The Weil proof and the geometry of the adeles class space

Alain Connes^{1}, Caterina Consani^{2}, and Matilde Marcolli^{3}

1 Collège de France, 3, rue d’Ulm, Paris, F-75005 France

  alain@connes.org

2 Mathematics Department, Johns Hopkins University, Baltimore, MD 21218 USA

  kc@math.jhu.edu

3 Max–Planck Institut für Mathematik, Vivatsgasse 7, Bonn, D-53111 Germany

  marcolli@mpim-bonn.mpg.de

Dedicated to Yuri Manin on the occasion of his 70th birthday

O simili o dissimili che sieno questi mondi non con minor raggione sarebe bene a l’uno l’essere che a l’altro Giordano Bruno – De l’infinito, universo e mondi

1 Introduction

This paper explores analogies between the Weil proof of the Riemann Hypothesis for function fields and the geometry of the adeles class space, which is the noncommutative space underlying the spectral realization of the zeros of the Riemann zeta function constructed in [10]. Our purpose is to build a dictionary between the algebro-geometric setting of algebraic curves, divisors, the Riemann–Roch formula, and the Frobenius map, around which the Weil proof is built, and the world of noncommutative spaces, cyclic cohomology and KK-theory, index formulae, and the thermodynamical notions of quantum statistical mechanics, which, as we already argued in [11], provide an analog of the Frobenius in characteristic zero via the scaling action on the dual system. The present work builds upon several previous results. The first input is the spectral realization of [10], where the adeles class space was first identified as the natural geometric space underlying the Riemann zeta function, where the Weil explicit formula acquires an interpretation as a trace formula. In [10] the analytic setting is that of Hilbert spaces, which provide the required positivity, but the spectral realization only involves the critical zeros. In [11], we provided a cohomological interpretation of the trace formula, using cyclic homology. In the setting of [11], the analysis is as developed by Ralph Meyer in [33] and uses spaces of rapidly decaying functions instead of Hilbert spaces. In this case, all zeros contribute to the trace formula, and the Riemann Hypothesis becomes equivalent to a positivity question. This mirrors more closely the structure of the two main steps in the Weil proof, namely the explicit formula and the positivity Tr(Z ∗ Z′) > 0 for correspondences (see below). The second main building block we need to use is the theory of endomotives and their quantum statistical mechanical properties we studied in [11]. Endomotives are a pseudo-abelian category of noncommutative spaces that naturally generalize the category of Artin motives. They are built from semigroup actions on projective limits of Artin motives. The morphisms in the category of endomotives generalize the notion of correspondence given by algebraic cycles in the product used in the theory of motives to the setting of étale groupoids, to account naturally for the presence of the semigroup actions. Endomotives carry a Galois action inherited from Artin motives and they have both an algebraic and an analytic manifestation. The latter provides the data for a quantum statistical mechanical system, via the natural time evolution associated by Tomita’s theory to a probability measure carried by the analytic endomotive. The main example that is of relevance to the Riemann zeta function is the endomotive underlying the Bost–Connes quantum statistical mechanical system of [5]. One can pass from a quantum statistical mechanical system to the “dual system” (in the sense of the duality of type III and type II factors in [6], [37]), which comes endowed with a scaling action induced by the time evolution. A general procedure described in [11] shows that there is a “restriction map” (defined as a morphism in the abelian category of modules over the cyclic category) from the dual system to a line bundle over the space of low temperature KMS states of the quantum statistical mechanical system. The cokernel of this map is not defined at the level of algebras, but it makes sense in the abelian category and carries a corresponding scaling action. We argued in [11] that the induced scaling action on the cyclic homology of this cokernel may be thought of as an analog of the action of Frobenius on étale cohomology. This claim is justified by the role that this scaling action of ℝ_{+}^{∗}, combined with the action of ℤ̂^{∗} carried by the Bost–Connes endomotive, has in the trace formula, see [10], [11] and §4 of [13]. Further evidence for the role of the scaling action as Frobenius is given in [20], where it is shown that, in the case of function fields, for a natural quantum statistical mechanical system that generalizes the Bost–Connes system to rank one Drinfeld modules, the scaling action on the dual system can be described in terms of the Frobenius and inertia groups. In the present paper we continue along this line of thought. We begin by reviewing the main steps in the Weil proof for function fields, where we highlight the main conceptual steps and the main notions that will need an analog in the noncommutative geometry setting. We conclude this part by introducing the main entries in our still tentative dictionary. The rest of the paper discusses in detail some parts of the dictionary and provides evidence in support of the proposed comparison. We begin this part by recalling briefly the properties of the Bost–Connes endomotive from [11] followed by the description of the “restriction map” corresponding to the inclusion of the ideles class group C_{𝕂} = 𝔸_{𝕂}^{∗}/𝕂^{∗} in the noncommutative adeles class space X_{𝕂} = 𝔸_{𝕂}/𝕂^{∗}. We discuss its relation to the exact sequence of Hilbert spaces of [10] that plays a crucial role in obtaining the spectral realization as an “absorption spectrum”. We then concentrate on the geometry of the adeles class space over an arbitrary global field and the restriction map in this general setting, viewed as a map of cyclic modules. We introduce the actions ϑ_{a} and ϑ_{m} (with a and m respectively for additive and multiplicative) of 𝔸_{𝕂}^{∗} on suitable function spaces on 𝔸_{𝕂} and on C_{𝕂} and the induced action on the cokernel of the restriction map in the category of cyclic modules. We prove the corresponding general form of the associated Lefschetz trace formula, as a cohomological reformulation of the trace formula of [10] using the analytical setting of [33]. The form of the trace formula and the positivity property that is equivalent, in this setting, to the Riemann Hypothesis for the corresponding L-functions with Grössencharakter, suggest by comparison with the analogous notions in the Weil proof a natural candidate for the analog of the Frobenius correspondence on the curve. This is given by the graph of the scaling action. We can also identify the analog of the degree and co-degree of a correspondence, and the analog of the self intersection of the diagonal on the curve, by looking at the explicit form of our Lefschetz trace formula. We also have a clear analog of the first step in the Weil proof of positivity, which consists of adjusting the degree by multiples of the trivial correspondences. This step is possible, with our notion of correspondences, due to a subtle failure of Fubini’s theorem that allows us to modify the degree by adding elements in the range of the “restriction map”, which play in this way the role of the trivial correspondences. This leaves open the more difficult question of identifying the correct analog of the principal divisors, which is needed in order to continue the dictionary. We then describe how to obtain a good analog of the algebraic points of the curve in the number field case (in particular in the case of 𝕂 = ℚ), in terms of the thermodynamical properties of the system. This refines the general procedure described in [11]. In fact, after passing to the dual system, one can consider the periodic orbits. We explain how, by the result of [10], these are the noncommutative spaces where the geometric side of the Lefschetz trace formula concentrates. We show that, in turn, these periodic orbits carry a time evolution and give rise to quantum statistical mechanical systems, of which one can consider the low temperature KMS states. To each periodic orbit one can associate a set of “classical points” and we show that these arise as extremal low temperature KMS states of the corresponding system. We show that, in the function field case, the space obtained in this way indeed can be identified, compatibly with the Frobenius action, with the algebraic points of the curve, albeit by a non-canonical identification. Passing to the dual system is the analog in characteristic zero of the transition from 𝔽_{q} to its algebraic closure 𝔽̄_{q}. Thus, the procedure of considering periodic orbits in the dual system and classical points of these periodic orbits can be seen as an analog, for our noncommutative space, of considering points defined over the extensions 𝔽_{q^{n}} of 𝔽_{q} in the case of varieties defined over finite fields (cf. [11] and §4 of [13]). We analyze the behavior of the adeles class space under field extensions and the functoriality question. We then finish the paper by sketching an analogy between some aspects of the geometry of the adeles class spaces and the theory of singularities, which may be useful in adapting to this context some of the techniques of vanishing and nearby cycles.

2 A look at the Weil proof

In this preliminary section, we briefly review some aspects of the Weil proof of the Riemann Hypothesis for function fields, with an eye on extending some of the basic steps and concept to a noncommutative framework, which is what we will be doing in the rest of the paper. In this section we let 𝕂 be a global field of positive characteristic p > 0. One knows that, in this case, there exists a smooth projective curve over a finite field 𝔽_{q}, with q = p^{r} for some r ∈ ℕ, such that

𝕂 = 𝔽_{q}(C)  (1)

is the field of functions of C. For this reason, a global field of positive characteristic is called a function field. We denote by Σ_{𝕂} the set of places of 𝕂. A place v ∈ Σ_{𝕂} is a Galois orbit of points of C(𝔽̄_{q}). The degree n_{v} = deg(v) is its cardinality, namely the number of points in the orbit of the Frobenius acting on the fiber of the natural map from points to places

C(𝔽̄_{q}) → Σ_{𝕂}.  (2)

This means that the fiber over v consists of n_{v} conjugate points defined over 𝔽_{q^{nv}}, the residue field of the local field 𝕂_{v}. The curve C over 𝔽_{q} has a zeta function of the form

Z_{C}(T) = exp (∑_{n=1}^{∞} \frac{#C(𝔽_{q^{n}})}{n} T^{n}) ,  (3)

with log Z_{C}(T) the generating function for the number of points of C over the fields 𝔽_{q^{n}}. It is customary to use the notation

ζ_{𝕂}(s) = ζ_{C}(s) = Z_{C}(q^{−s}).  (4)

It converges for ℜ(s) > 1. In terms of Euler product expansions one writes

ζ_{𝕂}(s) = ∏_{v∈Σ_{𝕂}}(1 − q^{−n_{v}s})^{−1}.  (5)

In terms of divisors of C, one has equivalently

ζ_{𝕂}(s) = ζ_{C}(s) = ∑_{D≥0} N(D)^{−s},  (6)

where the norm of the divisor D is N(D) = q^{deg(D)}. The Riemann–Roch formula for the curve C states that

ℓ(D) − ℓ(κ_{C} − D) = deg(D) − g + 1,  (7)

where κ_{C} is the canonical divisor on C, with degree deg(κ_{C}) = 2g − 2 and h^{0}(κ_{C}) = g, and ℓ(D) the dimension of H^{0}(D). Both deg(D) and N(D) are well defined on the equivalence classes obtained by adding principal divisors, that is,

D ∼ D′ ⟺ D − D′ = (f),  (8)

for some f ∈ 𝕂^{∗}. The Riemann–Roch formula (7) also implies that the zeta function ζ_{𝕂}(s) satisfies the functional equation

ζ_{𝕂}(1 − s) = q^{(1−g)(1−2s)}ζ_{𝕂}(s).  (9)

The zeta function ζ_{𝕂}(s) can also be written as a rational function

Z_{C}(T) = \frac{P (T)}{(1 − T)(1 − qT)} , T = q^{−s} ,  (10)

where P (T) is a polynomial of degree 2g and integer coefficients

P (T) = ∏_{j=1}^{2g}(1 − λ_{j}T).  (11)

In particular, one has

#C(𝔽_{q^{n}}) = q^{n} + 1 − ∑_{j=1}^{2g} λ_{j}.  (12)

Another important reformulation of the zeta function can be given in terms of étale cohomology. Namely, the coefficients #C(𝔽_{q^{n}}) that appear in the zeta function can be rewritten as

#C(𝔽_{q^{n}}) = #Fix(Fr^{n} : C̄ → C̄)  (13)

with C̄ = C ⊗_{𝔽_{q}} 𝔽̄_{q}. The Lefschetz fixed point formula for étale cohomology then shows that

#C(𝔽_{q^{n}}) = ∑_{i=0}^{2}(−1)^{i}Tr (Fr^{n}| H_{et}^{i}(C̄, ℚ_{ℓ})) .  (14)

Thus, the zeta function can be written in the form

ζ_{𝕂}(s) = ∏_{i=0}^{2} (exp (∑_{n=1}^{∞} Tr(Fr^{n}| H_{et}^{i}(C̄, ℚ_{ℓ}))\frac{q^{−sn}}{n} ))^{(−1)^{i}} .  (15)

The analog of the Riemann hypothesis for the zeta functions ζ_{𝕂}(s) of function fields was stated in 1924 by E. Artin as the property that the zeros lie on the line ℜ(s) = 1/2. Equivalently, it states that the complex numbers λ_{j} of (11), which are the eigenvalues of the Frobenius acting on H_{et}^{1}(C̄, ℚ_{ℓ}), are algebraic numbers satisfying

|λ_{j}| = \sqrt{q}.  (16)

The Weil proof can be formulated either using étale cohomology, or purely in terms of the Jacobian of the curve, or again (equivalently) in terms of divisors on C × C. We follow this last viewpoint and we recall in detail some of the main steps in the proof.

2.1 Correspondences and divisors

Correspondences Z, given by (non-vertical) divisors on C × C, form a ring under composition

Z_{1} ⋆ Z_{2} = (p_{13})_{∗}(p_{12}^{∗}Z_{1} • p_{23}^{∗}Z_{2}),  (17)

with p_{ij} : C × C × C → C × C the projections, and • the intersection product. The ring has an involution

Z′ = σ(Z)  (18)

where σ is the involution that exchanges the two copies of C in the product C × C. The degree d(Z) and the codegree d′(Z) are defined as the intersection numbers

d(Z) = Z • (P × C) and d′(Z) = Z • (C × P), ∀P ∈ C.  (19)

They satisfy the relations

d(Z′) = d′(Z), and d(Z_{1} ⋆ Z_{2}) = d(Z_{1})d(Z_{2}).  (20)

The correspondences P × C and C × P are called trivial correspondences. One can consider the abelian group Div_{tr}(C × C) generated by these trivial correspondence and take the quotient

C(C) := Div(C × C)/Div_{tr}(C × C).  (21)

It is always possible to change the degree and codegree of a correspondence Z by adding a multiple of the trivial correspondences P × C and C × P, so that, for any element in C we find a representative Z ∈ Corr with

d(Z) = d′(Z) = 0.  (22)

One also wants to consider correspondences up to linear equivalence,

Z_{1} ∼ Z_{2} ⟺ Z_{1} − Z_{2} = (f),  (23)

where (f) is a principal divisor on C × C. Thus, one can consider

Pic(C × C) = Div(C × C)/ ∼

and its quotient P(C) modulo the classes of the trivial correspondences. A correspondence Z is effective if it is given by an effective divisor on C × C, namely if it is a combination Z = ∑_{i} n_{i}Z_{i} of curves Z_{i} ⊂ C × C with coefficients n_{i} ≥ 0. We write Z ≥ 0 to mean its effectiveness. An effective correspondence Z ≥ 0 that is nonempty can be viewed as a multivalued map

Z : C → C, P ↦ Z(P),  (24)

with Z(P) = proj_{C}(Z • (P × C)), of which the divisor is the graph and with the product (17) given by the composition. The trace of a correspondence Z on C × C is the expression

Tr(Z) = d(Z) + d′(Z) − Z • Δ,  (25)

with Δ the diagonal (identity correspondence) and • the intersection product. This is well defined on P(C) since Tr(Z) = 0 for principal divisors and trivial correspondences. Consider a correspondence of degree g of the form Z = ∑ a_{n}Fr^{n}, given by a combination of powers of the Frobenius. Then Z can be made effective by adding a principal correspondence which is defined over 𝔽_{q} and which commutes with Fr. This can be seen as follows. The Riemann–Roch theorem ensures that one can make Z effective by adding a principal correspondence, over the field k(P), where k is the common field of definition of the correspondence Z and of the curve (cf. [35]) and P is a generic point. A correspondence of the form Z = ∑ a_{n}Fr^{n} is in fact defined over 𝔽_{q} hence the principal correspondence is also defined over 𝔽_{q}. As such it automatically commutes with Fr (cf. [38], p. 287). Notice however that, in general, it is not possible to modify a divisor D of degree one on C to an effective divisor in such a way that the added pricipal divisor has support on the same Frobenius orbit. An illustrative example is given in Chapter 4 of [13].

2.2 The explicit formula

The main steps in the Weil proof of RH for function fields are

1. The explicit formula

2. Positivity

Let 𝕂 be a global field and let 𝔸_{𝕂} denote its ring of adeles. Let Σ_{𝕂} denote the set of places of 𝕂. Let α be a non-trivial character of 𝔸_{𝕂} which is trivial on 𝕂 ⊂ 𝔸_{𝕂}. We write

α = ∏_{v∈Σ_{𝕂}} α_{v},  (26)

for the decomposition of α as a product of its restrictions to the local fields α_{v} = α|_{𝕂_{v}}. Consider the bicharacter

⟨z, λ⟩ := λ^{z}, for (z, λ) ∈ ℂ × ℝ_{+}^{∗}.  (27)

Let N denote the range of the norm | · | : C_{𝕂} → ℝ_{+}^{∗}. Then N^{⊥} ⊂ ℂ denotes the subgroup

N^{⊥} := {z ∈ ℂ| λ^{z} = 1, ∀λ ∈ N}.  (28)

Consider then the expression

∑_{ρ∈ℂ/N^{⊥}| L(χ̃,ρ)=0} f̂(χ̃, ρ),  (29)

with L(χ̃, ρ) the L-function with Grössencharakter χ, where χ̃ denotes the extension to C_{𝕂} of the character χ ∈ Ĉ_{𝕂,1}, the Pontrjagin dual of C_{𝕂,1}. Here f̂(χ̃, ρ) denotes the Fourier transform

f̂(χ̃, ρ) = ∫_{C_{𝕂}} f(u)χ̃(u) |u|^{ρ} d^{∗}u  (30)

of a test function f in the Schwartz space S(C_{𝕂}). In the case where N = q^{ℤ} (function fields), the subgroup N^{⊥} is nontrivial and given by

N^{⊥} = \frac{2πi}{log q} ℤ.  (31)

Since in the function field case the L-fuctions are functions of q^{−s}, there is a periodicity by N^{⊥}, hence we need to consider only ρ ∈ ℂ/N^{⊥}. In the number field case this does not matter, since N = ℝ_{+}^{∗} and N^{⊥} is trivial. The Weil explicit formula is the remarkable identity [43]

ĥ(0) + ĥ(1) − ∑_{ρ∈ℂ/N^{⊥}| L(χ̃,ρ)=0} ĥ(χ̃, ρ) = ∑_{v∈Σ_{𝕂}} ∫′_{(𝕂_{v}^{∗},α_{v})} \frac{h(u^{−1})}{|1 − u|} d^{∗}u. (32)

Here the Fourier transform ĥ is as in (30). The test function h has compact support and belongs to the Schwartz space S(C_{𝕂}). As soon as h(1) ≠ 0 the integrals in the right hand side are singular so that one needs to specify how to take their principal value. This was done in [43] and it was shown in [10] that the same principal value can in fact be defined in the following unified way.

Definition 2.1. For a local field K and a given (non-trivial) additive character β of K, one lets ϱ_{β} denote the unique distribution extending d^{∗}u at u = 0, whose Fourier transform

ϱ̂(y) = ∫_{K} ϱ(x)β(xy) dx  (33)

satisfies the vanishing condition ϱ̂(1) = 0.

Then by definition the principal value ∫′ is given by

∫′_{(K,β)} \frac{f(u^{−1})}{|1 − u|} d^{∗}u = ⟨ϱ_{β}, g⟩, with g(λ) = \frac{f((λ + 1)^{−1})}{|λ + 1|},  (34)

where ⟨ϱ_{β}, g⟩ denotes the pairing of the distribution ϱ_{β} and the function g(λ). This makes sense provided the support of f is compact which implies that g(λ) vanishes identically in a neighborhood of λ = −1. The Weil explicit formula is a far reaching generalization of the relation between primes and zeros of the Riemann zeta function, originally due to Riemann [34].

2.3 Riemann–Roch and positivity

Weil positivity is the statement that, if Z is a nontrivial correspondence in P(C) (i.e. as above a correspondence on C × C modulo trivial ones and up to linear equivalence), then

Tr(Z ⋆ Z′) > 0.  (35)

This is proved using the Riemann–Roch formula on C to show that one can achieve effectivity. In fact, using trivial correspondences to adjust the degree one can assume that d(Z) = g. Then the Riemann–Roch formula (7) shows that if D is a divisor on C of degree deg(D) = g then there are effective representatives in the linear equivalence class of D. The intersection of Z ⊂ C × C with P × C defines a divisor Z(P) on C with

deg(Z(P)) = d(Z) = g.

Thus, the argument above shows that there exists f_{P} ∈ 𝕂^{∗} such that Z(P) + (f_{P}) is effective. This determines an effective divisor Z + (f) on C × C.

