ.[n] , μ_{[n]}(g, y) = 0 if g ∉ Γ.[n] .  (1.99)

One has μ_{[n]}^{∗} μ_{[n]} = 1 and σ_{t}(μ_{[n]}) = n^{2it} μ_{[n]} , ∀t ∈ ℝ. Moreover, the range π(n) = μ_{[n]} μ_{[n]}^{∗} of μ_{[n]} is the characteristic function of the set of ℚ-lattices that are divisible by n, i.e. those of the form (Λ, n ϕ). Let ν_{[n]} = Φ(μ_{[n]}). These are unitary multipliers of B. Since they are eigenvectors for σ_{t}, the system (B, σ_{t}) has no non-zero KMS_{β} positive functional. This shows that the pushforward of φ by Φ vanishes and by Proposition 1.5 that, with the notation introduced above,

φ(f) = lim_{n} φ(f ∗ u_{n}) .

Thus, since (f ∗u_{n})(g, y) = f(g, y) h_{n}(y), we can assume that f(g, y) = 0 unless p(y) ∈ K, where K ⊂ X is a compact subset disjoint from F. Let x ∈ K and V as in lemma 1.29 and let h ∈ C_{c}(V). Then, upon applying the KMS_{β} condition (1.6) to the pair a, b with a = f and

b(1, y) := h(y) , b(g, y) = 0 , ∀g ∉ Γ ,

one gets φ(b ∗ f) = φ(f ∗ b). One has (b ∗ f)(g, y) = h(gy) f(g, y). Applying this to f ∗ b instead of f and using h(gy) h(y) = 0 , ∀y ∈ X we get φ(f ∗ b^{2}) = 0 and φ(f) = 0, using a partition of unity on K. □ We let Det be the continuous map from M_{2}(R) to R given by the determinant

[\begin{matrix}a & b \\ c & d \end{matrix}] ∈ M_{2}(R) ↦ a d − b c ∈ R .

For each n ∈ ℕ^{∗}, the composition π_{n} ∘ Det defines a projection π′(n), which is the characteristic function of the set of ℚ-lattices whose determinant is divisible by n. If a ℚ-lattice is divisible by n its determinant is divisible by n^{2} and one controls divisibility using the following family of projections π_{p}(k, l). Given a prime p and a pair (k, l) of integers k ≤ l, we let

π_{p}(k, l) := (π(p^{k}) − π(p^{k+1})) (π′(p^{k+l}) − π′(p^{k+l+1})).  (1.100)

This corresponds, when working modulo N = p^{b}, b > l, to matrices in the double class of

[\begin{matrix}a & 0 \\ 0 & d \end{matrix}] , v_{p}(a) = k , v_{p}(d) = l ,

where v_{p} is the p-adic valuation.

Lemma 1.31 • Let φ be a KMS_{β} state on (A, σ_{t}). Then, for any prime p and pair (k, l) of integers k < l, one has

φ(π_{p}(k, l)) = p^{−(k+l)β} p^{l−k} (1 + p^{−1}) (1 − p^{−β}) (1 − p^{1−β})

while for k = l one has

φ(π_{p}(l, l)) = p^{−2lβ} (1 − p^{−β}) (1 − p^{1−β}) .

• For distinct primes p_{j} one has

φ(∏ π_{p_{j}}(k_{j}, l_{j})) = ∏ φ(π_{p_{j}}(k_{j}, l_{j})).

Proof. For each n ∈ ℕ^{∗} we let ν_{n} ∈ M(A) be given by

ν_{n}(g, y) = 1 , ∀g ∈ Γ [\begin{matrix}n & 0 \\ 0 & 1 \end{matrix}] Γ , ν_{n}(g, y) = 0 otherwise .

One has σ_{t}(ν_{n}) = n^{it} ν_{n} , ∀t ∈ ℝ. The double class Γ [\begin{matrix}n & 0 \\ 0 & 1 \end{matrix}] Γ is the union

of the left Γ-cosets of the matrices [\begin{matrix}a & b \\ 0 & d \end{matrix}] where a d = n and gcd(a, b, d) = 1.

The number of these left cosets is

ω(n) := n ∏_{p prime, p|n} (1 + p^{−1})

and

ν_{n}^{∗} ∗ ν_{n}(1, y) = ω(n) , ∀y ∈ Y.  (1.101)

One has

ν_{n} ∗ ν_{n}^{∗}(1, y) = ∑_{h∈Γ\GL_{2}^{+}(ℚ), hy∈Y} ν_{n}(h^{−1}, hy)^{2} .

With y = (ρ, τ), the r.h.s. is independent of τ and only depends upon the SL_{2}(ℤ/nℤ) − GL_{2}(ℤ/nℤ) double class of ρ_{n} = p_{n}(ρ) ∈ M_{2}(ℤ/nℤ).

Let us assume that n = p^{l} is a prime power. We can assume that ρ_{n} = p_{n}(ρ) is of the form

ρ_{n} = [\begin{matrix}p^{a} & 0 \\ 0 & p^{b} \end{matrix}] , 0 ≤ a ≤ b ≤ l .

We need to count the number ω(a, b) of left Γ-cosets Γ h_{j} in the double class

Γ [\begin{matrix}n^{−1} & 0 \\ 0 & 1 \end{matrix}] Γ such that h_{j}y ∈ Y i.e. h_{j}ρ ∈ M_{2}(R). A full set of representatives

of the double class is given by h_{j} = (α_{j}^{t})^{−1} where the α_{j} are

α_{0} = [\begin{matrix}n & 0 \\ 0 & 1 \end{matrix}] α(s) = [\begin{matrix}1 & s \\ 0 & n \end{matrix}] , s ∈ \{0, 1, ..., n − 1\}

and for x ∈ {1, 2, ..., l − 1}, s ∈ ℤ/p^{l−x}ℤ prime to p

α(x, s) = [\begin{matrix}p^{x} & s \\ 0 & p^{l−x} \end{matrix}] .

The counting gives

• ω(a, b) = 0 if b < l.

• ω(a, b) = p^{a} if a < l, b ≥ l.

• ω(a, b) = p^{l}(1 + p^{−1}) if a ≥ l.

Let e_{p}(i, j), (i ≤ j) be the projection corresponding to a ≥ i, b ≥ j. Then for i < j one has

π_{p}(i, j) = e_{p}(i, j) − e_{p}(i + 1, j) − e_{p}(i, j + 1) + e_{p}(i + 1, j + 1)  (1.102)

while

π_{p}(j, j) = e_{p}(j, j) − e_{p}(j, j + 1).  (1.103)

The computation above gives

ν_{n} ∗ ν_{n}^{∗}(1, y) = p^{l}(1 + p^{−1})e_{p}(l, l) + ∑_{0}^{l−1} p^{k} (e_{p}(k, l) − e_{p}(k + 1, l)) , (1.104)

where we omit the variable y in the r.h.s. Let φ be a KMS_{β} state, and σ(k, l) := φ(e_{p}(k, l)). Then, applying the KMS_{β} condition to the pair (μ_{[p]}f, μ_{[p]}^{∗}) for f ∈ C(X), one gets

σ(k, l) = p^{−2kβ} σ(0, l − k) .

Let σ(k) = σ(0, k). Upon applying the KMS_{β} condition to (ν_{n}, ν_{n}^{∗}), one gets

p^{l}(1+p^{−1})p^{−lβ} = p^{l}(1+p^{−1})p^{−2lβ}+∑_{0}^{l−1} p^{k} (p^{−2kβ}σ(l−k)−p^{−2(k+1)β}σ(l−k−1) .

Since σ(0) = 1, this determines the σ(n) by induction on n and gives

σ(n) = a p^{n(1−β)} + (1 − a) p^{−2nβ},

with

a = (1 + p)\frac{p^{β} − 1}{p^{1+β} − 1} .

Combined with (1.102) and (1.103), this gives the required formulas for φ(π_{p}(k, l)) and the first part of the lemma follows. To get the second part, one proceeds by induction on the number m of primes p_{j}. The function f = ∏_{1}^{m−1} π_{p_{j}}(k_{j}, l_{j}) fulfills

f(h y) = f(y) , ∀y ∈ Y , ∀h ∈ Γ [\begin{matrix}n^{−1} & 0 \\ 0 & 1 \end{matrix}] Γ,

where n = p_{m}^{l}. Thus, when applying the KMS_{β} condition to (ν_{n} f, ν_{n}^{∗}), the above computation applies with no change to give the result. □

Let us now complete the proof of 3) of Theorem 1.26. Let φ be a KMS_{β} state. Proposition 1.30 shows that there is a probability measure μ on X such that

φ(f) = ∫_{X} f(1, x) dμ(x) , ∀f ∈ A .

With y = (ρ, τ) ∈ X, Lemma 1.31 shows that the probability φ(e_{p}(1, 1)) = σ(1, 1) that a prime p divides ρ is p^{−2β}. Since the series ∑ p^{−2β} converges (β > \frac{1}{2} would suffice here), it follows (cf. [44] Thm. 1.41) that, for almost all y ∈ X, ρ is only divisible by a finite number of primes. Next, again by Lemma 1.31, the probability that the determinant of ρ is divisible by p is

φ(e_{p}(0, 1)) = σ(1) = (1 + p)p^{−β} − p^{1−2β}.

For β > 2 the corresponding series ∑ ((1 + p)p^{−β} − p^{1−2β}) is convergent. Thus, we conclude that with probability one

ρ_{p} ∈ GL_{2}(ℤ_{p}) , for almost all p.

Moreover, since ∑ φ(π_{p}(k, l)) = 1, one gets with probability one

ρ_{p} ∈ GL_{2}(ℚ_{p}) , ∀p .

In other words, the measure μ gives measure one to finite idèles. (Notice that finite idèles form a Borel subset which is not closed.) However, when ρ is a finite idèle the corresponding ℚ-lattice is commensurable to a unique invertible ℚ-lattice. Then the KMS_{β} condition shows that the measure μ is entirely determined by its restriction to invertible ℚ-lattices, so that, for some probability measure ν,

φ = ∫ φ_{β,l} dν(l).

It follows that the Choquet simplex of extremal KMS_{β} states is the space of probability measures on the locally compact space

GL_{2}(ℚ)\GL_{2}(𝔸)/ℂ^{∗}

of invertible ℚ-lattices ^{3} and its extreme points are the φ_{β,l}. □

In fact Lemma 1.31 admits the following corollary:

Corollary 1.32 For β ≤ 1 there is no KMS_{β} state on (A, σ_{t}).

