NONCOMMUTATIVE GEOMETRY AND THE RIEMANN ZETA FUNCTION

Alain Connes

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 André 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 Grössencharakter.

1 Local class field theory and the classification of factors

Let K be a local field, i.e. a nondiscrete locally compact field. The action of K^{∗} = GL_{1}(K) on the additive group K by multiplication,

(1)  (λ, x) → λx  ∀ λ ∈ K^{∗} , x ∈ K ,

together with the uniqueness, up to scale, of the Haar measure of the additive group K, yield a homomorphism,

(2)  a ∈ K^{∗} → |a| ∈ ℝ_{+}^{∗} ,

from K^{∗} to ℝ_{+}^{∗}, called the module of K. Its range

(3)  Mod(K) = {|λ| ∈ ℝ_{+}^{∗}; λ ∈ K^{∗}}

is a closed subgroup of ℝ_{+}^{∗}.

The fields ℝ, ℂ and ℍ (of quaternions) are the only ones with Mod(K) = ℝ_{+}^{∗}, they are called Archimedian local fields.

Let K be a non Archimedian local field, then

(4)  R = {x ∈ K; |x| ≤ 1} ,

is the unique maximal compact subring of K and the quotient R/P of R by its unique maximal ideal is a finite field 𝔽_{q} (with q = p^{ℓ} a prime power). One has,

(5)  Mod(K) = q^{ℤ} ⊂ ℝ_{+}^{∗} .

Let K be commutative. An extension K ⊂ K′ of finite degree of K is called unramified iff the dimension of K′ over K is the order of Mod(K′) as a subgroup of 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 θ ∈ Aut_{K}(K′) such that,

(6)  θ(μ) = μ^{q} ,

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,

(7)  Γ ⊂ Mod(K) , Γ ≠ {1} .

Let then \overline{K} be an algebraic closure of K, K_{sep} ⊂ \overline{K} the separable algebraic closure, K_{ab} ⊂ K_{sep} the maximal abelian extension of K and K_{un} ⊂ K_{ab} the maximal unramified extension of K, i.e. the union of all unramified extensions of finite degree. One has,

(8)  K ⊂ K_{un} ⊂ K_{ab} ⊂ K_{sep} ⊂ \overline{K} ,

and the Galois group Gal(K_{un} : K) is topologically generated by θ called the Frobenius automorphism.

The correspondence (7) is given by,

(9)  K′ = {x ∈ K_{un} ; θ_{λ}(x) = x ∀ λ ∈ Γ} ,

with rather obvious notations so that θ_{q} is the θ of (6). Let then W_{K} be the subgroup of Gal(K_{ab} : K) whose elements induce on K_{un} an integral power of the Frobenius automorphism. One endows W_{K} with the locally compact topology dictated by the exact sequence of groups,

(10)  1 → Gal(K_{ab} : K_{un}) → W_{K} → Mod(K) → 1 ,

and the main result of local class field theory asserts the existence of a canonical isomorphism,

(11)  W_{K} →^{∼} K^{∗} ,

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,

(12)  A = M_{n}(D) ,

where D is a (central) division algebra over K and the symbol M_{n} stands for n × 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,

(13)  Br(K) →^{η} ℚ/ℤ ,

of the Brauer group of classes of central simple algebras over K (with tensor product as the group law), with the group ℚ/ℤ of roots of 1 in ℂ.

All the above discussion was under the assumption that K is non Archimedian. For Archimedian fields ℝ and ℂ the same questions have an idiotically simple answer. Since ℂ is algebraically closed one has K = \overline{K} and the whole picture collapses. For K = ℝ the only non trivial value of the Hasse invariant η is

(14)  η(ℍ) = −1 .

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 ∈ K^{∗}.

Let k be a global field, i.e. a discrete cocompact subfield of a (non discrete) locally compact semi-simple commutative ring A. (Cf. Iwasawa Ann. of Math. 57 (1953).) The topological ring A is canonically associated to k and called the Adele ring of k, one has,

(15)  A = ∏_{res} k_{v} ,

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 𝔽_{q}, one has

(16)  k ⊂ k_{un} ⊂ k_{ab} ⊂ k_{sep} ⊂ \overline{k} ,

where, as above \overline{k} is an algebraic closure of k, k_{sep} the separable algebraic closure, k_{ab} the maximal abelian extension and k_{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 Gal(k_{ab} : k) of those automorphisms which induce on k_{un} an integral power of θ,

(17)  θ(μ) = μ^{q}  ∀ μ root of 1 of order prime to p .

