# Knots, Primes and the adele class space

Alain Connes  Collège de France
3 Rue d’Ulm
75005 Paris, France  I.H.E.S., France<EMAIL_ADDRESS>and  Caterina Consani
Department of Mathematics
Johns Hopkins University
Baltimore, MD 21218, USA<EMAIL_ADDRESS>
###### Abstract.

We show that the scaling site $X_{\mathbb{Q}}$ and its periodic orbits $C_{p}$
of length $\log p$ offer a geometric framework for the well-known analogy
between primes and knots. The role of the maximal abelian cover of
$X_{\mathbb{Q}}$ is played by the quotient map $\pi:X_{\mathbb{Q}}^{ab}\to
X_{\mathbb{Q}}$ from the adele class space
$X_{\mathbb{Q}}^{ab}:={\mathbb{Q}}^{\times}\backslash{\mathbb{A}}_{\mathbb{Q}}$
to $X_{\mathbb{Q}}=X_{\mathbb{Q}}^{ab}/{\hat{\mathbb{Z}}^{\times}}$. The
inverse image $\pi^{-1}(C_{p})\subset X_{\mathbb{Q}}^{ab}$ of the periodic
orbit $C_{p}$ is canonically isomorphic to the mapping torus of the
multiplication by the Frobenius at $p$ in the abelianized étale fundamental
group $\pi_{1}^{et}({\rm Spec\,}\,\mathbb{Z}_{(p)})^{ab}$ of the spectrum of
the local ring $\mathbb{Z}_{(p)}$, thus exhibiting the linking of $p$ with all
other primes. In the same way as the Grothendieck theory of the étale
fundamental group of schemes is an extension of Galois theory to schemes, the
adele class space gives, as a covering of the scaling site, the corresponding
extension of the class field isomorphism for ${\mathbb{Q}}$ to schemes related
to ${\rm Spec\,}\,\mathbb{Z}$.

###### Key words and phrases:

Knots and Primes, Étale fundamental group, Scheme, Class field theory, Adele
class space, Scaling site, Baum-Connes map, Noncommutative Geometry

###### 1991 Mathematics Subject Classification:

11R37, 11M06, 11M55, 14A15, 14F20, 57K10

###### Contents

  1. 1 Introduction
  2. 2 Proof of Theorem 1.1
  3. 3 The semilocal space $\Gamma\backslash\left({\mathbb{Q}}_{p}\times{\mathbb{Q}}_{q}\times\mathbb{R}\right)$
  4. 4 The classifying space $\Gamma\backslash\left({\mathbb{A}}_{{\mathbb{Q}},S}\times\underline{E\Gamma}\right)$ and its codimension $1$ foliation