Thus, we can assume that Z is effective, hence we can write it as a multivalued function

P ↦ Z(P) = Q_{1} + ⋯ + Q_{g}.  (36)

The product Z ⋆ Z′ is of the form

Z ⋆ Z′ = d′(Z)Δ + Y,  (37)

where Δ is the diagonal in C × C and Y is the effective correspondence such that Y (P) is the divisor on C given by the sum of points in

{Q ∈ C| Q = Q_{i}(P) = Q_{j}(P), i ≠ j}.

One sees this from the description in terms of intersection product that it is given by the multivalued function

(Z ⋆ Z′)(Q) = ∑_{i,j} ∑_{P ∈U_{ij}(Q)} P,  (38)

where

U_{ij}(Q) = {P ∈ C| Q_{i}(P) = Q_{j}(P) = Q}.

One can separate this out in the contribution of the locus where Q_{i} = Q_{j} for i ≠ j and the part where i = j,

(Z ⋆ Z′)(Q) = U(Q) + Y (Q).

Notice that

#{P ∈ C| Q = Q_{i}(P), for some i = 1, …, g} = d′(Z).  (39)

Thus, for i = j we obtain that the divisor U(Q) = ∑_{i} ∑_{P ∈U_{ii}(Q)} P is just d′(Z)Δ(Q), while for i ≠ j one obtains the remaining term Y of (37). In the case g = 1, the effective correspondence Z(P) = Q(P) is single valued and the divisor (Z ⋆ Z′)(P) of (38) reduces to the sum of points in

U(Q) = {P ∈ C| Q(P) = Q}.

There are d′(Z) such points so one obtains

Z ⋆ Z′ = d′(Z)Δ, with Tr(Z ⋆ Z′) = 2d′(Z) ≥ 0,  (40)

since for g = 1 one has Δ • Δ = 0 and d′(Z) ≥ 0 since Z is effective. In the case of genus g > 1, the Weil proof proceeds as follows. Let κ_{C} be a choice of an effective canonical divisor for C without multiple points, and let {f_{1}, …, f_{g}} be a basis of the space H^{0}(κ_{C}). One then considers the function C → M_{g×g}(𝔽_{q}) to g × g matrices

P ↦ M(P), with M_{ij}(P) = f_{i}(Q_{j}(P)).  (41)

and the function K : C → 𝔽_{q} given by

K(P) = det(M(P))^{2}.  (42)

The function P ↦ K(P) of (42) is a rational function with (2g − 2)d′(Z) double poles. In fact, K(P) is a symmetric function of the Q_{j}(P), because of the squaring of the determinant. The composition P ↦ (Q_{j}(P)) ↦ K(P) is then a rational function of P ∈ C. The poles occur (as double poles) at those points P ∈ C for which some Q_{i}(P) is a component of κ_{C}. The canonical divisor κ_{C} has degree 2g − 2. This means that there are (2g − 2)d′(Z) such double poles. For Z ⋆ Z′ = d′(Z)Δ + Y as above, the intersection number Y • Δ satisfies the estimate

Y • Δ ≤ (4g − 4) d′(Z).  (43)

In fact, the rational function K(P) of (42) has a number of zeros equal to (4g − 4) d′(Z). On the other hand, Y • Δ counts the number of times that Q_{i} = Q_{j} for i ≠ j. Since each point P with Q_{i}(P) = Q_{j}(P) for i ≠ j produces a zero of K(P), one sees that Y • Δ satisfies the estimate (43). Notice that, for genus g > 1 the self intersection of the diagonal is the Euler characteristic

Δ • Δ = 2 − 2g = χ(C).  (44)

Moreover, we have

d(Z ⋆ Z′) = d(Z)d′(Z) = g d′(Z) = d′(Z ⋆ Z′).  (45)

Thus, using again the decomposition (37) and the definition of the trace of a correspondence (25), together with (44) and (45) one obtains

\begin{matrix}Tr(Z ⋆ Z′) = 2g d′(Z) + (2g − 2) d′(Z) − Y • Δ \\ ≥ (4g − 2) d′(Z) − (4g − 4) d′(Z) = 2d′(Z) ≥ 0. \end{matrix}  (46)

This gives the positivity (35).

In the Weil proof of RH for function fields, one concentrates on a particular type of correspondences, namely those that are of the form

Z_{n,m} = mΔ + nFr,  (47)

for n, m ∈ ℤ, with Fr the Frobenius correspondence. Notice that, while the correspondence depends linearly on n, m ∈ ℤ, the expression for the trace gives

Tr(Z_{n,m} ⋆ Z′_{n,m}) = 2gm^{2} + 2(1 + q − N)mn + 2gqn^{2},  (48)

where N = #C(𝔽_{q}). In particular, (48) depends quadratically on (n, m). In the process of passing from a correspondence of degree g to an effective correspondence, this quadratic dependence on (n, m) is contained in the multiplicity d′(Z). Notice, moreover, that the argument does not depend on the torsion part of the ring of correspondences.

2.4 A tentative dictionary

In the rest of the paper we illustrate some steps towards the creation of a dictionary relating the main steps in the Weil proof described above to the noncommutative geometry of the adeles class space of a global field. The noncommutative geometry approach has the advantage that it provides (see [10], [33], [11]) a Lefschetz trace formula interpretation for the Weil explicit formula and that it gives a parallel formulation for both function fields and number fields. Parts of the dictionary sketched below are very tentative at this stage, so we mostly concentrate, in the rest of the paper, on illustrating what we put in the first few lines of the dictionary, on the role of the scaling correspondence as Frobenius and its relation to the explicit formula.

Frobenius correspondence  Z(f) = ∫_{C_{𝕂}} f(g) Z_{g} d^{∗}g

Trivial correspondences  Elements of the range V

Adjusting the degree  Fubini step
by trivial correspondences  on the test functions

Correspondences  Bivariant elements Z(f) ⇒ Γ (f)

Degree of a correspondence  Pointwise index

Riemann–Roch  Index theorem

Effective correspondences  Epimorphism of C^{∗}-modules

deg Z(P) ≥ g ⇒ Z + (f) effective  d(Γ) > 0 ⇒ Γ + K onto

Lefschetz formula  Bivariant Chern of Γ (f)

(by localization on the graph Z(f))

3 Quantum statistical mechanics and arithmetic

The work of Bost–Connes [5] first revealed the presence of an interesting interplay between quantum statistical mechanics and Galois theory. More recently, several generalizations [12], [14], [15], [11], [20], [25], [26] have confirmed and expanded this viewpoint. The general framework of interactions between noncommutative geometry and number theory described in [30], [31], [13], [32] recast these phenomena into a broader picture, of which we explore in this paper but one of many facets. The basic framework that combines quantum statistical mechanics and Galois theory can be seen as an extension, involving noncommutative spaces, of the category of Artin motives. In the setting of pure motives (see [28]), Artin motives correspond to the subcategory generated by zero dimensional objects, with morphisms given by algebraic cycles in the product (in this case without the need to specify with respect to which equivalence relation). Endomotives were introduced in [11] as noncommutative spaces of the form

A_{𝕂} = A ⋊ S,  (1)

where A is an inductive limit of reduced finite dimensional commutative algebras over the field 𝕂, i.e. a projective limit of Artin motives, and S is a unital abelian semigroup of algebra endomorphisms ρ : A → A. The crossed product (1) is obtained by adjoining to A new generators U_{ρ} and U_{ρ}^{∗}, for ρ ∈ S, satisfying the relations

\begin{matrix}U_{ρ}^{∗}U_{ρ} = 1, & U_{ρ}U_{ρ}^{∗} = ρ(1), & ∀ρ ∈ S \\ U_{ρ_{1} ρ_{2}} = U_{ρ_{1}} U_{ρ_{2}}, & U_{ρ_{2} ρ_{1}}^{∗} = U_{ρ_{1}}^{∗} U_{ρ_{2}}^{∗}, & ∀ρ_{1}, ρ_{2} ∈ S \\ U_{ρ} a = ρ(a) U_{ρ}, & a U_{ρ}^{∗} = U_{ρ}^{∗} ρ(a), & ∀ρ ∈ S, ∀a ∈ A. \end{matrix}  (2)

The algebras (1) have the following properties: the algebra A is unital; the image e = ρ(1) ∈ A is an idempotent, for all ρ ∈ S; each ρ ∈ S is an isomorphism of A with the compressed algebra eAe. A general construction given in [11] based on self maps of algebraic varieties provides a large class of examples over different fields 𝕂. We are mostly interested here in the case where 𝕂 is a number field and for part of our discussion below we will concentrate on a special case (the Bost–Connes endomotive) over the field 𝕂 = ℚ. Endomotives form a pseudo-abelian category where morphisms are correspondences given by A_{𝕂}–B_{𝕂}-bimodules that are finite and projective as right modules. These define morphisms in the additive KK-category and in the abelian category of cyclic modules. In fact, in addition to the algebraic form described above, endomotives also have an analytic structure given by considering, instead of the 𝕂-algebra (1) the C^{∗}-algebra

C(X) ⋊ S,  (3)

where X denotes the totally disconnected Hausdorff space X = X(𝕂̄) of algebraic points of the projective limit of Artin motives. This follows by first showing that the semigroup S acts by endomorphisms of the C^{∗}-algebra of continuous functions C(X) and then passing to the norm completion of the algebraic crossed product as in the general theory of semi-groups crossed product C^{∗}-algebras (see [27]; see also [13], Chapter 4, §2.2). The C^{∗}-completion is taken with respect to the norm

∥f∥ = sup_{x∈X} ∥π_{x}(f)∥,  (4)

where, for each x ∈ X, one lets π_{x} be the representation by left convolution on the Hilbert space ℓ^{2}(G_{x}) of the countable fiber G_{x} over x ∈ X of the source map of the groupoid G associated to the action of S on X.

There is a canonical action of the Galois group G = Gal(𝕂̄/𝕂) by automorphisms of the C^{∗}-algebra (3) globally preserving C(X), coming from the action by composition of G on X(𝕂̄) = Hom(A, 𝕂̄). We refer the reader to [11] for a more detailed discussion of algebraic and analytic endomotives and the properties of morphisms in the corresponding categories. If the endomotive is “uniform” in the sense specified in [11], the space X comes endowed with a probability measure μ that induces a state φ on the C^{∗}-algebra (3). The general Tomita theory of modular automorphism groups in the context of von Neumann algebras [36] shows that there is a natural time evolution for which the state φ is KMS_{1}. We refer the reader to Chapter 4 §4.1 of [13] for a more detailed discussion of this step and the necessary von Neumann algebra background. Here the KMS_{1} condition means the following. Given a system of an algebra of observable with a time evolution, there is a good notion of thermodynamic equilibrium states given by the KMS condition. A state on a unital C^{∗}-algebra is a continuous linear functional φ : A → ℂ with φ(1) = 1 and φ(a^{∗}a) ≥ 0 for all a ∈ A. The KMS condition at inverse temperature β for a state φ is the property that, for all a, b ∈ A there exists a function F_{a,b}(z), which is holomorphic on the strip I_{β} = {z ∈ ℂ | 0 < ℑ(z) < β} ⊂ ℂ and continuous on I_{β} ∪ ∂I_{β}, satisfying F_{a,b}(t) = φ(aσ_{t}(b)) and F_{a,b}(t + iβ) = φ(σ_{t}(b)a), for all t ∈ ℝ. For a more detailed account of the quantum statistical mechanical formalism, the notion of KMS state and the properties of such states, we refer the reader to Chaper 3 of [13], §2. One can then consider the set Ω_{β} of low temperature (large β) KMS states for the same quantum statistical mechanical system obtained from a uniform endomotive in the way described above. One also associates to the system (A, σ) of the C^{∗}-algebra with the time evolution its dual system (Â, θ), where the algebra Â = A⋊_{σ} ℝ is obtained by taking the crossed product with the time evolution and θ is the scaling action of ℝ_{+}^{∗}

θ_{λ}(∫ x(t) U_{t} dt) = ∫ λ^{it} x(t) U_{t} dt.  (5)

One then constructs an ℝ_{+}^{∗} equivariant map

π : Â_{β} → C(Ω̃_{β}, L^{1}),  (6)

from a suitable subalgebra Â_{β} ⊂ Â of the dual system to functions on a principal ℝ_{+}^{∗}-bundle Ω̃_{β} over the low temperature KMS states of the system, with values in trace class operators. Since traces define morphisms in the cyclic category, the map (6) can be used to construct a morphism δ = (Tr ∘ π)^{♮} at the level of cyclic modules

Â_{β}^{♮} ⟶^{(Tr∘π)^{♮}} C(Ω̃_{β})^{♮}.  (7)

This map can be loosely thought of as a “restriction map” corresponding to the inclusion of the “classical points” in the noncommutative space. One can then consider the cokernel of this map in the abelian category of cyclic modules. In [11] we denoted the procedure described above “cooling and distillation” of endomotives. We refer the reader to [11] for the precise technical hypotheses under which this procedure can be performed. Here we only gave an impressionistic sketch aimed at recalling briefly the main steps involved.

3.1 The Bost–Connes endomotive

The main example of endomotive we will consider here in relation to the geometry of the adeles class space is the Bost–Connes system. This can be constructed as an endomotive over 𝕂 = ℚ, starting from the projective system X_{n} = Spec(A_{n}), with A_{n} = ℚ[ℤ/nℤ] the group ring of ℤ/nℤ. The inductive limit is the group ring A = ℚ[ℚ/ℤ] of ℚ/ℤ. The endomorphism ρ_{n} associated to an element n ∈ S of the (multiplicative) semigroup S = ℕ = ℤ_{>0} is given on the canonical basis e_{r} ∈ ℚ[ℚ/ℤ], r ∈ ℚ/ℤ, by

ρ_{n}(e_{r}) = \frac{1}{n} ∑_{ns=r} e_{s}  (8)

The corresponding analytic endomotive is the crossed product C^{∗}-algebra

A = C^{∗}(ℚ/ℤ) ⋊ ℕ.

The Galois action is given by composing a character χ : A_{n} → ℚ̄ with an element g of the Galois group G = Gal(ℚ̄/ℚ). Since χ is determined by the n-th root of unity χ(e_{1/n}), one obtains the cyclotomic action. In the case of the Bost–Connes endomotive, the state φ on A induced by the measure μ on X = ℤ̂ is of the form

φ(f) = ∫ℤ̂ f(1, ρ) dμ(ρ),  (9)

and the modular automorphism group restricts to the C^{∗}-algebra as the time evolution of the BC system, cf. [5], [11] and §4 of [13]. The dual system of the Bost–Connes system is best described in terms of commensurability classes of ℚ-lattices. In [12] the Bost–Connes system is reinterpreted as the noncommutative space describing the relation of commensurability for 1-dimensional ℚ-lattices up to scaling. One can also consider the same equivalence relation without dividing out by the scaling action. If we let G_{1} denote the groupoid of the commensurability relation on 1-dimensional ℚlattices and G_{1}/ℝ_{+}^{∗} the one obtained after moding out by scaling, we identify the C^{∗}-algebra of the Bost–Connes system with C^{∗}(G_{1}/ℝ_{+}^{∗}) (cf. [12]). The algebra Â of the dual system is then obtained in the following way (cf. [11]). There is a C^{∗}-algebra isomorphism ι : Â → C^{∗}(G_{1}) of the form

ι(X)(k, ρ, λ) = ∫_{ℝ} x(t)(k, ρ) λ^{it} dt  (10)

for (k, ρ, λ) ∈ G_{1} and X = ∫ x(t) U_{t} dt ∈ Â.

3.2 Scaling as Frobenius in characteristic zero

In the general setting described in [11] one denotes by D(A, φ) the cokernel of the morphism (7), viewed as a module in the cyclic category. The notation is meant to recall the dependence of the construction on the initial data of an analytic endomotive A and a state φ. The cyclic module D(A, φ) inherits a scaling action of ℝ_{+}^{∗} and one can consider the induced action on the cyclic homology HC_{0}(D(A, φ)). We argued in [11] that this cyclic homology with the induced scaling action plays a role analogous to the role played by the Frobenius action on étale cohomology in the algebro-geometric context. Our main supporting evidence is the Lefschetz trace formula for this action that gives a cohomological interpretation of the spectral realization of the zeros of the Riemann zeta function of [10]. We return to discuss the Lefschetz trace formula for the more general case of global fields in §6 below.

The main results of [10] show that we have the following setup. There is an exact sequence of Hilbert spaces

0 → L_{δ}^{2}(𝔸_{ℚ}/ℚ^{∗})_{0} → L_{δ}^{2}(𝔸_{ℚ}/ℚ^{∗}) → ℂ^{2} → 0,  (11)

which defines the subspace L_{δ}^{2}(𝔸_{ℚ}/ℚ^{∗})_{0} by imposing the conditions f(0) = 0 and f̂(0) = 0 and a suitable decay condition imposed by the weight δ. The space L_{δ}^{2}(𝔸_{ℚ}/ℚ^{∗})_{0} fits into another exact sequence of Hilbert spaces of the form

0 → L_{δ}^{2}(𝔸_{ℚ}/ℚ^{∗})_{0} →^{𝔈} L_{δ}^{2}(C_{ℚ}) → H → 0  (12)

where the map 𝔈 is defined by

𝔈(f)(g) = |g|^{1/2} ∑_{q∈ℚ^{∗}} f(qg), ∀g ∈ C_{ℚ} = 𝔸_{ℚ}^{∗}/ℚ^{∗}.  (13)

The map is equivariant with respect to the actions of C_{ℚ} i.e.

𝔈 ∘ ϑ_{a}(γ) = |γ|^{1/2}ϑ_{m}(γ) ∘ 𝔈  (14)

where (ϑ_{a}(γ)ξ)(x) = ξ(γ^{−1}x) for ξ ∈ L_{δ}^{2}(𝔸_{ℚ}/ℚ^{∗})_{0} and similarly ϑ_{m}(γ) is the regular representation of C_{𝕂}. We showed in [11] that the map 𝔈, translated from the context of Hilbert spaces to that of nuclear spaces as in [33], has a natural interpretation in terms of the “cooling and distillation process” for the BC endomotive. In fact, we showed in [11] that, if (A, σ) denotes the BC system, then the following properties hold.

1. For β > 1 there is a canonical isomorphism

Ω̃_{β} ≃ ℤ̂^{∗} × ℝ_{+}^{∗} ≃ C_{ℚ}  (15)

of Ω̃_{β} with the space of invertible 1-dimensional ℚ-lattices.

2. For X ∈ Â and f = ι(X) ∈ C^{∗}(G_{1}), the cooling map (7) takes the form

δ(X)(u, λ) = ∑_{n∈ℕ=ℤ_{>0}} f(1, nu, nλ), ∀(u, λ) ∈ C_{ℚ} ≃ Ω̃_{β}. (16)

One can compare directly the right hand side of (16) with the map 𝔈 (up to the normalization by |j|^{1/2}) written as in (13) by considering a function f(ρ, v) = f(1, ρ, v) and its unique extension f̃ to adeles where f is extended by 0 outside ℤ̂ × ℝ^{∗} and one requires the parity

f̃(−u, −λ) = f(u, λ) .  (17)

This gives then

∑_{n∈ℕ} f(1, nu, nλ) = \frac{1}{2} ∑_{q∈ℚ^{∗}} f̃(q j), where j = (u, λ) ∈ C_{ℚ}.  (18)

4 The adeles class space

Let 𝕂 be a global field, with 𝔸_{𝕂} its ring of adeles.

Definition 4.1. The adeles class space of a global field 𝕂 is the quotient 𝔸_{𝕂}/𝕂^{∗}.

When viewed from the classical standpoint this is a “bad quotient” due to the ergodic nature of the action which makes the quotient ill behaved topologically. Thus, following the general philosophy of noncommutative geometry, we describe it by a noncommutative algebra of coordinates, which allows one to continue to treat the quotient as a “nice quotient” in the context of noncommutative geometry. A natural choice of the algebra is the cross product

C_{0}(𝔸_{𝕂}) ⋊ 𝕂^{∗} with the smooth subalgebra S(𝔸_{𝕂}) ⋊ 𝕂^{∗}.  (1)

A better description can be given in terms of groupoids. Consider the groupoid law G_{𝕂} = 𝕂^{∗} ⋉ 𝔸_{𝕂} given by

(k, x) ∘ (k′, y) = (kk′, y), ∀k, k′ ∈ 𝕂^{∗}, and ∀x, y ∈ 𝔸_{𝕂} with x = k′y, (2)

with the composition (2) defined whenever the source s(k, x) = x agrees with the range r(k′, y) = k′y.

Lemma 4.2. The algebras (1) are, respectively, the groupoid C^{∗}-algebra C^{∗}(G_{𝕂}) and its dense subalgebra S(G_{𝕂}).

