ℚ-LATTICES: QUANTUM STATISTICAL MECHANICS AND GALOIS THEORY

ALAIN CONNES AND MATILDE MARCOLLI

1. Introduction

This paper summarizes the main aspects of our joint work [4] on quantum statistical mechanics of ℚ-lattices, with a view towards its relations to class field theory investigated in our joint work with Ramachandran [5].

The noncommutative geometry of the space of ℚ-lattices modulo the equivalence relation of commensurability provides a setting that unifies several phenomena involving the interaction of noncommutative geometry and number theory. These include, in the 1-dimensional case, the Bost–Connes (BC) system [1] with arithmetic spontaneous symmetry breaking and its dual space under the duality given by taking the crossed product with the time evolution. The latter is the noncommutative space underlying the construction of the spectral realization of the zeros of the Riemann zeta function in [3]. The corresponding space in the 2-dimensional case contains in its algebra of coordinates the modular Hecke algebras of [6] [7]. The noncommutative compactifications of modular curves of [14] also appear here as a stratum in the compactification of the space of commensurability classes of 2-dimensional ℚ-lattices.

Moreover, an interesting and difficult problem is the generalization of the results of [1] to other number fields. (For an overview of existing results in this direction we refer the reader to the “further developments” section of [4] and the references quoted therein.)

The space of commensurability classes of 2-dimensional ℚ-lattices up to scaling, which is the main object of this paper, provides a new approach to the problem, for the case of quadratic fields. In fact, while the BC system is closely related to the Kronecker–Weber construction of the maximal abelian extension of ℚ, we shall see that the 2-dimensional system introduced in [4] is naturally related to the Galois theory of the modular field, which in turn lies at the heart of the explicit class field theory problem for imaginary quadratic fields. A generalization of the results of [1] to imaginary quadratic fields is being investigated in [5]. Moreover, the fact that the noncommutative modular curves of [14] appear in the compactification suggests the possible existence of a path towards the case of real quadratic fields, along the lines of Manin’s real multiplication program [13].

The fundamental notions in all that follows are those of ℚ-lattices and commensurability.

Definition 1.1. A ℚ-lattice in ℝ^{n} consists of a pair (Λ, ϕ) of a lattice Λ ⊂ ℝ^{n} (a cocompact free abelian subgroup of ℝ^{n} of rank n) together with a system of labels of

Figure 1. Generic and invertible 2-dimensional ℚ-lattices

its torsion points given by a homomorphism of abelian groups

ϕ : ℚ^{n}/ℤ^{n} ⟶ ℚΛ/Λ.

Two ℚ-lattices are commensurable,

(Λ_{1}, ϕ_{1}) ∼ (Λ_{2}, ϕ_{2}),

iff ℚΛ_{1} = ℚΛ_{2} and

ϕ_{1} = ϕ_{2} mod Λ_{1} + Λ_{2}

Commensurability defines an equivalence relation among ℚ-lattices. By definition a ℚ-lattice is invertible when ϕ is an isomorphism. Two invertible ℚ-lattices are commensurable if and only if they are equal. While most ℚ-lattices are not commensurable to an invertible one, the set of invertible ℚ-lattices gives a cross-section of the equivalence relation One has u^{∗}u = 1 and uu^{∗} = e. We say that Adu defined The condition (??) ensures that (Adu)^{∗}φ is KMS condition shows that the induced action of an inner endomorphism on KMS states is trivial. One needs to be especially careful in defining the action of endomorphisms by (??). In fact, there are cases where for KMS_{∞} states one finds only φ(e) = 0, yet it is still possible to define an interesting action of endomorphisms by a procedure of “warming up and cooling down”. For this to work one needs sufficiently favorable conditions, namely that the “warming up” map

(1.1)  W_{β}(φ)(a) = \frac{Tr(π_{φ}(a) e^{−β H})}{Tr( e^{−β H})}

gives a homeomorphism W_{β} : E_{∞} → E_{β} for all β sufficiently large. One can then define the action by

(1.2)  (ρ^{∗}φ)(a) = lim_{β→∞} (ρ^{∗}W_{β}(φ)) (a),

for all φ ∈ E_{∞} and all a ∈ A.

2. The Bost–Connes system

In [1] Bost and Connes constructed a C^{∗}-dynamical system (A, σ_{t}) with spontaneous symmetry breaking, which encodes the arithmetic of the cyclotomic field ℚ^{cycl}, that is, of the maximal abelian extension of ℚ by the Kronecker–Weber theorem.

The algebra A of the Bost–Connes system is generated by two types of operators. The first type consists of phase operators e(r), parameterized by elements r ∈ ℚ/ℤ. These can be represented on the Fock space generated by occupation numbers |n⟩ as the operators

(2.1)  e(r)|n⟩ = α(ζ_{r}^{n})|n⟩.

Here we denote by ζ_{a/b} = ζ_{b}^{a} the abstract roots of unity generating ℚ^{cycl} and by α : ℚ^{cycl} ↪ ℂ an embedding that identifies ℚ^{cycl} with the subfield of ℂ generated by the concrete roots of unity.

The operators (2.1) are familiar in the theory of quantum optics, where they are used to define the quantized optical phase as a state

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

where v_{N} is a superposition of occupation states

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

In such quantization of the phase, N is chosen as a scale at which the phase is discretized. One needs then to ensure that the results are consistent over changes of scale.