##  1\. Introduction

The adele class space of [2] delivers both a geometric interpretation of the
explicit Riemann-Weil formulas as a trace formula, as well as a spectral
realization of the zeros of the Riemann zeta function and of $L$-functions
with Grössencharakter. There is a well known analogy, due to Barry Mazur111on
a suggestion of David Mumford, between knots and primes [11, 12, 13, 14, 15,
16]. We show that the periodic orbits $C_{p}$ of length $\log p$ in the
Riemann sector
$X_{\mathbb{Q}}:={\mathbb{Q}}^{\times}\backslash{\mathbb{A}}_{\mathbb{Q}}/{\hat{\mathbb{Z}}^{\times}}$
of the adele class space provide a geometric realization of this conjectural
relation. By construction $X_{\mathbb{Q}}$ is obtained as the quotient of
${\mathbb{Q}}^{\times}\backslash{\mathbb{A}}_{\mathbb{Q}}$ by the action of
the compact subgroup ${\hat{\mathbb{Z}}^{\times}}$ of idele classes, acting by
multiplication. The space $X_{\mathbb{Q}}$ admits a simple description in
terms of rank one groups [5]. Namely, it is the union of the space of such
groups, up to isomorphism, and the space of rank one subgroups of
$\mathbb{R}$. This latter space is a prototype of a noncommutative space.
The action of the scaling group $\mathbb{R}_{+}^{*}$ on $X_{\mathbb{Q}}$ gives
a Hasse-Weil interpretation of the Riemann zeta function 222completed with its
archimedean local factor [3, 4] and coincides with the action of the Frobenius
automorphisms on the points of the arithmetic site over the tropical semifield
$\mathbb{R}_{+}^{max}$ [5]. To each prime $p$ corresponds the periodic orbit
$C_{p}\subset X_{\mathbb{Q}}$ of length $\log p$ for the action of
$\mathbb{R}_{+}^{*}$: $C_{p}$ consists precisely of rank one subgroups of
$\mathbb{R}$ which are isomorphic to the additive group of the ring
$\mathbb{Z}[1/p]$. We refer to [6] for the structure of $C_{p}$ inherited from
the scaling site : $C_{p}=\mathbb{R}_{+}^{*}/p^{\mathbb{Z}}$ appears as an
elliptic curve in characteristic one, similar to the Jacobi elliptic curve
${\mathbb{C}}^{*}/q^{\mathbb{Z}}$. The Riemann-Roch formula for $C_{p}$
involves real valued dimensions.
The role of the maximal abelian cover $X_{\mathbb{Q}}^{ab}$ of
$X_{\mathbb{Q}}$ is played by the adele class space
${\mathbb{Q}}^{\times}\backslash{\mathbb{A}}_{\mathbb{Q}}$ together with the
canonical quotient map $\pi$:

$\pi:X_{\mathbb{Q}}^{ab}:={\mathbb{Q}}^{\times}\backslash{\mathbb{A}}_{\mathbb{Q}}\to
X_{\mathbb{Q}}={\mathbb{Q}}^{\times}\backslash{\mathbb{A}}_{\mathbb{Q}}/{\hat{\mathbb{Z}}^{\times}}.$
(1)

Let $\mathbb{Z}_{(p)}$ be the ring $\mathbb{Z}$ localized at $p$,
$r:\mathbb{Z}_{(p)}\to{\mathbb{F}}_{p}$ the residue morphism, and $r^{*}$ the
induced embedding of schemes

$r^{*}:{\rm Spec\,}\,{\mathbb{F}}_{p}\hookrightarrow{\rm
Spec\,}\,\mathbb{Z}_{(p)}.$ (2)

Our first result involves the map $\pi_{1}^{et}(r^{*})$ at the level of the
étale fundamental groups $\pi_{1}^{et}$ of schemes. One lets $Frob_{p}$ be the
canonical generator of $\pi_{1}^{et}({\rm
Spec\,}\left({\mathbb{F}}_{p}\right))\simeq{\rm
Gal}(\bar{\mathbb{F}}_{p}/{\mathbb{F}}_{p})$ given by the Frobenius.
Theorem 1.1 shows that the monodromy obtained by lifting the periodic orbit
$C_{p}$ in the maximal abelian cover $X_{\mathbb{Q}}^{ab}$ coincides with the
multiplication by $r^{*}\left\\{Frob_{p}\right\\}$ in the abelianized étale
fundamental group $\pi_{1}^{et}({\rm Spec\,}\,\mathbb{Z}_{(p)})$.

######  Theorem 1.1.

Let $p$ be a prime and $\left\\{Frob_{p}\right\\}\in\pi_{1}^{et}({\rm
Spec\,}\left({\mathbb{F}}_{p}\right))$ be the canonical generator. The inverse
image $\pi^{-1}(C_{p})\subset X_{\mathbb{Q}}^{ab}$ of the periodic orbit
$C_{p}$ is canonically isomorphic to the mapping torus333in the ordinary
topological sense of the multiplication by $r^{*}\left\\{Frob_{p}\right\\}$ in
the abelianized étale fundamental group $\pi_{1}^{et}({\rm
Spec\,}\,\mathbb{Z}_{(p)})^{ab}$. The canonical isomorphism is equivariant for
the action of the idele class group.