Proof. The product in the groupoid algebra is given by the associative convolution product

(f_{1} ∗ f_{2}) (k, x) = ∑_{s∈𝕂^{∗}} f_{1}(k s^{−1}, s x)f_{2}(s, x),  (3)

and the adjoint is given by f^{∗}(k, x) = \overline{f(k^{−1}, k x)}. The functions (on the groupoid) associated to f ∈ S(𝔸_{K}) and U_{k} are given, respectively, by

\begin{matrix}f(1, x) = f(x) & and & f(k, x) = 0 & ∀k ≠ 1 \\ U_{k}(k, x) = 1 & and & U_{g}(k, x) = 0 & ∀g ≠ k. \end{matrix}  (4)

The product f U_{k} is then the convolution product of the groupoid.

The algebra S(G_{𝕂}) is obtained by considering finite sums of the form

∑_{k∈𝕂^{∗}} f_{k} U_{k}, for f_{k} ∈ S(𝔸_{𝕂}).  (5)

The product is given by the convolution product

(U_{k} f U_{k}^{∗})(x) = f(k^{−1}x),  (6)

for f ∈ S(𝔸_{𝕂}), k ∈ 𝕂^{∗}, and x ∈ 𝔸_{𝕂}.

4.1 Cyclic module

We can associate to the algebra S(G_{𝕂}) of the adeles class space an object in the category of Λ-modules. This means that we consider the cyclic module S(G_{𝕂})^{♮} and the two cyclic morphisms

ε_{j} : S(G_{𝕂})^{♮} → ℂ  (7)

given by

ε_{0}(∑ f_{k} U_{k}) = f_{1}(0) and ε_{1}(∑ f_{k} U_{k}) = ∫_{𝔸_{𝕂}} f_{1}(x) dx  (8)

and in higher degree by

ε_{j}^{♮}(a^{0} ⊗ ⋯ ⊗ a^{n}) = ε_{j}(a^{0} ⋯ a^{n}).  (9)

The morphism ε_{1} is given by integration on 𝔸_{𝕂} with respect to the additive Haar measure. This is 𝕂^{∗} invariant, hence it defines a trace on S(G_{𝕂}). In the case of 𝕂 = ℚ, this corresponds to the dual trace τ_{φ} for the KMS_{1}-state φ associated to the time evolution of the BC system. The morphism ε_{0} here takes into account the fact that we are imposing a vanishing condition at 0 ∈ 𝔸_{𝕂} (cf. [11] and [13] Chapter 4). In fact, the Λ-module we associate to S(G_{𝕂}) is given by

S(G_{𝕂})_{0}^{♮} := Ker ε_{0}^{♮} ∩ Ker ε_{1}^{♮}.  (10)

Note that since S(G_{𝕂}) is non-unital, the cyclic module S(G_{𝕂})^{♮} is obtained using the adjunction of a unit to S(G_{𝕂}).

4.2 The restriction map

Consider the ideles 𝔸_{𝕂}^{∗} = GL_{1}(𝔸_{𝕂}) of 𝕂 with their natural locally compact topology induced by the map

𝔸_{𝕂}^{∗} ∋ g ↦ (g, g^{−1}).  (11)

We can see the ideles class group C_{𝕂} = 𝔸_{𝕂}^{∗}/𝕂^{∗} as a subspace of the adeles class space X_{𝕂} = 𝔸_{𝕂}/𝕂^{∗} in the following way.

Lemma 4.3. The pairs ((k, x), (k′, y)) ∈ G_{𝕂} such that both x and y are in 𝔸_{𝕂}^{∗} form a full subgroupoid of G_{𝕂} which is isomorphic to 𝕂^{∗} ⋉ 𝔸_{𝕂}^{∗}.

Proof. Elements of 𝔸_{𝕂} whose orbit under the 𝕂^{∗} action contains an idele are also ideles. Thus, we obtain a groupoid that is a full subcategory of G_{𝕂}.

This implies the existence of a restriction map. Consider the map

ρ : S(𝔸_{K}) ∋ f ↦ f|_{𝔸_{𝕂}^{∗}}.  (12)

We denote by C_{ρ}(𝔸_{𝕂}^{∗}) ⊂ C(𝔸_{𝕂}^{∗}) the range of ρ.

Corollary 4.4. The restriction map ρ of (12) extends to an algebra homomorphism

ρ : S(G_{𝕂}) → C_{ρ}(𝔸_{𝕂}^{∗}) ⋊ 𝕂^{∗}.  (13)

Proof. The map (12) induced by the inclusion 𝔸_{𝕂}^{∗} ⊂ 𝔸_{𝕂} is continuous and 𝕂^{∗} equivariant hence the map

ρ( ∑_{k∈𝕂^{∗}} f_{k} U_{k}) = ∑_{k∈𝕂^{∗}} ρ(f_{k}) U_{k}

is an algebra homomorphism.

The action of 𝕂^{∗} on 𝔸_{𝕂}^{∗} is free and proper so that we have an equivalence of the locally compact groupoids 𝕂^{∗} ⋉ 𝔸_{𝕂}^{∗} and 𝔸_{𝕂}^{∗}/𝕂^{∗} = C_{𝕂}. We use the exact sequence of locally compact groups

1 → 𝕂^{∗} → 𝔸_{𝕂}^{∗} →^{p} C_{𝕂} → 1  (14)

to parameterize the orbits of 𝕂^{∗} as the fibers p^{−1}(x) for x ∈ C_{𝕂}. By construction the Hilbert spaces

H_{x} = ℓ^{2}(p^{−1}(x)) , ∀x ∈ C_{K}  (15)

form a continuous field of Hilbert spaces over C_{𝕂}. We let L^{1}(H_{x}) be the Banach algebra of trace class operators in H_{x}, these form a continuous field over C_{𝕂}.

Proposition 4.5. The restriction map ρ of (12) extends to an algebra homomorphism

ρ : S(G_{𝕂}) → C(C_{𝕂}, L^{1}(H_{x})) .  (16)

Proof. Each p^{−1}(x) is globally invariant under the action of 𝕂^{∗} so the crossed product rules in C_{ρ}(𝔸_{𝕂}^{∗}) ⋊ 𝕂^{∗} are just multiplication of operators in H_{x}. To show that the obtained operators are in L^{1} we just need to consider monomials f_{k} U_{k}. In that case the only non-zero matrix elements correspond to k = xy^{−1}. It is enough to show that, for any f ∈ S(𝔸_{𝕂}), the function k ↦ f(k b) is summable. This follows from the discreteness of b 𝕂 ⊂ 𝔸_{𝕂} and the construction of the Bruhat–Schwartz space S(𝔸_{𝕂}), cf. [10]. In fact the associated operator is of finite rank when f has compact support. In general what happens is that the sum will look like the sum over ℤ of the values f(nb) of a Schwartz function f on ℝ.

In general the exact sequence (14) does not split and one does not have a natural C_{𝕂}-equivariant trivialization of the continuous field H_{x}. Thus it is important in the general case to keep the nuance between the algebras C(C_{𝕂}, L^{1}(H_{x})) and C(C_{𝕂}). We shall first deal with the special case 𝕂 = ℚ in which this issue does not arise.

4.3 The Morita equivalence and cokernel for 𝕂 = ℚ

The exact sequence (14) splits for 𝕂 = ℚ and admits a natural continuous section which corresponds to the open and closed fundamental domain Δ_{ℚ} = ℤ̂^{∗} × ℝ_{+}^{∗} ⊂ 𝔸_{ℚ}^{∗} for the action of ℚ^{∗} on ideles. This allows us to construct a cyclic morphism between the cyclic module associated, respectively, to the algebra C_{ρ}(𝔸_{ℚ}^{∗}) ⋊ ℚ^{∗} and to a suitable algebra C_{ρ}(C_{ℚ}) of functions on C_{ℚ}.

Lemma 4.6. The composition d_{ℚ} ∘ e_{ℚ} of the maps

e_{ℚ} : (k, hb) ↦ (b, (k, h)), and d_{ℚ}(k, h) = (kh, h)  (17)

with b ∈ Δ_{ℚ} and k, h ∈ ℚ^{∗}, gives an isomorphism of the locally compact groupoids

ℚ^{∗} ⋉ 𝔸_{ℚ}^{∗} ≃ Δ_{ℚ} × ℚ^{∗} × ℚ^{∗}.  (18)

Proof. The map e_{ℚ} realizes an isomorphism between the locally compact groupoids

ℚ^{∗} ⋉ 𝔸_{ℚ}^{∗} ≃ Δ_{ℚ} × (ℚ^{∗} ⋉ ℚ^{∗}),

where ℚ^{∗} ⋉ ℚ^{∗} is the groupoid of the action of ℚ^{∗} on itself by multiplication. The latter is isomorphic to the trivial groupoid ℚ^{∗} × ℚ^{∗} via the map d_{ℚ}.

We then have the following result.

Proposition 4.7. The map

∑_{k∈ℚ^{∗}} f_{k} U_{k} ↦ M_{b}(x, y) = f_{xy^{−1}}(x b),  (19)

for x, y ∈ ℚ^{∗} with k = xy^{−1} and b ∈ Δ_{ℚ}, defines an algebra homomorphism

C_{ρ}(𝔸_{ℚ}^{∗}) ⋊ ℚ^{∗} → C(Δ_{ℚ}, M_{∞}(ℂ))

to the algebra of matrix valued functions on Δ_{ℚ}. For any f ∈ S(G_{ℚ}) the element M_{b} obtained in this way is of trace class.

Proof. We use the groupoid isomorphism (17) to write k = xy^{−1} and khb = xb, for x = kh and y = h. The second statement follows from Proposition 4.5.

Let π = M ∘ρ : S(G_{𝕂}) → C(Δ_{ℚ}, M_{∞}(ℂ)) be the composition of the restriction map ρ of (13) with the algebra morphism (19). Since the trace Tr on M_{∞}(ℂ) gives a cyclic morphism one can use this to obtain a morphism of cyclic modules (Tr ∘ π)^{♮}, which we now describe explicitly. We let, in the number field case,

S (C_{𝕂}) = ∩_{β∈ℝ} μ^{β}S(C_{𝕂}),  (20)

where μ ∈ C(C_{𝕂}) is the module morphism from C_{𝕂} to ℝ_{+}^{∗}. In the function field case one can simply use for S (C_{𝕂}) the Schwartz functions with compact support.

Proposition 4.8. The map Tr∘π defines a morphism (Tr∘π)^{♮} of cyclic modules from S(G_{ℚ})_{0}^{♮} to the cyclic submodule S^{♮}(C_{ℚ}) ⊂ C(C_{ℚ})^{♮} whose elements are continuous functions whose restriction to the main diagonal belongs to S (C_{ℚ}).

Proof. By Proposition 4.7 the map π is an algebra homomorphism from S(G_{ℚ}) to C(Δ_{ℚ}, L^{1}) ∼ C(C_{ℚ}, L^{1}). We need to show that the corresponding cyclic morphism using Tr^{♮} lands in the cyclic submodule S^{♮}(C_{ℚ}). For simplicity we can just restrict to the case of monomials, where we consider elements of the form

Z = f_{k_{0}} U_{k_{0}} ⊗ f_{k_{1}} U_{k_{1}} ⊗ ⋯ ⊗ f_{k_{n}} U_{k_{n}}.  (21)

The matrix valued functions associated to the monomials f_{k_{j}} U_{k_{j}} as in Proposition 4.7 have matrix elements at a point b ∈ Δ_{ℚ} that are non zero only for x_{j+1} = x_{j}k_{j}^{−1} and are of the form

f_{k_{j}} U_{k_{j}} ↦ M_{b}(x_{j}, x_{j+1}) = f_{k_{j}}(x_{j}b).  (22)

Composing with the cyclic morphism Tr^{♮} gives

(Tr ∘ π)^{♮}(Z)(b_{0}, b_{1}, …, b_{n}) = ∑ ∏ M_{b_{j}}(x_{j}, x_{j+1})  (23)

where the x_{j} ∈ 𝕂^{∗} and x_{n+1} = x_{0}. Let γ_{0} = 1 and γ_{j+1} = k_{j}γ_{j}. Then we find that (Tr ∘ π)^{♮}(Z) = 0, unless ∏_{j} k_{j} = 1, i.e. γ_{n+1} = 1. In this case we obtain

Tr ∘ π(Z)(b_{0}, b_{1}, ⋯, b_{n}) = ∑_{k∈ℚ^{∗}} ∏_{j=0}^{n} f_{k_{j}}(γ_{j}^{−1}kb_{j}), ∀b_{j} ∈ Δ_{ℚ}. (24)

For n = 0 the formula (24) reduces to

Tr ∘ π(f)(b) = ∑_{k∈ℚ^{∗}} f(kb), ∀b ∈ Δ_{ℚ}, ∀f ∈ S(𝔸_{ℚ})_{0},  (25)

where S(𝔸_{ℚ})_{0} = Kerε_{0} ∩ Kerε_{1} ⊂ S(𝔸_{ℚ}). This gives an element of S (C_{ℚ}), by Lemma 2 Appendix 1 of [10]. In general, (24) gives a continuous function of n + 1 variables on C_{ℚ}, and its restriction to the main diagonal belongs to S (C_{ℚ}).

Since the category of cyclic modules is an abelian category, we can consider the cokernel in the category of Λ-modules of the cyclic morphism (Tr ∘ π)^{♮}, with π the composite of (13) and (19). This works nicely for 𝕂 = ℚ but makes use of the splitting of the exact sequence (14).

4.4 The cokernel of ρ for general global fields

To handle the general case in a canonical manner one just needs to work directly with C(C_{𝕂}, L^{1}(H_{x})) instead of C(C_{𝕂}) and express at that level the decay condition of the restrictions to the diagonal in the cyclic submodule S^{♮}(C_{ℚ}) of Proposition 4.8.

Definition 4.9. We define S^{♮}(C_{𝕂}, L^{1}(H_{x})) to be the cyclic submodule of the cyclic module C(C_{𝕂}, L^{1}(H_{x}))^{♮}, whose elements are continuous functions such that the trace of the restriction to the main diagonal belongs to S (C_{𝕂}).

Note that for T ∈ C(C_{𝕂}, L^{1}(H_{x}))^{♮} of degree n, T (x_{0}, …, x_{n}) is an operator in H_{x_{0}} ⊗…⊗H_{x_{n}}. On the diagonal, x_{j} = x for all j, the trace map corresponding to Tr^{♮} is given by

Tr^{♮}(T_{0} ⊗ T_{1} ⊗ … ⊗ T_{n}) = Tr(T_{0} T_{1} … T_{n}) .  (26)

This makes sense since on the diagonal all the Hilbert spaces H_{x_{j}} are the same. The argument of Proposition 4.8 extends to the general case and shows that the cyclic morphism ρ^{♮} of the restriction map ρ lands in S^{♮}(C_{𝕂}, L^{1}(H_{x})).

Definition 4.10. We define H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}) to be the cokernel of the cyclic morphism

ρ^{♮} : S(G_{𝕂})_{0}^{♮} → S^{♮}(C_{𝕂}, L^{1}(H_{x}))

Moreover, an important issue arises, since the ranges of continuous linear maps are not necessarily closed subspaces. In order to preserve the duality between cyclic homology and cyclic cohomology we shall define the cokernel of a cyclic map T : A^{♮} → B^{♮} as the quotient of B^{♮} by the closure of the range of T. In a dual manner, the kernel of the transposed map T^{t} : B^{♯} → A^{♯} is automatically closed and is the dual of the above. The choice of the notation H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}) is explained by the fact that we consider this a first cohomology group, in the sense that it is a cokernel in a sequence of cyclic homology groups for the inclusion of the ideles class group in the adeles class space (dually for the restriction map of algebras), hence we can think of it as giving rise to an H^{1} in the relative cohomology sequence of an inclusion of C_{𝕂} in the noncommutative space 𝔸_{𝕂}/𝕂^{∗}. We can use the result of [7], describing the cyclic (co)homology in terms of derived functors in the category of cyclic modules, to write the cyclic homology as

HC_{n}(A) = Tor_{n}(ℂ^{♮}, A^{♮}).  (27)

Thus, we obtain a cohomological realization of the cyclic module H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}) by setting

H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}) := Tor(ℂ^{♮}, H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂})).  (28)

We think of this as an H^{1} because of its role as a relative term in a cohomology exact sequence of the pair (𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}). We now show that H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}) carries an action of C_{𝕂}, which we can view as the abelianization W_{𝕂}^{ab} ∼ C_{𝕂} of the Weil group. This action is induced by the multiplicative action of C_{𝕂} on 𝔸_{𝕂}/𝕂^{∗} and on itself. This generalizes to global fields the action of C_{ℚ} = ℤ̂^{∗} × ℝ_{+}^{∗} on HC_{0}(D(A, φ)) for the Bost– Connes endomotive (cf. [11]).

Proposition 4.11. The cyclic modules S(G_{𝕂})_{0}^{♮} and S^{♮}(C_{𝕂}, L^{1}(H_{x})) are endowed with an action of 𝔸_{𝕂}^{∗} and the morphism ρ^{♮} is 𝔸_{𝕂}^{∗}-equivariant. This induces an action of C_{𝕂} on H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}).

Proof. For γ ∈ 𝔸_{𝕂}^{∗} one defines an action by automorphisms of the algebra A = S(G_{𝕂}) by setting

ϑ_{a}(γ)(f)(x) := f(γ^{−1}x), for f ∈ S(𝔸_{𝕂}),  (29)
ϑ_{a}(γ)( ∑_{k∈𝕂^{∗}} f_{k} U_{k}) := ∑_{k∈𝕂^{∗}} ϑ_{a}(γ)(f_{k}) U_{k} .  (30)

This action is inner for γ ∈ 𝕂^{∗} and induces an outer action

C_{𝕂} → Out(S(G_{𝕂})) .  (31)

Similarly, the continuous field H_{x} = ℓ^{2}(p^{−1}(x)) over C_{𝕂} is 𝔸_{𝕂}^{∗}-equivariant for the action of 𝔸_{𝕂}^{∗} on C_{𝕂} by translations, and the equality

(V (γ)ξ)(y) := ξ(γ^{−1} y) , ∀y ∈ p^{−1}(γx) , ξ ∈ ℓ^{2}(p^{−1}(x)) ,  (32)

defines an isomorphism H_{x} ⟶^{V (γ)} H_{γx}. One obtains then an action of 𝔸_{𝕂}^{∗} on C(C_{𝕂}, L^{1}(H_{x})) by setting

ϑ_{m}(γ)(f)(x) := V (γ) f(γ^{−1} x) V (γ^{−1}), ∀f ∈ C(C_{𝕂}, L^{1}(H_{x})) .  (33)

The morphism ρ is 𝔸_{𝕂}^{∗}-equivariant, so that one obtains an induced action on the cokernel H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}). This action is inner for γ ∈ 𝕂^{∗} and thus induces an action of C_{𝕂} on H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}).

We denote by

C_{𝕂} ∋ γ ↦ \underline{ϑ}_{m}(γ)  (34)

the induced action on H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}). We have a non-canonical isomorphism

C_{𝕂} ≃ C_{𝕂,1} × N,  (35)

where N ⊂ ℝ_{+}^{∗} is the range of the norm |·| : C_{𝕂} → ℝ_{+}^{∗}. For number fields this is N = ℝ_{+}^{∗}, while for function fields in positive characteristic N ≃ ℤ is the subgroup q^{ℤ} ⊂ ℝ_{+}^{∗} with q = p^{ℓ} the cardinality of the field of constants. We denote by Ĉ_{𝕂,1} the group of characters of the compact subgroup C_{𝕂,1} ⊂ C_{𝕂}, i.e. the Pontrjagin dual of C_{𝕂,1}. Given a character χ of C_{𝕂,1}, we let χ̃ denote the unique extension of χ to C_{𝕂} which is equal to one on N. One obtains a decomposition of H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}) according to projectors associated to characters of C_{𝕂,1}.

Proposition 4.12. Characters χ ∈ Ĉ_{𝕂,1} determine a canonical direct sum decomposition

\begin{matrix}H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}) = ⨁_{χ∈Ĉ_{𝕂,1}} H_{χ}^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}) \\ H_{χ}^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}) = \{ξ| \underline{ϑ}_{m}(γ) ξ = χ(γ) ξ, ∀γ ∈ C_{𝕂,1}\}. \end{matrix}  (36)

where \underline{ϑ}_{m}(γ) denotes the induced action (34) on H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}).

Proof. The action of 𝔸_{𝕂}^{∗} on H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}) induces a corresponding action of C_{𝕂} on H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}).

We can then reformulate the result of [11] based on the trace formula of [10] in the formulation of [33] in terms of the cohomology H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}) in the following way.