Proof. Indeed the value of φ(π_{p}(k, l)) provided by the lemma is strictly negative for β < 1 and vanishes for β = 1. In the latter case this shows that the measure μ is supported by {0} × ℍ ⊂ Y and one checks that no such measure fulfills the KMS condition for β = 1. □ In fact the measure provided by Lemma 1.31 allows us to construct a specific KMS_{β} state on (A, σ_{t}) for 1 < β ≤ 2. We shall analyze this range of values in Chapter III in connection with the renormalization group. To get some feeling about what happens when β → 2 from above, we shall show that, on functions f which are independent of τ, the states φ_{β,l} converge weakly to the KMS_{2} state φ of (1.93), independently of the choice of the invertible ℚlattice l. Namely, we have

φ_{β,l}(f) → ∫_{M_{2}(R)} f(a) da .

Using the density of functions of the form f ∘ p_{N} among left Γ-invariant continuous functions on M_{2}(R), this follows from:

Lemma 1.33 For N ∈ ℕ, let Γ(N) be the congruence subgroup of level N and

Z_{β} = ∑_{Γ(N)\M_{2}(ℤ)^{+}} Det(m)^{−β} .

When β → 2 one has, for any function f on M_{2}(ℤ/Nℤ),

Z_{β}^{−1} ∑_{Γ(N)\M_{2}(ℤ)^{+}} f(p_{N}(m)) Det(m)^{−β} → N^{−4} ∑_{M_{2}(ℤ/Nℤ)} f(a) .

Proof. For x ∈ M_{2}(ℤ/Nℤ) we let

h(x) = lim_{β→2} Z_{β}^{−1} ∑_{m∈Γ(N)\M_{2}(ℤ)^{+}, p_{N} (m)=x} Det(m)^{−β}

be the limit of the above expression, with f the characteristic function of the subset {x} ⊂ M_{2}(ℤ/Nℤ). We want to show that

h(x) = N^{−4} , ∀x ∈ M_{2}(ℤ/Nℤ) .  (1.105)

3cf. e.g. [39] for the standard identification of the set of invertible ℚ-lattices with the above double quotient

Since p_{N} is a surjection SL_{2}(ℤ) → SL_{2}(ℤ/Nℤ) and Γ(N) a normal subgroup of Γ, one gets

h(γ_{1} xγ_{2}) = h(x) , ∀γ_{j} ∈ SL_{2}(ℤ/Nℤ) .  (1.106)

Thus, to prove (1.105) we can assume that x is a diagonal matrix

x = [\begin{matrix}n & 0 \\ 0 & n ℓ \end{matrix}] ∈ M_{2}(ℤ/Nℤ) .

Dividing both n and N by their g.c.d. k does not affect the validity of (1.105), since all m ∈ Γ(N)\M_{2}(ℤ)^{+} with p_{N}(m) = x are of the form k m′, while Det(m)^{−β} = k^{−2β} Det(m′)^{−β}. This shows that (1.105) holds for n = 0 and allows us to assume that n is coprime to N. Let then r be the g.c.d. of ℓ and

N. One can then assume that x = [\begin{matrix}n & 0 \\ 0 & n′ r \end{matrix}], with r|N and with n and n′

coprime to N. Let Δ ⊂ SL_{2}(ℤ/Nℤ) be the diagonal subgroup. The left coset Δ x ⊂ M_{2}(ℤ/Nℤ) only depends on r and the residue δ ∈ (ℤ/N′ℤ)^{∗} of n n′ modulo N′ = N/r. It is the set of all diagonal matrices of the form

y = [\begin{matrix}n_{1} & 0 \\ 0 & n_{2} r \end{matrix}] , n_{1} ∈ (ℤ/Nℤ)^{∗} , n_{1} n_{2} = δ (N′) .

Let Γ_{Δ}(N) ⊂ Γ be the inverse image of Δ by p_{N}. By (1.106) h is constant on Δ x, hence

h(x) = lim_{β→2} Z_{β}^{−1} ∑_{m∈Γ_{Δ}(N)\M_{2}(ℤ)^{+}, p_{N} (m)∈Δ x} Det(m)^{−β} .  (1.107)

In each left coset m ∈ Γ_{Δ}(N)\M_{2}(ℤ)^{+} with p_{N}(m) ∈ Δ x one can find a unique

triangular matrix [\begin{matrix}a & b \\ 0 & d \end{matrix}] with a > 0 coprime to N, d > 0 divisible by r,

a d/r = δ (N′) and b = N b′ with 0 ≤ b′ < d. Thus, we can rewrite (1.107) as

h(x) = lim_{β→2} Z_{β}^{−1} ∑_{Y} d (ad)^{−β},  (1.108)

where Y is the set of pairs of positive integers (a, d) such that

p_{N}(a) ∈ (ℤ/Nℤ)^{∗} , r|d , a d = r δ (N).

To prove (1.105) we assume first that r < N and write N = N_{1} N_{2}, where N_{1} is coprime to N′ and N_{2} has the same prime factors as N′. One has r = N_{1} r_{2}, with r_{2}|N_{2}. An element of ℤ/N_{2}ℤ is invertible iff its image in ℤ/N′ℤ is invertible. To prove (1.105) it is enough to show that, for any of the r_{2} lifts δ_{2} ∈ ℤ/N_{2}ℤ of δ, one has

lim_{β→2} Z_{β}^{−1} ∑_{Y′} d (ad)^{−β} = N_{1} N^{−4},  (1.109)

where Y′ is the set of pairs of positive integers (a, d) such that

p_{N}(a) ∈ (ℤ/Nℤ)^{∗} , p_{N_{2}}(a d) = δ_{2}.

We let 1_{N} be the trivial Dirichlet character modulo N. Then when X_{N_{2}} varies among Dirichlet characters modulo N_{2} one has

∑_{Y′} d (ad)^{−β} = φ(N_{2})^{−1} ∑ X_{N_{2}}(δ_{2})^{−1} L(1_{N_{1}} × X_{N_{2}}, β) L(X_{N_{2}}, β − 1),

where φ is the Euler totient function. Only the trivial character X_{N_{2}} = 1_{N_{2}} contributes to the limit (1.109), since the other L-functions are regular at 1. Moreover, the residue of L(1_{N_{2}}, β − 1) at β = 2 is equal to \frac{φ(N_{2})}{N_{2}} so that, when β → 2, we have

∑_{Y′} d (ad)^{−β} ∼ N_{2}^{−1} L(1_{N}, 2) (β − 2)^{−1} .

By construction one has

Z_{β} ∼ |Γ : Γ(N)| ζ(2) (β − 2)^{−1} ,

where the order of the quotient group Γ : Γ(N) is N^{3} ∏_{p|N} (1 − p^{−2}) ([51]). Since

L(1_{N}, s) = ∏_{p|N} (1 − p^{−s}) ζ(s)

one gets (1.109). A similar argument handles the case r = N. □

The states φ_{β,l} converge when β → ∞ and their limits restrict to C_{c}(X) ⊂ A as characters given by evaluation at l:

φ_{∞,l}(f) = f(l) , ∀f ∈ C_{c}(X).

These characters are all distinct and we thus get a bijection of the space

GL_{2}(ℚ)\GL_{2}(𝔸)/ℂ^{∗}

of invertible ℚ-lattices with the space E_{∞} of extremal KMS_{∞} states. We shall now describe the natural symmetry group of the above system, from an action of the quotient group S

S := ℚ^{∗}\GL_{2}(𝔸_{f})

as symmetries of our dynamical system. Here the finite adèlic group of GL_{2} is given by

GL_{2}(𝔸_{f}) = ∏_{res} GL_{2}(ℚ_{p}),

where in the restricted product the p-component lies in GL_{2}(ℤ_{p}) for all but finitely many p’s. It satisfies

GL_{2}(𝔸_{f}) = GL_{2}^{+}(ℚ) GL_{2}(R).

The action of the subgroup

GL_{2}(R) ⊂ S

is defined in a straightforward manner using the following right action of GL_{2}(R) on ℚ-lattices:

(Λ, ϕ).γ = (Λ, ϕ ∘ γ) , ∀γ ∈ GL_{2}(R) .

By construction this action preserves the commensurability relation for pairs of ℚ-lattices and preserves the value of the ratio of covolumes for such pairs. We can view it as the action

(ρ, τ).γ = (ρ ∘ γ, τ)

of GL_{2}(R) on Y = M_{2}(R)×ℍ, which commutes with the left action of GL_{2}^{+}(ℚ). Thus, this action defines automorphisms of the dynamical system (A, σ_{t}) by

θ_{γ}(f)(g, y) := f(g, y.γ) , ∀f ∈ A , γ ∈ GL_{2}(R) ,

and one has

θ_{γ_{1}} θ_{γ_{2}} = θ_{γ_{1}γ_{2}} , ∀γ_{j} ∈ GL_{2}(R) .

The complementary action of GL_{2}^{+}(ℚ) is more subtle and is given by endomorphisms of the dynamical system (A, σ_{t}), following Definition 1.3. For m ∈ M_{2}(ℤ)^{+}, let m̃ = Det(m) m^{−1} ∈ M_{2}(ℤ)^{+}. The range R_{m} of the map ρ → ρ m̃ only depends on L = m(ℤ^{2}). Indeed if m_{j} ∈ M_{2}(ℤ)^{+} fulfill m_{1}(ℤ^{2}) = m_{2}(ℤ^{2}) then m_{2} = m_{1} γ for some γ ∈ Γ, hence M_{2}(R) m̃_{1} = M_{2}(R) m̃_{2}. Let then

e_{L} ∈ C(X)  (1.110)

be the characteristic function of Γ\(R_{m} × ℍ) ⊂ Γ\(M_{2}(R) × ℍ), for any m such that m(ℤ^{2}) = L. Equivalently, it is the characteristic function of the open and closed subset E_{L} ⊂ X of ℚ-lattices of the form (Λ, ϕ ∘ m̃). One has

e_{L} e_{L′} = e_{L∩L′} ,  e_{ℤ^{2}} = 1 .  (1.111)

For l = (Λ, ϕ) ∈ E_{L} ⊂ X and m ∈ M_{2}(ℤ)^{+}, m(ℤ^{2}) = L we let

l ∘ m̃^{−1} := (Λ, ϕ ∘ m̃^{−1}) ∈ X.

This map preserves commensurability of ℚ-lattices. On Y it is given by

(ρ, τ) ∘ m̃^{−1} := (ρ ∘ m̃^{−1}, τ) , ∀(ρ, τ) ∈ R_{m} × ℍ

and it commutes with the left action of GL_{2}^{+}(ℚ). The formula

θ_{m}(f)(g, y) := f(g, y ∘ m̃^{−1}) , ∀y ∈ R_{m} × ℍ,  (1.112)

extended by θ_{m}(f)(g, y) = 0 for y ∉ R_{m} × ℍ, defines an endomorphism θ_{m} of A that commutes with the time evolution σ_{t}. Notice that θ_{m}(1) = e_{L} ∈ M(A) is a multiplier of A and that θ_{m} lands in the reduced algebra A_{e_{L}}, so that (1.112) is unambiguous. Thus one obtains an action of the semigroup M_{2}(ℤ)^{+} by endomorphisms of the dynamical system (A, σ_{t}), fulfilling Definition 1.3.