The other operators that generate the Bost–Connes algebra can be thought of as implementing the changes of scales in the optical phases in a consistent way. These operators are isometries μ_{n} parameterized by positive integers n ∈ ℕ^{×} = ℤ_{>0}. The changes of scale are described by the action of the μ_{n} on the e(r) by

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

In addition to this compatibility condition, the operators e(r) and μ_{n} satisfy other simple relations. These give a presentation of the algebra A of the form ([1], [10]):

• 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, the relation (2.2) holds.

One considers on the algebra A the time evolution given by

(2.3)  σ_{t}(μ_{n}) = n^{it}μ_{n}, σ_{t}(e(r)) = e(r).

2.1. Hecke algebra. The fact that (2.3) defines a natural time evolution is best understood by describing the algebra A as a Hecke algebra for the pair of groups (Γ_{0}, Γ) = (P_{ℤ}, P_{ℚ}), where P is the ax + b group. This is the way the algebra A was introduced in [1]. Whenever the inclusion Γ_{0} ⊂ Γ has the property that the left Γ_{0} orbits of any γ ∈ Γ/Γ_{0} are finite (same for right orbits on the left coset), one can consider the Hecke algebra of the pair (Γ_{0}, Γ) given by functions on Γ_{0}\Γ/Γ_{0} with the convolution product

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

and the involution f^{∗}(γ) := \overline{f(γ^{−1})}. The Hecke algebra defined this way has a regular representation on the Hilbert space ℓ^{2}(Γ_{0}\Γ)

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

The canonical time evolution on the corresponding von Neumann algebra is determined by the ratio of the length of left and right Γ_{0} orbits,

(2.5)  σ_{t}(f)(γ) = (\frac{L(γ)}{R(γ)})^{−it} f(γ),

where, for γ ∈ Γ/Γ_{0} we set L(γ) = #Γ_{0}γ and R(γ) = L(γ^{−1}). In the case of the pair (Γ_{0}, Γ) of parabolic subgroups (P_{ℤ}^{+}, P_{ℚ}^{+}) of GL_{2}^{+}(ℚ), the Hecke algebra (2.4) gives the Bost–Connes algebra and the time evolution (2.5) is given by (2.3).

2.2. 1-dimensional ℚ-lattices. We now return to the point of view of ℚ-lattices. As showed in [4], the algebra A of the Bost–Connes system has a natural interpretation as the noncommutative algebra of coordinates of the space L_{1}/ℝ_{+}^{∗} of 1-dimensional ℚ-lattices (up to scaling) modulo commensurability. In fact, a 1-dimensional ℚ-lattice can always be written in the form

(2.6)  (Λ, ϕ) = (λ ℤ, λ ρ)

for some λ > 0 and some

(2.7)  ρ ∈ Hom(ℚ/ℤ, ℚ/ℤ) = \underleftarrow{lim} ℤ/nℤ = ℤ̂.

By considering lattices up to scaling, we eliminate the factor λ > 0 so that 1dimensional ℚ-lattices up to scale are completely specified by the choice of the element ρ ∈ ℤ̂. Thus, the algebra of coordinates of the space of 1-dimensional ℚ-lattices up to scale is the commutative C^{∗}-algebra

(2.8)  C(ℤ̂) ≃ C^{∗}(ℚ/ℤ).

The identification in (2.8) results from the fact that ℤ̂ is the Pontrjagin dual of ℚ/ℤ. The equivalence relation of commensurability is implemented by the action of the semigroup ℕ^{×} on ℚ-lattices. The corresponding action on the algebra (2.8) is by

(2.9)  α_{n}(f)(ρ) = \{ \begin{matrix}f(n^{−1}ρ) & ρ ∈ nℤ̂ \\ 0 & otherwise. \end{matrix}

Thus, the quotient of the space of 1-dimensional ℚ-lattices up to scale by the commensurability relation and its algebra of coordinates of is given by the semigroup crossed product

(2.10)  C^{∗}(ℚ/ℤ) ⋊ ℕ^{×}.

This is another description of the Bost–Connes algebra, as (2.10) has the right set of generators and relations, with (2.2) implementing the semigroup action (2.9)

2.3. Structure of KMS states. The Bost–Connes algebra has irreducible representations on the Hilbert space H = ℓ^{2}(ℕ^{×}). These are parameterized by elements α ∈ ℤ̂^{∗} = GL_{1}(ℤ̂). Any such element defines an embedding α : ℚ^{cycl} ↪ ℂ and the corresponding representation is of the form

(2.11)  \begin{matrix}π_{α}(e(r)) ϵ_{k} = & α(ζ_{r}^{k}) ϵ_{k} \\ π_{α}(μ_{n}) ϵ_{k} = & ϵ_{nk} \end{matrix}

The Hamiltonian implementing the time evolution (2.9) on H is

(2.12)  H ϵ_{k} = log k ϵ_{k}

Thus, the partition function of the Bost–Connes system is the Riemann zeta function

(2.13)  Z(β) = Tr (e^{−βH}) = ∑_{k=1}^{∞} k^{−β} = ζ(β).

Bost and Connes showed in [1] that KMS states have the following structure, with a phase transition at β = 1.