Proposition 4.13. The induced representation of C_{𝕂} on H_{χ}^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}) gives the spectral realization of the zeros of the L-function with Grössencharakter χ.

This result is a variant of Corollary 2 of [10], the proof is similar and essentially reduces to the result of [33]. There is a crucial difference with [10] in that all zeros (including those not located on the critical line) now appear due to the choice of the function spaces. To see what happens it is simpler to deal with the dual spaces i.e. to compute the cyclic cohomology HC^{0}. Its elements are cyclic morphisms T from H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}) to ℂ^{♮} and they are determined by the map T^{0} in degree 0. The cyclic morphism property then shows that T^{0} defines a trace on S^{♮}(C_{𝕂}, L^{1}(H_{x})) which vanishes on the range of ρ^{♮}. The freeness of the action of 𝕂^{∗} on 𝔸_{𝕂}^{∗} then ensures that these traces are given by continuous linear forms on S (C_{𝕂}) which vanish on the following subspace of S (C_{𝕂}) which is the range of the restriction map, defined as follows.

Definition 4.14. Let V ⊂ S (C_{𝕂}) denote the range of the map Tr ∘ ρ, that is,

V = {h ∈ S (C_{𝕂})| h(x) = ∑_{k∈𝕂^{∗}} ξ(kx), with ξ ∈ S(𝔸_{𝕂})_{0}},  (37)

where S(𝔸_{𝕂})_{0} = Kerε_{0} ∩ Kerε_{1} ⊂ S(𝔸_{𝕂}).

We have seen above in the case 𝕂 = ℚ (cf. [10]) that the range of Tr ∘ ρ is indeed contained in S (C_{𝕂}). Moreover, we have the following results about the action \underline{ϑ}_{m}(γ), for γ ∈ C_{𝕂}, on H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}). Suppose given f ∈ S (C_{𝕂}). We define a corresponding operator

\underline{ϑ}_{m}(f) = ∫_{C_{𝕂}} f(γ) \underline{ϑ}_{m}(γ) d^{∗}γ,  (38)

acting on the complex vector space H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}). Here d^{∗}γ is the multiplicative Haar measure on C_{𝕂}. We have the following description of the action of \underline{ϑ}_{m}(f).

Lemma 4.15. For f ∈ S (C_{𝕂}), the action of the operator \underline{ϑ}_{m}(f) of (38) on H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}) is the action induced on the quotient of S (C_{𝕂}) by V ⊂ S (C_{𝕂}) of the action of ϑ_{m}(f) on S (C_{𝕂}) by convolution product

ϑ_{m}(f)ξ(u) = ∫_{C_{𝕂}} ξ(g^{−1}u)f(g) d^{∗}g = (f ⋆ ξ)(u).  (39)

Proof. One first shows that one can lift f to a function f̃ on 𝔸_{𝕂}^{∗} such that

∑_{k∈𝕂^{∗}} f̃(kx) = f(x)

and that convolution by f̃ i.e.

∫ f̃(γ)ϑ_{a}(γ)d^{∗}γ

leaves S(G_{𝕂}) globally invariant. This means showing that that S(𝔸_{𝕂})_{0} is stable under convolution by the lift of S (C_{𝕂}). Then (39) follows directly from the definition of the actions (33), (30), (34) and the operator (38).

For f ∈ S (C_{𝕂}) and χ̃ the extension of a character χ ∈ Ĉ_{𝕂,1} to C_{𝕂} and f̂(χ̃, ρ) the Fourier transform (30), the operators \underline{ϑ}_{m}(f) of (38) satisfy the spectral side of the trace formula. Namely, we have the following result.

Theorem 4.16. For any f ∈ S (C_{𝕂}), the operator \underline{ϑ}_{m}(f) defined in (38) acting on H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}) is of trace class. The trace is given by

Tr(\underline{ϑ}_{m}(f)|H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂})) = ∑_{ρ∈ℂ/N^{⊥}| L(χ̃,ρ)=0} f̂(χ̃, ρ),  (40)

with f̂(χ̃, ρ) the Fourier transform (30).

Proof. Due to the different normalization of the summation map, the representation \underline{ϑ}_{m}(γ) considered here differs from the action W (γ) considered in [10] by

\underline{ϑ}_{m}(γ) = |γ|^{1/2} W (γ).  (41)

This means that we have

\underline{ϑ}_{m}(f) = ∫_{C_{𝕂}} f(γ) \underline{ϑ}_{m}(γ) d^{∗}γ = ∫_{C_{𝕂}} h(γ) W (γ) d^{∗}γ,  (42)

where

h(γ) = |γ|^{1/2} f(γ).  (43)

We then have, for W (h) = ∫_{C_{𝕂}} h(γ) W (γ) d^{∗}γ,

TrW (h) = ∑_{ρ∈ℂ/N^{⊥}| L(χ̃, \frac{1}{2} +ρ)=0} ĥ(χ̃, ρ).  (44)

Note that unlike in [10] all zeros contribute, including those that might fail to be on the critical line, and they do with their natural multiplicity. This follows from the choice of function space as in [33]. The Fourier transform ĥ(χ̃, ρ) satisfies

ĥ(χ̃, ρ) = ∫_{C_{𝕂}} h(u)χ̃(u) |u|^{ρ} d^{∗}u = ∫_{C_{𝕂}} f(u)χ̃(u) |u|^{ρ+1/2} d^{∗}u = f̂(χ̃, ρ + 1/2),

(45)

where h and f are related as in (43). Thus, the shift by 1/2 in (44) is absorbed in (45) and this gives the required formula (40).

4.5 Trace pairing and vanishing

The commutativity of the convolution product implies the following vanishing result.

Lemma 4.17. Suppose given an element f ∈ V ⊂ S (C_{𝕂}), where V is the range of the reduction map as in Definition 4.14. Then one has

\underline{ϑ}_{m}(f)|_{H^{1}(𝔸_{𝕂}/𝕂^{∗},C_{𝕂})} = 0.  (46)

Proof. The result follows by showing that, for f ∈ V, the operator \underline{ϑ}_{m}(f) maps any element ξ ∈ S (C_{𝕂}) to an element in V, hence the induced map on the quotient of S (C_{𝕂}) by V is trivial. Since V is a submodule of S (C_{𝕂}) for the action of S (C_{𝕂}) by convolution we obtain

\underline{ϑ}_{m}(f)ξ = f ⋆ ξ = ξ ⋆ f ∈ V,

where ⋆ is the convolution product of (39).

This makes it possible to define a trace pairing as follows.

Remark 4.18. The pairing

f_{1} ⊗ f_{2} ↦ ⟨f_{1}, f_{2}⟩_{H^{1}} := Tr(\underline{ϑ}_{m}(f_{1} ⋆ f_{2})|H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}))  (47)

descends to a well defined pairing on H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}) ⊗ H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}).

5 Primitive cohomology

The aim of this section is to interpret the motivic construction described in the previous paragraph as the noncommutative version of a classical construction in algebraic geometry. In motive theory, realizations of (mixed) motives appear frequently in the form of kernels/cokernels of relevant homomorphisms. The primitive cohomology is the example we shall review hereafter.

If Y is a compact Kähler variety, a Kähler cocycle class [ω] ∈ H^{2}(Y, ℝ) determines the Lefschetz operator (i ∈ ℤ_{≥0}):

L : H^{i}(Y, ℝ) → H^{i+2}(Y, ℝ), L(a) := [ω] ∪ a.

Let n = dim Y. Then, the primitive cohomology is defined as the kernel of iterated powers of the Lefschetz operator

H^{i}(Y, ℝ)_{prim} := Ker(L^{n−i+1} : H^{i}(Y, ℝ) → H^{2n−i+2}(Y, ℝ)).

In particular, for i = n we have

H^{n}(Y, ℝ)_{prim} := Ker(L : H^{n}(Y, ℝ) → H^{n+2}(Y, ℝ)).

Let assume, from now on, that j : Y ↪ X is a smooth hyperplane section of a smooth, projective complex algebraic variety X. Then, it is a classical result of geometric topology that L = j^{∗} ∘ j_{∗}, where

j_{∗} : H^{i}(Y, ℝ) → H^{i+2}(X, ℝ)

is the Gysin homomorphism: the Poincaré dual of the restriction homomorphism

j^{∗} : H^{2n−i}(X, ℝ) → H^{2n−i}(Y, ℝ).

In fact, because the class of L comes from an integral class, the equality L = j^{∗} ∘ j_{∗} holds already in integral cohomology. For i = n, the above description of the Lefschetz operator together with the Lefschetz theorem of hyperplane sections imply that

H^{n}(Y, ℝ)_{prim} ≅ Ker(j_{∗} : H^{n}(Y, ℝ) → H^{n+2}(X, ℝ)) =: H^{n}(Y, ℝ)_{van}

where by H^{i}(Y, ℝ)_{van} we denote the vanishing cohomology

H^{i}(Y, ℝ)_{van} := Ker(j_{∗} : H^{i}(Y, ℝ) → H^{i+2}(X, ℝ)).

Now, we introduce the theory of mixed Hodge structures in this set-up. Let U := X ∖Y be the open space which is the complement of Y in X and let denote by k : U ↪ X the corresponding open immersion. Then, one knows that R^{i}j_{∗}ℤ = 0 unless i = 0, 1 so that the Leray spectral sequence for j:

E_{2}^{p,q} = H^{q}(X, R^{p}k_{∗}ℤ) ⇒ H^{p+q}(U, ℤ)

coincides with the long exact sequence (of mixed Hodge structures)

… →^{∂} H^{i−2}(Y, ℤ)(−1) →^{j_{∗}} H^{i}(X, ℤ) → H^{i}(U, ℤ) →^{∂} …

The boundary homomorphism ∂ in this sequence is known to coincide ([21], § 9.2) with the residue homomorphism

Res : H^{i+1}(U, ℤ) → H^{i}(Y, ℤ)(−1)

whose description, with complex coefficients, is derived from a corresponding morphism of filtered complexes (Poincaré residue map). This morphism fits in the following exact sequence of filtered complexes of Hodge modules

0 → Ω_{X}^{·} → Ω_{X}^{·}(log Y) →^{res} j_{∗}Ω_{Y}^{·} [−1] → 0

res(α ∧ \frac{dt}{t}) = α_{|Y} .

One knows that Res is a homomorphism of Hodge structures, hence the Hodge filtration on H^{n+1}(U, ℂ) ≅ ℍ^{n+1}(X, Ω_{X}^{·}(logY)) determines a corresponding filtration on the (twisted) vanishing cohomology

H^{n}(Y, ℂ)(n)_{van} = Ker(j_{∗} : H^{n}(Y, ℂ)(n) → H^{n+2}(X, ℂ)(n+1)) ≅ H^{n+1}(U)(n+1).

In degree i = n, one also knows that the excision exact sequence (of Hodge structures) becomes the short exact sequence

0 → H^{n}(X, ℂ) →^{j^{∗}} H^{n}(Y, ℂ) → H_{c}^{n+1}(U, ℂ) → 0.

Therefore, it follows by the Poincaré duality isomorphism

H_{c}^{n+1}(U, ℂ)^{∗} ≅ H^{n+1}(U, ℂ)(n + 1)

that

(Coker(j^{∗} : H^{n}(X, ℂ) → H^{n}(Y, ℂ)))^{∗} ≅ H_{c}^{n+1}(U, ℂ)^{∗} ≅ H^{n}(Y, ℂ)(n)_{van}. (1)

When j : Y ↪ X is a singular hypersurface or a divisor in X with (local) normal crossings (i.e.: Y = ⋃_{i} Y_{i}, dim Y_{i} = n = dim X −1, Y locally described by an equation x_{i_{1}} ⋯ x_{i_{r}} = 0, {i_{1}, … i_{r}} ⊆ {1, … n + 1}, {x_{1}, … x_{n+1}} = system of local coordinates in X), the notion of the Gysin homomorphism is lost. One then replaces the vanishing cohomology by the primitive cohomology, whose definition extends to this general set-up and is given, in analogy to (1), as

H^{n}(Y, ℂ)_{prim} := Coker(j^{∗} : H^{n}(X, ℂ) → H^{n}(Y, ℂ)) ⊆ H_{c}^{n+1}(U, ℂ).

One also knows that the primitive cohomology is motivic (cf. [22] and [3] for interesting examples). Following the classical construction that we have just reviewed, we like to argue now that the definition of the cyclic module H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}) (as in Definition 4.10), which is based on a noncommutative version of a restriction map “from adeles to ideles” defined in the category of Λ-modules, should be interpreted as the noncommutative analogue of a primitive motive (a cyclic primitive module). The cohomological realization of such motive (i.e. its cyclic homology) is given by the group H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}) = Tor(ℂ^{♮}, H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂})) (cf. (28)) which therefore can be interpreted as a noncommutative version of a primitive cohomology.

6 A cohomological Lefschetz trace formula

6.1 Weil’s explicit formula as a trace formula

As in §2.2 above, let α be a non-trivial character of 𝔸_{𝕂} which is trivial on 𝕂 ⊂ 𝔸_{𝕂}. It is well known ([44] VII-2) that for such a character α there exists a differental idele a = (a_{v}) ∈ 𝔸_{𝕂}^{∗} such that

α_{v}(x) = e_{𝕂_{v}}(a_{v} x), ∀x ∈ 𝕂_{v},  (1)

where, for a local field K, the additive character e_{K} is chosen in the following way.

• If K = ℝ then e_{ℝ}(x) = e^{−2πix}, for all x ∈ ℝ.

• If K = ℂ then e_{ℂ}(z) = e^{−2πi(z+z̄)}, for all z ∈ ℂ.

• If K is a non-archimedean local field with maximal compact subring O, then the character e_{K} satisfies Ker e_{K} = O.

The notion of differental idele can be thought of as an extension of the canonical class of the algebraic curve C, from the setting of function fields 𝔽_{q}(C) to arbitrary global fields 𝕂. For instance, one has

|a| = q^{2−2g} or |a| = D^{−1},  (2)

respectively, for the case of a function field 𝔽_{q}(C) and of a number field. In the number field case D denotes the discriminant.

In [11] we gave a cohomological formulation of the Lefschetz trace formula of [10], using the version of the Riemann–Weil explicit formula as a trace formula given in [33] in the context of nuclear spaces, rather than the semi-local Hilbert space version of [10].

Theorem 6.1. For f ∈ S (C_{𝕂}) let \underline{ϑ}_{m}(f) be the operator (38) acting on the space H^{1} = H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}). Then the trace is given by

Tr(\underline{ϑ}_{m}(f)|H^{1}) = f̂(0) + f̂(1) − (log |a|) f(1) − ∑_{v∈Σ_{𝕂}} ∫′_{(𝕂_{v}^{∗},e_{𝕂v})} \frac{f(u^{−1})}{|1 − u|} d^{∗}u. (3)

The formula (3) is obtained in [11] first by showing that the Lefschetz trace formula of [10] in the version of [33] can be formulated equivalently in the form

Tr(\underline{ϑ}_{m}(f)|H^{1}) = f̂(0) + f̂(1) − ∑_{v∈𝕂_{v}} ∫′_{𝕂_{v}^{∗}} \frac{f(u^{−1})}{|1 − u|} d^{∗}u,  (4)

where one uses the global character α to fix the local normalizations of the principal values in the last term of the formula. We then compute this principal value using the differental idele in the form

∫′_{(𝕂_{v}^{∗},α_{v})} \frac{f(u^{−1})}{|1 − u|} d^{∗}u = (log |a_{v}|) f(1) + ∫′_{(𝕂_{v}^{∗},e_{𝕂v})} \frac{f(u^{−1})}{|1 − u|} d^{∗}u.  (5)

6.2 Weil Positivity and the Riemann Hypothesis

We introduce an involution for elements f ∈ S (C_{𝕂}) by setting

f^{∗}(g) = \overline{f(g^{−1})}.  (6)

We also consider a one parameter group z ↦ Δ^{z} of automorphisms of the convolution algebra S (C_{𝕂}), with the convolution product (39) by setting

Δ^{z}(f)(g) = |g|^{z} f(g),  (7)

for f ∈ S (C_{𝕂}) and z ∈ ℂ. Since (7) is given by multiplication by a character, it satisfies

Δ^{z}(f ⋆ h) = Δ^{z}(f) ⋆ Δ^{z}(f), ∀f, h ∈ S (C_{𝕂}).  (8)

We consider also the involution

f ↦ f^{♯} = Δ^{−1} f^{∗}, with f^{♯}(g) = |g|^{−1}\overline{f(g^{−1})}.  (9)

The reformulation, originally due to A. Weil, of the Riemann Hypothesis in our setting is given by the following statement.

Proposition 6.2. The following two conditions are equivalent:

• All L-functions with Grössencharakter on 𝕂 satisfy the Riemann Hypothesis.

• The trace pairing (47) satisfies the positivity condition

⟨Δ^{−1/2} f, Δ^{−1/2} f^{∗}⟩ ≥ 0, ∀f ∈ S (C_{𝕂}).  (10)

Proof. Let W (γ) = |γ|^{−1/2} \underline{ϑ}_{m}(γ). Then, by [42] the RH for L-functions with Grössencharakter on 𝕂 is equivalent to the positivity

Tr(W (f ⋆ f^{∗})) ≥ 0, ∀f ∈ S (C_{𝕂}).  (11)

Thus, in terms of the representation \underline{ϑ}_{m} we are considering here, we have

W (f) = \underline{ϑ}_{m}(Δ^{−1/2} f).

Using the multiplicative property (8) of Δ^{z} we rewrite (11) in the equivalent form (10).

In terms of the involution (9) we can reformulate Proposition 6.2 in the following equivalent way.

Corollary 6.3. The following conditions are equivalent

• All L-functions with Grössencharakter on 𝕂 satisfy the Riemann Hypothesis.

• The trace pairing (47) satisfies ⟨f, f^{♯}⟩ ≥ 0, for all f ∈ S (C_{𝕂}).

Proof. In (10) we write Δ^{−1/2}f = h. This gives

Δ^{−1/2}f^{∗} = Δ^{−1/2}(Δ^{1/2}h)^{∗} = Δ^{−1}h^{∗} = h^{♯}

and the result follows, since Δ^{−1/2} is an automorphism of S (C_{𝕂}).

The vanishing result of Lemma 4.17, for elements in the range V ⊂ S (C_{𝕂}) of the reduction map Tr ∘ ρ from adeles, gives then the following result.

Proposition 6.4. The elements f ⋆f^{♯} considered in Corollary 6.3 above have the following properties.

1. The trace pairing ⟨f, f^{♯}⟩ vanishes for all f ∈ V, i.e. when f is the restriction Tr ∘ ρ of an element of S(G_{𝕂}).

2. By adding elements of V one can make the values

f ⋆ f^{♯}(1) = ∫_{C_{𝕂}} |f(g)|^{2} |g| d^{∗}g < ϵ  (12)

for arbitrarily small ϵ > 0.

Proof. (1) The vanishing result of Lemma 4.17 shows that \underline{ϑ}_{m}(f)|_{H^{1}(𝔸_{𝕂}/𝕂^{∗},C_{𝕂})} = 0 for all f ∈ V. Thus, the trace pairing satisfies ⟨f, h⟩ = 0, for f ∈ V and for all h ∈ S (C_{𝕂}). In particular this applies to the case h = f^{♯}.

(2) This follows from the surjectivity of the map 𝔈 for the weight δ = 0 (cf. Appendix 1 of [10]).

Proposition 6.4 shows that the trace pairing admits a large radical given by all functions that extend to adeles. Thus, one can divide out this radical and work with the cohomology H^{1}(𝔸_{𝕂}/𝕂^{∗}, C_{𝕂}) described above.

7 Correspondences

To start building the dictionary between the Weil proof and the noncommutative geometry of the adeles class space, we begin by reformulating the trace formula discussed above in more intersection theoretic language, so as to be able to compare it with the setup of §2.1 above. We also discuss in this section the analog of moding out by trivial correspondence.

7.1 The scaling correspondence as Frobenius

To the scaling action

ϑ_{a}(γ)(ξ)(x) = ξ(γ^{−1}x), for γ ∈ C_{𝕂} and ξ ∈ S(𝔸_{𝕂}),

one associates the graph Z_{γ} given by the pairs (x, γ^{−1}x). These should be considered as points in the product 𝔸_{𝕂}/𝕂^{∗} × 𝔸_{𝕂}/𝕂^{∗} of two copies of the adeles class space. Thus, the analog in our context of the correspondences Z = ∑_{n} a_{n}Fr^{n} on C × C is given by elements of the form

Z(f) = ∫_{C_{𝕂}} f(g) Z_{g} d^{∗}g,  (1)

for some f ∈ S (C_{𝕂}). With this interpretation of correspondences, we can then make sense of the terms in the trace formula in the following way.