Figure 1. Lifting the periodic orbit $C_{p}$

Our second result gives the geometry underlying the analogy with the linking
number of two knots $K,L\subset S^{3}$ in the three sphere. By definition, the
linking number $\operatorname{lk}\left(K,L\right)$ is the monodromy obtained
by lifting the first knot $K$ in the maximal abelian cover
$\left(S^{3}-L\right)^{ab}$ of the complement of the second knot $L$.
Equivalently, $\operatorname{lk}\left(K,L\right)$ is the image in
$\pi_{1}\left(S^{3}-L\right)^{ab}\simeq\mathbb{Z}$ of the canonical generator
of $\pi_{1}(K)$

$\ \
\pi_{1}(K)\longrightarrow\pi_{1}\left(S^{3}-L\right)^{ab}\simeq\mathbb{Z}.$
(3)

In the analogy between knots and primes the role of the knots $K,L$ is played
by two distinct primes $p,q$, while the sphere is replaced by ${\rm
Spec\,}\,\mathbb{Z}$ and $X_{L}:=S^{3}-L$ is replaced by ${\rm
Spec\,}\,\mathbb{Z}[1/q]$.
Given two primes $p\neq q$ we consider the inverse image of the periodic orbit
$C_{p}$ in the semilocal adele class space444the other places do not play a
role associated to the set of places $S=\\{p,q,\infty\\}$. The semilocal adele
class space is
$\Gamma\backslash\left({\mathbb{Q}}_{p}\times{\mathbb{Q}}_{q}\times\mathbb{R}\right)$
where $\Gamma:=\\{\pm p^{m}q^{n}\mid m,n\in\mathbb{Z}\\}$.

######  Theorem 1.2.

The inverse image of the periodic orbit $C_{p}$ in the semilocal adele class
space
$\Gamma\backslash\left({\mathbb{Q}}_{p}\times{\mathbb{Q}}_{q}\times\mathbb{R}\right)$
associated to $S=\\{p,q,\infty\\}$ is the mapping torus of the canonical
generator $\left\\{Frob_{p}\right\\}$ acting by multiplication in the
abelianized étale fundamental group $\pi_{1}^{et}({\rm
Spec\,}\,\mathbb{Z}[1/q])^{ab}$ of the complement of $q\in{\rm
Spec\,}\,\mathbb{Z}$.

Thus the periodic orbit $C_{p}$ plays the role of the first knot, and the
monodromy of its lift in the maximal abelian cover of the complement of
$q\in{\rm Spec\,}\,\mathbb{Z}$ is the element $p\in\mathbb{Z}_{q}^{*}$. In the
analogy between knots and primes developed in [12, 15, 16],
$p\in\mathbb{Z}_{q}^{*}$ plays the role of the linking number555and allows one
to compare the quadratic reciprocity with the antisymmetry of the linking
number of $p$ with $q$.
Section 3 provides a detailed description of the natural stratification of the
semilocal adele class space
${\mathbb{A}}_{{\mathbb{Q}},S}=\Gamma\backslash\left({\mathbb{Q}}_{p}\times{\mathbb{Q}}_{q}\times\mathbb{R}\right)$
that generates a spectral sequence for the $K$-theory of the associated
$C^{*}$-algebras.
Finally, section 4 describes the canonical codimension one foliation of the
three dimensional manifold
$\Gamma\backslash\left({\mathbb{A}}_{{\mathbb{Q}},S}\times\underline{E\Gamma}\right)=\Gamma\backslash\left({\mathbb{Q}}_{p}\times{\mathbb{Q}}_{q}\times\mathbb{R}\times\mathbb{R}^{2}\right)$
source of the Baum-Connes map [1] which computes the $K$-theory of the
$C^{*}$-algebra
$\mathcal{A}=C_{0}({\mathbb{A}}_{{\mathbb{Q}},S})\rtimes\Gamma$.