• In the range β ≤ 1 there is a unique KMS_{β} state. Its restriction to ℚ[ℚ/ℤ] is of the form

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

• For 1 < β ≤ ∞ the set of extremal KMS states E_{β} can be identified with ℤ̂^{∗}. It has a free and transitive action of this group induced by an action on A by automorphisms. The extremal KMS_{β} state corresponding to α ∈ ℤ̂^{∗} is of the form

(2.14)  φ_{β,α}(x) = \frac{1}{ζ(β)} Tr (π_{α}(x) e^{−βH}) .

• At β = ∞ the Galois group Gal(ℚ^{cycl}/ℚ) acts on the values of states φ ∈ E_{∞} on an arithmetic subalgebra A_{ℚ} ⊂ A. These have the property that φ(A_{ℚ}) ⊂ ℚ^{cycl} and that the isomorphism (class field theory isomorphism) θ : Gal(ℚ^{cycl}/ℚ) →^{≅} ℤ̂^{∗} intertwines the Galois action on values with the action of ℤ̂^{∗} by symmetries, namely,

(2.15)  γ φ(x) = φ(θ(γ) x),

for all φ ∈ E_{∞}, for all γ ∈ Gal(ℚ^{cycl}/ℚ) and for all x ∈ A_{ℚ} Here the arithmetic subalgebra can be taken as the algebra over ℚ generated by the e(r) and μ_{n}, μ_{n}^{∗}, or equivalently as the Hecke algebra of compactly supported ℚ-valued functions on Γ_{0}\Γ with the convolution product (2.4). As we shall see, a different description of the arithmetic subalgebra is given in [4] in terms of homogeneous weight zero functions of ℚ-lattices. The choice of an “arithmetic subalgebra” corresponds to endowing the noncommutative space A with an arithmetic structure. The subalgebra corresponds to the rational functions and the values of KMS_{∞} states at elements of this subalgebra should be thought of as “values of rational functions at classical points” (cf. [5]). 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. Moreover, these span the maximal abelian extension of ℚ and 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. In general, the fact that the Galois action on the values of states would preserve positivity (i.e. would give values of other states) is a very unusual property. We refer to such states as “fabulous states”.

2.4. Noncommutative Geometry and Class Field Theory. The main result of Bost–Connes [1] on the structure of KMS states for the system described above suggests the possibility of a connection between noncommutative geometry and class field theory. If 𝕂 is a number field with [𝕂 : ℚ] = n, and 𝕂̄ is an algebraic closure of 𝕂, then one has the Galois group Gal(𝕂̄/𝕂). This group of symmetries is a very beautiful object, and quite mysterious even in the case of 𝕂 = ℚ. On the other hand, one can consider a smaller field than 𝕂̄, namely the maximal abelian extension 𝕂^{ab} of 𝕂. This has the property that

Gal(𝕂^{ab}/𝕂) = Gal(𝕂̄/𝕂)^{ab}.

The Kronecker–Weber theorem shows that for 𝕂 = ℚ

ℚ^{ab} = ℚ^{cycl} and Gal(ℚ^{ab}/ℚ) ≃ ℤ̂^{∗}.

Finding an analogous result for more general number fields is the content of Hilbert’s 12th problem, the problem of explicit class field theory. For a number field 𝕂 one knows that there is an identification (the class field theory isomorphism)

(2.16)  θ : C_{𝕂}/D_{𝕂} ⟶^{≃} Gal(𝕂^{ab}/𝕂),

where C_{𝕂} = 𝔸_{𝕂}^{∗}/𝕂^{∗} is the group of idèle classes and D_{𝕂} the connected component of the identity in C_{𝕂}. In the explicit class field theory problem one wants to obtain an explicit set of generators for 𝕂^{ab} and an explicit description of the action of Gal(𝕂^{ab}/𝕂). Remarkably, a complete solution to Hilbert’s 12th problem exists only for ℚ and for the imaginary quadratic fields ℚ(\sqrt{−d}). The first challenge is posed by the case of real quadratic fields ℚ(\sqrt{d}). It is natural to ask whether noncommutative geometry can provide some new insight on the Hilbert 12th problem, at least for the case of real quadratic fields. A series of beautiful reflections on this theme is given in Manin’s real multiplication project [13].

The Bost–Connes system has also an adèlic description [1], where the algebra A is Morita equivalent to the crossed product

(2.17)  C_{0}(𝔸_{f}) ⋊ ℚ_{+}^{∗}

(cf. [11]) with 𝔸_{f} = ℤ̂ ⊗ ℚ the finite adèles of ℚ. The set of extremal KMS states below critical temperature can also be described as the adèlic quotient

(2.18)  E_{∞} ≃ GL_{1}(ℚ)\GL_{1}(𝔸)/ℝ_{+}^{∗},

with 𝔸 = 𝔸_{f} × ℝ the full adèles of ℚ. Given a number field 𝕂 with [𝕂 : ℚ] = n, there is an embedding 𝕂^{∗} ↪ GL_{n}(ℚ) of its multiplicative group in GL_{n}(ℚ). Such embedding induces an embedding of GL_{1}(𝔸_{𝕂,f}) where 𝔸_{𝕂,f} = 𝔸_{f} ⊗ 𝕂 are the finite adèles of 𝕂 into GL_{n}(𝔸_{f}). This suggests a possible strategy to develop an approach to explicit class field theory via the construction of “fabulous states” for quantum statistical mechanical systems associated to other number fields, by studying GL_{n} analogs of the Bost– Connes system. This was done (especially in the case of GL_{2}) in [4]. In the case of GL_{2}, one sees that the geometry of modular curves and the algebra of modular forms appear naturally. These are the main ingredients also in the solution of the explicit class field theory problem for imaginary quadratic fields (cf. [16]).

