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\begin{document}
\title{NONCOMMUTATIVE GEOMETRY AND THE RIEMANN ZETA FUNCTION}
\author{Alain Connes}
\date{ \ }
\maketitle
According to my first teacher Gustave Choquet one does, by openly facing
a well known unsolved problem, run the risk of being remembered more by
one's failure than anything else. After reaching a certain age, I
realized that waiting ``safely'' until one reaches the end-point of one's
life is an equally selfdefeating alternative.
In this paper I shall first look back at my early work on the
classification of von Neumann algebras and cast it in the unusual light
of Andr\'{e} Weil's Basic Number Theory.
I shall then explain that this leads to a natural spectral interpretation
of the zeros of the Riemann zeta function and a geometric framework in
which the Frobenius, its eigenvalues and the Lefschetz formula
interpretation of the explicit formulas continue to hold even for number
fields. We shall then prove the positivity of the Weil distribution
assuming the validity of the analogue of the Selberg trace formula. The
latter remains unproved and is equivalent to $RH$ for all $L$-functions
with Gr\"ossencharakter.
\section{Local class field theory and the classification of factors}
Let $K$ be a {\it local} field, i.e. a nondiscrete locally compact field.
The action of $K^* = GL_1 (K)$ on the additive group $K$ by
multiplication,
$$
(\lambda , x) \rightarrow \lambda x \qquad \forall \, \lambda \in K^* \,
, \ x \in K \, , \leqno (1)
$$
together with the uniqueness, up to scale, of the Haar measure of the
additive group $K$, yield a homomorphism,
$$
a \in K^* \rightarrow \vert a \vert \in \Rb_+^* \, , \leqno (2)
$$
from $K^*$ to $\Rb_+^*$, called the {\it module} of $K$. Its range
$$
{\rm Mod} (K) = \{ \vert \lambda \vert \in \Rb_+^* \, ; \ \lambda \in K^*
\} \leqno (3)
$$
is a closed subgroup of $\Rb_+^*$.
The fields $\Rb$, $\Cb$ and $\Hb$ (of quaternions) are the only ones with
${\rm Mod} (K) = \Rb_+^*$, they are called Archimedian local fields.
Let $K$ be a non Archimedian local field, then
$$
R = \{ x \in K \, ; \ \vert x \vert \leq 1 \} \, , \leqno (4)
$$
is the unique maximal compact subring of $K$ and the quotient $R/P$ of
$R$ by its unique maximal ideal is a finite field $\Fb_q$ (with
$q=p^{\ell}$ a prime power). One has,
$$
{\rm Mod} (K) = q^{\Zb} \subset \Rb_+^* \, . \leqno (5)
$$
Let $K$ be commutative. An extension $K \subset K'$ of finite degree of
$K$ is called {\it unramified} iff the dimension of $K'$ over $K$ is the
order of ${\rm Mod} (K')$ as a subgroup of ${\rm Mod} (K)$. When this is
so, the field $K'$ is commutative, is generated over $K$ by roots of
unity of order prime to $q$, and is a cyclic Galois extension of $K$ with
Galois group generated by the automorphism $\theta \in {\rm Aut}_K (K')$
such that,
$$
\theta (\mu) = \mu^q \, , \leqno (6)
$$
for any root of unity of order prime to $q$ in $K'$.
The unramified extensions of finite degree of $K$ are classified by the
subgroups,
$$
\Gamma \subset {\rm Mod} (K) \, , \ \Gamma \not= \{ 1 \} \, . \leqno (7)
$$
Let then $\overline K$ be an algebraic closure of $K$, $K_{\rm sep}
\subset
\overline K$ the separable algebraic closure, $K_{\rm ab} \subset K_{\rm
sep}$ the maximal abelian extension of $K$ and $K_{\rm un} \subset K_{\rm
ab}$ the maximal unramified extension of $K$, i.e. the union of all
unramified extensions of finite degree. One has,
$$
K \subset K_{\rm un} \subset K_{\rm ab} \subset K_{\rm sep} \subset
\overline K \, , \leqno (8)
$$
and the Galois group ${\rm Gal} (K_{\rm un} : K)$ is topologically
generated by $\theta$ called the Frobenius automorphism.
The correspondence (7) is given by,
$$
K' = \{ x \in K_{\rm un} \, ; \ \theta_{\lambda} (x) = x \quad \forall \,
\lambda \in \Gamma \} \, , \leqno (9)
$$
with rather obvious notations so that $\theta_q$ is the $\theta$ of (6).
Let then $W_K$ be the subgroup of ${\rm Gal} (K_{\rm ab} : K)$ whose
elements induce on $K_{\rm un}$ an integral power of the Frobenius
automorphism. One endows $W_K$ with the locally compact topology dictated
by the exact sequence of groups,
$$
1 \rightarrow {\rm Gal} (K_{\rm ab} : K_{\rm un}) \rightarrow W_K
\rightarrow {\rm Mod} (K) \rightarrow 1 \, , \leqno (10)
$$
and the main result of local class field theory asserts the existence of
a canonical isomorphism,
$$
W_K \stackrel{\sim}{\rightarrow} K^* \, , \leqno (11)
$$
compatible with the module.
The basic step in the construction of the isomorphism (11) is the
classification of finite dimensional central simple algebras $A$ over
$K$. Any such algebra is of the form,
$$
A = M_n (D) \, , \leqno (12)
$$
where $D$ is a (central) division algebra over $K$ and the symbol $M_n$
stands for $n \times n$ matrices.
Moreover $D$ is the crossed product of an unramified extension $K'$ of
$K$ by a 2-cocycle on its cyclic Galois group. Elementary group
cohomology then yields the isomorphism,
$$
{\rm Br} (K) \stackrel{\eta}{\rightarrow} \Qb / \Zb \, , \leqno (13)
$$
of the Brauer group of classes of central simple algebras over $K$ (with
tensor product as the group law), with the group $\Qb / \Zb$ of roots of
1 in $\Cb$.
All the above discussion was under the assumption that $K$ is non
Archimedian. For Archimedian fields $\Rb$ and $\Cb$ the same questions
have an idiotically simple answer. Since $\Cb$ is algebraically closed
one has $K = \overline K$ and the whole picture collapses. For $K = \Rb$
the only non trivial value of the Hasse invariant $\eta$ is
$$
\eta (\Hb) = -1 \, . \leqno (14)
$$
A Galois group $G$ is by construction totally disconnected so that a
morphism from $K^*$ to $G$ is necessarily trivial on the connected
component of $1 \in K^*$.
Let $k$ be a {\it global} field, i.e. a discrete cocompact subfield of a
(non discrete) locally compact semi-simple commutative ring $A$. (Cf.
Iwasawa {\it Ann. of Math.} {\bf 57} (1953).) The topological ring $A$ is
canonically associated to $k$ and called the Adele ring of $k$, one has,
$$
A = \prod_{\rm res}^{} k_v \, , \leqno (15)
$$
where the product is the restricted product of the local fields $k_v$
labelled by the places of $k$.
When the characteristic of $k$ is $p>1$ so that $k$ is a function field
over $\Fb_q$, one has
$$
k \subset k_{\rm un} \subset k_{\rm ab} \subset k_{\rm sep} \subset
\overline k \, , \leqno (16)
$$
where, as above $\overline k$ is an algebraic closure of $k$, $k_{\rm
sep}$ the separable algebraic closure, $k_{\rm ab}$ the maximal abelian
extension and $k_{\rm un}$ is obtained by adjoining to $k$ all roots of
unity of order prime to $p$.