The main theorem of global class field theory asserts the existence of a canonical isomorphism,

(18)  W_{k} ≃ C_{k} = GL_{1}(A)/GL_{1}(k) ,

of locally compact groups.

When k is of characteristic 0, i.e. is a number field, one has a canonical isomorphism,

(19)  Gal(k_{ab} : k) ≃ C_{k}/D_{k} ,

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 [28], “La recherche d’une interprétation pour C_{k} si k est un corps de nombres, analogue en quelque manière à l’interprétation par un groupe de Galois quand k est un corps de fonctions, me semble constituer l’un des problèmes fondamentaux de la théorie des nombres à l’heure actuelle ; il se peut qu’une telle interprétation renferme la clef de l’hypothèse de Riemann …”.

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, “Is there a non trivial Brauer theory of central simple algebras over ℂ.” As we shall see shortly the approximately finite dimensional simple central algebras over ℂ provide a satisfactory answer to this question. They are classified by their module,

(20)  Mod(M) ⊂_{∼} ℝ_{+}^{∗} ,

which is a virtual closed subgroup of ℝ_{+}^{∗}.

Let us now explain this statement with more care. First we exclude the trivial case M = M_{n}(ℂ) of matrix algebras. Next Mod(M) is a virtual subgroup of ℝ_{+}^{∗}, in the sense of G. Mackey, i.e. an ergodic action of ℝ_{+}^{∗}. All ergodic flows appear and M_{1} is isomorphic to M_{2} iff Mod(M_{1}) ≅ Mod(M_{2}).

The birth place of central simple algebras is as the commutant of isotypic representations. When one works over ℂ 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,

(A) The definition of the invariant Mod(M) for arbitrary factors (central von Neumann algebras).

(B) The equivalence of all possible notions of approximate finite dimensionality.

(C) The proof that Mod is a complete invariant and that all virtual subgroups are obtained.

The module of a factor M was first defined ([6]) as a closed subgroup of ℝ_{+}^{∗} by the equality

(21)  S(M) = ⋂_{φ} Spec(Δ_{φ}) ⊂ ℝ_{+}

where φ varies among (faithful, normal) states on M, i.e. linear forms φ : M → ℂ such that,

(22)  φ(x^{∗}x) ≥ 0  ∀ x ∈ M , φ(1) = 1 ,

while the operator Δ_{φ} is the modular operator ([24])

(23)  Δ_{φ} = S_{φ}^{∗} S_{φ} ,

which is the module of the involution x → x^{∗} in the Hilbert space attached to the sesquilinear form,

(24)  ⟨x, y⟩ = φ(y^{∗}x) , x, y ∈ M .

In the case of local fields the module was a group homomorphism ((2)) from K^{∗} to ℝ_{+}^{∗}. The counterpart for factors is the group homomorphism, ([6])

(25)  δ : ℝ → Out(M) = Aut(M)/Int(M) ,

from the additive group ℝ viewed as the dual of ℝ_{+}^{∗} for the pairing,

(26)  (λ, t) → λ^{it}  ∀ λ ∈ ℝ_{+}^{∗} , t ∈ ℝ ,

to the group of automorphism classes of M modulo inner automorphisms. The virtual subgroup,

(27)  Mod(M) ⊂_{∼} ℝ_{+}^{∗} ,

is the flow of weights ([25] [15] [8]) of M. It is obtained from the module δ as the dual action of ℝ_{+}^{∗} on the abelian algebra,

(28)  C = Center of M >◁_{δ} ℝ ,

where M >◁_{δ} ℝ is the crossed product of M by the modular automorphism group δ.

This takes care of (A), to describe (B) let us simply state the equivalence ([5]) of the following conditions

(29)

M is the closure of the union of an increasing sequence of finite dimensional algebras.

(30)

M is complemented as a subspace of the normed space of all operators in a Hilbert space.

The condition (29) is obviously what one would expect for an approximately finite dimensional algebra. Condition (30) is similar to amenability for discrete groups and the implication (30) ⇒ (29) is a very powerful tool.

We refer to [5] [15] [12] for (C) and we just describe the actual construction of the central simple algebra M associated to a given virtual subgroup,

(31)  Γ ⊂_{∼} ℝ_{+}^{∗} .