Definition 7.1. For a correspondence of the form (1) we define degree and codegree by the following prescription

d(Z(f)) := f̂(1) = ∫_{C_{𝕂}} f(u) |u| d^{∗}u,  (2)

d′(Z(f)) := d(Z(f^{♯})) = ∫_{C_{𝕂}} f(u) d^{∗}u = f̂(0).  (3)

Here the Fourier transform f̂ is as in (30), with the trivial character χ = 1. Notice that, with this definition of degree and codegree we find

d(Z_{g}) = |g|, and d′(Z_{g}) = 1.  (4)

Thus, the term f̂(1) + f̂(0) in the trace formula of Theorem 6.1 match the term d(Z) + d′(Z) in Weil’s formula for the trace of a correspondence as in (25). The term

− ∫′_{(𝕂_{v}^{∗},α_{v})} \frac{f(u^{−1})}{|1 − u|} d^{∗}u  (5)

of (4) in turn can be seen as the remaining term −Z • Δ in (25). In fact, the formula (5) describes, using distributions, the local contributions to the trace of the intersections between the graph Z(f) and the diagonal Δ. This was proved in [10], Section VI and Appendix III. It generalizes the analogous formula for flows on manifolds of [23], which in turn can be seen as a generalization of the usual Atiyah–Bott Lefschetz formula for a diffeomorphism of a smooth compact manifold [2]. When we separate out the contribution log |a| h(1), as in passing from (4) to (3), and we rewrite the trace formula as in Theorem 6.1. This corresponds to separating the intersection Z • Δ into a term that is proportional to the self intersection Δ • Δ and a remaning terms where the intersection is transverse. To see this, we notice that the term log |a|, for a = (a_{v}) a differental idele, is of the form (2). Indeed one sees that, in the function field case the term

− log |a| = − log q^{2−2g} = (2g − 2) log q = −Δ • Δ log q

is proportional to the self intersection of the diagonal, which brings us to consider the value log |a| = − log D with the discriminant of a number field as the analog in characteristic zero of the self intersection of the diagonal. In these intersection theoretic terms we can reformulate the positivity condition (cf. [4]) equivalent to the Riemann Hypothesis in the following way.

Proposition 7.2. The following two conditions are equivalent

• All L-functions with Grössencharakter on 𝕂 satisfy the Riemann Hypothesis.

• The estimate

Z(f) •_{trans} Z(f) ≤ 2d(Z(f))d′(Z(f)) − Δ • Δ f ⋆ f^{♯}(1)  (6)

holds for all f ∈ S (C_{𝕂}).

Proof. As in the Weil proof one separates the terms Z ⋆ Z′ = d′(Z)Δ + Y, where Y has transverse intersection with the diagonal, here we can write an identity

Tr(\underline{ϑ}_{m}(f ⋆ f^{♯})|H^{1}) =: Z(f) • Z(f) = Δ • Δ f ⋆ f^{♯}(1) + Z(f) •_{trans} Z(f) (7)

where the remaning term Z(f) •_{trans} Z(f) which represents the transverse intersection is given by the local contributions given by the principal values over (𝕂_{v}^{∗}, e_{𝕂_{v}}) in the formula (3) for Tr(\underline{ϑ}_{m}(f ⋆ f^{♯})|H^{1}). The formula (3) for Tr(\underline{ϑ}_{m}(f ⋆ f^{♯})|H^{1}) gives a term of the form − log |a| f ⋆ f^{♯}(1), with

f ⋆ f^{♯}(1) = ∫_{C_{𝕂}} |f(g)|^{2} |g| d^{∗}g.

We rewrite this term as −Δ • Δ f ⋆ f^{♯}(1) according to our interpretation of log |a| as self-intersection of the diagonal. This matches the term (2g −2)d′(Z) in the estimate for Tr(Z ⋆ Z′) in the Weil proof. The first two terms in the formula (3) for Tr(\underline{ϑ}_{m}(f ⋆ f^{♯})|H^{1}) are of the form

{ \atop f ⋆ f^{♯}}(0) + { \atop f ⋆ f^{♯}}(1) = 2f̂(0)f̂(1) = 2d′(Z(f))d(Z(f)).  (8)

This matches the term 2gd′(Z) = 2d(Z)d′(Z) in the expression for Tr(Z ⋆ Z′) in the Weil proof. With this notation understood, we see that the positivity Tr(\underline{ϑ}_{m}(f ⋆f^{♯})|H^{1}) ≥ 0 corresponds indeed to the estimate (6).

7.2 Fubini’s theorem and the trivial correspondences

As we have seen in recalling the main steps in the Weil proof, a first step in dealing with correspondences is to use the freedom to add multiples of the trivial correspondences in order to adjust the degree. We describe an analog, in our noncommutative geometry setting, of the trivial correspondences and of this operation of modifying the degree. In view of the result of Proposition 6.4 above, it is natural to regard the elements f ∈ V ⊂ S (C_{𝕂}) as those that give rise to the trivial correspondences Z(f). Here, as above, V is the range of the reduction map from adeles.

The fact that it is possible to arbitrarily modify the degree d(Z(f)) = f̂(1) of a correspondence by adding to f an element in V depends on the subtle fact that we deal with a case where the Fubini theorem does not apply.

In fact, consider an element ξ ∈ S(𝔸_{𝕂})_{0}. We know that it satisfies the vanishing condition

∫_{𝔸_{𝕂}} ξ(x) dx = 0.

Thus, at first sight it would appear that, for the function on C_{𝕂} defined by f(x) = ∑_{k∈𝕂^{∗}} ξ(kx),

f̂(1) = ∫_{C_{𝕂}} f(g)|g|d^{∗}g  (9)

should also vanish, since we have f(x) = ∑_{k∈𝕂^{∗}} ξ(kx) and for local fields (but not in the global case) the relation between the additive and multiplicative Haar measures is of the form dg = |g|d^{∗}g. This, however, is in general not the case. To see more clearly what happens, let us just restrict to the case 𝕂 = ℚ and assume that the function ξ(x) is of the form

ξ = 1ℤ̂ ⊗ η,

with 1ℤ̂ the characteristic function of ℤ̂ and with η ∈ S(ℝ)_{0}. We then have C_{ℚ} = ℤ̂^{∗} × ℝ_{+}^{∗} and the function f is of the form

f(u, λ) = ∑_{n∈ℤ,n≠0} η(nλ), ∀λ ∈ ℝ_{+}^{∗} , u ∈ ℤ̂^{∗} .  (10)

We can thus write (9) in this case as

f̂(1) = ∫_{ℤ̂^{∗}×ℝ_{+}^{∗}} f(u, λ)du dλ = ∫_{ℝ} ∑_{n∈ℕ} η(nλ) dλ  (11)

Moreover since η ∈ S(ℝ)_{0} we have for all n,

∫_{ℝ} η(nλ)dλ = 0 .  (12)

It is however not necessarily the case that we can apply Fubini’s theorem and write

∫_{ℝ} ∑_{n∈ℕ} η(nλ) dλ = ∑_{n} ∫_{ℝ} η(nλ)dλ = 0  (13)

since as soon as η ≠ 0 one has

∑_{n=1}^{∞} ∫_{ℝ} |η(nλ)|dλ = (∫_{ℝ} |η(λ)|dλ) ∑_{n=1}^{∞} \frac{1}{n} = ∞

so that Fubini’s theorem does not apply and one cannot interchange the integral and the sum in (13). Thus, one can in general have f̂(1) ≠ 0, even though ∑_{n} ∫_{ℝ} η(nλ)dλ = 0. In fact, we have the following result.

Lemma 7.3. Given f ∈ S (C_{𝕂}), it is possible to change arbitrarily the value of the degree d(Z(f)) = f̂(1) by adding elements of V.

Proof. It suffices to exhibit an element f ∈ V such that f̂(1) ≠ 0, as then by linearity one can obtain the result. We only treat the case 𝕂 = ℚ. We take η ∈ S(ℝ)_{0} given by

η(x) = πx^{2}(πx^{2} − \frac{3}{2}) e^{−πx^{2}}

One finds that, up to normalization, the Fourier transform f̂ is given by

f̂(is) = ∫_{ℝ_{+}^{∗}} ∑_{n∈ℕ} η(nλ) λ^{is}d^{∗}λ = s(s + i)ζ^{∗}(is)

where ζ^{∗} is the complete zeta function,

ζ^{∗}(z) = π^{−z/2} Γ (\frac{z}{2}) ζ(z).  (14)

This function has a simple pole at z = 1 thus one gets that f̂(1) ≠ 0.

An important question, in order to proceed and build a dictionary that parallels the main steps in the Weil proof, is to identify the correct notion of principal divisors. To this purpose, we show that we have at least a good analog for the points of the curve, in terms of states of some thermodynamical systems, that extend from the function field setting to the number field case.

8 Thermodynamics and geometry of the primes

Let 𝕂 be a global field, with 𝔸_{𝕂} the ring of adeles and C_{𝕂} the ideles classes, as above. We denote by C_{𝕂,1} ⊂ C_{𝕂} the kernel of the norm | · | : C_{𝕂} → ℝ_{+}^{∗}. The origin (cf. [10]) of the terms in the geometric side of the trace formula (Theorem 6.1) comes from the Lefschetz formula by Atiyah-Bott [2] and its adaptation by Guillemin-Sternberg (cf. [23]) to the distribution theoretic trace for flows on manifolds, which is a variation on the theme of [2]. For the action of C_{𝕂} on the adele class space X_{𝕂} the relevant periodic points are

P = {(x, u) ∈ X_{𝕂} × C_{𝕂} | u x = x}  (1)

and one has (cf. [10])

Proposition 8.1. Let (x, u) ∈ P, with u ≠ 1. There exists a place v ∈ Σ_{𝕂} such that

x ∈ X_{𝕂,v} = {x ∈ X_{𝕂} | x_{v} = 0}  (2)

The isotropy subgroup of any x ∈ X_{𝕂,v} contains the cocompact subgroup

𝕂_{v}^{∗} ⊂ C_{𝕂} , 𝕂_{v}^{∗} = {(k_{w}) | k_{w} = 1 ∀w ≠ v}  (3)

The spaces X_{𝕂,v} are noncommutative spaces, as such they are described by the following noncommutative algebras:

Definition 8.2. Let 𝔸_{𝕂,v} ⊂ 𝔸_{𝕂} denote the closed 𝕂^{∗}-invariant subset of adeles

𝔸_{𝕂,v} = {a = (a_{w})_{w∈Σ_{𝕂}} | a_{v} = 0}.  (4)

Let G_{𝕂,v} denote the closed subgroupoid of G_{𝕂} given by

G_{𝕂,v} = {(k, x) ∈ G_{𝕂} | x_{v} = 0},  (5)

and let A_{v} = S(G_{𝕂,v}) be the corresponding groupoid algebra.

Since the inclusion 𝔸_{𝕂,v} ⊂ 𝔸_{𝕂} is 𝕂^{∗}-equivariant and proper, it extends to an algebra homomorphism

ρ_{v} : S(G_{𝕂}) → S(G_{𝕂,v})  (6)

which plays the role of the restriction map to the periodic orbit X_{𝕂,v}. We shall now determine the classical points of each of the X_{𝕂,v}. Taken together these will form the following locus inside the adeles class space, which we refer to as the “periodic classical points” of X_{𝕂} = 𝔸_{𝕂}/𝕂^{∗}.

Definition 8.3. Let 𝕂 be a global field. For a place v ∈ Σ_{𝕂} consider the adele

a^{(v)} = (a_{w}^{(v)}), with a_{w}^{(v)} = \{ \begin{matrix}1 & w ≠ v \\ 0 & w = v. \end{matrix}  (7)

The set of periodic classical points of the adeles class space 𝔸_{𝕂}/𝕂^{∗} is defined as the union of orbits

Ξ_{𝕂} := ⋃_{v∈Σ_{𝕂}} C_{𝕂}a^{(v)}.  (8)

8.1 The global Morita equivalence

In order to deal with states rather than weights, we perform a global Morita equivalence, obtained by reducing the groupoid G_{𝕂} by a suitable open set. The set 𝔸_{𝕂}^{(1)} of (9) that we use to reduce the groupoid G_{𝕂} will only capture part of the classical subspace C_{𝕂}, but since our main focus is on the geometry of the complement of this subspace (the cokernel of the reduction map), this will not be a problem.

Lemma 8.4. Let 𝕂 be a global field. Let W ⊂ 𝔸_{𝕂} be a neighborhood of 0 ∈ 𝔸_{𝕂}. Then for x ∈ 𝔸_{𝕂} one has 𝕂^{∗}x∩W ≠ ∅, unless x ∈ 𝔸_{𝕂}^{∗} is an idele. For x ∈ 𝔸_{𝕂}^{∗}, the orbit 𝕂^{∗}x is discrete in 𝔸_{𝕂}.

Proof. One can assume that W is of the form

W = {a = (a_{w})| |a_{w}| < ε ∀w ∈ S and |a_{w}| ≤ 1 ∀w ∉ S},

for S a finite set of places and for some ε > 0. Multiplying by a suitable idele one can in fact assume that S = ∅, so that we have

W = {a = (a_{w})| |a_{w}| ≤ 1 ∀w ∈ Σ_{𝕂}}.

One has |x_{v}| ≤ 1 except on a finite set F ⊂ Σ_{𝕂} of places. Moreover, if x is not an idele, one can also assume that

∏_{v∈F} |x_{v}| < δ

for any fixed δ. Thus, − log |x_{v}| is as large as one wants and there exists k ∈ 𝕂^{∗} such that k x ∈ W. This is clear in the function field case because of the Riemann Roch formula (7). In the case of ℚ one can first multiply x by an integer to get |x_{v}| ≤ 1 for all finite places, then since this does not alter the product of all |x_{v}| one gets |x_{∞}| < 1 and x ∈ W. The case of more general number fields is analogous. In the case of ideles, one can assume that x = 1 and then the second statement follows from the discreteness of 𝕂 in 𝔸_{𝕂}.

We consider the following choice of a neighborhood of zero.

Definition 8.5. Consider the open neighborhood of 0 ∈ 𝔸_{𝕂} defined by

𝔸_{𝕂}^{(1)} = ∏_{w∈Σ_{𝕂}} 𝕂_{w}^{(1)} ⊂ 𝔸_{𝕂}  (9)

where for any place we let 𝕂_{w}^{(1)} be the interior of {x ∈ 𝕂_{w}; |x| ≤ 1}. Let G_{𝕂}^{(1)} denote the reduction of the groupoid G_{𝕂} by the open subset 𝔸_{𝕂}^{(1)} ⊂ 𝔸_{𝕂} of the units and let S(G_{𝕂}^{(1)}) denote the corresponding (smooth) groupoid algebra.

The algebra S(G_{𝕂}^{(1)}) is a subalgebra of S(G_{𝕂}) where one simply extends the function f(k, x) by zero outside of the open subgroupoid G_{𝕂}^{(1)} ⊂ G_{𝕂}. With this convention, the convolution product of S(G_{𝕂}^{(1)}) is simply given by the convolution product of S(G_{𝕂}) of the form

(f_{1} ⋆ f_{2})(k, x) = ∑_{h∈𝕂^{∗}} f_{1}(kh^{−1}, hx)f_{2}(h, x).

We see from Lemma 8.4 above that the only effect of the reduction to G_{𝕂}^{(1)} is to remove from the noncommutative space 𝔸_{𝕂}/𝕂^{∗} all the elements of C_{𝕂} whose class modulo 𝕂^{∗} does not intersect G_{𝕂}^{(1)} (i.e. in particular those whose norm is greater than or equal to one). We then have the following symmetries for the algebra S(G_{𝕂}^{(1)}).

Proposition 8.6. Let J^{+} denote the semi-group of ideles j ∈ 𝔸_{𝕂}^{∗} such that j𝔸_{𝕂}^{(1)} ⊂ 𝔸_{𝕂}^{(1)}. The semigroup J^{+} acts on the algebra S(G_{𝕂}^{(1)}) by endomorphisms obtained as restrictions of the automorphisms of S(G_{𝕂}) of the form

ϑ_{a}(j)(f)(k, x) = f(k, j^{−1}x), ∀(k, x) ∈ G_{𝕂} , j ∈ J^{+}.  (10)

Let 𝕂 = ℚ and C_{ℚ}^{+} ⊂ C_{ℚ} be the semigroup C_{ℚ}^{+} = {g ∈ C_{ℚ}| |g| < 1}. The semi-group C_{ℚ}^{+} acts on S(G_{ℚ}^{(1)}) by the endomorphisms

F (g) = ϑ_{a}(ḡ)  (11)

with ḡ the natural lift of g ∈ C_{ℚ}^{+} to ℤ̂^{∗} × ℝ_{+}^{∗}.

Proof. By construction ϑ_{a}(j) is an automorphism of S(G_{𝕂}). For a function f with support B in the open set G_{𝕂}^{(1)} the support of the function ϑ_{a}(j)(f) is jB = {(k, jx)|(k, x) ∈ B} ⊂ G_{𝕂}^{(1)} so that ϑ_{a}(j)(f) still has support in G_{𝕂}^{(1)}. For 𝕂 = ℚ let ḡ ∈ ℤ̂^{∗} ×ℝ_{+}^{∗} be the natural lift of an element g ∈ C_{ℚ}^{+}. Then the archimedean component ḡ_{∞} is of absolute value less than 1 so that ḡ ∈ J^{+}. The action of ϑ_{a}(ḡ) by endomorphisms of S(G_{ℚ}^{(1)}) induces a corresponding action of C_{ℚ}^{+}.

Remark 8.7. For m a positive integer, consider the element g = (1, m^{−1}) ∈ C_{ℚ}^{+}. Both g = (1, m^{−1}) and m̃ = (m, 1) are in J^{+} and have the same class in the idele class group C_{ℚ}, since m g = m̃. Thus the automorphisms ϑ_{a}(g) and ϑ_{a}(m̃) of S(G_{𝕂}) are inner conjugate. Since the open set 𝔸_{𝕂}^{(1)} ⊂ 𝔸_{𝕂} is not closed its characteristic function is not continuous and does not define a multiplier of S(G_{𝕂}). It follows that the endomorphism F (g) is inner conjugate to the endomorphism ϑ_{a}(m̃) only in the following weaker sense. There exists a sequence of elements u_{n} of S(G_{𝕂}^{(1)}) such that for any f ∈ S(G_{𝕂}^{(1)}) with compact support

F (g)(f) = u_{n} ϑ_{a}(m̃)(f) u_{n}^{∗},

holds for all n large enough.

8.2 The valuation systems

We now explain why the orbits C_{𝕂}a^{(v)} appear indeed as the set of classical points, in the sense of the low temperature KMS states, of the noncommutative spaces X_{𝕂,v}. The notion of classical points obtained from low temperature KMS states is discussed at length in [15] (cf. also [12], [13], [14]). The noncommutative space X_{𝕂,v} is described by the the restricted groupoid

G(v) = 𝕂^{∗} ⋉ 𝔸_{𝕂,v}^{(1)} = {(g, a) ∈ 𝕂^{∗} ⋉ 𝔸_{𝕂,v} | a and ga ∈ 𝔸_{𝕂,v}^{(1)}} . (12)

We denote by φ the positive functional on C^{∗}(𝕂^{∗} ⋉ 𝔸_{𝕂,v}^{(1)}) given by

φ(f) = ∫_{𝔸_{𝕂,v}^{(1)}} f(1, a) da  (13)

Proposition 8.8. The modular automorphism group of the functional φ on the crossed product C^{∗}(G(v)) is given by the time evolution

σ_{t}^{v}(f)(k, x) = |k|_{v}^{it} f(k, x), ∀t ∈ ℝ, ∀f ∈ C^{∗}(𝕂^{∗} ⋉ 𝔸_{𝕂,v}^{(1)}) .  (14)

Proof. We identify elements of C_{c}(𝕂^{∗} ⋉ 𝔸_{𝕂,v}^{(1)}) with functions f(g, a) of ele-

ments g ∈ 𝕂^{∗} and a ∈ 𝔸_{𝕂,v}^{(1)}. The product is simply of the form

f_{1} ∗ f_{2}(g, a) = ∑_{r} f_{1}(g r^{−1}, ga)f_{2}(r, a).