One defines the Weil group $W_k$ as above as the subgroup of ${\rm Gal}
(k_{\rm ab} : k)$ of those automorphisms which induce on $k_{\rm un}$ an
integral power of $\theta$,
$$
\theta (\mu) = \mu^q \qquad \forall \, \mu \ \hbox{root of 1 of order
prime to} \ p \, . \leqno (17)
$$
The main theorem of global class field theory asserts the existence of a
canonical isomorphism,
$$
W_k \simeq C_k = GL_1 (A) / GL_1 (k) \, , \leqno (18)
$$
of locally compact groups.
When $k$ is of characteristic 0, i.e. is a number field, one has a
canonical isomorphism,
$$
{\rm Gal} (k_{\rm ab} : k) \simeq C_k / D_k \, , \leqno (19)
$$
where $D_k$ is the connected component of identity in the Idele class
group $C_k = GL_1 (A) / GL_1 (k)$, but because of the Archimedian places
of $k$ there is no interpretation of $C_k$ analogous to the Galois group
interpretation for function fields. According to A.~Weil \cite{W4}, ``La
recherche d'une interpr\'{e}tation pour $C_k$ si $k$ est un corps de nombres,
analogue en quelque mani\`{e}re \`{a} l'interpr\'{e}tation par un groupe de Galois
quand $k$ est un corps de fonctions, me semble constituer l'un des
probl\`{e}mes fondamentaux de la th\'{e}orie des nombres \`{a} l'heure actuelle~; il
se peut qu'une telle interpr\'{e}tation renferme la clef de l'hypoth\`{e}se de
Riemann~$\ldots$''.
\smallskip
Galois groups are by construction
projective limits of the finite groups attached to finite extensions. To
get connected groups one clearly needs to relax this finiteness condition
which is the same as the finite dimensionality of the central simple
algebras.
Since Archimedian places of $k$ are responsible for the non triviality of $D_k$
it is natural to ask the following preliminary question,
\smallskip
\noindent ``Is there a non trivial Brauer theory of central simple algebras over $\Cb$.''
\smallskip
\noindent As we shall see shortly the {\it approximately finite
dimensional} simple central algebras over $\Cb$ provide a satisfactory answer to this question.
They are classified by their
module,
$$
{\rm Mod} (M) \mathop{\subset}_{\sim} \ \Rb_+^* \, , \leqno (20)
$$
which is a virtual closed subgroup of $\Rb_+^*$.
Let us now explain this statement with more care. First we exclude the
trivial case $M = M_n (\Cb)$ of matrix algebras. Next ${\rm Mod} (M)$ is
a virtual subgroup of $\Rb_+^*$, in the sense of G.~Mackey, i.e. an ergodic
action of $\Rb_+^*$.
All ergodic flows appear and $M_1$ is isomorphic to $M_2$ iff ${\rm Mod}
(M_1) \cong {\rm Mod} (M_2)$.
The birth place of central simple algebras is as the commutant
of isotypic representations. When one works over $\Cb$ it is natural to
consider unitary representations in Hilbert space so that we shall
restrict our attention to algebras $M$ which appear as commutants of
unitary representations. They are called von Neumann algebras. The terms
central and simple keep their usual algebraic meaning.
The classification involves three independent parts,
\begin{itemize}
\item[(A)] The definition of the invariant ${\rm Mod} (M)$ for arbitrary
factors (central von Neumann algebras).
\item[(B)] The equivalence of all possible notions of approximate finite
dimensionality.
\item[(C)] The proof that Mod is a complete invariant and that all
virtual subgroups are obtained.
\end{itemize}
The module of a factor $M$ was first defined (\cite{Co_2}) as a closed
subgroup of $\Rb_+^*$ by the equality
$$
S(M) = \bigcap_{\varphi} \ {\rm Spec} (\Delta_{\varphi}) \subset \Rb_+
\leqno (21)
$$
where $\varphi$ varies among (faithful, normal) states on $M$, i.e.
linear forms $\varphi : M \rightarrow \Cb$ such that,
$$
\varphi (x^* x) \geq 0 \qquad \forall \, x \in M \, , \ \varphi (1) = 1
\, , \leqno (22)
$$
while the operator $\Delta_{\varphi}$ is the {\it modular operator}
(\cite{T})
$$
\Delta_{\varphi} = S_{\varphi}^* \, S_{\varphi} \, , \leqno (23)
$$
which is the {\it module} of the involution $x \rightarrow x^*$ in the
Hilbert space attached to the sesquilinear form,
$$
\langle x,y \rangle = \varphi (y^* x) \, , \ x,y \in M \, . \leqno (24)
$$
In the case of local fields the module was a group homomorphism $((2))$
from $K^*$ to $\Rb_+^*$. The counterpart for factors is the group
homomorphism, (\cite{Co_2})
$$
\delta : \Rb \rightarrow {\rm Out} (M) = {\rm Aut} (M) / {\rm Int} (M) \,
, \leqno (25)
$$
from the additive group $\Rb$ viewed as the dual of $\Rb_+^*$ for the
pairing,
$$
(\lambda , t) \rightarrow \lambda^{it} \qquad \forall \, \lambda \in
\Rb_+^* \, , \ t \in \Rb \, , \leqno (26)
$$
to the group of automorphism classes of $M$ modulo inner automorphisms.
The virtual subgroup,
$$
{\rm Mod} (M) \mathop{\subset}_{\sim} \ \Rb_+^* \, , \leqno (27)
$$
is the {\it flow of weights} (\cite{Ta} \cite{K} \cite{CT}) of $M$. It is obtained
from the module $\delta$ as the dual action of $\Rb_+^*$ on the abelian
algebra,
$$
C = \hbox{Center of} \ M \, {\textstyle \semi_{\delta}} \ \Rb \, , \leqno
(28)
$$
where $M \, \semi_{\delta} \ \Rb$ is the crossed product of $M$ by the
modular automorphism group $\delta$.
This takes care of (A), to describe (B) let us simply state the
equivalence (\cite{Co_1}) of the following conditions
$$
\matrix{
&M \ \hbox{is the closure of the union of an increasing sequence of} \cr
&\hbox{finite dimensional algebras.} \hfill \cr
} \leqno (29)
$$
$$
\matrix{
&M \ \hbox{is complemented as a subspace of the normed space of} \cr
&\hbox{all operators in a Hilbert space.} \hfill \cr
} \leqno (30)
$$
The condition (29) is obviously what one would expect for an
approximately finite dimensional algebra. Condition (30) is similar to
{\it amenability} for discrete groups and the implication (30)
$\Rightarrow$ (29) is a very powerful tool.