##  2\. Proof of Theorem 1.1

Let $\mathbb{A}^{f}$ be the ring of finite adeles of ${\mathbb{Q}}$. Let
$(a,\lambda)\in\mathbb{A}^{f}\times\mathbb{R}={\mathbb{A}}_{\mathbb{Q}}$ be an
adele with $\lambda>0$. Let $\Phi$ be the map from
$\left(\mathbb{A}^{f}/\hat{\mathbb{Z}}^{\times}\right)\times\mathbb{R}_{+}^{\times}$
to subgroups of $\mathbb{R}$ defined by

$\Phi(a,\lambda):=\lambda H_{a},\ \ H_{a}:=\\{q\in\mathbb{Q}\mid
aq\in\hat{\mathbb{Z}}\\}.$

Then ([5], Lemma 3.7) $\Phi$ is a bijection between the subset of
$X_{\mathbb{Q}}={\mathbb{Q}}^{\times}\backslash{\mathbb{A}}_{\mathbb{Q}}/{\hat{\mathbb{Z}}^{\times}}$
formed of adele classes with non-zero archimedean component, and the set of
non-zero subgroups of $\mathbb{R}$ whose elements are pairwise commensurable.
Let
$\tilde{\pi}:{\mathbb{A}}_{\mathbb{Q}}\to{\mathbb{Q}}^{\times}\backslash{\mathbb{A}}_{\mathbb{Q}}\to
X_{\mathbb{Q}}$ be the projection.

${{\mathbb{A}}_{\mathbb{Q}}}$${{\mathbb{Q}}^{\times}\backslash{\mathbb{A}}_{\mathbb{Q}}}$${X_{\mathbb{Q}}}$$\scriptstyle{\tilde{\pi}}$$\scriptstyle{\pi}$

Then $\tilde{\pi}^{-1}(C_{p})\subset{\mathbb{A}}_{\mathbb{Q}}$ is the
saturation ${\mathbb{Q}}^{\times}F$, of the subset

$F=\\{a=(a_{v}),\ \ a_{\infty}>0,\ a_{p}=0,\
a_{v}\in\mathbb{Z}_{v}^{*}\,,\,\leavevmode\nobreak\ \forall
v\notin\\{p,\infty\\}\\}\subset{\mathbb{A}}_{\mathbb{Q}}.$

The residual diagonal action of ${\mathbb{Q}}^{\times}$ on $F$ is given by
$p^{\mathbb{Z}}\subset{\mathbb{Q}}^{\times}$. Thus one obtains

$\pi^{-1}(C_{p})=\left(\mathbb{R}_{+}^{*}\times\prod_{q\neq
p}\mathbb{Z}_{q}^{*}\right)/p^{\mathbb{Z}}$ (4)

where $p$ is diagonally embedded in the product. This quotient is by
construction the mapping torus of the multiplication by $p$ in the compact
group $\prod_{q\neq p}\mathbb{Z}_{q}^{*}$. We now interpret these terms in the
language of étale abelianized fundamental groups. This derives from the
following facts:

  * •

The abelianized étale fundamental group666The abelianization takes care of the
need to specify the base point $\pi_{1}^{et}({\rm Spec\,}\,\mathbb{Z}_{(p)})$
is canonically isomorphic to $\prod_{q\neq p}\mathbb{Z}_{q}^{*}$.

  * •

The image $\pi_{1}^{et}(r^{*})\left\\{Frob_{p}\right\\}$ in the étale
abelianized fundamental group $\pi_{1}^{et}({\rm
Spec\,}\,\mathbb{Z}_{(p)})\simeq\prod_{q\neq p}\mathbb{Z}_{q}^{*}$ is equal to
$p$ diagonally embedded in $\prod_{q\neq p}\mathbb{Z}_{q}^{*}$ (see (2)).

The first fact follows from [9] (Corollary 6.17), together with the
determination of the maximal abelian extension of ${\mathbb{Q}}$ in which $p$
is unramified as obtained by adjoining all roots of unity of order prime to
$p$777following the local to global proof of the Kronecker-Weber theorem. Its
Galois group is $\prod_{q\neq p}\mathbb{Z}_{q}^{*}$. The second fact follows
since the action of $\left\\{Frob_{p}\right\\}$ on roots of unity is given by
raising to the power $p$.