The additive Haar measure da on 𝔸_{𝕂,v} satisfies the scaling property

d(ka) = |k|_{v}^{−1}da , ∀k ∈ 𝕂^{∗} ,  (15)

since the product measure da × da_{v} on 𝔸_{𝕂} = 𝔸_{𝕂,v} × 𝕂_{v} is invariant under the scaling by k ∈ 𝕂^{∗}, while the additive Haar measure da_{v} on 𝕂_{v} gets multiplied by |k|_{v}, namely d(ka_{v}) = |k|_{v}da_{v}. We then check the KMS_{1} condition, for φ associated to the additive Haar measure, as follows,

φ(f_{1} ∗ f_{2}) = ∑_{r} ∫_{𝔸_{𝕂,v}^{(1)}} f_{1}(r^{−1}, r a)f_{2}(r, a) da

= ∑_{r} ∫_{𝔸_{𝕂,v}^{(1)}} f_{2}(k^{−1}, k b)f_{1}(k, b) |k|_{v}^{−1} db = φ(f_{2} ∗ σ_{i}(f_{1})) ,

using the change of variables k = r^{−1}, a = kb and da = |k|_{v}^{−1} db.

It is worthwhile to observe that these automorphisms extend to the global algebra. Let G_{𝕂}^{(1)} be the groupoid 𝕂^{∗} ⋉ 𝔸_{𝕂}^{(1)} of Definition 8.5.

Lemma 8.9. Let 𝕂 be a global field and v ∈ Σ_{𝕂} a place. The map

d_{v}(k, x) = log |k|_{v} ∈ ℝ  (16)

defines a homomorphism of the groupoid G_{𝕂}^{(1)} to the additive group ℝ and the time evolution

σ_{t}^{v}(f)(k, x) = |k|_{v}^{it} f(k, x), ∀t ∈ ℝ, ∀f ∈ S(G_{𝕂}^{(1)})  (17)

generates a 1-parameter group of automorphisms of the algebra S(G_{𝕂}^{(1)}).

The following result shows that the nontrivial part of the dynamics σ_{t}^{v} concentrates on the algebra S(G(v)) with G(v) as in (12).

Proposition 8.10. The morphism ρ_{v} of (6) restricts to a σ_{t}^{v}-equivariant morphism S(G_{𝕂}^{(1)}) → S(G(v)). Moreover, the restriction of the one parameter group σ_{t}^{v} to the kernel of ρ_{v} is inner.

Proof. For the first statement note that the proper inclusion 𝔸_{𝕂,v} ⊂ 𝔸_{𝕂} restricts to a proper inclusion 𝔸_{𝕂,v}^{(1)} ⊂ 𝔸_{𝕂}^{(1)}. For the second statement, notice that the formula

h_{v}(x) = log |x|_{v}, ∀x ∈ 𝔸_{𝕂}^{(1)},  (18)

defines the multipliers e^{ith_{v}} of the kernel of ρ_{v}. Indeed e^{ith_{v}} is a bounded continuous function on 𝔸_{𝕂}^{(1)} ∖ 𝔸_{𝕂,v}^{(1)}. We can then check that the 1-cocycle d_{v} is the coboundary of h_{v}. In fact, we have

h_{v}(k x) − h_{v}(x) = d_{v}(k, x), ∀(k, x) ∈ G_{𝕂}^{(1)} ∖ G(v).  (19)

We now recall that, for an étale groupoid like G(v), every unit y ∈ G(v)^{(0)} defines, by

(π_{y}(f)ξ)(γ) = ∑_{γ_{1}γ_{2}=γ} f(γ_{1})ξ(γ_{2}),  (20)

a representation π_{y} by left convolution of the algebra of G(v) in the Hilbert space H_{y} = ℓ^{2}(G(v)_{y}), where G(v)_{y} denotes the set of elements of the groupoid G(v) with source y. By construction the unitary equivalence class of the representation π_{y} is unaffected when one replaces y by an equivalent z ∈ G(v)^{(0)} i.e. one assumes that there exists γ ∈ G(v) with range and source y and z. Thus we can think of the label y of π_{y} as living in the quotient space X_{𝕂, v} of equivalence classes of elements of G(v)^{(0)}. The relation between Ξ_{𝕂, v} and X_{𝕂, v} is then the following.

Theorem 8.11. For y ∈ X_{𝕂, v}, the representation π_{y} is a positive energy representation if and only if y ∈ Ξ_{𝕂, v}.

Proof. Let first y ∈ G(v)^{(0)} ∩ Ξ_{𝕂, v}. Thus one has y ∈ 𝔸_{𝕂,v}^{(1)}, y_{w} ≠ 0 for all w and |y_{w}| = 1 for all w ∉ S where S is a finite set of places. We can identify G(v)_{y} with the set of k ∈ 𝕂^{∗} such that k y ∈ 𝔸_{𝕂,v}^{(1)}. We extend y to the adele ỹ = y × 1 whose component at the place v is equal to 1 ∈ 𝕂_{v}. Then ỹ is an idele. Thus by Lemma 8.4 the number of elements of the orbit 𝕂^{∗}ỹ in a given compact subset of 𝔸_{𝕂} is finite. It follows that log |k|_{v} is lower bounded on G(v)_{y}. Indeed otherwise there would exist a sequence k_{n} ∈ 𝕂^{∗} ∩ G(v)_{y} such that |k_{n}|_{v} → 0. Then k_{n} ỹ ∈ 𝔸_{𝕂}^{(1)} for all n large enough and this contradicts the discreteness of 𝕂^{∗}ỹ. In the representation π_{y} the time evolution σ_{t} is implemented by the Hamiltonian H_{y} given by

(H_{y} ξ)(k, y) = log |k|_{v} ξ(k, y).  (21)

Namely, we have

π_{y}(σ_{t}(f)) = e^{itH_{y}}π_{y}(f)e^{−itH_{y}}, ∀f ∈ C_{c}(G(v)) .  (22)

Thus since log |k|_{v} is lower bounded on G(v)_{y} we get that the representation π_{y} is a positive energy representation. Let then y ∈ G(v)^{(0)} ∖ Ξ_{𝕂, v}. We shall show that log |k|_{v} is not lower bounded on G(v)_{y}, and thus that π_{y} is not a positive energy representation. We consider as above the adele ỹ = y×1 whose component at the place v is equal to 1 ∈ 𝕂_{v}. Assume that log |k|_{v} is lower bounded on G(v)_{y}. Then there exists ϵ > 0 such that, for k ∈ 𝕂^{∗},

k y ∈ 𝔸_{𝕂,v}^{(1)} ⇒ |k|_{v} ≥ ϵ .

This shows that the neighborhood of 0 ∈ 𝔸_{𝕂} defined as

W = {a ∈ 𝔸_{𝕂}; |a_{v}| < ϵ , a_{w} ∈ 𝕂_{w}^{(1)} , ∀w ≠ v}

does not intersect 𝕂^{∗}ỹ. Thus by Lemma 8.4 we get that ỹ is an idele and y ∈ Ξ_{𝕂, v}.

The specific example of the Bost-Connes system combined with Theorem 8.11 shows that one can refine the recipe of [15] (cf. also [12], [13], [14]) for taking “classical points” of a noncommutative space. The latter recipe only provides a notion of classical points that can be thought of, by analogy with the positive characteristic case, as points defined over the mysterious “field with one element” 𝔽_{1} (see e.g. [29]). To obtain instead a viable notion of the points defined over the maximal unramified extension 𝔽̄_{1}, one performs the following sequence of operations.

X ⟶^{Dual System} X̂ ⟶^{P eriodic Orbits} ∪ X̂_{v} ⟶^{Classical P oints} ∪ Ξ_{v}  (23)

which make sense in the framework of endomotives of [11]. Note in particular that the dual system X̂ is of type II and as such does not have a non-trivial time evolution. Thus it is only by restricting to the periodic orbits that one passes to noncommutative spaces of type III for which the cooling operation is non-trivial. In the analogy with geometry in non-zero characteristic, the set of points X(𝔽̄_{q}) over 𝔽̄_{q} of a variety X is indeed obtained as the union of the periodic orbits of the Frobenius.

Remark 8.12. Theorem 8.11 does not give the classification of KMS_{β} states for the quantum statistical system (C^{∗}(𝕂^{∗} ⋉ 𝔸_{𝕂,v}^{(1)}), σ_{t}). It just exhibits extremal KMS_{β} states but does not show that all of them are of this form.

8.3 The curve inside the adeles class space

In the case of a function field 𝕂 = 𝔽_{q}(C), the set of periodic classical points of the adeles class space 𝔸_{𝕂}/𝕂^{∗} is (non-canonically) isomorphic to the algebraic points C(𝔽̄_{q}). In fact, more precisely the set of algebraic points C(𝔽̄_{q}) is equivariantly isomorphic to the quotient Ξ_{𝕂}/C_{𝕂,1} where C_{𝕂,1} ⊂ C_{𝕂} is the kernel of the norm | · | : C_{𝕂} → ℝ_{+}^{∗}, and Ξ_{𝕂} is as in (8).

Proposition 8.13. For 𝕂 = 𝔽_{q}(C) a function field, the orbits of Frobenius on C(𝔽̄_{q}) give an equivariant identification

Ξ_{𝕂}/C_{𝕂,1} ≃ C(𝔽̄_{q}),  (24)

between Ξ_{𝕂}/C_{𝕂,1} with the action of q^{ℤ} and C(𝔽̄_{q}) with the action of the group of integer powers of the Frobenius.

Proof. At each place v ∈ Σ_{𝕂} the quotient group of the range N of the norm | · | : C_{𝕂} → ℝ_{+}^{∗} by the range N_{v} of | · | : 𝕂_{v} → ℝ_{+}^{∗} is the finite cyclic group

N/N_{v} = q^{ℤ}/q^{n_{v}ℤ} ≃ ℤ/n_{v}ℤ,  (25)

where n_{v} is the degree of the place v ∈ Σ_{𝕂}. The degree n_{v} is the same as the cardinality of the orbit of the Frobenius acting on the fiber of the map (2) from algebraic points in C(𝔽̄_{q}) to places in Σ_{𝕂}. Thus, one can construct in this way an equivariant embedding

C(𝔽̄_{q}) ↪ (𝔸_{𝕂}/𝕂^{∗})/C_{𝕂,1}  (26)

obtained, after choosing a point in each orbit, by mapping the orbit of the integer powers of the Frobenius in C(𝔽̄_{q}) over a place v to the orbit of C_{𝕂}/C_{𝕂,1} ∼ q^{ℤ} on the adele a^{(v)}.

Modulo the problem created by the fact that the identification above is noncanonical and relies upon the choice of a point in each orbit, it is then possible to think of the locus Ξ_{𝕂}, in the number field case, as a replacement for C(𝔽̄_{q}) inside the adeles class space 𝔸_{𝕂}/𝕂^{∗}. In the case of 𝕂 = ℚ, the quotient Ξ_{ℚ}/C_{ℚ,1} appears as a union of periodic orbits of period log p under the action of C_{ℚ}/C_{ℚ,1} ∼ ℝ, as in Figure 1. What matters, however, is not the space Ξ_{ℚ}/C_{ℚ,1} in itself but the way it sits inside 𝔸_{ℚ}/ℚ^{∗}. Without taking into account the topology induced by 𝔸_{𝕂} the space Ξ_{𝕂} would just be a disjoint union of orbits without any interesting global structure, while it is the embedding in the adeles class space that provides the geometric setting underlying the Lefschetz trace formula of [10] and its cohomological formulation of [11].

Fig. 1. The classical points Ξ_{ℚ}/C_{ℚ,1} of the adeles class space 𝔸_{ℚ}/ℚ^{∗}.

8.4 The valuation systems for 𝕂 = ℚ

We concentrate again on the specific case of 𝕂 = ℚ to understand better the properties of the dynamical systems σ_{t}^{p} associated to the finite primes p ∈ Σ_{ℚ}. We know that, in the case of the BC system, the KMS state at critical temperature β = 1 is given by the additive Haar measure on finite adeles [5]. Thus, one expects that, for the systems associated to the finite primes, the additive Haar measure of 𝔸_{ℚ,p} should play an analogous role.

Definition 8.14. Let 𝔸_{ℚ,p}^{∗} ⊂ 𝔸_{ℚ,p}^{(1)} be the subspace

𝔸_{ℚ,p}^{∗} = {x ∈ 𝔸_{ℚ,p}| |x_{w}| = 1 ∀w ≠ p, ∞ and p^{−1} ≤ |x_{∞}| < 1}.  (27)

As above G(p) denotes the reduction of the groupoid G_{ℚ,p} by the open subset 𝔸_{ℚ,p}^{(1)} ⊂ 𝔸_{ℚ,p}, namely

G(p) = {(k, x) ∈ G_{ℚ,p} | x ∈ 𝔸_{ℚ,p}^{(1)}, kx ∈ 𝔸_{ℚ,p}^{(1)}}.  (28)

Notice that the set 𝔸_{ℚ,p}^{(1)} meets all the equivalence classes in 𝔸_{ℚ,p} by the action of ℚ^{∗}. In fact, given x ∈ 𝔸_{ℚ,p}, one can find a representative y with y ∼ x in 𝔸_{ℚ,p}/ℚ^{∗}, such that y ∈ ℤ̂ × ℝ. Upon multiplying y by a suitable power of p, one can make y_{∞} as small as required, and in particular one can obtain in this way a representative in 𝔸_{ℚ,p}^{(1)}. Let us assume that |y_{w}| = 1 for all finite places w ≠ p and that y_{∞} > 0. Then there exists a unique n ∈ ℕ ∪ {0} such that p^{n} y ∈ 𝔸_{ℚ,p}^{∗}. Given a prime p we define the function f_{p}(λ, β) for λ ∈ (1, p] and β > 1 by

f_{p}(λ, β) = ∑ c_{k} p^{−kβ}  (29)

where the c_{k} ∈ {0, … p − 1} are the digits of the expansion of λ in base p. There is an ambiguous case where all digits c_{k} are equal to 0 for k > m while c_{m} > 0, since the same number

λ = ∑ c_{k} p^{−k}

is obtained using the same c_{j} for j < m, c_{m} − 1 instead of c_{m} and c_{j} = p − 1 for j > m. In that case, for β > 1, (29) gives two different values and we choose the value coming from the second representation of λ, i.e. the lower of the two. These coefficients c_{k} of the expansion of λ in base p are then given by

c_{k} = ⌈λp^{k} − 1⌉ − p ⌈λp^{k−1} − 1⌉ ,  (30)

where ⌈x⌉ = inf_{n∈ℤ}{n ≥ x} denotes the ceiling function. Note that, for β > 1, the function f_{p}(λ, β) is discontinuous (cf. Figures 2 and 3) at any point (λ, β) where the expansion of λ in base p is ambiguous, i.e. λ ∈ ℕ p^{−k}. Moreover for β = 1 one gets

f_{p}(λ, 1) = λ , ∀λ ∈ (1, p] .  (31)

Fig. 2. Graphs of the functions f_{p}(λ, β) as functions of β for p = 3, λ = n/27. The gray regions are the gaps in the range of f_{p}.

We then obtain the following result.

Theorem 8.15. Let (C^{∗}(G(p)), σ_{t}^{p}) be the C^{∗}-dynamical system associated to the groupoid (28) with the time evolution (17). Then the following properties hold:

1. For any y ∈ 𝔸_{ℚ,p}^{∗} the corresponding representation π_{y} has positive energy.

2. Let H_{y} denote the Hamiltonian implementing the time evolution in the representation π_{y}, for y ∈ 𝔸_{ℚ,p}^{∗} with y_{∞} = λ^{−1} and λ ∈ (1, p]. Then the partition function is given by

Z_{p}(λ, β) = Tr(e^{−βH_{y}}) = 2 \frac{1 − p^{−β}}{1 − p^{1−β}} f_{p}(λ, β) .  (32)

3. The functionals

ψ_{β, y}(a) = Tr(e^{−βH_{y}} π_{y}(a)) , ∀a ∈ C^{∗}(G(p))  (33)

satisfy the KMS_{β} condition for σ_{t}^{p} and depend weakly continuously on the parameter y ∈ 𝔸_{ℚ,p}^{∗}.

Proof. (1) This follows from Theorem 8.11. For y ∈ 𝔸_{ℚ,p}^{∗} one has

r ∈ ℚ^{∗}, ry ∈ 𝔸_{ℚ,p}^{(1)} ⟹ r = p^{−k}m,  (34)

for some k ≥ 0 and some integer m prime to p and such that |r y_{∞}| < 1. This implies

|m| < p^{k+1}  (35)

and one finds

|r|_{p} = p^{k} ≥ 1 and log |r|_{p} ≥ 0,  (36)

In fact, the argument above shows that the spectrum of the Hamiltonian H_{y} implementing the time evolution σ_{t}^{p} in the representation π_{y} is given by

Spec(H_{y}) = {k log p}_{k∈ℕ∪{0}},  (37)

hence π_{y} is a positive energy representation.

(2) We begin by the special case with y_{∞} = p^{−1}. Then λ = p and f_{p}(λ, β) = \frac{p−1}{1−p^{−β}} since all digits of λ = p are equal to p − 1. We want to show that the partition function is given by

Tr(e^{−βH_{y}}) = 2 \frac{p − 1}{1 − p^{1−β}} .  (38)

The multiplicity of an eigenvalue k log p of H_{y} is the number of integers m ≠ 0 ∈ ℤ that are prime to p and such that p^{−k} |m| y_{∞} < 1. Since we are assuming that y_{∞} = p^{−1}, this gives |m| < p^{k+1}. Thus, the multiplicity is just 2 (p^{k+1} − p^{k}). The factor 2 comes from the sign of the integer m. The factor (p^{k+1} − p^{k}) corresponds to subtracting from the number p^{k+1} of positive integers m ≤ p^{k+1} the number p^{k} of those that are multiples of p. We now pass to the general case. For x > 0, ⌈x − 1⌉ is the cardinality of (0, x) ∩ ℕ. The same argument used above shows that the multiplicity of the eigenvalue k log p is given by the counting

2 (⌈λp^{k} − 1⌉ − ⌈λp^{k−1} − 1⌉) .

Thus

Tr(e^{−βH_{y}}) = 2 ∑_{k=0}^{∞} (⌈λp^{k} − 1⌉ − ⌈λp^{k−1} − 1⌉) p^{−kβ} .  (39)

One has the following equalities of convergent series,

∑_{k=0}^{∞} (⌈λp^{k} − 1⌉ − ⌈λp^{k−1} − 1⌉) p^{−kβ} = ∑_{k=0}^{∞}⌈λp^{k} − 1⌉ (p^{−kβ} − p^{−(k+1)β})

so that,

Tr(e^{−βH_{y}}) = 2 (1 − p^{−β}) ∑_{k=0}^{∞}⌈λp^{k} − 1⌉ p^{−kβ} .  (40)

Similarly

∑_{k=0}^{∞} (⌈λp^{k} − 1⌉ − p ⌈λp^{k−1} − 1⌉) p^{−kβ} = ∑_{k=0}^{∞}⌈λp^{k} − 1⌉ (p^{−kβ} − p p^{−(k+1)β})

which gives

f_{p}(λ, β) = (1 − p^{1−β}) ∑_{k=0}^{∞}⌈λp^{k} − 1⌉ p^{−kβ} ,  (41)

since the coefficients c_{k} of the expansion of λ in base p are given by (30). Combining (40) with (41) gives (32).

(3) It follows from (22) and the finiteness of the partition function (32) that the functionals (33) fulfill the KMS_{β} condition. In terms of functions on the groupoid G(p) one has

ψ_{β, y}(f) = ∑ f(1, n p^{−k} y) p^{−kβ} , ∀f ∈ C_{c}(G(p))  (42)

where the sum is absolutely convergent. Each of the terms in the sum gives a weakly continuous linear form thus one obtains the required continuity.

Remark 8.16. The partition function Z_{p}(λ, β) is a discontinuous function of the parameter λ and this might seem to contradict the third statement of Theorem 8.15. It would if the algebra C^{∗}(G(p)) were unital since, in that case, the partition function is given by evaluation on the unit and weak continuity

Fig. 3. Graph of the function Z_{p}(λ, β) as a function of λ for p = 3, β = 1.2.

implies that it is continuous. In our case C^{∗}(G(p)) is not unital, and the partition function is expressed as a supremum of the form

Z_{p}(λ, β) = sup{ψ_{β, y}(a^{∗}a)|a ∈ C^{∗}(G(p)) , ||a|| ≤ 1} .

In particular it shows that Z_{p}(λ, β) is lower semi-continuous as a function of λ.