Among the approximately finite dimensional factors (central von Neumann algebras), only two are not simple. The first is the algebra

(32)  M_{∞}(ℂ) ,

of all operators in Hilbert space. The second factor is the unique approximately finite dimensional factor of type II_{∞}. It is

(33)  R_{0,1} = R ⊗ M_{∞}(ℂ) ,

where R is the unique approximately finite dimensional factor with a finite trace τ_{0}, i.e. a state such that,

(34)  τ_{0}(xy) = τ_{0}(yx)  ∀ x, y ∈ R .

The tensor product of τ_{0} by the standard semifinite trace on M_{∞}(ℂ) yields a semi-finite trace τ on R_{0,1}. There exists, up to conjugacy, a unique one parameter group of automorphisms θ_{λ} ∈ Aut(R_{0,1}), λ ∈ ℝ_{+}^{∗} such that,

(35)  τ(θ_{λ}(a)) = λτ(a)  ∀ a ∈ Domain τ , λ ∈ ℝ_{+}^{∗} .

Let first Γ ⊂ ℝ_{+}^{∗} be an ordinary closed subgroup of ℝ_{+}^{∗}. Then the corresponding factor R_{Γ} with modulo Γ is given by the equality:

(36)  R_{Γ} = {x ∈ R_{0,1} ; θ_{λ}(x) = x ∀ λ ∈ Γ} ,

in perfect analogy with (9).

A virtual subgroup Γ⊂_{∼} ℝ_{+}^{∗} is by definition an ergodic action α of ℝ_{+}^{∗} on an abelian von Neumann algebra A, and the formula (36) easily extends to,

(37)  R_{Γ} = {x ∈ R_{0,1} ⊗ A; (θ_{λ} ⊗ α_{λ}) x = x ∀ λ ∈ ℝ_{+}^{∗}} .

(This reduces to (36) for the action of ℝ_{+}^{∗} on the algebra A = L^{∞}(X) where X is the homogeneous space X = ℝ_{+}^{∗}/Γ.)

The pair (R_{0,1}, θ_{λ}) arises very naturally in geometry from the geodesic flow of a compact Riemann surface (of genus > 1). Let V = S^{∗}Σ be the unit cosphere bundle of such a surface Σ, 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 θ_{λ} of the von Neumann algebra L^{∞}(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,

(38)  (T_{1} + T_{2})_{f} = (T_{1})_{f} + (T_{2})_{f} , (T_{1} T_{2})_{f} = (T_{1})_{f} (T_{2})_{f} ,

i.e. are effected pointwise.

One proves that,

(39)  L^{∞}(V, F) ≃ R_{0,1} ,

and that the geodesic flow θ_{λ} satisfies (35). Indeed the foliation (V, F) admits up to scale a unique transverse measure Λ and the trace τ is given (cf. [4]) by the formal expression,

(40)  τ(T) = ∫ Trace(T_{f}) dΛ(f) ,

since the geodesic flow satisfies θ_{λ}(Λ) = λΛ 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_{∞} such that,

(41)  Mod(R_{∞}) = ℝ_{+}^{∗} ,

arises from the codimension 1 foliation of V = S^{∗}Σ generated by F and the geodesic flow.

In fact this relation between the classification of central simple algebras over ℂ and the geometry of foliations goes much deeper. For instance using cyclic cohomology together with the following simple fact,

(42)

“A connected group can only act trivially on a homotopy invariant cohomology theory”, one proves (cf. [4]) that for any codimension one foliation F of a compact manifold V with non vanishing Godbillon-Vey class one has,

(43)  Mod(M) has finite covolume in ℝ_{+}^{∗} ,

where M = L^{∞}(V, F) and a virtual subgroup of finite covolume is a flow with a finite invariant measure.

2 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 ℂ. We shall now describe a particularly interesting example (cf. [3]) of Hecke algebra intimately related to arithmetic, and defined over ℚ.

Let Γ_{0} ⊂ Γ be an almost normal subgroup of a discrete group Γ, i.e. one assumes,

(1)  Γ_{0} ∩ s Γ_{0} s^{−1} has finite index in Γ_{0}  ∀ s ∈ Γ .