The above proof elucidates the geometric aspects of the well-known analogy
between primes and knots through the role of the scaling site
$X_{\mathbb{Q}}$, its periodic orbits and their liftings to the adele class
space of the rationals viewed as an abelian cover of $X_{\mathbb{Q}}$.

##  3\. The semilocal space
$\Gamma\backslash\left({\mathbb{Q}}_{p}\times{\mathbb{Q}}_{q}\times\mathbb{R}\right)$

Let $S$ be a finite set of places of ${\mathbb{Q}}$, with $\infty\in S$. We
briefly recall the construction of the semilocal adele class space associated
to $S$. One replaces the ring of adeles by the locally compact ring
${\mathbb{A}}_{{\mathbb{Q}},S}=\prod_{v\in S}{\mathbb{Q}}_{v}$ in which
${\mathbb{Q}}$ embeds diagonally. Let ${\mathcal{O}}_{{\mathbb{Q}},S}$ be the
subring of ${\mathbb{Q}}$ given by rational numbers whose denominator only
involves primes $p\in S$. In other words,
${\mathcal{O}}_{{\mathbb{Q}},S}=\\{q\in{\mathbb{Q}}\,|\,|q|_{v}\leq 1\,,\
\forall v\notin S\\}\,.$ The group $\Gamma$ of invertible elements of
${\mathcal{O}}_{{\mathbb{Q}},S}$ is

$\Gamma={\rm GL}_{1}({\mathcal{O}}_{{\mathbb{Q}},S})=\\{\pm
p_{1}^{n_{1}}\cdots p_{k}^{n_{k}}\,:\,p_{j}\in
S\setminus\\{\infty\\}\,,\,n_{j}\in\mathbb{Z}\\}.$ (5)

The semilocal adele class space is the quotient
$X_{{\mathbb{Q}},S}^{ab}:=\Gamma\backslash{\mathbb{A}}_{{\mathbb{Q}},S}$. The
map

$\pi_{S}:X_{{\mathbb{Q}},S}^{ab}\to
X_{{\mathbb{Q}},S}=\Gamma\backslash{\mathbb{A}}_{{\mathbb{Q}},S}/\left(\prod_{v\in
S\setminus\infty}\mathbb{Z}_{v}^{*}\right)$ (6)

plays the role of the maximal abelian cover of the double quotient
$X_{{\mathbb{Q}},S}$, the semilocal analogue of $X_{\mathbb{Q}}$. The group

$J_{S}=\Gamma\backslash{\rm
GL}_{1}({\mathbb{A}}_{{\mathbb{Q}},S})\simeq\mathbb{R}_{+}^{*}\times\prod_{v\in
S\setminus\infty}\mathbb{Z}_{v}^{*}$ (7)

is the analogue of the idele class group and acts naturally on
$X_{{\mathbb{Q}},S}^{ab}$ by multiplication.
We let $p\neq q$ be two primes and $S=\\{p,q,\infty\\}$. The ring
${\mathcal{O}}_{{\mathbb{Q}},S}$ is $\mathbb{Z}[1/p,1/q]$ and the abelianized
étale fundamental group of its spectrum is

$\pi_{1}^{et}({\rm
Spec\,}\,{\mathcal{O}}_{{\mathbb{Q}},S})^{ab}=\mathbb{Z}_{p}^{*}\times\mathbb{Z}_{q}^{*}.$
(8)

Next we describe the natural stratification of the space
$X_{{\mathbb{Q}},S}^{ab}$.