The precise qualitative properties of the partition functions Z_{p}(λ, β) are described by the following result

Proposition 8.17. As a function of λ ∈ (1, λ] the partition function Z_{p}(λ, β) satisfies for β > 1:

1. Z_{p} is strictly increasing.

2. Z_{p} is continuous on the left, and lower semi-continuous.

3. Z_{p} is discontinuous at any point of the form λ = m p^{−k} with a jump of 2 p^{−kβ} (for m prime to p).

4. The measure \frac{∂Z_{p}}{∂λ} is the sum of the Dirac masses at the points λ = m p^{−k}, m prime to p, with coefficients 2 p^{−kβ}.

5. The closure of the range of Z_{p} is a Cantor set.

Proof. (1) This follows from (40) which expresses Z_{p} as an absolutely convergent sum of multiples of the functions ⌈λp^{k} −1⌉. The latter are non-decreasing and jump by 1 at λ ∈ ℕ p^{−k} ∩ (1, p]. The density of the union of these finite sets for k ≥ 0 shows that Z_{p} is strictly increasing.

(2) This follows as above from (40) and the semi-continuity properties of the ceiling function.

(3) Let λ = m p^{−k} with m prime to p. Then for any j ≥ k one gets a jump of 2 (1 − p^{−β}) p^{−jβ} coming from (40) so that their sum gives

2 (1 − p^{−β}) ∑_{j=k}^{∞} p^{−jβ} = 2 p^{−kβ}

(4) This follows as above from (40) and from (3) which computes the discontinuity at the jumps.

(5) Recall that when writing elements of an interval in base p one gets a map from the cantor set to the interval. This map is surjective but fails to be injective due to the identifications coming from ∑_{0}^{∞}(p − 1) p^{−m} = p. The connectedness of the interval is recovered from these identifications. In our case the coefficients c_{k} of the expansion in base p of elements of (1, p] are such that c_{0} ∈ {1, …, p − 1} while c_{k} ∈ {0, …, p − 1} for k > 0. This is a Cantor set K in the product topology of K = {1, …, p−1}×∏_{ℕ} {0, …, p−1}. As shown in Figure 3, the discontinuities of the function Z_{p}(λ, β) as a function of λ replace the connected topology of (1, p] by the totally disconnected topology of K.

Fig. 4. Graphs of the functions Z_{p}(λ, β) as functions of β for p = 3, λ = n/27. The gray regions are the gaps in the range. All these functions have a pole at β = 1.

Remark 8.18. One can use (39) to define Z_{p}(λ, β) for any λ > 0, as

Z_{p}(λ, β) = 2 ∑_{−∞}^{∞} (⌈λp^{k} − 1⌉ − ⌈λp^{k−1} − 1⌉) p^{−kβ}.  (43)

This makes sense for ℜ(β) > 1 since ⌈λp^{k} − 1⌉ = 0 for k ≤ −\frac{log λ}{log p}. The extended function (43) satisfies

Z_{p}(pλ, β) = p^{β}Z_{p}(λ, β),

which suggests replacing Z_{p}(λ, β) with

ζ_{p}(λ, β) = λ^{−β}Z_{p}(λ, β)  (44)

so that

ζ_{p}(pλ, β) = ζ_{p}(λ, β).  (45)

This replacement Z_{p} ↦ ζ_{p} corresponds to the shift in the Hamiltonian H_{y} by

H_{y} ↦ H_{y} − log |y_{∞}|.

We can now refine Theorem 8.11 and consider the zero temperature KMS state of the system (C^{∗}(G(p)), σ_{t}^{p}) corresponding to the positive energy representation π_{y} for y ∈ Ξ_{ℚ, p}.

Proposition 8.19. As β → ∞ the vacuum states (zero temperature KMS states) of the system (C^{∗}(G(p)), σ_{t}^{p}) with Hamiltonian H_{y} have a degeneracy of 2⌈λ − 1⌉, where y_{∞} = λ^{−1}. There is a preferred choice of a vacuum state given by the evaluation at y ∈ 𝔸_{ℚ,p}^{∗}.

Proof. When we look at the orbit of y ∈ 𝔸_{ℚ,p}^{∗}, i.e. at the intersection ℚ^{∗}y ∩ 𝔸_{ℚ,p}^{(1)} and label its elements by pairs (k, m) as above, we find that all elements with k = 0 give a ground state. This degeneracy of the vacuum reflects the fact that the limit of the partition function as the temperature goes to 0 is not in general equal to 1. For instance, for y_{∞} = p^{−1}, one finds

lim_{β→∞} Tr(e^{−βH_{y}}) = lim_{β→∞} 2\frac{p − 1}{1 − p^{1−β}} = 2(p − 1).

More generally, one finds similarly the limit

lim_{β→∞} Tr(e^{−βH_{y}}) = 2⌈λ − 1⌉.

Among the 2⌈λ − 1⌉ vacuum states, the state given by evaluation at y ∈ 𝔸_{ℚ,p}^{∗} is singled out, since my ∉ 𝔸_{ℚ,p}^{∗} for m ≠ 1. It is then natural to consider, for each finite place p ∈ Σ_{ℚ}, the section

s_{p}(x) = ℚ^{∗}x ∩ 𝔸_{ℚ,p}^{∗}, ∀x ∈ C_{ℚ}a^{(p)} ⊂ 𝔸_{ℚ}/ℚ^{∗}  (46)

of the projection from 𝔸_{ℚ} to the orbit C_{ℚ}a^{(p)}.

Notice that s_{p} is discontinuous at the boundary of the domain 𝔸_{ℚ,p}^{∗}. Indeed when y_{∞} crosses the value p^{−1} the class in C_{ℚ}a^{(p)} varies continuously but the representative in 𝔸_{ℚ,p}^{∗} jumps discontinuously so that its archimedian component remains in the interval [p^{−1}, 1). This suggests to consider a cyclic covering of Ξ_{ℚ} which we now discuss in §8.5.

8.5 The cyclic covering Ξ̃_{ℚ} of Ξ_{ℚ}

By construction Ξ_{𝕂} is a subspace of the adeles class space X_{𝕂}. We shall now show, in the case 𝕂 = ℚ, that it admits a natural lift Ξ̃_{ℚ} to a subspace of 𝔸_{ℚ} which reduces the ambiguity group ℚ^{∗} to a cyclic group. One thus obtains a natural cyclic covering Ξ̃_{ℚ} ⊂ 𝔸_{ℚ} of Ξ_{ℚ}. We already saw above, in Proposition 8.19, that it is natural to choose representatives for the elements of the orbit C_{ℚ}a^{(p)}, for a finite prime p, in the subset of adeles given by

Ξ̃_{ℚ,p} := {y ∈ 𝔸_{ℚ}| y_{p} = 0 and |y_{ℓ}| = 1 for ℓ ≠ p, ∞ and y_{∞} > 0}. (47)

We extend this definition at ∞ by

Ξ̃_{ℚ,∞} := {y ∈ 𝔸_{ℚ}| |y_{w}| = 1 ∀w ≠ ∞ and y_{∞} = 0}.  (48)

Definition 8.20. The locus Ξ̃_{ℚ} ⊂ 𝔸_{ℚ} is defined as

Ξ̃_{ℚ} = ⋃_{v∈Σ_{ℚ}} Ξ̃_{ℚ,v} ⊂ 𝔸_{ℚ}  (49)

We then have the following simple fact.

Proposition 8.21. Let π be the projection from Ξ̃_{ℚ} to Ξ_{ℚ}, with π(x) the class of x modulo the action of ℚ^{∗}.

1. The map π : Ξ̃_{ℚ} → Ξ_{ℚ} is surjective.

2. Two elements in Ξ̃_{ℚ,v} have the same image in C_{ℚ}a^{(v)} iff they are on the same orbit of the following transformation T

T x = p x , ∀x ∈ Ξ̃_{ℚ,p} , T x = −x , ∀x ∈ Ξ̃_{ℚ,∞}  (50)

Proof. The first statement follows by lifting C_{ℚ} inside 𝔸_{ℚ}^{∗} as the subgroup ℤ̂^{∗} × ℝ_{+}^{∗}. Then any element of C_{ℚ}a^{(v)} has a representative in (ℤ̂^{∗} × ℝ_{+}^{∗})a^{(v)} The proof of the second statement is straightforward, since for a finite prime p the subgroup p^{ℤ} ⊂ ℚ^{∗} is the group of elements of ℚ^{∗} which leave Ξ̃_{ℚ,p} globally invariant.

8.6 Arithmetic subalgebra, Frobenius and monodromy

We now describe a natural algebra of coordinates B on Ξ_{ℚ}. The BC system of [5], as well as its arithmetic generalizations of [12] and [14], have the important property that they come endowed with an arithmetic structure given by an arithmetic subalgebra. The general framework of endomotives developed in [11] shows a broad class of examples where a similar arithmetic structure is naturally present. We consider here the issue of extending the construction of the “rational subalgebra” of the BC-system to the algebra S(G_{ℚ}^{(1)}) of §8.1.

In order to get a good geometric picture it is convenient to think in terms of ℚ-lattices rather than of adeles, as in [12]. Thus, we let L denote the set of 1-dimensional ℚ-lattices (as defined in [12]). We consider the map

ι : ℤ̂ × ℝ_{+}^{∗} → L , ι(ρ, λ) = (Λ, ϕ) = (λ^{−1}ℤ, λ^{−1}ρ)  (51)

which associates to an adele (ρ, λ) ∈ ℤ̂×ℝ_{+}^{∗} ⊂ 𝔸_{ℚ} the ℚ-lattice obtained using ρ to label the torsion points of ℝ/λ^{−1}ℤ. Replacing (ρ, λ) by (nρ, nλ), for a positive integer n ∈ ℕ, one obtains the pair (\frac{1}{n}Λ, ϕ), which is commensurable to (Λ, ϕ). Thus, the action of ℚ_{+}^{∗} corresponds to commensurability of ℚlattices under the map ι. Multiplying λ by a positive scalar corresponds to the scaling action of ℝ_{+}^{∗} on ℚ-lattices. Let us recall the definition of the “rational algebra” A_{ℚ} of [12] for the BC system, given in terms of ℚ-lattices. We let

ϵ_{a}(Λ, ϕ) = ∑_{y∈Λ+ϕ(a)} y^{−1},  (52)

for any a ∈ ℚ/ℤ. This is well defined, for ϕ(a) ≠ 0, using the summation lim_{N→∞} ∑_{−N}^{N}, and is zero by definition for ϕ(a) = 0. The function

φ_{a}(ρ, λ) = ϵ_{a}(ι(ρ, λ)), ∀(ρ, λ) ∈ ℤ̂ × ℝ_{+}^{∗},  (53)

is well defined and homogeneous of degree 1 in λ. Moreover, for fixed a ∈ ℚ/ℤ with denominator m, it only depends upon the projection of ρ on the finite group ℤ/mℤ, hence it defines a continuous function on ℤ̂ × ℝ_{+}^{∗}. Using the degree 1 homogeneity in λ, one gets that (53) extends by continuity to 0 on ℤ̂ × {0}. One gets functions that are homogeneous of weight zero by taking the derivatives of the functions φ_{a}. The functions

ψ_{a}(ρ, λ) = \frac{1}{2πi} \frac{d}{dλ}φ_{a}(ρ, λ), ∀(ρ, λ) ∈ ℤ̂ × ℝ_{+}^{∗},  (54)

are independent of λ hence they define continuous functions on 𝔸_{ℚ}^{(1)}. They are non trivial on Ξ̃_{ℚ,∞} = ℤ̂^{∗} × {0} ⊂ ℤ̂ × {0} and they agree there with the functions e_{a} of [12].

Proposition 8.22. Let B be the algebra generated by the φ_{a} and ψ_{a} defined in (53) and (54) above.

1. The expression

N(f) = \frac{1}{2πi} \frac{d}{dλ} f  (55)

defines a derivation N of B.

2. The algebra B is stable under the derivation Y that generates the 1parameter semigroup F (μ) of endomorphisms of S(G_{ℚ}^{(1)}) of (11) and one has, at the formal level, the relation

F (μ)N = μNF (μ).  (56)

3. For any element f ∈ B one has

α ∘ f(x) = f(α̃ x), ∀x ∈ Ξ̃_{ℚ,∞} and ∀α ∈ Gal(ℚ^{cycl}/ℚ),  (57)

where α̃ ∈ ℤ̂^{∗} ⊂ C_{ℚ} is the element of the idele class group associated to α ∈ Gal(ℚ^{cycl}/ℚ) by the class field theory isomorphism.

Proof. 1) By construction N is a derivation of the algebra of functions. Moreover (54) shows that N(φ_{a}) = ψ_{a}, while N(ψ_{a}) = 0. Thus, the derivation rule shows that B is stable under N.

2) The derivation generating the one parameter semigroup F (μ) is given, up to sign, by the grading operator

Y (f) = λ\frac{d}{dλ} f.  (58)

By construction, any of the φ_{a} is of degree one, i.e. Y (φ_{a}) = φ_{a} and ψ_{a} is of degree 0. Thus, again the derivation rule shows that B is stable under Y.

3) This only involves the functions ψ_{a}, since by construction the restriction of φ_{a} is zero on Ξ̃_{ℚ,∞}. The result then follows from the main result of [5] in the reformulation given in [12] (see also [13], Chapter 3). In fact, all these functions take values in the cyclotomic field ℚ^{cycl} ⊂ ℂ and they intertwine the action of the discontinuous piece ℤ̂^{∗} of C_{ℚ} with the action of the Galois group of ℚ^{cycl}.

This is in agreement with viewing the algebra B as the algebra of coordinates on Ξ̃_{ℚ}. Indeed, in the case of a global field 𝕂 of positive characteristic, the action of the Frobenius on the points of C(𝔽̄_{q}) (which have coordinates in 𝔽̄_{q}) corresponds to the Frobenius map

Fr : u ↦ u^{q}, ∀u ∈ 𝕂  (59)

of the function field 𝕂 of the curve C. The Frobenius endomorphism u ↦ u^{q} of 𝕂 is the operation that replaces a function f : C(𝔽̄_{q}) → 𝔽̄_{q} by its q-th power, i.e. the composition Fr∘f with the Frobenius automorphism Fr ∈ Gal(𝔽̄_{q}/𝔽_{q}). For f ∈ 𝕂, one has

Fr ∘ f = f^{q} = f ∘ Fr,  (60)

where on the right hand side Fr is the map that raises every coordinate to the power q. This corresponds to the interwtining with the Galois action discussed above.

Notice moreover that, as we have seen in Proposition 8.6, only the semigroup C_{ℚ}^{+} acts on the reduced system S(G_{ℚ}^{(1)}) and it acts by endomorphisms. It nevertheless acts in a bijective manner on the points of Ξ_{ℚ}. This is similar to what happens with the Frobenius endomorphism (59), which is only an endomorphism of the field of functions 𝕂̄, while it acts bijectively (as a Galois automorphism of the coordinates) on the points of C(𝔽̄_{q}). Further notice that there is a striking formal analogy between the operators F and N of Proposition 8.22 satisfying the relation (56) and the Frobenius and local monodromy operators introduced in the context of the “special fiber at arithmetic infinity” in Arakelov geometry (see [18], [19]). In particular, one should compare (56) with §2.5 of [19] that discusses a notion of Weil–Deligne group at arithmetic infinity.

9 Functoriality of the adeles class space

We investigate in this section the functoriality of the adele class space X_{𝕂} and of its classical subspace Ξ_{𝕂} ⊂ X_{𝕂}, for Galois extensions of the global field 𝕂. This issue is related to the question of functoriality. Namely, given a finite algebraic extension 𝕃 of the global field 𝕂, we want to relate the adele class spaces of both fields. Assume the extension is a Galois extension. In general, we do not expect the relation between the adeles class spaces to be canonical, in the sense that it will involve a symmetry breaking choice on the Galois group G = Gal(𝕃/𝕂) of the extension. More precisely, the norm map

𝔫(a) = ∏_{σ∈G} σ(a) ∈ 𝔸_{𝕂}, ∀a ∈ 𝔸_{𝕃}  (1)

appears to be the obvious candidate that relates the two adeles class spaces. In fact, since 𝔫(𝕃) ⊂ 𝕂, the map (1) passes to the quotient and gives a natural map from X_{𝕃} = 𝔸_{𝕃}/𝕃^{∗} to X_{𝕂} = 𝔸_{𝕂}/𝕂^{∗} that looks like the covering required by functoriality. However, the problem is that the norm map fails to be surjective in general, hence it certainly does not qualify as a covering map. In fact, this problem already occurs at the level of the idele class group C_{𝕂}, namely the norm map fails to be a surjection from C_{𝕃} to C_{𝕂}. The correct object to consider is the Weil group W_{𝕃,𝕂}. This is an extension of C_{𝕃} by the Galois group G = Gal(𝕃/𝕂), which is not a semi-direct product. The corresponding non-trivial 2-cocycle is called the “fundamental class”. One has a natural morphism t, called the transfer, from W_{𝕃,𝕂} to C_{𝕂}. The transfer satisfies the following two properties.

• The morphism t restricts to the norm map from C_{𝕃} to C_{𝕂}.

• The morphism t is surjective on C_{𝕂}

Thus, the correct way to understand the relation between the adeles class spaces X_{𝕃} and X_{𝕂} is by extending the construction of the Weil group and of the the transfer map.

One obtains in this way n copies of the adele class space X_{𝕃} of 𝕃 and a map to X_{𝕂} which is now a covering from G × Ξ_{𝕃} → Ξ_{𝕂}. This space has a natural action of the Weil group. We explain this in more detail in what follows.

9.1 The norm map

We begin by recalling the well known properties of the norm map that are relevant to our set-up. Thus, we let 𝕃 ⊃ 𝕂 be a finite Galois extension of 𝕂 of degree n, with G = Gal(𝕃/𝕂) the Galois group. Since the adeles depend naturally on the field, one has a canonical action of G on 𝔸_{𝕃}. If v ∈ Σ_{𝕂} is a place of 𝕂, there are m_{v} places of 𝕃 over v and they are permuted transitively by the action of G. Let G_{w} be the isotropy subgroup of w. Then G_{w} is the Galois group G_{w} = Gal(𝕃_{w}/𝕂_{v}). One has a canonical embedding of 𝔸_{𝕂} as the fixed points of the action of G on 𝔸_{𝕃} by

𝔸_{𝕂} = 𝔸_{𝕃}^{G}, (a_{v}) ↦ (a_{π(w)}), with π : Σ_{𝕃} → Σ_{𝕂}.  (2)

The norm map 𝔫 : 𝔸_{𝕃} → 𝔸_{𝕂} is then defined as in (1). By [44] IV 1, Corollary 3, it is given explicitly by

𝔫(x) = z , z_{v} = ∏_{w|v} 𝔫_{𝕃_{w}/𝕂_{v}}(x_{w}) , ∀x ∈ 𝔸_{𝕃} .  (3)

Here the notation w|v means that w is a place of 𝕃 over the place v ∈ Σ_{𝕂}. Also 𝔫_{𝕃_{w}/𝕂_{v}} is the norm map of the extension 𝕃_{w}/𝕂_{v}. When restricted to principal adeles of 𝕃 it gives the norm map from 𝕃 to 𝕂. When restricted to the subgroup 𝕃_{w}^{∗} = (…, 1, …, y, …, 1, …) ⊂ 𝔸_{𝕃}^{∗}, it gives the norm map of the extension 𝕃_{w}/𝕂_{v}. For nontrivial extensions this map is never surjective, but its restriction 𝔫 : O(𝕃_{w})^{∗} → O(𝕂_{v})^{∗} is surjective when the extension is unramified, which is the case for almost all places v ∈ Σ_{𝕂} (cf. [44], Theorem 1 p.153). In such cases, the module of the subgroup 𝔫(𝕃_{w}^{∗}) ⊂ 𝕂_{v}^{∗} is a subgroup of index the order of the extension 𝕂_{v} ⊂ 𝕃_{w}. The restriction of the norm map to the idele group 𝔸_{𝕃}^{∗} is very far from surjective to 𝔸_{𝕂}^{∗} and its range is a subgroup of infinite index. The situation is much better with the idele class groups since (cf. [44], Corollary p.153) the norm map is an open mapping 𝔫 : C_{𝕃} → C_{𝕂} whose range is a subgroup of finite index.