We refer to \cite{Co_1} \cite{K} \cite{Ha} for (C) and we just describe the
actual construction of the central simple algebra $M$ associated to a
given virtual subgroup,
$$
\Gamma \mathop{\subset}_{\sim} \Rb_+^* \, . \leqno (31)
$$
Among the approximately finite dimensional factors (central von Neumann algebras), only two are not
simple. The first is the algebra
$$
M_{\infty} (\Cb) \, , \leqno (32)
$$
of all operators in Hilbert space. The second factor is the unique approximately finite dimensional
factor of type II$_{\infty}$. It is
$$
R_{0,1} = R \otimes M_{\infty} (\Cb) \, , \leqno (33)
$$
where $R$ is the unique approximately finite dimensional factor with a finite trace $\tau_0$, i.e.
a state such that,
$$
\tau_0 (xy) = \tau_0 (yx) \qquad \forall \, x,y \in R \, . \leqno (34)
$$
The tensor product of $\tau_0$ by the standard semifinite trace on
$M_{\infty} (\Cb)$ yields a semi-finite trace $\tau$ on $R_{0,1}$. There
exists, up to conjugacy, a unique one parameter group of automorphisms
$\theta_{\lambda} \in {\rm Aut} (R_{0,1})$, $\lambda \in \Rb_+^*$ such
that,
$$
\tau (\theta_{\lambda} (a)) = \lambda \tau (a) \qquad \forall \, a \in
{\rm Domain} \, \tau \, , \ \lambda \in \Rb_+^* \, . \leqno (35)
$$
Let first $\Gamma \subset \Rb_+^*$ be an ordinary closed subgroup of
$\Rb_+^*$. Then the corresponding factor $R_{\Gamma}$ with modulo
$\Gamma$ is given by the equality:
$$
R_{\Gamma} = \{ x \in R_{0,1} \, ; \ \theta_{\lambda} (x) = x \quad
\forall \, \lambda \in \Gamma \} \, , \leqno (36)
$$
in perfect analogy with (9).
A virtual subgroup $\Gamma {\displaystyle \mathop{\subset}_{\sim}} \,
\Rb_+^*$ is by definition an ergodic action $ \alpha$ of $\Rb_+^*$ on an abelian
von Neumann algebra $A$, and the formula (36) easily extends to,
$$
R_{\Gamma} = \{ x \in R_{0,1} \otimes A \, ; \ (\theta_{\lambda} \otimes
\alpha_{\lambda}) \, x = x \quad \forall \, \lambda \in \Rb_+^* \} \, .
\leqno (37)
$$
(This reduces to (36) for the action of $\Rb_+^*$ on the algebra $A =
L^{\infty} (X)$ where $X$ is the homogeneous space $X = \Rb_+^* /
\Gamma$.)
The pair $(R_{0,1} , \theta_{\lambda})$ arises very naturally in geometry
from the geodesic flow of a compact Riemann surface (of genus $>1$). Let
$V = S^* \Sigma$ be the unit cosphere bundle of such a surface $\Sigma$,
and $F$ be the stable foliation of the geodesic flow. The latter defines
a one parameter group of automorphisms of the foliated manifold $(V,F)$
and thus a one parameter group of automorphisms $\theta_{\lambda}$ of the
von Neumann algebra $L^{\infty} (V,F)$.
This algebra is easy to describe, its elements are random operators $T =
(T_f)$, i.e. bounded measurable families of operators $T_f$ parametrized
by the leaves $f$ of the foliation. For each leaf $f$ the operator $T_f$ acts in
the Hilbert space $L^2 (f)$ of square integrable densities on the
manifold $f$. Two random operators are identified if they are equal for
almost all leaves $f$ (i.e. a set of leaves whose union in $V$ is
negligible). The algebraic operations of sum and product are given by,
$$
(T_1 + T_2)_f = (T_1)_f + (T_2)_f \, , \ (T_1 \, T_2)_f = (T_1)_f \, (T_2)_f
\, , \leqno (38)
$$
i.e. are effected pointwise.
One proves that,
$$
L^{\infty} (V,F) \simeq R_{0,1} \, , \leqno (39)
$$
and that the geodesic flow $\theta_{\lambda}$ satisfies (35). Indeed the
foliation $(V,F)$ admits up to scale a unique transverse measure
$\Lambda$ and the trace $\tau$ is given (cf. \cite{C}) by the formal
expression,
$$
\tau (T) = \int {\rm Trace} (T_f) \, d\Lambda (f) \, , \leqno (40)
$$
since the geodesic flow satisfies $\theta_{\lambda} (\Lambda) = \lambda
\Lambda$ are obtains (35) from simple geometric considerations. The
formula (37) shows that most approximately finite dimensional factors already arise from
foliations, for instance the unique approximately finite dimensional factor $R_{\infty}$ such that,
$$
{\rm Mod} (R_{\infty}) = \Rb_+^* \, , \leqno (41)
$$
arises from the codimension 1 foliation of $V = S^* \Sigma$ generated by
$F$ and the geodesic flow.
In fact this relation between the classification of central simple
algebras over $\Cb$ and the geometry of foliations goes much deeper. For
instance using cyclic cohomology together with the following simple fact,
$$
\matrix{
&\hbox{``A connected group can only act trivially on a homotopy} \cr
&\hbox{invariant cohomology theory'',} \hfill \cr
} \leqno (42)
$$
one proves (cf. \cite{C}) that for any codimension one foliation $F$ of
a compact manifold $V$ with non vanishing Godbillon-Vey class one has,
$$
{\rm Mod} (M) \ \hbox{has finite covolume in} \ \Rb_+^* \, , \leqno (43)
$$
where $M = L^{\infty} (V,F)$ and a virtual subgroup of finite covolume is
a flow with a finite invariant measure.
\section{Global class field theory and spontaneous symmetry breaking}
In the above discussion of approximately finite dimensional central
simple algebras, we have been working locally over $\Cb$. We shall now
describe a particularly interesting example (cf. \cite{B-C}) of Hecke algebra intimately
related to arithmetic, and defined over $\Qb$.
Let $\Gamma_0 \subset \Gamma$ be an almost normal subgroup of a discrete
group $\Gamma$, i.e. one assumes,
$$
\Gamma_0 \cap s \, \Gamma_0 \, s^{-1} \ \hbox{has finite index in} \
\Gamma_0 \qquad \forall \, s \in \Gamma \, . \leqno (1)
$$
Equivalently the orbits of the left action of $\Gamma_0$ on $\Gamma /
\Gamma_0$ are all finite. One defines the Hecke algebra,
$$
{\cal H} (\Gamma , \Gamma_0) \, , \leqno (2)
$$
as the convolution algebra of integer valued $\Gamma_0$ biinvariant
functions with finite support. For any field $k$ one lets,
$$
{\cal H}_k (\Gamma , \Gamma_0) = {\cal H} (\Gamma , \Gamma_0)
\otimes_{\Zb} k \, , \leqno (3)
$$
be obtained by extending the coefficient ring from $\Zb$ to $k$. We let
$\Gamma = P_{\Qb}^+$ be the group of $2 \times 2$ rational matrices,
$$
\Gamma = \left\{ \left[ \matrix{1 &b \cr 0 &a} \right] \, ; \ a \in \Qb^+
\, , \ b \in \Qb \right\} \, , \leqno (4)
$$
and $\Gamma_0 = P_{\Zb}^+$ be the subgroup of integral matrices,
$$
\Gamma_0 = \left\{ \left[ \matrix{1 &n \cr 0 &1} \right] \, ; \ n \in \Zb
\right\} \, . \leqno (5)
$$
One checks that $\Gamma_0$ is almost normal in $\Gamma$.
To obtain a central simple algebra over $\Cb$ in the sense of the
previous section we just take the commutant of the right regular
representation of $\Gamma$ on $\Gamma_0 \backslash \Gamma$, i.e. the weak
closure of ${\cal H}_{\Cb} (\Gamma , \Gamma_0)$ in the Hilbert space,
$$
\ell^2 (\Gamma_0 \backslash \Gamma) \, , \leqno (6)
$$
of $\Gamma_0$ left invariant function on $\Gamma$ with norm square,
$$
\Vert \xi \Vert^2 = \sum_{\gamma \, \in \, \Gamma_0 \backslash \Gamma} \vert \xi
(\gamma) \vert^2 \, . \leqno (7)
$$
This central simple algebra over $\Cb$ is approximately finite
dimensional and its module is $\Rb_+^*$ so that it is the same as
$R_{\infty}$ of (41).