Proposition 1.34 The above actions of the group GL_{2}(R) ⊂ S and of the semigroup M_{2}(ℤ)^{+} ⊂ S assemble to an action of the group S = ℚ^{∗}\GL_{2}(𝔸_{f}) as symmetries of the dynamical system (A, σ_{t}).

Proof. The construction above applies to give an action by endomorphisms of the semigroup GL_{2}(𝔸_{f}) ∩ M_{2}(R), which contains both GL_{2}(R) and M_{2}(ℤ)^{+}. It remains to show that the sub-semigroup ℕ^{×} ⊂ M_{2}(ℤ)^{+} acts by inner endomorphisms of (A, σ_{t}). Indeed for any n ∈ ℕ^{∗}, the endomorphism θ_{[n]} (where [n] =

[\begin{matrix}n & 0 \\ 0 & n \end{matrix}] ∈ M_{2}(ℤ)^{+}) is inner and implemented by the multiplier μ_{[n]} ∈ M(A)

which was defined in (1.99) above i.e. one has

θ_{n}(f) = μ_{[n]} f μ_{[n]}^{∗} , ∀f ∈ A .

□

1.9 The subalgebra A_{ℚ} and the Modular Field

The strategy outlined in §1.6 allows us to find, using Eisenstein series, a suitable arithmetic subalgebra A_{ℚ} of the algebra of unbounded multipliers of the basic Hecke C^{∗}-algebra A of the previous section. The extremal KMS_{∞} states φ ∈ E_{∞} extend to A_{ℚ} and the image φ(A_{ℚ}) generates, in the generic case, a specialization F_{φ} ⊂ ℂ of the modular field F. The state φ will then intertwine the symmetry group S of the system (A, σ_{t}) with the Galois group of the modular field i.e. we shall show that there exists an isomorphism θ of S with Gal(F_{φ}/ℚ) such that

α ∘ φ = φ ∘ θ^{−1}(α) , ∀α ∈ Gal(F_{φ}/ℚ) .  (1.113)

Let us first define A_{ℚ} directly without any reference to Eisenstein series and check directly its algebraic properties. We let Z ⊂ Γ\GL_{2}^{+}(ℚ) ×_{Γ} Y be as above and f ∈ C(Z) be a function with finite support in the variable g ∈ Γ\GL_{2}^{+}(ℚ). Such an f defines an unbounded multiplier of the C^{∗}-algebra A with the product given as above by

(f_{1} ∗ f_{2})(g, y) := ∑_{h∈Γ\GL_{2}^{+}(ℚ),hy∈Y} f_{1}(gh^{−1}, hy) f_{2}(h, y).

One has Y = M_{2}(R) × ℍ and we write f(g, y) = f(g, ρ, z), with (g, ρ, z) ∈ GL_{2}^{+}(ℚ) × M_{2}(R) × ℍ. In order to define the arithmetic elements f ∈ A_{ℚ} we first look at the way f depends on ρ ∈ M_{2}(R). Let as above p_{N} : M_{2}(R) → M_{2}(ℤ/Nℤ) be the canonical projection. It is a ring homomorphism. We say that f has level N iff f(g, ρ, z) only depends upon (g, p_{N}(ρ), z) ∈ GL_{2}^{+}(ℚ) × M_{2}(ℤ/Nℤ) × ℍ. Then specifying f amounts to assigning the finitely many continuous functions f_{g,m} ∈ C(ℍ) with m ∈ M_{2}(ℤ/Nℤ) and

f(g, ρ, z) = f_{g,p_{N} (ρ)}(z).

The invariance condition

f(g γ, y) = f(g, γ y) , ∀γ ∈ Γ , g ∈ GL_{2}^{+}(ℚ) , y ∈ Y  (1.114)

then shows that

f_{g,m}|γ = f_{g,m} , ∀γ ∈ Γ(N) ∩ g^{−1}Γg,

with standard notations for congruence subgroups and for the slash operation in weight 0 (cf. (1.141)). We denote by F the field of modular functions which are rational over ℚ^{ab}, i.e. the union of the fields F_{N} of modular functions of level N rational over ℚ(e^{2πi/N}). Its elements are modular functions h(τ) whose q^{\frac{1}{N}} -expansion has all its coefficients in ℚ(e^{2πi/N}) (cf. [51]). The first requirement for arithmetic elements is that

f_{g,m} ∈ F ∀(g, m).  (1.115)

This condition alone, however, is not sufficient. In fact, the modular field F_{N} of level N contains (cf. [51]) a primitive N-th root of 1. Thus, the condition (1.115) alone allows the algebra A_{ℚ} to contain the cyclotomic field ℚ^{ab} ⊂ ℂ, but this would prevent the existence of “fabulous states”, because the “fabulous” property would not be compatible with ℂ-linearity. We shall then impose an additional condition, which forces the spectrum of the corresponding elements of A_{ℚ} to contain all Galois conjugates of such a root, so that no such element can be a scalar. This is, in effect, a consistency condition on the roots of unity that appear in the coefficients of the q-series, when ρ is multiplied on the left by a diagonal matrix. Consider elements g ∈ GL_{2}^{+}(ℚ) and α ∈ GL_{2}(ℤ/Nℤ), respectively of the form

g = r [ \begin{matrix}n & 0 \\ 0 & 1 \end{matrix} ] and α = [ \begin{matrix}k & 0 \\ 0 & 1 \end{matrix} ] ,  (1.116)

with k prime to N and n|N.

Definition 1.35 We shall say that f of level N_{0} is arithmetic (f ∈ A_{ℚ}) iff for any multiple N of N_{0} and any pair (g, α) as in (1.116) we have f_{g,m} ∈ F_{N} for all m ∈ M_{2}(ℤ/Nℤ) and the q-series of f_{g,α m} is obtained from the q-series for f_{g,m} by raising to the power k the roots of unity that appear as coefficients.

The arithmetic subalgebra A_{ℚ} enriches the structure of the noncommutative space to that of a “noncommutative arithmetic variety”. As we shall prove in Theorem 1.39, a generic ground state φ of the system, when evaluated on A_{ℚ} generates an embedded copy F_{φ} of the modular field in ℂ. Moreover, there exists a unique isomorphism θ = θ_{φ} of the symmetry group S of the system with Gal(F_{φ}/ℚ), such that

θ(σ) ∘ φ = φ ∘ σ , ∀σ ∈ S .

A first step towards this result is to show that the arithmeticity condition is equivalent to a covariance property under left multiplication of ρ by elements α ∈ GL_{2}(R), in terms of Galois automorphisms. The condition is always satisfied for α ∈ SL_{2}(R). For each g ∈ GL_{2}(𝔸_{f}), we let Gal(g) ∈ Aut(F) be its natural action on F, written in a covariant way so that

Gal(g_{1} g_{2}) = Gal(g_{1}) ∘ Gal(g_{2}).

With the standard contravariant notation f ↦ f^{g} (cf. e.g. [28]) we let, for all f ∈ F,

Gal(g)(f) := f g̃ , g̃ = Det(g) g^{−1} .  (1.117)

Lemma 1.36 For any α ∈ SL_{2}(R) one has

f_{g,α m} = Gal(α)f_{g′,m},

where g α = α′ g′ is the decomposition of g α as a product in GL_{2}(R).GL_{2}(ℚ).

Proof. Notice that the decomposition α′ g′ is not unique, but the left invariance

f(γ g′, ρ, τ) = f(g′, ρ, τ) , ∀γ ∈ Γ

shows that the above condition is well defined. Let p_{N} : M_{2}(R) → M_{2}(ℤ/Nℤ) be the projection. Then f_{g,α m} = f_{g,p_{N} (α)p_{N} (m)}, for f of level N. Let γ ∈ Γ be such that p_{N}(γ) = p_{N}(α). Then

f_{g,αm}(τ) = f(g, γm, τ) = f(gγ, m, γ^{−1}(τ)).

Thus, for g′ = gγ, one obtains the required condition. □

Lemma 1.37 A function f is in A_{ℚ} iff condition (1.115) is satisfied and

f_{g,α m} = Gal(α)f_{g′,m}, ∀α ∈ GL_{2}(R),  (1.118)

where g α = α′ g′ is the decomposition of g α as a product in GL_{2}(R).GL_{2}(ℚ).

Proof. By Lemma 1.36, the only nontrivial part of the covariance condition

(1.118) is the case of diagonal matrices δ = [ \begin{matrix}u & 0 \\ 0 & 1 \end{matrix} ] with u ∈ GL_{1}(R).

To prove (1.118) we can assume that g = g_{0} γ with g_{0} diagonal as in (1.116) and γ ∈ Γ. Let then γ α = δ α_{1} with δ as above and α_{1} ∈ SL_{2}(R). One has

f_{g_{0},δα_{1}m} = Gal(δ) f_{g_{0},α_{1}m}

by Definition 1.35 since Gal(δ) is given by raising the roots of unity that appear as coefficients of the q-expansion to the power k where u is the residue of k modulo N (cf. [51] (6.2.1) p.141). One then has

f_{g,α m} = Gal(γ^{−1})f_{g_{0},γαm} = Gal(γ^{−1}δ) f_{g_{0},α_{1}m}

and by Lemma 1.36, with g_{0}α_{1} = α′_{1}g′_{0} we get

f_{g,α m} = Gal(γ^{−1}δα_{1}) f_{g′_{0},m} = Gal(α) f_{g′_{0},m}

Moreover

g α = g_{0} γ α = g_{0} δ α_{1} = δ g_{0} α_{1} = δ α′_{1} g′_{0}

which shows that g′ = g′_{0} One checks similarly that the converse holds. □

Proposition 1.38 A_{ℚ} is a subalgebra of the algebra of unbounded multipliers of A, globally invariant under the action of the symmetry group S.

Proof. For each generic value of τ ∈ ℍ the evaluation map

h ∈ F ↦ I_{τ} (h) = h(τ) ∈ ℂ

gives an isomorphism of F with a subfield F_{τ} ⊂ ℂ and a corresponding action Gal_{τ} of GL_{2}(𝔸_{f}) by automorphisms of F_{τ}, such that

Gal_{τ} (g)(I_{τ} (h)) = I_{τ} (Gal(g)(h)).  (1.119)

We first rewrite the product as

(f_{1} ∗ f_{2})(g, ρ, τ) = ∑_{g_{1}∈Γ\GL_{2}^{+}(ℚ),g_{1}ρ∈M_{2}(R)} f_{1}(gg_{1}^{−1}, g_{1}ρ, g_{1}(τ)) f_{2}(g_{1}, ρ, τ) .