3. The GL_{2} system

In this section we will describe the main features of the GL_{2} analog of the Bost– Connes system, according to the results of [4]. In the following, to avoid confusion, we use the notation A_{1} and A_{1,ℚ} for the C^{∗}- algebra of the Bost–Connes system and its arithmetic subalgebra and A_{2} and A_{2,ℚ} for the analogs in the GL_{2} case. Any 2-dimensional ℚ-lattice can be written in the form

(Λ, ϕ) = (λ(ℤ + ℤτ), λρ),

for some λ ∈ ℂ^{∗}, some τ ∈ ℍ, and some ρ ∈ M_{2}(ℤ̂) = Hom(ℚ^{2}/ℤ^{2}, ℚ^{2}/ℤ^{2}). Thus, the space of 2-dimensional ℚ-lattices up to the scale factor λ ∈ ℂ^{∗} and up to isomorphisms, is given by

(3.1)  M_{2}(ℤ̂) × ℍ mod Γ = SL(2, ℤ).

The commensurability relation giving the space L_{2}/ℂ^{∗} is implemented by the partially defined action of GL_{2}^{+}(ℚ). More precisely, we proceed as follows. We choose a basis {e_{1} = 1, e_{2} = −i} of ℂ as a vector space over ℝ, with respect to which we define the action of GL_{2}^{+}(ℝ) on ℂ. If we set Λ_{0} = ℤe_{1} + ℤe_{2} = ℤ + iℤ, an element ρ ∈ M_{2}(ℤ̂) defines a homomorphism

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

Consider the quotient of the space

(3.2)  Ũ := {(g, ρ, α) ∈ GL_{2}^{+}(ℚ) × M_{2}(ℤ̂) × GL_{2}^{+}(ℝ) : gρ ∈ M_{2}(ℤ̂)}

by the action of Γ × Γ given by

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

The groupoid R_{2} of the equivalence relation of commensurability on 2-dimensional ℚ-lattices (not considered up to scaling for the moment) is a locally compact groupoid, which can be parameterized by the quotient of (3.2) by Γ × Γ via the map r : Ũ → R_{2},

(3.4)  r(g, ρ, α) = ((α^{−1}g^{−1}Λ_{0}, α^{−1}ρ), (α^{−1}Λ_{0}, α^{−1}ρ)) .

We then consider the quotient by scaling. Upon identifying ℂ^{∗} ⊂ GL_{2}^{+}(ℝ) by

a + ib ∈ ℂ^{∗} ↦ (\begin{matrix}a & b \\ −b & a \end{matrix}) ∈ GL_{2}^{+}(ℝ),

the quotient GL_{2}^{+}(ℝ)/ℂ^{∗} can be identified with the hyperbolic plane ℍ in the usual way

α = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}^{+}(ℝ) ↦ \frac{ai + b}{ci + d} ∈ ℍ.

If (Λ_{k}, ϕ_{k}) k = 1, 2 are a pair of commensurable 2-dimensional ℚ-lattices, then for any λ ∈ ℂ^{∗}, the ℚ-lattices (λΛ_{k}, λϕ_{k}) are also commensurable, with

r(g, ρ, αλ^{−1}) = λr(g, ρ, α).

However, the action of ℂ^{∗} on ℚ-lattices is not free due to the presence of lattices (such as Λ_{0}) with nontrivial automorphisms. Thus, the quotient Z = R_{2}/ℂ^{∗} is no longer a groupoid. Still, one can define a convolution algebra for Z by restricting the convolution product of R_{2} to homogeneous functions of weight zero, where a function f has weight k if it satisfies

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

The space Z is the quotient of the space

(3.5)  U := {(g, ρ, z) ∈ GL_{2}^{+}(ℚ) × M_{2}(ℤ̂) × ℍ|gρ ∈ M_{2}(ℤ̂)}

by the action of Γ × Γ. Here the space M_{2}(ℤ̂) × ℍ has a partially defined action of GL_{2}^{+}(ℚ) given by

g(ρ, z) = (gρ, g(z)),

where g(z) denotes action as fractional linear transformation. Thus, the quotient L_{2}/ℂ^{∗} of the space of 2-dimensional ℚ-lattices up to scale by the relation of commensurability is a noncommutative space whose algebra of coordinates is a Hecke algebra obtained as follows. Consider the space C_{c}(Z) of continuous compactly supported functions on Z. These can be seen, equivalently, as functions on U as in (3.5) invariant under the Γ × Γ action (g, ρ, z) ↦ (γ_{1}gγ_{2}^{−1}, γ_{2}z). One endows C_{c}(Z) with the convolution product

(3.6)  (f_{1} ∗ f_{2})(g, ρ, z) = ∑_{s∈Γ\GL_{2}^{+}(ℚ):sρ∈M_{2}(ℤ̂)}f_{1}(gs^{−1}, sρ, s(z))f_{2}(s, ρ, z)

and the involution f^{∗}(g, ρ, z) = \overline{f(g^{−1}, gρ, g(z))}. The time evolution is given by

(3.7)  σ_{t}(f)(g, ρ, z) = det(g)^{it} f(g, ρ, z).