In particular its modular automorphism group is highly non trivial and
one can compute it explicitly for the state $\varphi$ associated to the
vector $\xi_0 \in \ell^2 (\Gamma_0 \backslash \Gamma)$ corresponding to
the left coset $\Gamma_0$.
The modular automorphism group $\sigma_t^{\varphi}$ leaves the dense
subalgebra ${\cal H}_{\Cb}$ $(\Gamma , \Gamma_0) \subset R_{\infty}$
globally invariant and is given by the formula,
$$
\sigma_t^{\varphi} (f)(\gamma) = L (\gamma)^{-it} \, R (\gamma)^{it} \,
f(\gamma) \qquad \forall \, \gamma \in \Gamma_0 \backslash \Gamma /
\Gamma_0 \leqno (8)
$$
for any $f \in {\cal H}_{\Cb} (\Gamma , \Gamma_0)$. Here we let,
$$
\matrix{
&L(\gamma) = \ \hbox{Cardinality of the image of} \ \Gamma_0 \, \gamma \,
\Gamma_0 \ {\rm in} \ \Gamma / \Gamma_0 \hfill \cr
&R(\gamma) = \ \hbox{Cardinality of the image of} \ \Gamma_0 \, \gamma \,
\Gamma_0 \ {\rm in} \ \Gamma_0 \backslash \Gamma \, . \cr
} \leqno (9)
$$
This is enough to make contact with the formalism of quantum statistical
mechanics which we now briefly describe. As many of the mathematical
frameworks legated to us by physicists it is characterized ``not by this
short lived novelty which can too often only influence the mathematician
left to his own devices, but this infinitely fecund novelty which springs
from the nature of things'' (J.~Hadamard).
A quantum statistical system is given by,
\smallskip
\noindent 1) The $C^*$ algebra of observables $A$,
\smallskip
\noindent 2) The time evolution $(\sigma_t)_{t \in \Rb}$ which is a one
parameter group of automorphisms of $A$.
An equilibrium or KMS (for Kubo-Martin and Schwinger) state, at inverse
temperature $\beta$ is a state $\varphi$ on $A$ which fulfills the
following condition,
\medskip
\noindent (10) For any $x,y \in A$ there exists a bounded holomorphic
function (continuous on the closed strip), $F_{x,y} (z)$, $0 \leq {\rm
Im} \, z \leq \beta$ such that
$$
\matrix{
&F_{x,y} (t) = \varphi (x \, \sigma_t (y)) \qquad \hfill &\forall \, t
\in \Rb \hfill \cr
&F_{x,y} (t+i\beta) = \varphi (\sigma_t (y) x) \qquad &\forall \, t \in
\Rb \, . \cr
}
$$
For fixed $\beta$ the KMS$_{\beta}$ states form a Choquet simplex and
thus decompose uniquely as a statistical superposition from the pure
phases given by the extreme points. For interesting systems with
nontrivial interaction, one expects in general that for large temperature
$T$, (i.e. small $\beta$ since $\beta = \frac{1}{T}$ up to a conversion
factor) the disorder will be predominant so that there will exist only
one KMS$_{\beta}$ state. For low enough temperatures some order should
set in and allow for the coexistence of distinct thermodynamical phases
so that the simplex $K_{\beta}$ of KMS$_{\beta}$ states should be non
trivial. A given symmetry group $G$ of the system will necessarily act
trivially on $K_{\beta}$ for large $T$ since $K_{\beta}$ is a point, but
acts in general non trivially on $K_{\beta}$ for small $T$ so that it is
no longer a symmetry of a given pure phase. This phenomenon of {\it
spontaneous symmetry breaking} as well as the very particular properties
of the critical temperature $T_c$ at the boundary of the two regions are
corner stones of statistical mechanics.
In our case we just let $A$ be the $C^*$ algebra which is the {\it norm}
closure of ${\cal H}_{\Cb} (\Gamma , \Gamma_0)$ in the algebra of
operators in $\ell^2 (\Gamma_0 \backslash \Gamma)$. We let $\sigma_t \in
{\rm Aut} (A)$ be the unique extension of the automorphisms
$\sigma_t^{\varphi}$ of (8).
For $\beta = 1$ it is tautological that $\varphi$ is a KMS$_{\beta}$
state since we obtained $\sigma_t^{\varphi}$ precisely this way
(\cite{T}). One proves (\cite{B-C}) that for any $\beta \leq 1$ (i.e.
for $T=1$) there exists one and only one KMS$_{\beta}$ state.
The compact group $G$,
$$
G = C_{\Qb} / D_{\Qb} \, , \leqno (11)
$$
quotient of the Idele class group $C_{\Qb}$ by the connected component of
identity $D_{\Qb} \simeq \Rb_+^*$, acts in a very simple and natural
manner as symmetries of the system $(A,\sigma_t)$. (To see this one notes
that the right action of $\Gamma$ on $\Gamma_0 \backslash \Gamma$ extends
to the action of $P_{\cal A}$ on the restricted product of the trees of
$SL(2,\Qb_p)$ where ${\cal A}$ is the ring of finite Adeles (cf.
\cite{B-C}).
For $\beta > 1$ this symmetry group $G$ of our system, is spontaneously
broken, the compact convex sets $K_{\beta}$ are non trivial and have the same
structure as $K_{\infty}$, which we now describe. First some terminology, a
KMS$_{\beta}$ state for $\beta = \infty$ is called a {\it ground state} and the
KMS$_{\infty}$ condition is equivalent to {\it positivity of energy} in the
corresponding Hilbert space representation.
Remember that ${\cal H}_{\Cb} (\Gamma , \Gamma_0)$ contains ${\cal H}_{\Qb}
(\Gamma , \Gamma_0)$ so,
$$
{\cal H}_{\Qb} (\Gamma , \Gamma_0) \subset A \, . \leqno (12)
$$
By \cite{B-C} theorem 5 and proposition 24 one has,
\medskip
\noindent {\bf Theorem.} {\it Let ${\cal E} (K_{\infty})$ be the set of extremal
KMS$_{\infty}$ states.
\smallskip
\noindent {\rm a)} The group $G$ acts freely and transitively on ${\cal E}
(K_{\infty})$ by composition, $\varphi \rightarrow \varphi \circ g^{-1}$,
$\forall \, g \in G$.
\smallskip
\noindent {\rm b)} For any $\varphi \in {\cal E} (K_{\infty})$ one has,
$$
\varphi ({\cal H}_{\Qb}) = \Qb_{\rm ab} \, ,
$$
and for any element $\alpha \in {\rm Gal} (\Qb_{\rm ab} : \Qb)$ there exists a
unique extension of $\alpha \circ \varphi$, by continuity, as a state of $A$.
One has $\alpha \circ \varphi \in {\cal E} (K_{\infty})$.