######  .

Let $p\neq q$ be two primes, $S=\\{p,q,\infty\\}$ and
$\pi_{S}:X_{{\mathbb{Q}},S}^{ab}\to X_{{\mathbb{Q}},S}$ as in (6). Then
$(i)$ The orbits of the action of the group $J_{S}$ on
$X_{{\mathbb{Q}},S}^{ab}$ are indexed by the subsets of $S=\\{p,q,\infty\\}$
as follows

$\Omega_{Z}:=\Gamma\backslash\\{(a_{v})\in{\mathbb{A}}_{{\mathbb{Q}},S}\mid
a_{v}=0\,,\,\leavevmode\nobreak\ \forall v\in Z,\ a_{v}\neq
0\,,\,\leavevmode\nobreak\ \forall v\notin Z\\}\,,\,\leavevmode\nobreak\
\forall Z\subset S.$

$(ii)$ The orbit $\Omega_{\\{p\\}}$ associated to the subset $\\{p\\}\subset
S$ is the inverse image $\pi_{S}^{-1}(C_{p})$ of the periodic orbit $C_{p}$ of
length $\log p$ (and a similar result holds for the subset $\\{q\\}\subset
S$).
$(iii)$ The orbit $\pi_{S}^{-1}(C_{p})$ is equivariantly isomorphic to the
mapping torus of the multiplication by $p$ in the compact group
$\mathbb{Z}_{q}^{*}$.

###### Proof.

$(i)$ Follows from the transitivity of the action of ${\rm
GL}_{1}({\mathbb{Q}}_{v})$ on ${\mathbb{Q}}_{v}\setminus\\{0\\}$ for any $v$.
$(ii)$ The periodic orbit $C_{p}$ of length $\log p$ for the action of the
scaling group $\mathbb{R}_{+}^{*}$ in $X_{{\mathbb{Q}},S}$ is equal to
$\pi_{S}(\Omega_{\\{p\\}})\subset X_{{\mathbb{Q}},S}$ and is identical to the
quotient $p^{\mathbb{Z}}\backslash\mathbb{R}_{+}^{*}$. This description shows
that $C_{p}$ appears in the boundary of the free orbit
$\pi_{S}(\Omega_{\emptyset})\subset X_{{\mathbb{Q}},S}$ as shown in Figure 2.
$(iii)$ One has $\Omega_{\\{p\\}}=\pi_{S}^{-1}(C_{p})$. The action of $\\{\pm
q^{n}\\}$ on
$\\{0\\}\times\left({\mathbb{Q}}_{q}\setminus\\{0\\}\right)\times\mathbb{R}^{*}$
admits $\\{0\\}\times\mathbb{Z}_{q}^{*}\times\mathbb{R}_{+}^{*}$ as
fundamental domain. This domain is preserved under the action of
$p^{\mathbb{Z}}$. The quotient

$p^{\mathbb{Z}}\backslash\left(\mathbb{Z}_{q}^{*}\times\mathbb{R}_{+}^{*}\right)$

is the mapping torus of the multiplication by $p$ in the compact group
$\mathbb{Z}_{q}^{*}$. ∎

There are $8$ orbits $\Omega_{Z}$, $Z\subset S=\\{p,q,\infty\\}$. Proposition
3 describes two of them, the remaining six orbits are described as follows:

  1. (1)

The orbit $\Omega_{\emptyset}\subset X_{{\mathbb{Q}},S}^{ab}$ is the idele
class group $J_{S}$ of (7).

  2. (2)

The orbit $\Omega_{\\{\infty\\}}\subset X_{{\mathbb{Q}},S}^{ab}$ is the
quotient $\left(\mathbb{Z}_{p}^{*}\times\mathbb{Z}_{q}^{*}\right)/\pm 1$.

  3. (3)

The orbit $\Omega_{\\{p,q\\}}$ is the quotient
$(p^{\mathbb{Z}}q^{\mathbb{Z}})\backslash\mathbb{R}_{+}^{*}$

  4. (4)

The orbit $\Omega_{\\{p,\infty\\}}$ is the quotient of
$\mathbb{Z}_{q}^{*}/\\{\pm 1\\}$ by the action of $\mathbb{Z}$ by
multiplication by powers of $p$.

  5. (5)

The orbit $\Omega_{\\{q,\infty\\}}$ is the quotient of
$\mathbb{Z}_{p}^{*}/\\{\pm 1\\}$ by the action of $\mathbb{Z}$ by
multiplication by powers of $q$.

  6. (6)

The orbit $\Omega_{\\{p,q,\infty\\}}$ is a single point.