For ρ ∈ M_{2}(ℤ̂) let

(3.8)  G_{ρ} := {g ∈ GL_{2}^{+}(ℚ) : gρ ∈ M_{2}(ℤ̂)}

and consider the Hilbert space H_{ρ} = ℓ^{2}(Γ\G_{ρ}).

A 2-dimensional ℚ-lattice L = (Λ, ϕ) = (ρ, z) determines a representation of the Hecke algebra by bounded operators on H_{ρ}, setting

(3.9)  (π_{L}(f)ξ) (g) = ∑_{s∈Γ\G_{ρ}} f(gs^{−1}, sρ, s(z)) ξ(s).

In particular, when the ℚ-lattice L = (Λ, ϕ) is invertible one obtains

H_{ρ} ≅ ℓ^{2}(Γ\M_{2}^{+}(ℤ)).

In this case, the Hamiltonian implementing the time evolution (3.7) is given by the operator

(3.10)  H ϵ_{m} = log det(m) ϵ_{m}.

Thus, in the special case of invertible ℚ-lattices (3.9) yields a positive energy representation. In general for ℚ-lattices which are not commensurable to an invertible one, the corresponding Hamiltonian H is not bounded below. The Hecke algebra (3.6) admits a C^{∗}-algebra completion A_{2}, where the norm is the sup over all representations π_{L}. The partition function for this GL_{2} system is given by

(3.11)  Z(β) = ∑_{m∈Γ\M_{2}^{+}(ℤ)} det(m)^{−β} = ∑_{k=1}^{∞} σ(k) k^{−β} = ζ(β)ζ(β − 1),

where σ(k) = ∑_{d|k} d. This already hints to the fact that the system might have more than one phase transition. In fact, the form of the partition function suggests the possibility that two distinct phase transitions might happen at β = 1 and β = 2.

4. KMS states and symmetries

The structure of KMS states for the GL_{2}-system is analysed in [4]. The main result is the following.

Theorem 4.1. The KMS_{β} states of the GL_{2}-system have the following properties: (1) In the range β ≤ 1 there are no KMS states.

(2) In the range β > 2 the set of extremal KMS states is given by the classical Shimura variety

(4.1)  E_{β} ≅ GL_{2}(ℚ)\GL_{2}(𝔸)/ℂ^{∗}.

This shows that the extremal KMS states at sufficiently low temperature are parameterized by the invertible ℚ-lattices. The explicit expression for these extremal KMS_{β} states is obtained as

(4.2)  φ_{β,L}(f) = \frac{1}{Z(β)} ∑_{m∈Γ\M_{2}^{+}(ℤ)} f(1, mρ, m(z)) det(m)^{−β}

where L = (ρ, z) is an invertible ℚ-lattice. The difficult part of the proof is to show that indeed all extremal KMS_{β} states are of this form. When β → 1 from above, the different pure phases merge, so it is reasonable to expect that in the intermediate range 1 < β < 2 there will be a unique KMS_{β} state. Thus, the system exhibits two distinct phase transitions at β = 2 and β = 1.

The main step in the proof of Theorem 4.1 is the construction of a subalgebra generated by projections π_{p}(k, l), where p is a prime number and k, l are integers with k ≤ l, with the following properties

• If φ is a KMS_{β} state for the GL_{2} system, then it satisfies

\begin{matrix}φ(π_{p}(k, l)) = & p^{−(k+l)β}p^{l−k}(1 + p^{−1})(1 − p^{−β})(1 − p^{1−β}), & k < l \\ φ(π_{p}(l, l)) = & p^{−2lβ}(1 − p^{−β})(1 − p^{1−β}) & k = l. \end{matrix}

• If p_{j} are distinct prime numbers, then

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

In particular these properties show that there cannot be any KMS state in the range 0 < β < 1.

4.1. Symmetries. In the range 2 < β ≤ ∞, there is a very interesting action of symmetries on the KMS states of the GL_{2}-system. The symmetry group of A_{2} (including both automorphisms and endomorphisms) can be identified with the group

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

Here the group GL_{2}(ℤ̂) acts by automorphisms,

(4.4)  θ_{γ}(f)(g, ρ, z) = f(g, ργ, z).

Geometrically, this is the group of deck transformations of coverings of modular curves. In fact, when we consider the (compact) modular curve X(n) over the cyclotomic field ℚ(ζ_{n}), these form a tower over the base X(1) = ℙ^{1} over ℚ, and the group GL_{2}(ℤ/nℤ)/±1 is the group of automorphisms of the projection X(n) → X(1), (cf. [15], [5]) so that one obtains the automorphism group

(4.5)  GL_{2}(ℤ̂)/±1 = \underleftarrow{lim}_{n} GL_{2}(ℤ/nℤ)/\{±1\}.