\smallskip
\noindent {\rm c)} The map $\alpha \rightarrow (\alpha \circ
\varphi)\varphi^{-1} \in G = C_k / D_k$ defined for $\alpha \in {\rm Gal} (\Qb_{\rm ab} :
\Qb)$ is the isomorphism of global class field theory (I.19).}
\medskip
This last map is independent of the choice of $\varphi$. What is quite
remarkable in this result is that the existence of the subalgebra ${\cal
H}_{\Qb} \subset {\cal H}_{\Cb}$ allows to bring into action the Galois group of
$\Cb$ on the {\it values of states}. Since the Galois group of $\Cb : \Qb$ is
(except for $z \rightarrow \overline z$) formed of {\it discontinuous}
automorphisms it is quite surprising that its action can actually be compatible
with the characteristic {\it positivity} of states. It is by no means clear how
to extend the above construction to arbitrary number fields $k$ while preserving
the three results of the theorem. There is however an easy computation which relates
the above construction to an object which makes sense for any global field $k$.
Indeed if we let as above $R_{\infty}$ be the weak closure of ${\cal H}_{\Cb}
(\Gamma , \Gamma_0)$ in $\ell^2 (\Gamma_0 \backslash \Gamma)$, we can compute
the associated pair $(R_{0,1} , \theta_{\lambda})$ of section I.
The $C^*$ algebra closure of ${\cal H}_{\Cb}$ is Morita equivalent (cf. M. Laca) to
the crossed product $C^*$ algebra,
$$
C_0 ({\cal A}) \semi \, \Qb_+^* \, , \leqno (13)
$$
where ${\cal A}$ is the locally compact space of finite Adeles. It follows
immediately that,
$$
R_{0,1} = L^{\infty} (\Qb_A) \semi \, \Qb^* \, , \leqno (14)
$$
i.e. the von Neumann algebra crossed product of the $L^{\infty}$ functions on
Adeles of $\Qb$ by the action of $\Qb^*$ by multiplication.
The one parameter group of automorphisms, $\theta_{\lambda} \in {\rm Aut}
(R_{0,1})$, is obtained as the restriction to,
$$
D_{\Qb} = \Rb_+^* \, , \leqno (15)
$$
of the obvious action of the Idele class group $C_{\Qb}$,
$$
(g,x) \rightarrow g \, x \qquad \forall \, g \in C_{\Qb} \, , \ x \in A_{\Qb} /
{\Qb^*} \, , \leqno (16)
$$
on the space $X = A_{\Qb }/ {\Qb^*}$ of Adele classes.
Our next goal will be to show that the latter space is intimately related to the
{\it zeros} of the Hecke $L$-functions with Grossencharakter.
(We showed in \cite{B-C} that the partition function of the above system is the
Riemann zeta function.)
\section{Weil positivity and the Trace formula}
Global fields $k$ provide a natural context for the Riemann Hypothesis on the
zeros of the zeta function and its generalization to Hecke $L$-functions. When
the characteristic of $k$ is non zero this conjecture was proved by A.~Weil.
His proof relies on the following dictionary (put in modern language) which
provides a geometric meaning, in terms of algebraic geometry over finite fields,
to the function theoretic properties of the zeta functions. Recall that $k$ is a
function field over a curve $\Sigma$ defined over $\Fb_q$,
$$
\matrix{
\hbox{Algebraic Geometry} &\qquad &\hbox{Function Theory} \cr
\cr
\hbox{Eigenvalues of action of} &\qquad &\hbox{Zeros of} \ {\zeta} \cr
\hbox{Frobenius on} \ H_{\rm et}^1 (\overline \Sigma , \Qb_{\ell}) &\qquad & \cr
\cr
\hbox{Poincar\'{e} duality in} &\qquad &\hbox{Functional equation} \cr
\ell\hbox{-adic cohomology} &\qquad & \cr
\cr
\hbox{Lefschetz formula for} &\qquad &\hbox{Explicit formulas} \cr
\hbox{the Frobenius} &\qquad & \cr
\cr
\hbox{Castelnuovo positivity} &\qquad &\hbox{Riemann Hypothesis}
}
$$
We shall describe a third column in this dictionary, which will make sense for
any global field. It is based on the geometry of the Adele class space,
$$
X = A/k^* \, , \ A = \hbox{Adeles of} \ k \, . \leqno (1)
$$
This space is of the same nature as the space of leaves of the horocycle
foliation (section I) and the same geometry will be used to analyse it.
Our spectral interpretation of the zeros of zeta involves Hilbert space. The
reasons why Hilbert space (apparently invented by Hilbert for this purpose)
should be involved are manifold, let us mention three,
\smallskip
\noindent (A) Let $N(E)$ be the number of zeros of the Riemann zeta function
satisfying $0 < {\rm Im} \rho < E$, then (\cite{R})
$$
N(E) = \langle N(E) \rangle + N_{\rm osc} (E) \, , \leqno (2)
$$
where the smooth function $\langle N(E) \rangle$ is given by
$$
\langle N(E) \rangle = \frac{E}{2\pi} \, \left( \log \, \frac{E}{2\pi} - 1
\right) + \frac{7}{8} + o(1) \, , \leqno (3)
$$
while the oscillatory part is
$$
N_{\rm osc} (E) = \frac{1}{\pi} \ {\rm Im} \log {\zeta} \left( \frac{1}{2} + i
\, E \right) \, . \leqno (4)
$$
The numbers $x_j = \langle N (\rho_j) \rangle$ where $\rho_n$ is the imaginary
part of the $n^{\rm th}$ zero are of average density one and behave like the
eigenvalues of a random Hermitian matrix. This was discovered by H.~Montgomery \cite{M}
who conjectured (and proved for suitable test functions) that when $M
\rightarrow \infty$, with $\alpha , \beta > 0$,
$$
\# \, \{ (i,j) \in \{ 1, \ldots ,M \}^2 \, ; \ x_i - x_j \in [\alpha , \beta] \}
\sim M \int_{\alpha}^{\beta} 1 - \left( \frac{\sin \pi u}{\pi u} \right)^2 \, du
\leqno (5)
$$
which is exactly what happens in the Gaussian Unitary Ensemble. Numerical tests
by A. Odlyzko \cite{O} and further theoretical work by Katz-Sarnak \cite{KS} and J.~Keating give
overwhelming evidence that zeros of zeta should be the eigenvalues of a hermitian
matrix.
\smallskip
\noindent (B) The equivalence between $RH$ and the positivity of the Weil
distribution on the Idele class group $C_k$ shows that Hilbert space is
implicitly present.
\smallskip
\noindent (C) The deep arithmetic significance of the work of A.~Selberg on the
spectral analysis of the Laplacian on $L^2 (G/\Gamma)$ where $\Gamma$ is an
arithmetic subgroup of a semi simple Lie group $G$.
Direct atempts (cf. \cite{B}) to construct the Polya-Hilbert space giving a
spectral realization of the zeros of $\zeta$ using quantum mechanics, meet the
following $-$ sign problem: Let $H$ be the Hamiltonian of the quantum mechanical
system obtained by quantizing the classical system,
$$
(X, F_t) \leqno (6)
$$
where $X$ is phase space and $t \in \Rb \rightarrow F_t$ the Hamiltonian flow.