The proof that (f_{1} ∗ f_{2})_{g,m} ∈ F is the same as in Proposition 2 of ([10]). It remains to be shown that condition (1.118) is stable under convolution. Thus we let α ∈ GL_{2}(R) and we want to show that f_{1} ∗ f_{2} fulfills (1.118). We let g′ ∈ GL_{2}(ℚ) and β ∈ GL_{2}(R) with g α = β g′. By definition, one has

(f_{1} ∗ f_{2})(g, α ρ, τ) = ∑_{g_{1}∈Γ\GL_{2}^{+}(ℚ),g_{1}αρ∈M_{2}(R)} f_{1}(gg_{1}^{−1}, g_{1} α ρ, g_{1}(τ)) f_{2}(g_{1}, α ρ, τ).

We let g_{1}α = α′g′_{1} be the decomposition of g_{1}α, and use 1.118 to write the r.h.s. as

(f_{1} ∗ f_{2})(g, α ρ, τ) = ∑ f_{1}(gg_{1}^{−1}, g_{1} α ρ, g_{1}(τ)) Gal_{τ} (α)(f_{2}(g′_{1}, ρ, τ)) ,

with Gal_{τ} as in (1.119). The result then follows from the equality

f_{1}(gg_{1}^{−1}, g_{1} α ρ, g_{1}(τ)) = Gal_{τ} (α)(f_{1}(g′g_{1}^{′ −1}, g′_{1} ρ, g′_{1}(τ))),  (1.120)

which we now prove. The equality g_{1}α = α′g′_{1} together with (1.118) shows that

f_{1}(gg_{1}^{−1}, g_{1} α ρ, g_{1}(τ)) = Gal_{g_{1}(τ)}(α′)(f_{1}(g′g_{1}^{′ −1}, g′_{1} ρ, g_{1}(τ))) ,

using gg_{1}^{−1} α′ = gg_{1}^{−1}(g_{1}α g_{1}^{′ −1}) = gα g_{1}^{′ −1} = β g′ g_{1}^{′ −1}.

For any h ∈ F, one has

I_{g_{1}(τ)}(Gal(α′)h) = I_{g_{1}(τ)}(Gal(g_{1})Gal(α)Gal(g_{1}^{′ −1})(h))

and, by construction of the Galois action [51],

I_{g_{1}(τ)} ∘ Gal(g_{1}) = I_{τ} ,

so that in fact

Gal_{g_{1}(τ)}(α′)I_{g_{1}(τ)}(h) = Gal_{τ} (α)I_{g′_{1}(τ)}(h)) .

This proves (1.120) and it shows that A_{ℚ} is a subalgebra of the algebra of unbounded multipliers of A. To prove the invariance under S is straightforward, since the endomorphisms are all acting on the ρ variable by right multiplication, which does not interfere with condition (1.118). □ In fact, modulo the nuance between “forms” and functions, the above algebra A_{ℚ} is intimately related to the modular Hecke algebra of [10]. We can now state the main result extending Theorem 1.7 to the two dimensional case.

Theorem 1.39 Let l = (ρ, τ) be a generic invertible ℚ-lattice and φ_{l} ∈ E_{∞} be the corresponding KMS_{∞} state. The image φ_{l}(A_{ℚ}) ⊂ ℂ generates the specialization F_{τ} ⊂ ℂ of the modular field F obtained for the modulus τ. The action of the symmetry group S of the dynamical system (A, σ_{t}) is intertwined by φ with the Galois group of the modular field F_{τ} by the formula

φ ∘ α = Gal_{τ} (ρ α ρ^{−1}) ∘ φ.

Proof. We first need to exhibit enough elements of A_{ℚ}. Let us first deal with functions f(g, ρ, τ) which vanish except when g ∈ Γ. By construction these are functions on the space X of ℚ-lattices

X = (Space of ℚ-lattices in ℂ)/ℂ^{∗} ∼ Γ\(M_{2}(R) × ℍ) .  (1.121)

To obtain such elements of A_{ℚ} we start with Eisenstein series and view them as functions on the space of ℚ-lattices. Recall that to a pair (ρ, τ) ∈ Y we associate the ℚ-lattice (Λ, ϕ) = θ(ρ, τ) by

Λ = ℤ + τ ℤ , ϕ(a) = ρ_{1}(a) − τρ_{2}(a) ∈ ℚΛ/Λ,  (1.122)

where ρ_{j}(a) = ∑ ρ_{jk}(a_{k}) ∈ ℚ/ℤ, for a = (a_{1}, a_{2}) ∈ (ℚ/ℤ)^{2}. The Eisenstein series are given by

E_{2k,a}(ρ, τ) = π^{−2k} ∑_{y∈Λ+ϕ(a)} y^{−2k}.  (1.123)

This is undefined when ϕ(a) ∈ Λ, but we shall easily deal with that point below. For k = 1 we let

X_{a}(ρ, τ) = π^{−2} ( ∑_{y∈Λ+ϕ(a)} y^{−2} − \begin{matrix}∑ & ′ \\ y∈Λ \end{matrix} y^{−2})  (1.124)

when ϕ(a) ∉ Λ and X_{a}(ρ, τ) = 0 if ϕ(a) ∈ Λ. This is just the evaluation of the Weierstrass ℘-function on ϕ(a). For γ ∈ Γ = SL_{2}(ℤ) we have X_{a}(γ ρ, γ τ) = (cτ + d)^{2} X_{a}(ρ, τ), which shows that the function c(τ)X_{a} is Γ-invariant on Y, where

c(τ) = −2^{7} 3^{5} \frac{g_{2}g_{3}}{Δ}  (1.125)

has weight −2 and no pole in ℍ. We use c as we used the covolume in the 1-dimensional case, to pass to modular functions. This corresponds in weight 2 to passing from division values of the Weierstrass ℘-function to the Fricke functions (cf. [28] §6.2)

f_{v}(τ) = −2^{7}3^{5} \frac{g_{2} g_{3}}{Δ} ℘(λ(v, τ)),  (1.126)

where v = (v_{1}, v_{2}) ∈ (ℚ/ℤ)^{2} and λ(v, τ) := v_{1} τ + v_{2}. Here g_{2}, g_{3} are the coefficients giving the elliptic curve E_{τ} = ℂ/Λ in Weierstrass form,

y^{2} = 4x^{3} − g_{2}x − g_{3},

with discriminant Δ = g_{2}^{3} − 27g_{3}^{2}. One has (up to powers of π)

g_{2} = 60 e_{4} , g_{3} = 140 e_{6},

where one defines the standard modular forms of even weight k ∈ 2 ℕ as

e_{k}(Λ) := π^{−k} ∑_{y∈Λ\{0}} y^{−k}

with q-expansion (q = e^{2πiτ})

e_{k} = \frac{2^{k}}{k!} B_{\frac{k}{2}} + (−1)^{k/2} \frac{2^{k+1}}{(k − 1)!} ∑_{1}^{∞} σ_{k−1}(N) q^{N},

where the B_{n} are the Bernoulli numbers and σ_{n}(N) is the sum of d^{n} over the divisors d of N. The e_{2n} for n ≥ 2 are in the ring ℚ[e_{4}, e_{6}] (cf [56]) thanks to the relation

\frac{1}{3}(m − 3)(4m^{2} − 1)e_{2m} = ∑_{2}^{m−2} (2r − 1)(2m − 2r − 1)e_{2r}e_{2m−2r}. (1.127)

Notice that X_{a} := c X_{a} ∈ C(M_{2}(R)×ℍ) = C(Y) is a continuous function on Y. The continuity of X_{a} as a function of ρ comes from the fact that it only involves the restriction ρ_{N} ∈ M_{2}(ℤ/Nℤ) of ρ to N-torsion elements a with Na = 0. We view X_{a} as a function on Z by

X_{a}(γ, ρ, τ) := X_{a}(ρ, τ) , ∀γ ∈ Γ,

while it vanishes for γ ∉ Γ. Let us show that X_{a} ∈ A_{ℚ}. Since the Fricke functions belong to the modular field F we only need to check (1.118). For α ∈ GL_{2}(R) and generic τ we want to show that

X_{a}(α ρ, τ) = Gal_{τ} (α)X_{a}(ρ, τ).

If ρ(a) = 0 both sides vanish, otherwise they are both given by Fricke functions f_{v}, f_{v′}, corresponding respectively to the labels (using (1.122))

v = s α ρ(a) , v′ = s ρ(a) , s = [\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}] .

Thus, v′ = s α s^{−1}(v) = (s^{−1} α^{t} s)^{t}(v) and the result follows from (1.117) and the equality

α̃ = Det(α) α^{−1} = s^{−1} α^{t} s,

with the Galois group GL_{2}(ℤ/nℤ)/ ± 1 of the modular field F_{n} over ℚ(j) acting on the Fricke functions by permutation of their labels:

f_{v}^{σ(u)} = f_{u^{t}v} , ∀u ∈ GL_{2}(ℤ/nℤ).

This shows that X_{a} ∈ A_{ℚ} and it suffices to show that, with the notation of the theorem, φ_{l}(A_{ℚ}) generates F_{τ}, since the modular field F is the field generated over ℚ by all the Fricke functions. It already contains ℚ(j) at level 2 and it contains in fact ℚ^{ab}(j). Let us now display elements T_{r_{1},r_{2}} ∈ A_{ℚ}, r_{j} ∈ ℚ_{+}^{∗}, r_{1}|r_{2}, associated to the classical Hecke correspondences. We let C_{r_{1},r_{2}} ⊂ Γ\GL_{2}(ℚ)^{+} be the finite

subset given by the double class of [\begin{matrix}r_{1} & 0 \\ 0 & r_{2} \end{matrix}] in Γ\GL_{2}(ℚ)^{+}/Γ. We then define

T_{r_{1},r_{2}}(g, ρ, τ) = 1 if g ∈ C_{r_{1},r_{2}} , gρ ∈ M_{2}(R) , T_{r_{1},r_{2}}(g, ρ, τ) = 0 otherwise .

One needs to check (1.118), but if g α = α′ g′ is the decomposition of g α as a product in GL_{2}(R).GL_{2}(ℚ)^{+}, then g′ belongs to the double coset of g ∈ Γ\GL_{2}(ℚ)^{+}/Γ, which gives the required invariance. It is not true that the T_{r_{1},r_{2}} ∈ A_{ℚ} fulfill the relations of the Hecke algebra H(GL_{2}(ℚ)^{+}, Γ) of double cosets, but this holds when r_{1}, r_{2} are restricted to vary among positive integers. To see this one checks that the map

τ(f)(g, y) := f(g) if g ∈ M_{2}(ℤ)^{+} , τ(f)(g, y) = 0 otherwise

defines an isomorphism

τ : H(M_{2}(ℤ)^{+}, Γ) → A_{ℚ}  (1.128)

of the standard Hecke algebra H(M_{2}(ℤ)^{+}, Γ) of Γ-biinvariant functions (with Γ-finite support) on M_{2}(ℤ)^{+} with a subalgebra H ⊂ A_{ℚ}. Notice that it is only because the condition h y ∈ Y of definition (1.89) is now automatically satisfied that τ is a homomorphism.