On the other hand, the group GL_{2}^{+}(ℚ) in (4.3) acts by endomorphisms,

(4.6)  θ_{m}(f)(g, ρ, z) = \{ \begin{matrix}f(g, ρm̃^{−1}, z) & ρ ∈ m M_{2}(ℤ̂) \\ 0 & otherwise \end{matrix}

where m̃ = det(m)m^{−1}. The subgroup ℚ^{∗} ↪ GL_{2}(𝔸_{f}) acts by inner, hence the group of symmetries of the set of extremal states E_{β} is of the form

(4.7)  S = ℚ^{∗}\GL_{2}(𝔸_{f}).

In the case of E_{∞} states (defined as weak limits) the action of GL_{2}^{+}(ℚ) is more subtle to define. In fact, (4.6) does not directly induce a nontrivial action on E_{∞}. However, there is a nontrivial action induced by the action on E_{β} states for sufficiently large β. The action on the KMS_{∞} states is obtained by a “warming up and cooling down procedure”, as in (1.1) and (1.2).

4.2. Lattice functions. In the case of the BC system, the arithmetic subalgebra A_{1,ℚ} can be regarded as the algebra generated by the μ_{n}, μ_{n}^{∗} and by homogeneous functions of weight zero on 1-dim ℚ-lattices obtained as a normalization of the functions

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

by covolume, namely, by the functions e_{k,a} := c^{k} ϵ_{k,a}, where c(Λ) is proportional to the covolume |Λ| and satisfies

(2π\sqrt{−1}) c(ℤ) = 1.

It is natural therefore to expect that the analogous A_{2,ℚ} of the GL_{2} system will involve Eisenstein series

(4.9)  E_{2k,a}(Λ, ϕ) = ∑_{y∈Λ+ϕ(a)} y^{−2k}

and

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

normalized to weight zero, in a similar fashion. This points to the fact that modular functions should appear naturally as the rational subalgebra of the GL_{2} system. This can also be noticed from the fact that the group of symmetries S described in (4.7) is in fact the Galois group of the field of modular functions, by a deep arithmetic result of Shimura [16]. As we shall see below, in fact A_{2,ℚ} will turn out to be a subalgebra of unbounded multipliers of A_{2}. Modular functions will appear naturally from a simple set of conditions specifying the arithmetic nature of these multipliers.

4.3. The modular field. We recall briefly some basic facts and results about the modular field. Let F denote the field of modular functions over ℚ^{ab}, namely the union of the fields F_{N} of modular functions of level N rational over the cyclotomic field ℚ(ζ_{n}), that is, such that the q-expansion in powers of q^{1/N} = exp(2πiτ/N) has all coefficients in ℚ(e^{2πi/N}). The action of the Galois group ℤ̂^{∗} ≃ Gal(ℚ^{ab}/ℚ) on the coefficients determines a homomorphism

(4.11)  cycl : ℤ̂^{∗} → Aut(F).

The modular field has an explicit set of generators given by the Fricke functions ([16], [12]). If ℘ is the Weierstrass ℘-function, which gives the parameterization

w ↦ (1, ℘(w; τ, 1), ℘′(w; τ, 1))

of the elliptic curve

y^{2} = 4x^{3} − g_{2}(τ)x − g_{3}(τ)

by the quotient ℂ/(ℤ + ℤτ), then the Fricke functions are homogeneous functions of 1-dimensional lattices of weight zero, parameterized by v ∈ ℚ^{2}/ℤ^{2}, of the form

(4.12)  f_{v}(z) = −2^{7}3^{5} \frac{g_{2}(z)g_{3}(z)}{Δ(z)} ℘(λ_{z}(v); z, 1),

where Δ(z) = g_{2}^{3} − 27g_{3}^{2} is the discriminant and λ_{z}(v) := v_{1}z + v_{2}.

The following important result of Shimura completely determines the Galois group of the modular field:

Aut(F) ≅ ℚ^{∗}\GL_{2}(𝔸_{f}).

If τ ∈ ℍ is a generic point, then the evaluation map f ↦ f(τ) is an embedding F ↪ ℂ. We denote by F_{τ} the image in ℂ. This yields an identification

(4.13)  θ_{τ} : Gal(F_{τ} /ℚ) →^{≃} ℚ^{∗}\GL_{2}(𝔸_{f}).

4.4. Arithmetic subalgebra. We determine a natural arithmetic subalgebra A_{2,ℚ} of unbounded multipliers of A_{2}. Unbounded multipliers on A_{2} are endowed with the same convolution product (3.6).

Elements of A_{2,ℚ} are continuous functions on Z (cf. (3.5)), with finite support in the variable g ∈ Γ\GL_{2}^{+}(ℚ). For convenience we adopt the notation

f_{(g,ρ)}(z) = f(g, ρ, z)

so that f_{(g,ρ)} ∈ C(ℍ). Let p_{N} : M_{2}(ℤ̂) → M_{2}(ℤ/Nℤ) be the canonical projection. We say that f is of level N if

f_{(g,ρ)} = f_{(g,p_{N} (ρ))} ∀(g, ρ).

Then f is completely determined by the functions

f_{(g,m)} ∈ C(ℍ), for m ∈ M_{2}(ℤ/Nℤ).

Notice that the invariance

f(gγ, ρ, z) = f(g, γρ, γ(z)),

for all γ ∈ Γ and for all (g, ρ, z) ∈ U, implies that we have

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

so that f is invariant under a congruence subgroup.

Thus, we define the arithmetic A_{2,ℚ} as follows.