Let $N(E)$ be the number of eigenvalues $\lambda$ of $H$ such that $0 \leq
\lambda \leq E$. Then, as for ${\zeta}$,
$$
N(E) = \langle N(E) \rangle + N_{\rm osc} (E) \, , \leqno (7)
$$
where $\langle N(E) \rangle$ is essentially a volume in phase space, while the
oscillatory part admits a heuristic asymptotic expansion (cf. \cite{B}) of the
form,
$$
N_{\rm osc} (E) \sim \frac{1}{\pi} \ \sum_{\gamma} \sum_{m \, = \, 1}^{\infty} \
\frac{1}{m} \ \frac{1}{2 sh \left( \frac{m \lambda_{\gamma}}{2} \, \right)} \ \sin
(T_{\gamma}^{\#} \, m \, E) \leqno (8)
$$
where the $\gamma$ are the periodic orbits of the flow $F$, the
$T_{\gamma}^{\#}$
are their periods and the $\lambda_{\gamma}$ the unstability exponents of these
orbits.
One can compare (\cite{B}) (8) with the equally heuristic asymptotic expansion
of
(4) using the Euler product of ${\zeta}$ which gives, using $-\log (1-x) =
{\displaystyle \sum_{m=1}^{\infty} \ \frac{x^m}{m}}$,
$$
N_{\rm osc} (E) \simeq -\frac{1}{\pi} \ \sum_{p} \sum_{m=1}^{\infty} \
\frac{1}{m}
\ \frac{1}{p^{m/2}} \ \sin ((\log p) \, m \, E) \, . \leqno (9)
$$
Comparing (8) and (9) one gets precious information on the hypothetical
``Riemann
flow'' of M.~Berry. The periodic orbits $\gamma$ should be labelled by the
primes $p$, the periods should be the $\log p$ as well as the unstability
exponents $\lambda_p$. Also, in order to avoid duplication of orbits, the flow
should not be ``time reversal symmetric'', i.e. non isomorphic to the time
reversed:
$$
(X,F_{-t}) \, . \leqno (10)
$$
There is however a fundamental mismatch between (8) and (9) which is the overall
$-$ sign in front of (9) and no adjustment of Maslov phases can account for it.
The very same $-$ sign appears in the Riemann-Weil explicit formula,
$$
\sum_{L(\chi,\rho) \, = \, 0} \widehat h (\chi , \rho) - \widehat h (0) -
\widehat h (1) = - \sum_v \int'_{k_v^*} \ \frac{h(u^{-1})}{\vert 1-u \vert} \
d^*
u \, , \leqno (11)
$$
where $h$ is a test function on the Idele class group $C_k$, $\widehat h$ is its
Fourier transform,
$$
\widehat h (x,z) = \int_{C_k} h(u) \, \chi (u) \, \vert u \vert^z \, d^* u \, ,
\leqno (12)
$$
and the finite values $\int'$ are suitably normalized. If we use the above
dictionary when ${\rm char} (k) \not= 0$, the geometric origin of this $-$ sign
becomes clear, the formula (11) is the Lefschetz formula,
$$
\# \ \hbox{of fixed points of} \ \varphi = {\rm Trace} \, \varphi / H^0 - {\rm
Trace} \, \varphi / H^1 + {\rm Trace} \, \varphi / H^2 \leqno (13)
$$
in which the space $H_{\rm et}^1 (\overline \Sigma , \Qb_{\ell})$ which provides
the spectral realization of the zeros appears with a $-$ sign. This indicates
that the spectral realization of zeros of zeta should be of cohomological nature
or to be more specific, that the Polya-Hilbert space should appear as the last
term of an exact sequence of Hilbert spaces,
$$
0 \rightarrow {\cal H}_0 \stackrel{T}{\rightarrow} {\cal H}_1 \rightarrow {\cal
H} \rightarrow 0 \, . \leqno (14)
$$
The example we have in mind for (14) is the assembled Euler complex for a
Riemann
surface, where ${\cal H}_0$ is the {\it codimension $2$ subspace} of
differential
forms of even degree orthogonal to harmonic forms, where ${\cal H}_1$ is the
space of 1-forms and where $T = d+d^*$ is the sum of the de Rham coboundary with
its adjoint $d^*$.
Since we want to obtain the spectral interpretation not only for zeta functions
but for all $L$-functions with Gr\"ossencharakter we do not expect to have only
an action of $\Zb$ for ${\rm char} (k) > 0$ corresponding to the Frobenius, or
of
the group $\Rb_+^*$ if ${\rm char} (k) = 0$, but to have the equivariance of
(14)
with respect to a natural action of the Idele class group $C_k = GL_1 (A) /
k^*$.
Let $X = A/k^*$ be the Adele class space. Our basic idea is to take for ${\cal
H}_0$ a suitable completion of the codimension 2 subspace of functions on $X$
such that,
$$
f(0) = 0 \, , \ \int f \, dx = 0 \, , \leqno (15)
$$
while ${\cal H}_1 = L^2 (C_k)$ and $T$ is the restriction map coming from the
inclusion $C_k \rightarrow X$, multiplied by $\vert a \vert^{1/2}$,
$$
(Tf) (a) = \vert a \vert^{1/2} \, f(a) \, . \leqno (16)
$$
The action of $C_k$ is then the obvious one, for ${\cal H}_0$
$$
(U(g)f) (x) = f (g^{-1} \, x) \qquad \forall \, g \in C_k \leqno (17)
$$
using the action II.15 of $C_k$ on $X$, and similarly the regular representation
$V$ for ${\cal H}_1$.
This idea works but there are two subtle points; first since $X$ is a delicate
quotient space the function spaces for $X$ are naturally obtained by starting
with function spaces on $A$ and moding out by the ``gauge transformations''
$$
f \rightarrow f_q \, , \ f_q (x) = f(xq) \, , \quad \forall \, q \in k^* \, .
\leqno (18)
$$
Here the natural function space is the Bruhat-Schwarz space ${\cal S} (A)$ and
by
(15) the codimension 2 subspace,
$$
{\cal S} (A)_0 = \left\{ f \in {\cal S} (A) \, ; \ f(0) = 0 \, , \ \int f \, dx
=
0 \right\} \, . \leqno (19)
$$
The restriction map $T$ is then given by,
$$
T(f) (a) = \vert a \vert^{1/2} \ \sum_{q \, \in \, k^*} \ f(aq) \qquad \forall
\,
a \in C_k \, . \leqno (20)
$$
The corresponding function $T(f)$ belongs to ${\cal S} (C_k)$ and all functions
$f-f_q$ are in the kernel of $T$.
The second subtle point is that since $C_k$ is abelian and non compact, its
regular representation does not contain any finite dimensional subrepresentation
so that the Polya-Hilbert space cannot be a subrepresentation (or unitary
quotient) of $V$. There is an easy way out (which we shall improve shortly)
which
is to replace $L^2 (C_k)$ by $L_{\delta}^2 (C_k)$ using the polynomial weight
$(\log^2 \vert a \vert)^{\delta /2}$, i.e. the norm,
$$
\Vert \xi \Vert_{\delta}^2 = \int_{C_k} \vert \xi (a)\vert^2 \, (1+\log^2 \vert
a
\vert)^{\delta /2} \, d^* a \, . \leqno (21)
$$
Let ${\rm char} (k) = 0$ so that ${\rm Mod} \, k = \Rb_+^*$ and $C_k = K \times
\Rb_+^*$ where $K$ is the compact group $C_{k,1} = \{ a \in C_k \, ; \ \vert a
\vert = 1\}$.