Let us now show the intertwining equality

φ_{l} ∘ α = Gal_{τ} (ρ α ρ^{−1}) ∘ φ_{l} , ∀α ∈ S .  (1.129)

One has

φ_{l}(f) = f(1, ρ, τ) , ∀f ∈ A_{ℚ}

It is enough to prove (1.129) for α ∈ GL_{2}(R) and for α ∈ GL_{2}(ℚ). For α ∈ GL_{2}(R), the state φ_{l} ∘ α is given simply by

(φ_{l} ∘ α)(f) = f(1, ρ α, τ) , ∀f ∈ A_{ℚ} ,

and using (1.118) one gets (1.129) in that case. Let m ∈ M_{2}^{+}(ℤ). Then the state φ_{l} ∘ m is more tricky to obtain, since it is not the straight composition but the 0-temperature limit of the states obtained by composition of the KMS_{β} state φ_{l,β} with the endomorphism θ_{m} defined in (1.112). Indeed, the range of θ_{m} is the reduced algebra by the projection e_{L}, with L = m(ℤ^{2}), on which any of the zero temperature states vanishes identically. Let us first show that for finite β we have

φ_{l,β} ∘ θ_{m} = φ_{l,β}(e_{L}) φ_{l′,β},  (1.130)

where L = m(ℤ^{2}) and l′ is given by

l′ = (ρ′, m^{′ −1}(τ)) , ρ m = m′ ρ′ ∈ M_{2}^{+}(ℤ).GL_{2}(R) .  (1.131)

By (1.97) we have

φ_{β,l}(θ_{m}(f)) = Z^{−1} ∑_{Γ\M_{2}(ℤ)^{+}} f(1, μ ρ m̃^{−1}, μ(τ)) Det(μ)^{−β} ,

where μ ∈ M_{2}(ℤ)^{+} is subject to the condition μ ρ m̃^{−1} ∈ M_{2}(R). The other values of μ a priori involved in the summation (1.97) do not contribute, since they correspond to the orthogonal of the support of θ_{m}(f). One has Det(m) = Det(m′) by construction, hence

ρ m̃^{−1} = ρ m Det(m)^{−1} = Det(m′)^{−1} m′ ρ′ = m̃′^{−1} ρ′ .

Therefore the condition μ ρ m̃^{−1} ∈ M_{2}(R) holds iff μ = ν m̃′ for some ν ∈ M_{2}(ℤ)^{+}. Thus, since Det(μ) = Det(ν) · Det(m̃′), we can rewrite, up to multiplication by a scalar,

φ_{β,l}(θ_{m}(f)) = Z^{′ −1} ∑_{Γ\M_{2}(ℤ)^{+}} f(1, ν ρ′, ν m̃′(τ)) Det(ν)^{−β} .

This proves (1.130). It remains to show that on A_{ℚ} we have

φ_{l′}(f) = Gal_{τ} (ρ m ρ^{−1}) ∘ φ_{l}(f) , ∀f ∈ A_{ℚ} .

Both sides only involve the values of f on invertible ℚ-lattices, and there, by (1.118) one has

f(1, α, τ) = Gal_{τ} (α)f(1, 1, τ) , ∀α ∈ GL(2, R) .

Thus, we obtain

φ_{l′}(f) = f(1, ρ′, m^{′ −1}(τ)) = I_{m^{′ −1}(τ)}(Gal(ρ′)f) = I_{τ} (Gal(m′ ρ′)f) .

Since m′ ρ′ = ρ m, this gives Gal_{τ} (ρ m ρ^{−1}) ∘ φ_{l}(f) as required. □

We shall now work out the algebraic relations fulfilled by the X_{a} as extensions of the division formulas of elliptic functions. We first work with lattice functions of some weight k, or equivalently with forms f(g, y) dy^{k/2}, and then multiply them by a suitable factor to make them homogeneous of weight 0 under scaling. The functions of weight 2 are the generators, the higher weight ones will be obtained from them by universal formulas with modular forms as coefficients. The powers X_{a}^{m} of the function X_{a} are then expressed as universal polynomials with coefficients in the ring ℚ [e_{4}, e_{6}] in the following weight 2k functions (k > 1):

E_{2k,a}(ρ, τ) = π^{−2k} ∑_{y∈Λ+ϕ(a)} y^{−2k}.  (1.132)

These fulfill by construction ([56]) the relations

E_{2m,a} = X_{a}(E_{2m−2,a} − e_{2m−2}) + (1 − (\begin{matrix}2m \\ 2 \end{matrix})) e_{2m}  (1.133)

− ∑_{1}^{m−2} (\begin{matrix}2k + 1 \\ 2k \end{matrix}) e_{2k+2}(E_{2m−2k−2,a} − e_{2m−2k−2}) .

These relations dictate the value of E_{2k}(ρ, τ) when φ(a) ∈ Λ: one gets

E_{2k}(ρ, τ) = ν_{2k}(τ)  if φ(a) ∈ Λ,  (1.134)

where ν_{2k} is a modular form of weight 2k obtained by induction from (1.133), with X_{a} replaced by 0 and E_{2m} by ν_{2m}. One has ν_{2k} ∈ ℚ [e_{4}, e_{6}] and the first values are

ν_{4} = −5 e_{4} , ν_{6} = −14 e_{6} , ν_{8} = \frac{45}{7} e_{4}^{2} , …  (1.135)

We shall now write the important algebraic relations between the functions X_{a}, which extend the division relations of elliptic functions from invertible ℚ-lattices to arbitrary ones. In order to work out the division formulas for the Eisenstein series E_{2m,a} we need to control the image of (\frac{1}{N} ℤ)^{2} = \frac{1}{N} ℤ^{2} under an arbitrary element ρ ∈ M_{2}(R). This is done as follows using the projections π_{L} defined in (1.65).

Lemma 1.40 Let N ∈ ℕ^{∗}, and ρ ∈ M_{2}(R). There exists a smallest lattice L ⊂ ℤ^{2} with L ⊃ Nℤ^{2}, such that π_{L}(ρ) = 1. One has

ρ(\frac{1}{N} ℤ^{2}) = \frac{1}{N} L .

Proof. There are finitely many lattices L with Nℤ^{2} ⊂ L ⊂ ℤ^{2}. Thus, the intersection of those L for which π_{L}(ρ) = 1 is still a lattice and fulfills π_{L}(ρ) = 1 by (1.66). Let L be this lattice, and let us show that ρ(\frac{1}{N} ℤ^{2}) ⊂ \frac{1}{N} L. Let m ∈ M_{2}(ℤ)^{+} be such that m(ℤ^{2}) = L. Then π_{L}(ρ) = 1 implies that ρ = mμ for some μ ∈ M_{2}(R). Thus ρ(\frac{1}{N} ℤ^{2}) ⊂ m(\frac{1}{N} ℤ^{2}) = \frac{1}{N} L. Conversely let L′ ⊂ ℤ^{2} be defined by ρ(\frac{1}{N} ℤ^{2}) = \frac{1}{N} L′. We need to show that π_{L′}(ρ) = 1, i.e. that there exists m′ ∈ M_{2}(ℤ)^{+} such that L′ = m′(ℤ^{2}) and m′^{−1} ρ ∈ M_{2}(R). Replacing ρ by γ_{1}ρ γ_{2} for γ_{j} ∈ SL_{2}(ℤ) does not change the problem, hence we can use this freedom to assume that the restriction of ρ to (\frac{1}{N} ℤ)^{2} is of the form

ρ_{N} = [\begin{matrix}d_{1} & 0 \\ 0 & d_{2} \end{matrix}] , d_{j} | N , d_{1} | d_{2} .  (1.136)

One then takes m′ = [\begin{matrix}d_{1} & 0 \\ 0 & d_{2} \end{matrix}] ∈ M_{2}(ℤ)^{+} and checks that L′ = m′(ℤ^{2}) while

m′^{−1}ρ belongs to M_{2}(R). □ Given an integer N > 1, we let S_{N} be the set of lattices

N ℤ^{2} ⊂ L ⊂ ℤ^{2},  (1.137)

which is the same as the set of subgroups of (ℤ/Nℤ)^{2}. For each L ∈ S_{N} we define a projection π(N, L) by

π(N, L) = π_{L} ∏_{L′∈S_{N} ,L′⫋L}(1 − π_{L′}) .  (1.138)

By Lemma 1.40 the range of π(N, L) is exactly the set of ρ ∈ M_{2}(R) such that

ρ(\frac{1}{N} ℤ^{2}) = \frac{1}{N} L .  (1.139)

The general form of the division relations is as follows.

Proposition 1.41 There exists canonical modular forms ω_{N,L,k} of level N and weight 2k, such that for all k and (ρ, τ) ∈ Y they satisfy

∑_{Na=0} X_{a}^{k}(ρ, τ) = ∑_{L∈S_{N}} π(N, L)(ρ) ω_{N,L,k}(τ) .

In fact, we shall give explicit formulas for the ω_{N,L,k} and show in particular that

ω_{N,L,k}(γ τ) = (cτ + d)^{2k} ω_{N,γ^{−1} L,k}(τ),

which implies that ω_{N,L,k} is of level N. We prove it for k = 1 and then proceed by induction on k. The division formulas in weight 2 involve the 1-cocycle on the group GL_{2}^{+}(ℚ) with values in Eisenstein series of weight 2 given in terms of the Dedekind η-function by (cf. [10])

μ_{γ} (τ) = \frac{1}{12πi} \frac{d}{dτ} log \frac{Δ|γ}{Δ} = \frac{1}{2πi} \frac{d}{dτ} log \frac{η^{4}|γ}{η^{4}} ,  (1.140)

where we used the standard ‘slash operator’ notation for the action of GL_{2}^{+}(ℝ) on functions on the upper half plane:

f|_{k} α (z) = Det (α)^{k/2} f(α · z) j(α, z)^{−k} ,  (1.141)

α = [\begin{matrix}a & b \\ c & d \end{matrix}] ∈ GL_{2}^{+}(ℝ), α · z = \frac{az + b}{cz + d} and j(α, z) = cz + d .

Since μ_{γ} = 0 for γ ∈ Γ, the cocycle property

μ_{γ_{1}·γ_{2}} = μ_{γ_{1}}|γ_{2} + μ_{γ_{2}}  (1.142)

shows that, for m ∈ M_{2}(ℤ)^{+}, the value of μ_{m^{−1}} only depends upon the lattice L = m(ℤ^{2}). We shall denote it by μ_{L}.