To be more precise one can describe for each orbit the corresponding
$C^{*}$-algebra up to Morita equivalence. For $(3)$ one obtains the
noncommutative torus associated to the irrational number $\log p/\log q$. For
$(4)$ one gets the cross product $C(\mathbb{Z}_{q}^{*}/\pm 1)\ltimes
p^{\mathbb{Z}}$ and similarly for $(5)$. Finally for $(6)$ one gets the group
ring $C^{*}(\Gamma)$. By construction the group $J_{S}$ acts by automorphisms
on these $C^{*}$-algebras.

(a)  (b)

Figure 2. (A) shows how the generic orbit (in green) for the action of
$\mathbb{R}_{+}^{*}$ on $X_{\mathbb{Q}}$ is dense in the periodic orbit
$C_{p}$. This holds for any $p$. (B) shows how the density of the generic
orbit in every $C_{p}$ imitates the density of the generic point of ${\rm
Spec\,}\,\mathbb{Z}$ (compare with [10], page 16).

### Proof of Theorem 1.2

This follows from Proposition 3 $(iii)$, together with the following facts :

  * •

The abelianized étale fundamental group $\pi_{1}^{et}({\rm
Spec\,}\,\mathbb{Z}[1/q])^{ab}$ is canonically isomorphic to
$\mathbb{Z}_{q}^{*}$.

  * •

The image $\pi_{1}^{et}(r^{*})\left\\{Frob_{p}\right\\}$ is equal to
$p\in\mathbb{Z}_{q}^{*}$.

As in the proof of Theorem 1.1, the first fact follows using [9] (Corollary
6.17), together with the determination of the maximal abelian extension of
${\mathbb{Q}}$ only ramified at $q$ (obtained by adjoining all roots of unity
of order a power of $q$). Its Galois group is $\mathbb{Z}_{q}^{*}$. The second
fact follows since the action of $\left\\{Frob_{p}\right\\}$ on roots of unity
is given by raising to the power $p$.

##  4\. The classifying space
$\Gamma\backslash\left({\mathbb{A}}_{{\mathbb{Q}},S}\times\underline{E\Gamma}\right)$
and its codimension $1$ foliation

Let $S=\\{p,q,\infty\\}$ as above. The $C^{*}$-algebra
$\mathcal{A}=C_{0}({\mathbb{A}}_{{\mathbb{Q}},S})\rtimes\Gamma$ of the
noncommutative space
$X_{{\mathbb{Q}},S}^{ab}:=\Gamma\backslash{\mathbb{A}}_{{\mathbb{Q}},S}$ is
not of type I (see [8], Lemma 2.28 Chapter II). The understanding of
$\mathcal{A}$ at the level of $K$-theory is obtained by considering the
assembly map [1] computing the $K$-theory of the $C^{*}$-algebra $\mathcal{A}$
in terms of the universal proper action $\underline{E\Gamma}$ of $\Gamma$.
Since $\Gamma$ is commutative the various definitions of the cross product
$C^{*}$-algebra all coincide with the reduced cross product, thus we drop the
lower index $r$ in the notation $C^{*}_{r}$. Moreover the assembly map is an
isomorphism. One obtains in this way an ordinary space
$Y=Y_{{\mathbb{Q}},S}=\Gamma\backslash\left({\mathbb{A}}_{{\mathbb{Q}},S}\times\underline{E\Gamma}\right)$
which resolves the noncommutative nature of the quotient
$\Gamma\backslash{\mathbb{A}}_{{\mathbb{Q}},S}$. The group $\Gamma$ is
$\mathbb{Z}^{2}\times\\{\pm 1\\}$ and the universal space for proper actions
$\underline{E\Gamma}$ is $\mathbb{R}^{2}$ where $\mathbb{Z}^{2}$ acts by
translations and $\\{\pm 1\\}$ acts trivially. Thus one obtains the following
three dimensional space, quotient by the diagonal action of $\Gamma$,

$Y=\Gamma\backslash\left({\mathbb{A}}_{{\mathbb{Q}},S}\times\underline{E\Gamma}\right)=\Gamma\backslash\left({\mathbb{Q}}_{p}\times{\mathbb{Q}}_{q}\times\mathbb{R}\times\mathbb{R}^{2}\right).$
(9)

We stress that the dimension three in the next proposition is due to taking a
pair of primes and has no relation to the homological dimension of ${\rm
Spec\,}\,\mathbb{Z}$.