Definition 4.1. A continuous function on Z is in the arithmetic subalgebra A_{2,ℚ} if it satisfies the following properties:

(1) The support of f in Γ\GL_{2}^{+}(ℚ) is finite.

(2) The function f is of finite level with

f_{(g,m)} ∈ F ∀(g, m).

(3) The function f satisfies the cyclotomic condition:

f_{(g,α(u)m)} = cycl(u) f_{(g,m)},

for all g ∈ GL_{2}^{+}(ℚ) diagonal and all u ∈ ℤ̂^{∗}, with

α(u) = (\begin{matrix}u & 0 \\ 0 & 1 \end{matrix})

and cycl as in (4.11).

If we took all but the last condition, this would allow the algebra A_{2,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. The cyclotomic condition forces the spectrum of the corresponding elements of A_{2,ℚ} to contain all Galois conjugates of any such root, so that these elements cannot be scalar. This is achieved via a simple an natural consistency condition on the roots of unity that appear in the coefficients of the q-series. The algebra A_{2,ℚ} defined by the properties above is a subalgebra of unbounded multipliers of A_{2}, which is globally invariant under the group of symmetries S.

4.5. Galois action on E_{∞}. Consider a state φ = φ_{∞,L} ∈ E_{∞}, where the invertible ℚ-lattice L = (ρ, τ) is generic, in the sense that τ ∈ ℍ is generic so that one has the identification (4.13).

Theorem 4.2. For φ_{∞,L} ∈ E_{∞} with L = (ρ, τ) generic, the values of the state on elements of the arithmetic subalgebra lie in the image in ℂ of the modular field,

(4.15)  φ(A_{2,ℚ}) ⊂ F_{τ} ,

and the isomorphism

(4.16)  θ_{φ} : Gal(F_{τ} /ℚ) ⟶^{≃} ℚ^{∗}\GL_{2}(𝔸_{f}),

given by

(4.17)  θ_{φ}(γ) = ρ^{−1} θ_{τ} (γ) ρ,

for θ_{τ} as in (4.13), intertwines the Galois action on the values of the state with the action of symmetries,

(4.18)  γ φ(f) = φ(θ_{φ}(γ)f), ∀f ∈ A_{2,ℚ}, ∀γ ∈ Gal(F_{τ} /ℚ).

5. Noncommutative Shimura varieties

This point of view is stressed in our joint work with Ramachandran [5]. With the notation Sh(G, X) := G(ℚ)\G(𝔸_{f}) × X, the Shimura variety associated to the tower of modular curves is described by the adèlic quotient

(5.1)

Sh(GL_{2}, ℍ^{±}) = GL_{2}(ℚ)\GL_{2}(𝔸_{f}) × ℍ^{±}
= GL_{2}^{+}(ℚ)\GL_{2}(𝔸_{f}) × ℍ = GL_{2}^{+}(ℚ)\GL_{2}(𝔸)/ℂ^{∗}.

The inverse limit \underleftarrow{lim} Γ\ℍ over congruence subgroups Γ ⊂ SL(2, ℤ) gives a connected component, while by taking congruence subgroups in SL(2, ℚ) one obtains the adèlic version Sh(GL_{2}, ℍ^{±}). The set of components of Sh(GL_{2}, ℍ^{±}) is given by

(5.2)  π_{0}(Sh(GL_{2}, ℍ^{±})) = Sh(GL_{1}, {±1}),

where

(5.3)  Sh(GL_{1}, {±1}) = GL_{1}(ℚ)\GL_{1}(𝔸_{f}) × {±1} = ℚ_{+}^{∗}\𝔸_{f}^{∗}

is the Shimura variety associated to the cyclotomic tower (cf. [5] [15]). As we shall see below, (5.3) can be thought of as the “set of classical points” of the noncommutative space of the BC system, where the algebra A_{1} is Morita equivalent to C_{0}(𝔸_{f})⋊ℚ_{+}^{∗}. The result of [1] shows in particular that, at zero temperature, the BC system settles onto its “commutative points” (extremal KMS_{∞} states) which form the classical Shimura variety (5.3).

Similarly, the results of Theorem 4.1 and 4.2 show the analogous behavior in the GL_{2} system. At zero temperature, the system settles onto its “commutative points” given by the Shimura variety (5.1).

This leads us naturally to think of the algebras of the BC system and of the GL_{2} system as noncommutative Shimura varieties. The first is associated to the adèlic quotient

(5.4)  Sh^{(nc)}({±1}, GL_{1}) := GL_{1}(ℚ)\(𝔸_{f} × {±1}) = GL_{1}(ℚ)\𝔸^{·}/ℝ_{+}^{∗}

with 𝔸^{·} := 𝔸_{f} × ℝ^{∗}.

This has a compactification, obtained by replacing 𝔸^{·} by 𝔸, as in [3],

(5.5)  \overline{Sh^{(nc)}}(\{±1\}, GL_{1}) = GL_{1}(ℚ)\𝔸/ℝ_{+}^{∗}.

The compactification consists of adding the trivial lattice (with a possibly nontrivial ℚ-structure).

The dual space (namely the principal ℝ_{+}^{∗}-bundle obtained by taking the crossed product by time evolution σ_{t}) is the space of adèle classes

(5.6)  L_{1} = GL_{1}(ℚ)\𝔸 → GL_{1}(ℚ)\𝔸/ℝ_{+}^{∗}

that gives the spectral realization of zeros of the Riemann ζ function in [3]. This dual space corresponds to considering commensurability classes of 1-dimensional ℚ-lattices (not up to scaling).