Lemma 1.42 For any integer N, the X_{a}, a ∈ ℚ/ℤ, fulfill the relation

∑_{N a=0} X_{a} = N^{2} ∑_{L∈S_{N}} π(N, L) μ_{L}

By construction the projections π(N, L), L ∈ S_{N} form a partition of unity,

∑_{L∈S_{N}} π(N, L) = 1 .

Thus to prove the lemma it is enough to evaluate both sides on ρ ∈ π(N, L). We can moreover use the equality

μ_{γ^{−1}L} = μ_{L}|γ , ∀γ ∈ Γ

to assume that L and ρ_{N} are of the form

L = [\begin{matrix}d_{1} & 0 \\ 0 & d_{2} \end{matrix}] ℤ^{2} , ρ_{N} = [\begin{matrix}d_{1} & 0 \\ 0 & d_{2} \end{matrix}] , d_{j} | N , d_{1} | d_{2} .

Let d_{2} = n d_{1}. The order of the kernel of ρ_{N} is d_{1} d_{2} and the computation of ∑_{N a=0} X_{a}(ρ, τ) gives

−N^{2}e_{2}(τ) + d_{1} d_{2} N^{2} ∑_{(a,b)∈ℤ^{2}\{0}} (a d_{1} − b d_{2}τ)^{−2}

which gives N^{2}(n e_{2}(n τ) − e_{2}(τ)) = N^{2}μ_{L}.

This proves the proposition for k = 1. Let us proceed by induction using (1.133) to express X_{a}^{k} as E_{2k,a} plus a polynomial of degree < k in X_{a} with coefficients in ℚ[e_{4}, e_{6}]. Thus, we only need to prove the equality

∑_{N a=0} E_{2k,a} = ∑_{L∈S_{N}} π(N, L) α_{N,L,k},

where the modular forms α_{N,L,k} are given explicitly as

α_{N,L,k} = N^{2k}d^{2}n^{k+1}e_{2k}|m^{−1} − Det(m) (e_{2k} − ν_{2k}),

with m ∈ M_{2}(ℤ)^{+}, m(ℤ^{2}) = L, and (d, dn) the elementary divisor of L. The proof is obtained as above by evaluating both sides on arbitrary ρ ∈ π(N, L). □ One can rewrite all the above relations in terms of the weight 0 elements

X_{a} := c X_{a} , E_{2k,a} := c^{k} E_{2k,a} ∈ A_{ℚ}.

In particular, the two basic modular functions c^{2} e_{4} and c^{3} e_{6} are replaced by

c^{2} e_{4} = \frac{1}{5} j (j − 1728) , c^{3} e_{6} = −\frac{2}{35} j (j − 1728)^{2}.

We can now rewrite the relations (1.133) in terms of universal polynomials

P_{n} ∈ ℚ(j)[X],

which express the generators E_{2k,a} in terms of X_{a} by

E_{2k,a} = P_{k}(X_{a}).

In fact, from (1.133) we see that the coefficients of P_{k} are themselves polynomials in j rather than rational fractions, so that

P_{n} ∈ ℚ[j, X].

The first ones are given by

P_{2} = X^{2} − j(j − 1728) , P_{3} = X^{3} − \frac{9}{5} Xj(j − 1728) + \frac{4}{5}j(j − 1728)^{2} , ⋯

1.10 The noncommutative boundary of modular curves

We shall explain in this section how to combine the dual of the GL_{2}-system described above with the idea, originally developed in the work of Connes– Douglas–Schwarz [12] and Manin–Marcolli [37], of enlarging the boundary of modular curves with a noncommutative space that accounts for the degeneration of elliptic curves to noncommutative tori. The GL_{2}-system described in the previous sections admits a “dual” system obtained by considering ℚ-lattices up to commensurability but no longer up to scaling. Equivalently this corresponds to taking the cross product of the GL_{2}-system by the action of the Pontrjagin dual of ℂ^{∗}, which combines the time evolution σ_{t} with an action by the group ℤ of integral weights of modular forms. The resulting space is the total space of the natural ℂ^{∗}-bundle. In adélic terms this “dual” noncommutative space L_{2} is described as follows,

Proposition 1.43 There is a canonical bijection from the space of GL_{2}(ℚ)- orbits of the left action of GL_{2}(ℚ) on M_{2}(𝔸_{f}) × GL_{2}(ℝ) to the space L_{2} of commensurability classes of two-dimensional ℚ-lattices.

Proof. The space of GL_{2}(ℚ) orbits on M_{2}(𝔸_{f}) × GL_{2}(ℝ) is the same as the space of GL_{2}^{+}(ℚ) orbits on M_{2}(R) × GL_{2}^{+}(ℝ). □ By the results of the previous sections, the classical space obtained by considering the zero temperature limit of the quantum statistical mechanical system describing commensurability classes of 2-dimensional ℚ-lattices up to scaling is the Shimura variety that represents the projective limit of all the modular curves

GL_{2}(ℚ)\GL_{2}(𝔸)/ℂ^{∗} .  (1.143)

Usually, the Shimura variety is constructed as the projective limit of the

Γ′\GL_{2}(ℝ)^{+}/ℂ^{∗}

over congruence subgroups Γ′ ⊂ Γ. This gives a connected component in (1.143). The other components play a crucial role in the present context, in that the existence of several connected components allows for non-constant solutions of the equation ζ^{n} = 1. Moreover all the components are permuted by the Galois covariance property of the arithmetic elements of the GL_{2} system. The total space of the natural ℂ^{∗}-bundle, i.e. the quotient

GL_{2}(ℚ)\GL_{2}(𝔸),  (1.144)

is the space of invertible 2-dimensional ℚ-lattices (not up to scaling). In the GL_{1} case, the analog of (1.144), i.e. the space of idèle classes

GL_{1}(ℚ)\GL_{1}(𝔸),

is compactified by first considering the noncommutative space of commensurability classes of ℚ-lattices not up to scaling

L = GL_{1}(ℚ)\𝔸^{·},

where 𝔸^{·} is the space of adèles with nonzero archimedean component. The next step, which is crucial in obtaining the geometric space underlying the spectral realization of the zeros of the Riemann zeta function, is to add an additional “stratum” that gives the noncommutative space of adèle classes

\overline{L} = GL_{1}(ℚ)\𝔸 ,

which will be analyzed in the next Chapter. Similarly, in the GL_{2} case, the classical space given by the Shimura variety (1.143) is first “compactified” by adding noncommutative “boundary strata” obtained by replacing GL_{2}(𝔸_{f}) in GL_{2}(𝔸) = GL_{2}(𝔸_{f})×GL_{2}(ℝ) by all matrices M_{2}(𝔸_{f}). As boundary stratum of the Shimura variety it corresponds to degenerating the invertible ℚ-structure ϕ on the lattice to a non-invertible one and yields the notion of ℚ-lattice. The corresponding space of commensurability classes of ℚ-lattices up to scaling played a central role in this whole chapter. The space of commensurability classes of 2-dimensional ℚ-lattices (not up to scaling) is

L_{2} = GL_{2}(ℚ)\(M_{2}(𝔸_{f}) × GL_{2}(ℝ)) .  (1.145)

On L_{2} we can consider not just modular functions but all modular forms as functions. One obtains in this way an antihomomorphism of the modular Hecke algebra of level one of [10] (with variable α ∈ GL_{2}(ℚ)^{+} restricted to M_{2}(ℤ)^{+}) to the algebra of coordinates on L_{2}. The further compactification at the archimedean place, corresponding to L ↪ \overline{L} in the GL_{1} case, now consists of replacing GL_{2}(ℝ) by matrices M_{2}(ℝ). This corresponds to degenerating the lattices to pseudo-lattices (in the sense of [34]) or in more geometric terms, to a degeneration of elliptic curves to noncommutative tori. It is this part of the “noncommutative compactification” that was considered in [12] and [37]. A ℚ-pseudolattice in ℂ is a pair (Λ, ϕ), with Λ = j(ℤ^{2}) the image of a homomorphism j : ℤ^{2} → ℓ, with ℓ ⊂ ℝ^{2} ≅ ℂ a real 1-dimensional subspace, and with a group homomorphism

ϕ : ℚ^{2}/ℤ^{2} → ℚΛ/Λ.

The ℚ-pseudolattice is nondegenerate if j is injective and is invertible if ϕ is invertible.

Proposition 1.44 Let ∂Y := M_{2}(R) × ℙ^{1}(ℝ). The map

(ρ, θ) ↦ (Λ, ϕ), Λ = ℤ + θℤ, ϕ(x) = ρ_{1}(x) − θρ_{2}(x)  (1.146)

gives an identification

Γ\∂Y ≃ (Space of ℚ-pseudolattices in ℂ)/ℂ^{∗}.  (1.147)

This space parameterizes the degenerations of 2-dimensional ℚ-lattices

λ(y) = (Λ, ϕ) where Λ = h̃(ℤ + i ℤ) and ϕ = h̃ ∘ ρ,  (1.148)

for y = (ρ, h) ∈ M_{2}(R) × GL_{2}(ℝ) and h̃ = h^{−1}Det (h), when h ∈ GL_{2}(ℝ) degenerates to a non-invertible matrix in M_{2}(ℝ).

Proof. ℚ-pseudolattices in ℂ are of the form

Λ = λ(ℤ + θℤ), ϕ(a) = λρ_{1}(a) − λθρ_{2}(a),  (1.149)

for λ ∈ ℂ^{∗} and θ ∈ ℙ^{1}(ℝ) and ρ ∈ M_{2}(R). The action of ℂ^{∗} multiplies λ, while leaving θ unchanged. This corresponds to changing the 1-dimensional linear subspace of ℂ containing the pseudo-lattice and rescaling it. The action of SL_{2}(ℤ) on ℙ^{1}(ℝ) by fractional linear transformations changes θ. The nondegenerate pseudolattices correspond to the values θ ∈ ℙ^{1}(ℝ) ∖ ℙ^{1}(ℚ) and the degenerate pseudolattices to the cusps ℙ^{1}(ℚ). For y = (ρ, h) ∈ M_{2}(R) × GL_{2}(ℝ) consider the ℚ-lattice (1.148), for h̃ = h^{−1}Det (h). Here we use the basis {e_{1} = 1, e_{2} = −i} of the ℝ-vector space ℂ to let GL_{2}^{+}(ℝ) act on ℂ as ℝ-linear transformations. These formulas continue to make sense when h ∈ M_{2}(ℝ) and the image Λ = h̃(ℤ + i ℤ) is a pseudolattice when the matrix h is no longer invertible. To see this more explicitly, consider the right action

m ↦ m · z  (1.150)

of ℂ^{∗} on M_{2}(ℝ) determined by the inclusion ℂ^{∗} ⊂ GL_{2}(ℝ) as in (1.80), The action of ℂ^{∗} on M_{2}(ℝ) ∖ {0} is free and proper. The map

ρ(α) = \{ \begin{matrix}α(i) & (c, d) ≠ (0, 0) \\ ∞ & (c, d) = (0, 0) \end{matrix}  with α = [ \begin{matrix}a & b \\ c & d \end{matrix} ]  (1.151)

defines an isomorphism

ρ : (M_{2}(ℝ) ∖ {0})/ℂ^{∗} → ℙ^{1}(ℂ),  (1.152)

equivariant with respect to the left action of GL_{2}(ℝ) on M_{2}(ℝ) and the action of GL_{2}(ℝ) on ℙ^{1}(ℂ) by fractional linear transformations. Moreover, this maps M_{2}(ℝ)^{+} to the closure of the upper half plane

\overline{ℍ} = ℍ ∪ ℙ^{1}(ℝ).  (1.153)

The rank one matrices in M_{2}(ℝ) map to ℙ^{1}(ℝ) ⊂ ℙ^{1}(ℂ). In fact, the isotropy group of m ∈ M_{2}(ℝ) is trivial if m ≠ 0, since m · z = m only has nontrivial solutions for m = 0, since z − 1 is invertible when nonzero. This shows that M_{2}(ℝ)\{0} is the total space of a principal ℂ^{∗}-bundle. □ Notice that, unlike the case of ℚ-lattices of (1.87), where the quotient Γ\Y can be considered as a classical quotient, here the space Γ\∂Y should be regarded as a noncommutative space with function algebra

C(∂Y) ⋊ Γ.