9.2 The Weil group and the transfer map

The Weil group W_{𝕃,𝕂} associated to the Galois extension 𝕂 ⊂ 𝕃 is an extension

1 → C_{𝕃} → W_{𝕃,𝕂} → G → 1  (4)

of C_{𝕃} by the Galois group G. One chooses a section s from G and lets a ∈ Z^{2}(G, C_{𝕃}) be the corresponding 2-cocycle so that

a_{α,β} = s_{αβ}^{−1} s_{α} s_{β}, ∀α, β ∈ G.  (5)

The algebraic rules in W_{𝕃,𝕂} are then given by

s_{α} s_{β} = s_{αβ} a_{α,β}, ∀α, β ∈ G  (6)

and

s_{α} x s_{α}^{−1} = α(x), ∀α ∈ G, ∀x ∈ C_{𝕃}.  (7)

The transfer homomorphism

t : W_{𝕃,𝕂} → C_{𝕂}  (8)

is then given by

t(x) = 𝔫(x), ∀x ∈ C_{𝕃} and t(s_{α}) = ∏_{β} a_{α,β}, ∀α ∈ G.  (9)

Its main properties are the following (see [40]).

• t is a surjective group morphism W_{𝕃,𝕂} → C_{𝕂}.

• Let W_{𝕃,𝕂}^{ab} be the abelian quotient of W_{𝕃,𝕂} by the closure of its commutator subgroup W_{𝕃,𝕂}^{c}. Then t induces an isomorphism of W_{𝕃,𝕂}^{ab} with C_{𝕂}.

9.3 The covering

We finally describe the resulting functoriality of the adeles class spaces in terms of a covering map obtained by extending the Weil group and transfer map described above. Let, as above, 𝕃 ⊃ 𝕂 be a finite Galois extension of 𝕂.

Lemma 9.1. The transfer map extends to a map

τ : G × X_{𝕃} → X_{𝕂}  (10)

of the adele class spaces.

Proof. We endow G × X_{𝕃} with a two sided action of G compatible with τ. By construction the norm map 𝔫 of (1) is well defined on 𝔸_{𝕃}. Since it is multiplicative and we have 𝔫(𝕃^{∗}) ⊂ 𝕂^{∗}, it induces a map of quotient spaces 𝔫 : X_{𝕃} → X_{𝕂}. By construction C_{𝕃} acts on X_{𝕃} and the actions by left and right multiplication coincide, so we use both notations. We define the map τ as

τ : G × X_{𝕃} → X_{𝕂}, τ(α, x) = t(s_{α})𝔫(x), ∀x ∈ X_{𝕃}, ∀α ∈ G.  (11)

This makes sense since t(s_{α}) ∈ C_{𝕂} and C_{𝕂} acts on X_{𝕂}. By construction, the restriction of τ to G × C_{𝕃} is the transfer map.

One identifies G×C_{𝕃} with W_{𝕃,𝕂} by the map which to (α, g) ∈ G×C_{𝕃} associates the element s_{α} g of W_{𝕃,𝕂}. In the following we use the notation

x^{g} = g^{−1}(x).  (12)

We have the following result.

Lemma 9.2. Let 𝕃 ⊃ 𝕂 be a finite Galois extension of 𝕂.

1. The expressions

s_{α}g(β, x) = (αβ, a_{α,β}g^{β}x), and (α, x)s_{β}g = (αβ, a_{α,β}x^{β}g) (13)

define a left and a right action of W_{𝕃,𝕂} on G × X_{𝕃}.

2. The map τ of (11) satisfies the equivariance property

τ(gxk) = t(g)τ(x)t(k), ∀x ∈ G × X_{𝕃}, and ∀g, k ∈ W_{𝕃,𝕂}. (14)

Proof. 1) We defined the rules (6) as the natural extension of the multiplication in W_{𝕃,𝕂} using

s_{α}gs_{β}h = s_{α}s_{β}g^{β}h = s_{αβ}a_{α,β}g^{β}h.  (15)

Thus, the proof of associativity in the group W_{𝕃,𝕂} extends and it implies that (13) defines a left and a right action of W_{𝕃,𝕂} and that these two actions commute.

2) The proof that the transfer map t is a group homomorphism extends to give the required equality, since the norm map is a bimodule morphism when extended to X_{𝕃}.

At the level of the classical points, we can then describe the covering map in the following way.

Proposition 9.3. Let 𝕃 and 𝕂 be as above.

1. The restriction of τ to G × Ξ_{𝕃} ⊂ G × X_{𝕃} defines a surjection

τ : G × Ξ_{𝕃} → Ξ_{𝕂}.  (16)

2. The map τ induces a surjection

τ : G × (Ξ_{𝕃}/C_{𝕃,1}) → Ξ_{𝕂}/C_{𝕂,1}.  (17)

Proof. 1) By construction Ξ_{𝕃} = ∪_{w∈Σ_{𝕃}}C_{𝕃}a^{(w)}, where a^{(w)} ∈ X_{𝕃} is the class, modulo the action of 𝕃^{∗}, of the adele with all entries equal to 1 except for a zero at w as in (7). Let π denote the natural surjection from Σ_{𝕃} to Σ_{𝕂}. One has

τ(1, a^{(w)}) = a^{(π(w))}, ∀w ∈ Σ_{𝕃}.  (18)

In fact, one has τ(1, a^{(w)}) = 𝔫(a^{(w)}). Moreover, by (3), the adele a = 𝔫(a^{(w)}) has components a_{z} = 1 for all z ≠ π(w) and a_{π(w)} = 0. Thus a = a^{(π(w))}. The equivariance of the map τ as in Lemma 9.2 together with the surjectivity of the transfer map from W_{𝕃,𝕂} to C_{𝕂} then show that we have

τ(W_{𝕃,𝕂}(1, a^{(w)})) = C_{𝕂} a^{(π(w))}, ∀w ∈ Σ_{𝕃}.

For s_{α}g ∈ W_{𝕃,𝕂}, one has

s_{α}g(1, a^{(w)}) = (α, ga^{(w)}),

since a_{α,1} = 1. Thus, W_{𝕃,𝕂}(1, a^{(w)}) = G × C_{𝕃}a^{(w)} and one gets

τ(G × C_{𝕃}a^{(w)}) = C_{𝕂}a^{(π(w))}, ∀w ∈ Σ_{𝕃}.

Since the map π is surjective we get the conclusion.

2) The transfer map satisfies t(C_{𝕃,1}) ⊂ C_{𝕂,1}. When restricted to the subgroup C_{𝕃} the transfer coincides with the norm map 𝔫 and in particular if |g| = 1 one has |𝔫(g)| = 1. Thus one obtains a surjection of the quotient spaces

τ : (G × Ξ_{𝕃})/C_{𝕃,1} → Ξ_{𝕂}/C_{𝕂,1}.

Moreover, the right action of the subgroup C_{𝕃,1} ⊂ W_{𝕃,𝕂} is given by

(α, x)g = (α, xg).

This means that we can identify

(G × Ξ_{𝕃})/C_{𝕃,1} ∼ G × (Ξ_{𝕃}/C_{𝕃,1}).

9.4 The function field case

Let 𝕂 = 𝔽_{q}(C) be a global field of positive characteristic, identified with the field of rational functions on a nonsingular curve C over 𝔽_{q}. We consider the extensions

𝕃 = 𝕂 ⊗_{𝔽_{q}} 𝔽_{q^{n}}.  (19)

The Galois group G is the cyclic group of order n with generator σ ∈ Gal(𝕃/𝕂) given by σ = id⊗Fr, where Fr ∈ Gal(𝔽_{q^{n}}/𝔽_{q}) is the Frobenius automorphism. Given a point x ∈ C(𝔽̄_{q}) we let n be the order of its orbit under the Frobenius. One then has x ∈ C(𝔽_{q^{n}}) and evaluation at x gives a well defined place w(x) ∈ Σ_{𝕃}. The projection π(w(x)) ∈ Σ_{𝕂} is a well defined place of 𝕂 which is invariant under x ↦ Fr(x). In the isomorphism of ℤ-spaces

ϑ_{𝕃} : C(𝔽̄_{q}) → Ξ_{𝕃}/C_{𝕃,1}

described in §8.3, we have no ambiguity for places corresponding to points x ∈ C(𝔽_{q^{n}}). To such a point we assign simply

ϑ_{𝕃}(x) = a^{(w(x))} ∈ Ξ_{𝕃}/C_{𝕃,1}.

We now describe what happens with these points of Ξ_{𝕃}/C_{𝕃,1} under the covering map τ. We first need to see explicitly why the surjectivity only occurs after crossing by G.

Proposition 9.4. Let 𝕂 and 𝕃 = 𝕂 ⊗_{𝔽_{q}} 𝔽_{q^{n}} be as above.

1. The image 𝔫(C_{𝕃}) ⊂ C_{𝕂} is the kernel of the morphism from C_{𝕂} to G =

ℤ/nℤ given by

g ↦ ρ(g) = log_{q} |g| mod n.

2. One has ρ(t(s_{σ})) = 1 mod n, where σ ∈ Gal(𝕃/𝕂) is the Frobenius generator.

Proof. Since 𝕃 is an abelian extension of 𝕂, one has the inclusions

𝕂 ⊂ 𝕃 ⊂ 𝕂^{ab} ⊂ 𝕃^{ab},  (20)

where 𝕂^{ab} is the maximal abelian extension of 𝕂. Using the class field theory isomorphisms

C_{𝕂} ∼ W (𝕂^{ab}/𝕂) and C_{𝕃} ∼ W (𝕃^{ab}/𝕃),

one can translate the proposition in terms of Galois groups. The result then follows using [40] p.502.

10 Vanishing cycles: an analogy

We begin by considering some simple examples that illustrate some aspects of the geometry of the adeles class space, by restricting to the semilocal case of a finite number of places. This will also illustrate more explicitly the idea of considering the adeles class space as a noncommutative compactification of the idele class group. We draw an analogy between the complement of the idele classes in the adele classes and the singular fiber of a degeneration. This analogy should be taken with a big grain of salt, since this complement is a highly singular space and it really makes sense only as a noncommutative space in the motivic sense described in sections 4 and 5 above.

10.1 Two real places

We first consider the example of the real quadratic field 𝕂 = ℚ(\sqrt{2}) and we restrict to its two real places v_{1} and v_{2}. Thus, we replace the adeles 𝔸_{𝕂} simply by the product 𝕂_{v_{1}} × 𝕂_{v_{2}} over the real places, which is just the product of two copies of ℝ. The ideles 𝔸_{𝕂}^{∗} are correspondingly replaced by 𝕂_{v_{1}}^{∗} ×𝕂_{v_{2}}^{∗} and the inclusion of ideles in adeles is simply given by the inclusion

(ℝ^{∗})^{2} ⊂ ℝ^{2}.  (1)

The role of the action of the group 𝕂^{∗} by multiplication is now replaced by the action by multiplication of the group U of units of 𝕂 = ℚ(\sqrt{2}). This group is

U = ℤ/2ℤ × ℤ

where the ℤ/2ℤ comes from ±1 and the ℤ is generated by the unit u = 3−2\sqrt{2}. Its action on ℝ^{2} is given by the transformation

S(x, y) = (ux, u^{−1}y).  (2)

Thus, in this case of two real places the semi-local version of the adeles class space is the quotient

X_{v_{1},v_{2}} := ℝ^{2}/U  (3)

of ℝ^{2} by the symmetry (x, y) ↦ (−x, −y) and the transformation S. Both of these transformations preserve the function

f̃: ℝ^{2} → ℝ, f̃(x, y) = xy,  (4)

which descends to a function

f : X_{v_{1},v_{2}} → ℝ.  (5)

Moreover one has

(x, y) ∈ (ℝ^{∗})^{2} ⊂ ℝ^{2} ⇔ f(x, y) ≠ 0

and the fiber of f over any non zero ε ∈ ℝ is easily identified with a one dimensional torus

f^{−1}(ε) ∼ ℝ_{+}^{∗}/u^{ℤ}, ∀ε ≠ 0  (6)

where one can use the map (x, y) ↦ |x| to obtain the required isomorphism. The fiber f^{−1}(0) of f over the point ε = 0, on the other hand, is no longer a one dimensional torus and it is singular. It is the union of three pieces

f^{−1}(0) = T_{1} ∪ T_{2} ∪ {0}  (7)

corresponding respectively to

• T_{1} is the locus x = 0, y ≠ 0, which is a torus T_{1} ∼ ℝ_{+}^{∗}/u^{ℤ} under the identification given by the map (x, y) ↦ |y|.

• T_{2} is the locus x ≠ 0, y = 0, which is also identified with a torus T_{2} ∼ ℝ_{+}^{∗}/u^{ℤ} under the analogous map (x, y) ↦ |x|.

• The last piece is the single point x = 0, y = 0.

One can see that at the naive level that the quotient topology on the singular fiber (7) looks as follows. For any point x ∈ T_{j} its closure is x̄ = {x} ∪ {0}. Moreover the point 0 is closed and the induced topology on its complement is the same as the disjoint union of two one dimensional tori T_{j}. In fact one can be more precise and see what happens by analyzing the C^{∗}-algebras involved. The C^{∗}-algebra A associated to the singular fiber is by construction the crossed product

A = C_{0}(f̃^{−1}(0)) ⋊ U  (8)

with f̃ as in (4). One lets

A_{j} = C_{0}(V_{j}) ⋊ U  (9)

where we use the restriction of the action of U to the subsets

V_{j} = {(x_{1}, x_{2}) | x_{j} = 0} ∼ ℝ.

Evaluation at 0 ∈ ℝ gives a homomorphism

ϵ_{j} : A_{j} → C^{∗}(U) .

Lemma 10.1. One has an exact sequence of the form

0 → C(T_{j}) ⊗ K → A_{j} →^{ϵ_{j}} C^{∗}(U) → 0,

where K is the algebra of compact operators. The C^{∗}-algebra A is the fibered product of the A_{j} over C^{∗}(U) using the morphisms ϵ_{j}.

Proof. The first statement follows using the fact that the action of U on ℝ^{∗} is free. Notice that A_{j} is not unital, so that it is not the unital algebra obtained from C(T_{j}) ⊗ K by adjoining a unit. Since the decomposition of f̃^{−1}(0) as the union of the V_{j} over their common point 0 is U-equivariant one gets the second statement.

After collapsing the spectrum of C^{∗}(U) to a point, the topology of the spectrum of A_{j} is the topology of T_{j} ∪ {0} described above. The topology of the spectrum of A is the topology of f^{−1}(0) of (7) described above.

10.2 A real and a non-archimedean place

We now consider another example, namely the case of 𝕂 = ℚ with two places v_{1}, v_{2}, where v_{1} = p is a non-archimedean place associated to a prime p and v_{2} = ∞ is the real place. Again, we replace adeles by the product 𝕂_{v_{1}} × 𝕂_{v_{2}} over the two places, which in this case is just the product

𝕂_{v_{1}} × 𝕂_{v_{2}} = ℚ_{p} × ℝ.  (10)

The ideles are correspondingly replaced by 𝕂_{v_{1}}^{∗} × 𝕂_{v_{2}}^{∗} = ℚ_{p}^{∗} × ℝ^{∗} and the inclusion is given by

ℚ_{p}^{∗} × ℝ^{∗} ⊂ ℚ_{p} × ℝ.  (11)

The role of the action of the group 𝕂^{∗} by multiplication is now replaced by the action by multiplication by the group U of elements of 𝕂^{∗} = ℚ^{∗} which are units outside the above two places. This group is

U = ℤ/2ℤ × ℤ,  (12)

where the ℤ/2ℤ comes from ±1 and the cyclic group is p^{ℤ} generated by p ∈ 𝕂^{∗} = ℚ^{∗}. The action of U of (12) on ℝ × ℚ_{p} is given by the transformation

S(x, y) = (px, py).  (13)

By comparison with the previous case of 𝕂 = ℚ(\sqrt{2}), notice how in that case (cf. (2)) the pair (u, u^{−1}) was just the image of the element 3 − 2\sqrt{2} under the diagonal embedding of 𝕂 in 𝕂_{v_{1}} × 𝕂_{v_{2}}. In the present case, the role of the adeles class space X_{𝕂} = 𝔸_{𝕂}/𝕂^{∗} is then played by its semi-local version

X_{p,∞} = (ℚ_{p} × ℝ)/U  (14)

quotient of ℚ_{p}×ℝ by the symmetry (x, y) → (−x, −y) and the transformation S. Both of these transformations preserve the function

f̃: ℚ_{p} × ℝ → ℝ_{+}, f̃(x, y) = |x|_{p} |y| ∈ ℝ_{+},  (15)

which descends to a function

f : X_{p,∞} → ℝ_{+}.  (16)

Moreover, one has

(x, y) ∈ ℚ_{p}^{∗} × ℝ^{∗} ⊂ ℚ_{p} × ℝ ⇔ f(x, y) ≠ 0

and the fiber of f over any non zero ε ∈ ℝ_{+} is easily identified with ℤ_{p}^{∗}

f^{−1}(ε) ∼ ℤ_{p}^{∗}, ∀ε ≠ 0.

In fact, one can use the fundamental domain

ℤ_{p}^{∗} × ℝ_{+}^{∗}

for the action of U on ℚ_{p}^{∗} × ℝ^{∗} to obtain the required isomorphism. The fiber f^{−1}(0) of f over the point ε = 0 is no longer ℤ_{p}^{∗} and once again it is singular. It is again described as the union of three pieces

f^{−1}(0) = T_{p} ∪ T_{∞} ∪ {0},  (17)

which have, respectively, the following description.

• T_{p} is the locus x = 0, y ≠ 0, which is identified with a torus T_{p} ∼ ℝ_{+}^{∗}/p^{ℤ}, using the map (x, y) ↦ |y|.

• T_{∞} is the locus x ≠ 0, y = 0, which gives the compact space T_{∞} ∼ ℤ_{p}^{∗}/±1 obtained as quotient of ℚ_{p}^{∗} by the action of U.

• The remaining piece is the point x = 0, y = 0.

The description of the topology of f^{−1}(0) is similar to what happens in the case of ℚ(\sqrt{2}) analyzed above. What is not obvious in this case is how the totally disconnected fiber f^{−1}(ε) ∼ ℤ_{p}^{∗} can tie in with the torus T_{p} ∼ ℝ_{+}^{∗}/p^{ℤ} when ε → 0. To see what happens, we use the map

X_{p,∞} ∋ (x, y) ↦ g(x, y) = class of |y| ∈ ℝ_{+}^{∗}/p^{ℤ}.  (18)

This is well defined on the open set y ≠ 0. It is continuous and passes to the quotient. Thus, when a sequence (x_{n}, y_{n}) ∈ X_{p,∞} converges to a point (0, y) ∈ T_{p}, y ≠ 0, one has g(0, y) = lim_{n} g(x_{n}, y_{n}). The point then is simply that we have the relation

g(x, y) = f(x, y) ∈ ℝ_{+}^{∗}/p^{ℤ}.  (19)

In other words, g(x_{n}, y_{n}) = ε_{n} with (x_{n}, y_{n}) in the fiber f^{−1}(ε_{n}) and the point of the singular fiber T_{p} towards which (x_{n}, y_{n}) ∈ X_{p,∞} converges depends only on the value of ε_{n} in ℝ_{+}^{∗}/p^{ℤ}.

Fig. 5. The limit cycle of a foliation.

This phenomenon is reminiscent of the behavior of holonomy in the context of foliations, using a logarithmic scale ℝ_{+}^{∗}/p^{ℤ} ∼ ℝ/(ℤ log p). It corresponds to what happens in the limit cycle of the foliation associated to a flow as depicted in Figure 5. As we argued in [11] (see also §3.2 and §7.1 here above), the role of Frobenius in characteristic zero is played by the one parameter group Fr(t) with t ∈ ℝ which corresponds to the action of ℝ on the adele class space X_{ℚ} = 𝔸_{ℚ}/ℚ^{∗} given in the above logarithmic scale, namely

Fr(t)(a) = e^{t} a, ∀a ∈ X_{ℚ}.  (20)

Its orbit over p ∈ Σ_{ℚ} is of length log p and it corresponds, in the simplified picture of X_{p,∞}, to the component T_{p} of the singular fiber f^{−1}(0).

10.3 Singularities of maps

The simple examples described above illustrate how one can use the function f(x) = |x| in general, and see the place where it vanishes as the complement of C_{𝕂} in the adeles class space X_{𝕂}. This provides a way of thinking of the inclusion of C_{𝕂} in X_{𝕂} in terms of the notions of “singular fiber” and “generic fiber” as seen in the examples above. The generic fiber appears to be typically identified with C_{𝕂,1}, with the union of the generic fibers giving C_{𝕂} as it should. This suggests the possibility of adapting to our noncommutative geometry context some aspects of the well developed theory of nearby and vanishing cycles. A brief dictionary summarizing this analogy is given here below.

Total space  Adele class space X_{𝕂} = 𝔸_{𝕂}/𝕂^{∗}

The map f  f(x) = |x|

Singular fiber  X_{𝕂} ∖ C_{𝕂} = f^{−1}(0)

Union of generic fibers  C_{𝕂} = f^{−1}({0}^{c})

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