######  .

$(i)$ The space $Y$ is locally compact of dimension $3$.
$(ii)$ The action of $\mathbb{R}^{2}$ by translations on the space $Y$ defines
a codimension $1$ lamination of $Y$ whose space of leaves is
$X_{{\mathbb{Q}},S}^{ab}$.

###### Proof.

$(i)$ The properness of the action of $\Gamma$ on $\underline{E\Gamma}$
implies that its action on
$\left({\mathbb{A}}_{{\mathbb{Q}},S}\times\underline{E\Gamma}\right)$ is also
proper. Thus $Y$ is Hausdorff and since
$\left({\mathbb{A}}_{{\mathbb{Q}},S}\times\underline{E\Gamma}\right)$ is
locally compact, $Y$ is locally compact. The dimension of
${\mathbb{Q}}_{p}\times{\mathbb{Q}}_{q}\times\mathbb{R}\times\mathbb{R}^{2}$
is equal to $3$ and is unaltered when passing to the quotient.
$(ii)$ By construction the orbits of the action by translation of
$\mathbb{R}^{2}$ on
$\left({\mathbb{Q}}_{p}\times{\mathbb{Q}}_{q}\times\mathbb{R}\times\mathbb{R}^{2}\right)$
define a lamination which is invariant under the action of $\Gamma$ and hence
descends to the proper quotient $Y$. The space of leaves of the first
lamination is
$\left({\mathbb{Q}}_{p}\times{\mathbb{Q}}_{q}\times\mathbb{R}\right)$ and its
quotient by $\Gamma$ is $X_{{\mathbb{Q}},S}^{ab}$.∎

The simplest meaningful computation of the $K$-theory of the involved
$C^{*}$-algebras is for the cross product $A$ associated to the union in
$X_{{\mathbb{Q}},S}$ of the generic orbit with the three periodic orbits
$C_{p},C_{q},C_{\infty}$. One obtains that $K_{0}(A)\simeq\mathbb{Z}^{3}$
reflects the presence of the three periodic orbits, while
$K_{1}(A)\simeq\mathbb{Z}^{2}$ reflects the one-dimensionality of the periodic
orbits $C_{p},C_{q}$.

######  Remark 4.1.

$(i)$ The theory of the étale fundamental group for schemes can be viewed as
the extension of Galois theory to schemes ([9]). The classical abelian class
field theory provides an isomorphism of the Galois group of the maximal
abelian extension of a global field with an adelic group. It is natural to
wonder if this isomorphism can be extended to compare the behavior of the
étale fundamental group of schemes with an adelic geometric construction.
Theorems 1.1 and 1.2 answer positively to this question for schemes related to
${\rm Spec\,}\,\mathbb{Z}$ in terms of the geometry of the adele class space
viewed as a covering $\pi:X_{\mathbb{Q}}^{ab}\to X_{\mathbb{Q}}$ of the
scaling site. This extension of the class field theory isomorphism to schemes
ought to play a role in the understanding of the subtle relation between the
scaling site and ${\rm Spec\,}\,\mathbb{Z}$.

$(ii)$ The scaling site which plays a central role in the present paper admits
a natural complexification $\mathscr{C}_{\mathbb{Q}}$ [7], whose role is to
pass from characteristic one to characteristic zero. It replaces the real
half-line by the pro-étale cover of the punctured unit disk in the complex
plane. Then $\mathscr{C}_{\mathbb{Q}}$ is interpreted as the moduli space of
elliptic curves with a triangular structure. The extrapolation of the findings
of the present paper to the context of $\mathscr{C}_{\mathbb{Q}}$ has the
potential to formulate the results in crystalline terms in place of the étale
one.

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