The usual algebro–geometric compactification of a modular curve Y_{Γ′} = Γ′\ℍ, for Γ′ a finite index subgroup of Γ, is obtained by adding the cusp points Γ\ℙ^{1}(ℚ),

X_{Γ′} = Y_{Γ′} ∪ { cusps } = Γ′\(ℍ ∪ ℙ^{1}(ℚ)) .  (1.154)

Replacing GL_{2}(ℝ) by M_{2}(ℝ) in (1.145) corresponds to replacing the cusp points ℙ^{1}(ℚ) by the full boundary ℙ^{1}(ℝ) of ℍ. Since Γ does not act discretely on ℙ^{1}(ℝ), the quotient is best described by noncommutative geometry, as the cross product C^{∗}-algebra C(ℙ^{1}(ℝ)) ⋊ Γ′ or, up to Morita equivalence,

C(ℙ^{1}(ℝ) × ℙ) ⋊ Γ,  (1.155)

with ℙ the coset space ℙ = Γ/Γ′. The noncommutative boundary of modular curves defined this way retains a lot of the arithmetic information of the classical modular curves. Various results of [37] show, from the number theoretic point of view, why the irrational points of ℙ^{1}(ℝ) in the boundary of ℍ should be considered as part of the compactification of modular curves. The first such result is that the classical definition of modular symbols (cf. [36]), as homology classes on modular curves defined by geodesics connecting cusp points, can be generalized to “limiting modular symbols”, which are asymptotic cycles determined by geodesics ending at irrational points. The properties of limiting modular symbols are determined by the spectral theory of the Ruelle transfer operator of a dynamical system, which generalizes the Gauss shift of the continued fraction expansion by taking into account the extra datum of the coset space ℙ. Manin’s modular complex (cf. [36]) gives a combinatorial presentation of the first homology of modular curves, useful in the explicit computation of the intersection numbers obtained by pairing modular symbols to cusp forms. It is shown in [37] that the modular complex can be recovered canonically from the K-theory of the C^{∗}-algebra (1.155). Moreover, Mellin transforms of cusp forms of weight two for the congruence subgroups Γ_{0}(p), with p prime, can be obtained by integrating along the boundary ℙ^{1}(ℝ) certain “automorphic series” defined in terms of the continued fraction expansion and of modular symbols. These extensions of the theory of modular symbols to the noncommutative boundary appear to be interesting also in relation to the results of [10], where the pairing with modular symbols is used to give a formal analog of the Godbillon– Vey cocycle and to obtain a rational representative for the Euler class in the group cohomology H^{2}(SL_{2}(ℚ), ℚ). The fact that the arithmetic information on modular curves is stored in their noncommutative boundary (1.155) is interpreted in [38] as an instance of the physical principle of holography. Noncommutative spaces arising at the boundary of Shimura varieties have been further investigated by Paugam [41] from the point of view of Hodge structures. This noncommutative boundary stratum of modular curves representing degenerations of lattices to pseudo-lattices has been proposed by Manin ([34] [35]) as a geometric space underlying the explicit class field theory problem for real quadratic fields. In fact, this is the first unsolved case of the Hilbert 12th problem. Manin developed in [34] a theory of real multiplication, where noncommutative tori and pseudolattices should play for real quadratic fields a role parallel to the one that lattices and elliptic curves play in the construction of generators of the maximal abelian extensions of imaginary quadratic fields. The picture that emerges from this “real multiplication program” is that the cases of ℚ (Kronecker–Weber) and of both imaginary and real quadratic fields should all have the same underlying geometry, related to different specializations of the GL_{2} system. The relation of the GL_{2} system and explicit class field theory for imaginary quadratic fields is analyzed in [13].

1.11 The BC algebra and optical coherence

It is very natural to look for concrete physical realizations of the phase transition exhibited by the BC system. An attempt in this direction has been proposed in [43], in the context of the physical phenomenon of quantum phase locking in lasers. This interpretation relates the additive generators e(r) of the BC algebra (cf. Proposition 1.6) with the quantum phase states, which are a standard tool in the theory of optical coherence (cf. e.g. [32]), but it leaves open the interpretation of the generators μ_{n}. Since on a finite dimensional Hilbert space isometries are automatically unitary, this rules out nontrivial representations of the μ_{n} in a fixed finite dimensional space. After recalling the basic framework of phase states and optical coherence, we interpret the action of the μ_{n} as a “renormalization” procedure, relating the quantum phase states at different scales. There is a well known analogy (cf. [46] §21-3) between the quantum statistical mechanics of systems with phase transitions, such as the ferromagnet or the Bose condensation of superconducting liquid Helium, and the physics of lasers, with the transition to single mode radiation being the analog of “condensation”. The role of the inverse temperature β is played in laser physics by the “population inversion” parameter, with critical value at the inversion threshold. The injected signal of the laser acts like the external field responsible for the symmetry breaking mechanism. Given these identifications, one in fact obtains similar forms in the two systems for both thermodynamic potential and statistical distribution. The phase locking phenomenon is also analogous in systems with phase transitions and lasers, with the modes in the laser assuming same phase and amplitude above threshold being the analog of Cooper pairs of electrons acquiring the same energy and phase below critical temperature in superconductors. In a laser cavity typically many longitudinal modes of the radiation are oscillating simultaneously. For a linewidth Δν around a frequency ν_{0} for the active medium in the cavity of length L and frequency spacing δν = c/2L, the number

Figure 1.3: Output pulse train in lasers above threshold.

of oscillating modes is N = ⌈Δν/δν⌉ and the field output of the laser is

E(x, t) = ∑_{n=−N/2}^{N/2} A_{n} exp(−2πiν_{n}(t − x/c) + 2πiθ_{n}),  (1.156)

with all the beat frequencies between adjacent modes ν_{n} − ν_{n−1} = δν. Due to noise in the cavity all these modes are uncorrelated, with a random distribution of amplitudes A_{n} and phases θ_{n}. A mode locking phenomenon induced by the excited lasing atoms is responsible for the fact that, above the threshold of population inversion, the phases and amplitudes of the frequency modes become locked together. The resulting field

E(x, t) = Ae^{2πiθ} exp(−2πiν_{0}(t − x/c)) (\frac{sin(πδν(N + 1)(t − x/c))}{sin(πδν(t − x/c))}) , (1.157)

shows many locked modes behaving like a single longitudinal mode oscillating inside the cavity (cf. Figure 1.3). This phenomenon accounts for the typical narrowness of the laser linewidth and monocromaticity of laser radiation.

Since the interaction of radiation and matter in lasers is essentially a quantum mechanical phenomenon, the mode locking should be modeled by quantum mechanical phase operators corresponding to the resonant interaction of many different oscillators. In the quantum theory of radiation one usually describes a single mode by the Hilbert space spanned by the occupation number states |n⟩, with creation and annihilation operators a^{∗} and a that raise and lower the occupation numbers and satisfy the relation [a, a^{∗}] = 1. The polar decomposition a = S\sqrt{N} of the annihilation operator is used to define a quantum mechanical phase operator, which is conjugate to the occupation number operator N = a^{∗}a. Similar phase operators are used in the modeling of Cooper pairs. This approach to the definition of a quantum phase has the drawback that the one sided shift S is not a unitary operator. This can also be seen in the fact that the inverse Cayley transform of S, which gives the cotangent of the phase, is a symmetric non self-adjoint operator.

The emission of lasers above threshold can be described in terms of coherent state excitations,

|α⟩ = exp(−|α|^{2}) ∑_{n} \frac{α^{n}}{(n!)^{1/2}} |n⟩,

which are eigenfunctions of the annihilation operators,

a |α⟩ = α |α⟩.

These are quantum mechanical analogs of classical electromagnetic waves as in (1.156) (1.157). One can show (cf. e.g. [32] §7.4) that the field excitation in a laser approaches a coherent state as the pumping increases to values above the population inversion threshold, with the phase diffusion governed by the equation of motion for quantum mechanical phase states. The problem in defining a proper quantum phase operator, due to lack of selfadjointness, has been overcome by the following approximation of the basic quantum operators on the Fock space H. One selects a scale, given by a positive integer N ∈ ℕ and cuts down H to a finite dimensional subspace by the phase state projector

P_{N} = ∑_{m} |θ_{m,N}⟩ ⟨θ_{m,N}|,

where the orthonormal vectors |θ_{m,N}⟩ in H are given by

|θ_{m,N}⟩ := \frac{1}{(N + 1)^{1/2}} ∑_{n=0}^{N} exp (2πi\frac{m n}{N + 1}) |n⟩.  (1.158)

These are eigenvectors for the phase operators, that affect discrete values given by roots of unity, replacing a continuously varying phase. This way, phase and occupation number behave like positions and momenta. An occupation number state has randomly distributed phase and, conversely, a phase state has a uniform distribution of occupation numbers. We now realize the ground states of the BC system as representations of the algebra A in the Fock space H of the physical system described above. Given an embedding ρ : ℚ^{ab} → ℂ, which determines the choice of a ground state, the generators e(r) and μ_{n} (cf. Proposition 1.6) act as

e(a/b) |n⟩ = ρ(ζ_{a/b}^{n}) |n⟩,
μ_{k} |n⟩ = |kn⟩.

In the physical system, the choice of the ground state is determined by the primitive N + 1-st root of unity

ρ(ζ_{N+1}) = exp(2πi/(N + 1)).