\medskip
\noindent {\bf Theorem.} {\it Let $\delta > 1$, ${\cal H}$ be the cokernel of
$T$
in $L_{\delta} (C_k)$ and $W$ the quotient representation of $C_k$. Let $\chi$ be
a
character of $K$, $\widetilde{\chi} = \chi \times 1$ the corresponding character
of $C_k$. Let ${\cal H}_{\chi} = \{ \xi \in {\cal H} \, ; \ W(g) \, \xi = \chi
(g) \, \xi \quad \forall \, g \in K \}$ and $D_{\chi} = \ \build\lim_{\epsilon
\rightarrow 0}^{} \, \frac{1}{\epsilon} \ (W (e^{\epsilon}) - 1)$. Then
$D_{\chi}$ is an unbounded closed operator with discrete spectrum, ${\rm Sp} \,
D_{\chi} \subset i \, \Rb$ is the set of imaginary parts of zeros of the $L$
function with Gr\"ossencharakter $\widetilde{\chi}$ which have real part $1/2$.
Moreover the spectral multiplicity of $\rho$ is the largest integer $n <
\frac{1+\delta}{2}$ in $\{ 1,\ldots , \hbox{multiplicity as a zero of} \ L \}$.}
\medskip
A similar result holds for ${\rm char} (k) > 0$. This allows to compute the
character of the representation $W$ as,
$$
{\rm Trace} (W(h)) = \sum_{{L \left( \chi , \frac{1}{2} + \rho \right) \, = \, 0
\atop \rho \, \in \, i \, \Rb / N^{\perp}}} \widehat h (\chi , \rho) \leqno (22)
$$
where $N = {\rm Mod} (k)$, $W(h) = \int W(g) \, h(g) \, d^* g$, $h \in {\cal S}
(C_k)$, $\widehat h$ is defined in (12) and the multiplicity is counted as in the
theorem.
This result is only preliminary because of the unwanted parameter $\delta$ which
artificially restricts the multiplicities. The restriction ${\rm Re} \, \rho =
\frac{1}{2}$ involves the same $\frac{1}{2}$ as in (16), and this has a natural
meaning. Indeed the natural Hilbert space norm for $L^2 (X)$, namely $\Vert \xi
\Vert^2 = \int_X \vert \xi (x) \vert^2 \, dx$ is naturally given upstairs on
${\cal S} (A)_0$ by:
$$
\Vert f \Vert^2 = \int_D \vert \Sigma \, f(xq)\vert^2 \, \vert x \vert \, d^* x
\, , \ \forall \, f \in {\cal S} (A)_0 \, , \leqno (23)
$$
where $D$ is a fundamental domain for $k^*$ acting on Ideles. For a local field
one has indeed the equality
$$
dx = \vert x \vert \, d^* x \, , \leqno (24)
$$
(up to normalization) between the additive Haar measure and the multiplicative
one. In the global case one has,
$$
dx = \ \build\lim_{\epsilon \rightarrow 0}^{} \ \epsilon \, \vert x
\vert^{1+\epsilon} \, d^* x \, , \leqno (25)
$$
and (23) ignores the divergent normalization constant which plays no role in the
computation of traces or of adjoint operators. The exponent $\frac{1}{2}$ in
(16)
turns $T$ into an isometry,
$$
T : L^2 (X)_0 \rightarrow L^2 (C_k) \, . \leqno (26)
$$
The analogue of the Hodge $*$ operation is given on ${\cal H}_0$ by the Fourier
transform,
$$
(Ff)(x) = \int_A f(y) \, \alpha (xy) \, dy \qquad \forall \, f \in {\cal S}
(A)_0
\leqno (27)
$$
which, because we take the quotient by (18), is independent of the choice of
additive character $\alpha$ of $A$ such that $\alpha \not= 1$ and $\alpha (q) =
1
\quad \forall \, q \in k$. Note also that $F^2 = 1$ on the quotient. On ${\cal
H}_1$ the Hodge $*$ is given by,
$$
(* \, \xi) (a) = \xi (a^{-1}) \qquad \forall \, a \in C_k \, . \leqno (28)
$$
The Poisson formula means exactly that $T$ commutes with the $*$ operation. This
is just a reformulation of the work of Tate and Iwasawa on the proof of the
functional equation, but we shall now see that if we follow the proof by
Atiyah-Bott (\cite{AB}) of the Lefschetz formula we do obtain a clear geometric
meaning for the Weil distribution. One can of course as in \cite{G} define inner
products on function spaces on $C_k$ using the Weil distribution, but as long as
the latter is put by hands and does not appear naturally one has very little chance to
understand why it should be positive. Now, let $\varphi$ be a diffeomorphism of
a smooth manifold $\Sigma$ and assume that the graph of $\varphi$ is transverse to
the diagonal, one can then easily define and compute (cf. \cite{AB}) the
distribution theoretic trace of the permutation $U$ of functions on $\Sigma$
associated to $\varphi$,
$$
(U \xi) (x) = \xi (\varphi (x)) \qquad \forall \, x \in \Sigma \, . \leqno (29)
$$
One has ``Trace'' $(U) = \int k(x,x) \, dx$, where $k(x,y) \, dy$ is the Schwarz
kernel associated to $U$, i.e. the distribution on $\Sigma \times \Sigma
$ such that,
$$
(U \xi) (x) = \int k (x,y) \, \xi (y) \, dy \, . \leqno (30)
$$
Now near the diagonal and in local coordinates one has,
$$
k(x,y) = \delta (y-\varphi (x)) \, , \leqno (31)
$$
where $\delta$ is the Dirac distribution. One then obtains,
$$
\hbox{``Trace''} \ (U) = \sum_{\varphi (x) \, = \, x} \ \frac{1}{\vert 1 -
\varphi' (x) \vert} \, , \leqno (32)
$$
where $\varphi'$ is the Jacobian of $\varphi$ and $\vert \ \vert$ stands for
the
absolute value of the determinant.
With more work (\cite{Gu}) one obtains a similar formula for the distributional
trace of the action of a flow,
$$
(U_t \, \xi) (x) = \xi (F_t (x)) \qquad \forall \, x \in \Sigma \, , \ t \in \Rb
\, . \leqno (33)
$$
It is given, under suitable transversality hypothesis, by
$$
\hbox{``Trace''} \ (U (h)) = \sum_{\gamma} \int_{I_{\gamma}} \ \frac{h(u)}{\vert
1 - (F_u)_* \vert} \ d^* u \, , \leqno (34)
$$
where $U(h) = \int h(t) \, U(t) \, dt$, $h$ is a test function on $\Rb$, the
$\gamma$ labels the periodic orbits of the flow, including the fixed points,
$I_{\gamma}$ is the corresponding isotropy subgroup, and $(F_u)_*$ is the
tangent
map to $F_u$ on the transverse space to the orbits, and finally $d^* u$ is the
unique Haar measure on $I_{\gamma}$ which is of covolume 1 in $(\Rb , dt)$.
Now it is truly remarkable that when one analyzes the periodic orbits of the
action of $C_k$ on $X$ one finds that not only it qualifies as a Riemann flow in
the above sense, but that (34) becomes,
$$
\hbox{``Trace''} \ (U (h)) = \sum_v \int_{k_v^*} \ \frac{h(u^{-1})}{\vert 1 - u
\vert} \ d^* u \, . \leqno (35)
$$
Thus, the isotropy subgroups $I_{\gamma}$ are parametrized by the places $v$ of
$k$ and coincide with the natural cocompact inclusion $k_v^* \subset C_k$ which
relates local to global in class field theory. The denominator $\vert 1-u \vert$
is for the module of the local field $k_v$ and the $u^{-1}$ in $h(u^{-1})$ comes
from the discrepancy between notations (16) and (28). It turns out that if one normalizes
the Haar measure $d^* u$ of modulated groups as in Weil \cite{W3}, by,
$$
\int_{1 \leq \vert u \vert \leq \Lambda} d^* u \sim \log \Lambda \qquad
\hbox{for} \ \Lambda \rightarrow \infty \, , \leqno (36)
$$
one gets the same covolume 1 condition as in (34).