In the case of the GL_{2}-system, similarly, we have a noncommutative Shimura variety

(5.7)  Sh^{(nc)}(ℍ^{±}, GL_{2}) := GL_{2}(ℚ)\(M_{2}(𝔸_{f}) × ℍ^{±}).

This also admits a compactification, now given by adding the boundary ℙ^{1}(ℝ) to ℍ^{±}, as in the noncommutative compactification of modular curves of [14],

(5.8) \overline{Sh^{(nc)}}(ℍ^{±}, GL_{2}) := GL_{2}(ℚ)\(M_{2}(𝔸_{f}) × ℙ^{1}(ℂ)) = GL_{2}(ℚ)\M_{2}(𝔸)/ℂ^{∗},

where ℙ^{1}(ℂ) = ℍ^{±} ∪ ℙ^{1}(ℝ). This corresponds to adding to the space of commensurability classes of 2-dimensional ℚ-lattices the pseudolattices (in the sense of [13]), here considered together with a ℚ-structure.

In this case we can also consider the dual system. This is a ℂ^{∗}-bundle

(5.9)  L_{2} = GL_{2}(ℚ)\M_{2}(𝔸).

On this dual space modular forms appear naturally instead of modular functions and the algebra of coordinates contains the modular Hecke algebra of Connes– Moscovici ([6], [7]) as arithmetic subalgebra.

The identification (5.2) then gives the compatibility between the GL_{1} and the GL_{2} system. At the level of the classical commutative spaces, this is given by the map

(5.10)  det × sign : Sh(ℍ^{±}, GL_{2}) → Sh({±1}, GL_{1}),

which corresponds in fact to passing to the set π_{0} of connected components.

6. Class Field Theory

In our joint work with Ramachandran [5], we use the GL_{2} system to extend the relation between noncommutative geometry and class field theory illustrated in the BC system for the case of ℚ to the next important case, that of imaginary quadratic fields. Thus, we assume that 𝕂 = ℚ(\sqrt{−d}). A point τ ∈ ℍ is a CM (complex multiplication) point for 𝕂 if we have 𝕂 = ℚ(τ). This is a non-generic case, in the sense of the properties of the modular field under the evaluation map. In fact, in this case, the evaluation F → F_{τ} ⊂ ℂ does not give an embedding. The image in ℂ of the modular field can be characterized ([16]) and is the maximal abelian extension of 𝕂,

(6.1)  F_{τ} ≃ 𝕂^{ab}.

The values {f(τ), f ∈ F } give a set of generators of 𝕂^{ab} and the Galois action is described explicitly in the following way ([16])

\xymatrix{1 \ar[r] & 𝕂^{∗} \ar[r] & GL_{1}(𝔸_{𝕂,f}) \ar[d] \ar[r]^{≃} & Gal(𝕂^{ab}/𝕂) \ar[r] & 1 \\ 1 \ar[r] & ℚ^{∗} \ar[r] & GL_{2}(𝔸_{f}) \ar[r]^{≃} & Aut(F) \ar[r] & 1,}

where 𝔸_{𝕂,f} = 𝔸_{f} ⊗ 𝕂. This gives a complete solution to the problem of explicit class field theory for imaginary quadratic fields. As in the BC system one sees the explicit class field theory of ℚ^{ab} appear in the symmetries of E_{∞} states, the class field theory for imaginary quadratic fields appears naturally in relation to the GL_{2} system. In fact, one can consider a special class of 2-dimensional ℚ-lattices, given by those that also have the similarly defined structure of a 1-dimensional 𝕂-lattice. The commensurability relation (compatible with the 𝕂-structure) gives a system A_{𝕂} which is closely related to both the original BC system and the GL_{2}-system and has properties in common with both. The arithmetic structure of the GL_{2}-system induces a corresponding arithmetic structure A_{𝕂,ℚ} on the A_{𝕂} system, which also inherits a natural time evolution. The Galois theory of KMS_{∞} states of the GL_{2}-system has a parallel result for the A_{𝕂} system, which mirrors the relation between the explicit class field theory of imaginary quadratic fields and the Galois theory of the modular field described above. (See [5] for details.) The next fundamental question in the direction of generalizations of the BC system to other number fields is how to approach the more complicated case of real quadratic fields, ℚ(\sqrt{d}), for which there is not yet a complete solution to the explicit class field theory problem. Manin’s real multiplication program [13] suggests that the right geometric setup may still be found within the GL_{2}-system, by looking at the boundary strata of \overline{Sh^{(nc)}}(ℍ^{±}, GL_{2}).

References

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

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

[15] J.S. Milne, Canonical models of Shimura curves, manuscript, 2003 (www.jmilne.org).

[16] G. Shimura, Arithmetic theory of automorphic functions, Iwanami Shoten and Princeton 1971.

A. Connes: Collège de France, 3, rue d’Ulm, Paris, F-75005 France

E-mail address: alain@connes.org

M. Marcolli: Max–Planck Institut für Mathematik, Vivatsgasse 7, Bonn, D-53111 Germany

E-mail address: marcolli@mpim-bonn.mpg.de