One can then write (1.158) in the form

|θ_{m,N}⟩ = e (\frac{m}{N + 1}) · v_{N}.  (1.159)

where we write v_{N} for the superposition of the first N + 1 occupation states

v_{N} := \frac{1}{(N + 1)^{1/2}} ∑_{n=0}^{N} |n⟩.

Any choice of a primitive N + 1-st root of unity would correspond to another ground state, and can be used to define analogous phase states. This construction of phase states brings in a new hidden group of symmetry, which is different from the standard rotation of the phase, and is the Galois group Gal(ℚ^{ab}/ℚ). This raises the question of whether such symmetries are an artifact of the approximation, or if they truly represent a property of the physical system. In the BC algebra, the operators μ_{n} act on the algebra generated by the e(r) by endomorphisms given by

μ_{n} P (e(r_{1}), …, e(r_{k})) μ_{n}^{∗} = \frac{π_{n}}{n^{k}} ∑_{ns=r} P (e(s_{1}), …, e(s_{k})),  (1.160)

for an arbitrary polynomial P in k-variables, with π_{n} = μ_{n}μ_{n}^{∗} and s = (s_{1}, …, s_{k}), r = (r_{1}, …, r_{k}) in (ℚ/ℤ)^{k}. In particular, this action has the effect of averaging over different choices of the primitive roots. The averaging on the right hand side of (1.160), involving arbitrary phase observables P (e(r_{1}), …, e(r_{k})), has physical meaning as statistical average over the choices of primitive roots. The left hand side implements this averaging as a renormalization group action. Passing to the limit N → ∞ for the phase states is a delicate process. It is known in the theory of optical coherence (cf. e.g. [33] §10) that one can take such limit only after expectation values have been calculated. The analogy between the laser and the ferromagnet suggests that this limiting procedure should be treated as a case of statistical limit, in the sense of [57]. In fact, when analyzing correlations near a phase transition, one needs a mechanism that handles changes of scale. In statistical mechanics, such mechanism exists in the form of a renormalization group, which expresses the fact that different length or energy scales are locally coupled. This is taken care here by the action of (1.160).

Bibliography

[1] J. Arledge, M. Laca, I. Raeburn, Semigroup crossed products and Hecke algebras arising from number fields, Doc. Math. 2 (1997) 115–138.

[2] M.W. Binder, Induced factor representations of discrete groups and their types, J. Functional Analysis, 115 (1993), 294–312.

[3] F. Boca and A. Zaharescu, Factors of type III and the distribution of prime numbers, Proc. London Math. Soc. (3) Vol.80 (2000), no. 1, 145–178.

[4] J.-B. Bost and A. Connes, Produits Euleriens et facteurs de type III, C. R. Acad. Sci. Paris Sér. I Math. Vol.315 (1992), no. 3, 279–284.

[5] J.B. Bost, A. Connes, Hecke algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. (New Series) Vol.1 (1995) N.3, 411–457.

[6] O. Bratteli, D.W. Robinson. Operator Algebras and Quantum Statistical Mechanics I and II, Springer, New York 1979 and 1981.

[7] B. Brenken, Hecke algebras and semigroup crossed product C*-algebras, Pacific J. Math., 187 (1999), 241–262.

[8] P.B. Cohen, A C^{∗}-dynamical system with Dedekind zeta partition function and spontaneous symmetry breaking, Journal de Théorie des Nombres de Bordeaux 11 (1999) 15–30.

[9] A. Connes, Trace formula in noncommutative geometry and the zeros of the Riemann zeta function, Selecta Math. (New Series) Vol.5 (1999) 29–106.

[10] A. Connes, H. Moscovici, Modular Hecke Algebras and their Hopf Symmetry, Moscow. Math. Journal, vol.4, n 1, (2004) 67–109.

[11] A. Connes, A survey of foliations and operator algebras, Proc. Sympos. Pure Math., 38, Amer. Math. Soc., Providence, R.I., (1982) 521–628.

[12] A. Connes, M. Douglas, A. Schwarz, Noncommutative geometry and matrix theory: compactification on tori, J. High Energy Phys. 1998, no. 2, Paper 3, 35 pp. (electronic).

[13] A. Connes, M. Marcolli, N. Ramachandran, KMS states and complex multiplication, in preparation.

[14] J. Dixmier, Les C^{∗}-algèbres et leurs Représentations, Gauthier-Villars, Paris 1964.

[15] H. Glöckner and G. Willis, Topologisation of Hecke pairs and Hecke C^{∗}- algebras, (preprint), University of Newcastle, Australia (2002).

[16] R. Haag, Local Quantum Physics, Springer, Berlin 1992.

[17] R. Haag, N. M. Hugenholtz, M. Winnink On the equilibrium states in quantum statistical mechanics, Comm. Math. Phys. 5 (1967), 215–236.

[18] R. Hall, Hecke C*-algebras, PhD Thesis, Pennsylvania State University (1999).

[19] D. Harari, E. Leichtnam, Extension du phénomène de brisure spontanée de symétrie de Bost–Connes au cas des corps globaux quelconques, Selecta Math. (New Series) Vol.3 (1997) 205–243.

[20] B. Julia, Statistical theory of numbers, in “Number Theory and Physics”, J.-M. Luck, P. Moussa and M. Waldschmidt (Eds.), Springer Verlag, Berlin, 1990.

[21] M. Laca, Semigroups of ∗-endomorphisms, Dirichlet series, and phase transitions, J. Funct. Anal. 152 (1998) 330–378.

[22] M. Laca, From endomorphisms to automorphisms and back: dilations and full corners, J. London Math. Soc. (2) 61 (2000), no. 3, 893–904.

[23] M. Laca and N. Larsen, Hecke algebras of semidirect products, Proc. Amer. Math. Soc. (to appear in 2003).

[24] M. Laca and I. Raeburn, Semigroup crossed products and the Toeplitz algebras of nonabelian groups, J. Funct. Anal., 139 (1996), 415–440.

[25] M. Laca and I. Raeburn, A semigroup crossed product arising in number theory. J. London Math. Soc. (2) 59 (1999), no. 1, 330–344.

[26] M. Laca and I. Raeburn, The ideal structure of the Hecke C*-algebra of Bost and Connes, Math. Ann. 318 (2000), no. 3, 433–451.

[27] M. Laca and M. van Frankenhuysen, Phase transitions on Hecke C*- algebras and class field theory (in preparation).

[28] S. Lang, Elliptic Functions, (Second Edition), Graduate Texts in Mathematics, Vol.112, Springer-Verlag 1987.

[29] N. Larsen, I. Putnam, I. Raeburn, The two-prime analogue of the Hecke C^{∗}-algebra of Bost and Connes, Indiana Univ. Math. J. Vol.51 (2002), no. 1, 171–186.

[30] N. Larsen and I. Raeburn, Representations of Hecke algebras and dilations of semigroup cross products, J. London Math. Soc. (2) 66 (2002), no. 1, 198–212.

[31] E. Leichtnam and V. Nistor, Cross product algebras and the homology of certain p-adic and adelic dynamical systems, K-Theory, 21 (2000), 1–23.

[32] R. Loudon, The Quantum Theory of Light, third edition, Oxford University Press, 2000.

[33] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, 1995.

[34] Yu.I. Manin, Real Multiplication and noncommutative geometry, preprint arXiv math.AG/0202109.

[35] Yu.I. Manin, Von Zahlen und Figuren, preprint arXiv math.AG/0201005.

[36] Yu.I. Manin, Parabolic points and zeta functions of modular curves, Math. USSR Izvestjia 6 (1972) N.1, 19–64.

[37] Yu.I. Manin, M. Marcolli, Continued fractions, modular symbols, and noncommutative geometry, Selecta Math. (New Series) Vol.8 (2002) N.3, 475– 520.

[38] Yu.I. Manin, M. Marcolli, Holography principle and arithmetic of algebraic curves, Adv. Theor. Math. Phys. Vol.3 (2001) N.5, 617–650.

[39] D. Mumford, Tata Lectures on Theta. I, Progress in Mathematics, Vo.28, Birkhäuser, Boston, MA, 1983.

[40] S. Neshveyev, Ergodicity of the action of the positive rationals on the group of finite adeles and the Bost-Connes phase transition theorem, Proc. Amer. Math. Soc. 130 (2002), no. 10, 2999–3003.

[41] F. Paugam Three examples of Noncommutative moduli spaces and related questions Preprint 2004.

[42] G. Pedersen, C^{∗}-algebras and their Automorphism Groups. Academic Press, New York 1979.

[43] M. Planat, Invitation to the “spooky” quantum phase–locking effect and its link to 1/f fluctuations, math-ph/0310082.

[44] W. Rudin, Real and Complex Analysis, McGraw-Hill, NewYork 1987.

[45] D. Ruelle, Statistical Mechanics. World Scientific Publishing Co. Inc., River Edge, NJ, 1999. Reprint of the 1989 edition.

[46] M. Sargent, M. Scully, W. Lamb, Laser Physics, Addison–Wesley, 1974.

[47] G. Schlichting, Polynomidentitäten und Permutationsdarstellungen lokalkompakter Gruppen, Invent. Math. 55 (1979) N.2, 97–106.

[48] G. Schlichting, Operationen mit periodischen Stabilisatoren, Arch.Math. 34 (1980) N.2, 97–99.

[49] B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, New York - Heidelberg - Berlin, 1974.

[50] J.P. Serre, Cours d’Arithmétique, Presse Universitaire de France, 1977.

[51] G. Shimura, Arithmetic Theory of Automorphic Functions, Iwanami Shoten and Princeton 1971.

[52] D. Spector, Supersymmetry and the Möbius inversion function, Commun. Math. Phys., Vol.127 (1990), 239–252.

[53] P. Stevenhagen, Hilbert’s 12th problem, complex multiplication and Shimura reciprocity, Advanced Studies in Pure Math. 30 (2001) “Class Field Theory – its centenary and prospect” pp. 161–176.

[54] M. Takesaki, Tomita’s theory of modular Hilbert algebras and its applications. Lecture Notes in Math., 28, Springer, 1970.

[55] K. Tzanev, Hecke C*-algebras and amenability. To appear in JOT.

[56] A. Weil, Elliptic functions according to Eisenstein and Kronecker, Springer 1976.

[57] K.G. Wilson, The renormalization group: critical phenomena and the Kondo problem, Rev. Mod. Phys. Vol. 47 (1975) N.4, 773–840.

A. Connes Collège de France, 3, rue Ulm, F-75005 Paris, France I.H.E.S. 35 route de Chartres F-91440 Bures-sur-Yvette, France Department of Mathematics, Vanderbilt University, TN-37240, USA email: connes@ihes.fr, alain@connes.org

M. Marcolli Max–Planck Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany email: marcolli@mpim-bonn.mpg.de