The transversality condition imposes the condition $h(1) = 0$. The
distributional
trace for the action of $C_k$ on $C_k$ by translations vanishes under the
condition $h(1) = 0$.
Remembering that ${\cal H}_0$ is the codimension 2 subspace of $L^2 (X)$
determined by the condition (15) and computing the characters of the
corresponding 1-dimensional representations gives,
$$
h \rightarrow \widehat h (0) + \widehat h (1) \, . \leqno (37)
$$
Thus equating the alternate sum of traces on ${\cal H}_0$, ${\cal H}_1$ with the
trace on the cohomology should thus provide the geometric understanding of the
Riemann-Weil explicit formula (11) and in fact of RH using (21) if it could be
justified for some value of $\delta$.
The trace of permutation matrices is positive and this explains the Hadamard
positivity,
$$
\hbox{``Trace''} \ (U (h)) \geq 0 \qquad \forall \, h \, , \ h(1) = 0 \, , \
h(u)
\geq 0 \quad \forall \, u \in C \leqno (38)
$$
(not to be confused with Weil postivity).
To eliminate the artificial parameter $\delta$ and give rigorous meaning, as a
Hilbert space trace, to the distribution ``trace'', one proceeds as in the
Selberg trace formula \cite{Se} and introduces a cutoff. In physics terminology the
divergence of the trace is both infrared and ultraviolet as is seen in the
simplest case of the action of $K^*$ on $L^2 (K)$ for a local field $K$. In this
local case one lets,
$$
R_{\Lambda} = \widehat{P}_{\Lambda} \, P_{\Lambda} \, , \ \Lambda \in \Rb_+ \, ,
\leqno (39)
$$
where $P_{\Lambda}$ is the orthogonal projection on the subspace,
$$
\{ \xi \in L^2 (K) \, ; \ \xi (x) = 0 \qquad \forall \, x , \vert x \vert >
\Lambda \} \, , \leqno (40)
$$
while $\widehat{P}_{\Lambda} = F \, P_{\Lambda} \, F^{-1}$, $F$ the Fourier
transform.
One proves (\cite{Co_4}) in this local case the following analogue of the Selberg
trace formula,
$$
\hbox{Trace} \, (R_{\Lambda} \, U(h)) = 2 \, h(1) \log' (\Lambda) + \int'
\frac{h(u^{-1})}{\vert 1-u \vert} \, d^* u + o(1) \leqno (41)
$$
where $h \in {\cal S} (K^*)$ has compact support, $2\log' (\Lambda) =
\int_{\lambda \, \in \, K^* , \, \vert \lambda \vert \, \in \, [\Lambda^{-1} ,
\Lambda]} d^* \lambda$, and the principal value $\int'$ is uniquely determined
by
the pairing with the unique distribution on $K$ which agrees with
$\frac{du}{\vert 1-u \vert}$ for $u \not= 1$ and whose Fourier transform
vanishes
at 1.
As it turns out this principal value agrees with that of Weil for the choice of
$F$ associated to the standard character of $K$.
Let $k$ be a global field and let first $S$ be a finite set of places of $k$
containing all the infinite places. To $S$ corresponds the following localized
version of the action of $C_k$ on $X$. One replaces $C_k$ by
$$
C_S = \prod_{v \in S} k_v^* / O_S^* \, , \leqno (42)
$$
where $O_S^* \subset k^*$ is the group of $S$-units. One replaces $X$ by
$$
X_S = \prod_{v \in S} k_v / O_S^* \, . \leqno (43)
$$
The Hilbert space $L^2 (X_S)$, its Fourier transform $F$ and the orthogonal
projection $P_{\Lambda}$, $\widehat{P}_{\Lambda} = F \, P_{\Lambda} \, F^{-1}$
continue to make sense, with
$$
{\rm Im} \, P_{\Lambda} = \{ \xi \in L^2 (X_S) \, ; \ \xi (x) = 0 \quad \forall
\, x \, , \ \vert x \vert > \Lambda \} \, . \leqno (44)
$$
As soon as $S$ contains more than $3$ elements, (e.g. $\{ 2,3,\infty \}$ for
$k=\Qb$) the space $X_S$ is an extremely delicate quotient space. It is thus
quite remarkable that the {\it trace formula} holds,
\medskip
\noindent {\bf Theorem.} {\it For any $h \in {\cal S}_c (C_S)$ one has, with
$R_{\Lambda} = \widehat{P}_{\Lambda} \, P_{\Lambda}$,
$$
\hbox{Trace} \, (R_{\Lambda} \, U(h)) = 2 \log' (\Lambda) \, h(1) + \sum_{v\in
S} \ \int'_{k_v^*} \ \frac{h(u^{-1})}{\vert 1-u \vert} \, d^* u + o(1)
$$
}
where the notations are as above and the finite values $\int'$ depend on the
additive character of $\Pi k_v$ defining the Fourier transform $F$. When ${\rm
Char} \, (k) = 0$ the projectors $P_{\Lambda}$, $\widehat{P}_{\Lambda}$ commute
on $L_{\chi}^2$ for $\Lambda$ large enough so that one can replace $R_{\Lambda}$ by
the orthogonal projection $Q_{\Lambda}$ on ${\rm Im} \, P_{\Lambda} \cap {\rm
Im} \, \widehat{P}_{\Lambda}$. The situation for ${\rm Char} \, (k) = 0$ is more
delicate since $P_{\Lambda}$ and $\widehat{P}_{\Lambda}$ do not commute (for
$\Lambda$ large) even in the local Archimedian case. But fortunately \cite{LPS} these
operators commute with a specific second order differential operator, whose
eigenfunctions, the Prolate Spheroidal Wave functions provide the right
filtration $Q_{\Lambda}$. This allows to replace $R_{\Lambda}$ by $Q_{\Lambda}$
and to state the global trace formula
$$
\hbox{Trace} \, (Q_{\Lambda} \, U(h)) = 2 \log' (\Lambda) \, h(1) + \sum_v
\int_{k_v^*} \frac{h(u^{-1})}{\vert 1-u \vert} \, d^* u + o(1) \, . \leqno (45)
$$
Our final result is that the validity of this trace formula implies (in fact is
equivalent to) the positivity of the Weil distribution, i.e. $RH$ for all
$L$-functions with Gr{\"o}ssencharakter. Moreover the filtration by
$Q_{\Lambda}$ allows to define the Adelic cohomology and to complete the
dictionary between the function theory and the geometry of the Adele class
space.
$$
\matrix{
\hbox{ Function Theory} &\qquad &\hbox{ Geometry} \cr
\cr
\hbox{Zeros and poles of Zeta} &\qquad &\hbox{Eigenvalues of action of $C_k$ } \cr
&\qquad &\hbox{on Adelic cohomology} \cr
\cr
\hbox{Functional Equation} &\qquad &* \ \hbox{operation} \cr
\cr
\hbox{Explicit formula} &\qquad &\hbox{Lefschetz formula} \cr
\cr
RH &\qquad &\hbox{Trace formula}
}
$$
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\end{document}