Equivalently the orbits of the left action of Γ_{0} on Γ/Γ_{0} are all finite. One defines the Hecke algebra,

(2)  H(Γ, Γ_{0}) ,

as the convolution algebra of integer valued Γ_{0} biinvariant functions with finite support. For any field k one lets,

(3)  H_{k}(Γ, Γ_{0}) = H(Γ, Γ_{0}) ⊗_{ℤ} k ,

be obtained by extending the coefficient ring from ℤ to k. We let Γ = P +_{ℚ} be the group of 2 × 2 rational matrices,

(4)  Γ = \{[ \begin{matrix}1 & b \\ 0 & a \end{matrix} ] ; a ∈ ℚ^{+} , b ∈ ℚ\} ,

and Γ_{0} = P +_{ℤ} be the subgroup of integral matrices,

(5)  Γ_{0} = \{[ \begin{matrix}1 & n \\ 0 & 1 \end{matrix} ] ; n ∈ ℤ\} .

One checks that Γ_{0} is almost normal in Γ.

To obtain a central simple algebra over ℂ in the sense of the previous section we just take the commutant of the right regular representation of Γ on Γ_{0}\Γ, i.e. the weak closure of H_{ℂ}(Γ, Γ_{0}) in the Hilbert space,

(6)  ℓ^{2}(Γ_{0}\Γ) ,

of Γ_{0} left invariant function on Γ with norm square,

(7)  ∥ξ∥^{2} = ∑_{γ ∈ Γ_{0}\Γ} |ξ(γ)|^{2} .

This central simple algebra over ℂ is approximately finite dimensional and its module is ℝ_{+}^{∗} so that it is the same as R_{∞} of (41).

In particular its modular automorphism group is highly non trivial and one can compute it explicitly for the state φ associated to the vector ξ_{0} ∈ ℓ^{2}(Γ_{0}\Γ) corresponding to the left coset Γ_{0}.

The modular automorphism group σ_{t}^{φ} leaves the dense subalgebra H_{ℂ} (Γ, Γ_{0}) ⊂ R_{∞} globally invariant and is given by the formula,

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

for any f ∈ H_{ℂ}(Γ, Γ_{0}). Here we let,

(9)

L(γ) = Cardinality of the image of Γ_{0} γ Γ_{0} in Γ/Γ_{0}
R(γ) = Cardinality of the image of Γ_{0} γ Γ_{0} in Γ_{0}\Γ .

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,

1) The C^{∗} algebra of observables A,

2) The time evolution (σ_{t})_{t∈ℝ} which is a one parameter group of automorphisms of A.

An equilibrium or KMS (for Kubo-Martin and Schwinger) state, at inverse temperature β is a state φ on A which fulfills the following condition,

(10) For any x, y ∈ A there exists a bounded holomorphic function (continuous on the closed strip), F_{x,y}(z), 0 ≤ Im z ≤ β such that

\begin{matrix}F_{x,y}(t) = φ(x σ_{t}(y)) & ∀ t ∈ ℝ \\ F_{x,y}(t + iβ) = φ(σ_{t}(y)x) & ∀ t ∈ ℝ . \end{matrix}

For fixed β the KMS_{β} 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 β since β = \frac{1}{T} up to a conversion factor) the disorder will be predominant so that there will exist only one KMS_{β} state. For low enough temperatures some order should set in and allow for the coexistence of distinct thermodynamical phases so that the simplex K_{β} of KMS_{β} states should be non trivial. A given symmetry group G of the system will necessarily act trivially on K_{β} for large T since K_{β} is a point, but acts in general non trivially on K_{β} for small T so that it is no longer a symmetry of a given pure phase. This phenomenon of 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 norm closure of H_{ℂ}(Γ, Γ_{0}) in the algebra of operators in ℓ^{2}(Γ_{0}\Γ). We let σ_{t} ∈ Aut(A) be the unique extension of the automorphisms σ_{t}^{φ} of (8).

For β = 1 it is tautological that φ is a KMS_{β} state since we obtained σ_{t}^{φ} precisely this way ([24]). One proves ([3]) that for any β ≤ 1 (i.e. for T = 1) there exists one and only one KMS_{β} state.

The compact group G,

(11)  G = C_{ℚ}/D_{ℚ} ,

quotient of the Idele class group C_{ℚ} by the connected component of identity D_{ℚ} ≃ ℝ_{+}^{∗}, acts in a very simple and natural manner as symmetries of the system (A, σ_{t}). (To see this one notes that the right action of Γ on Γ_{0}\Γ extends to the action of P_{A} on the restricted product of the trees of SL(2, ℚ_{p}) where A is the ring of finite Adeles (cf. [3]).

For β > 1 this symmetry group G of our system, is spontaneously broken, the compact convex sets K_{β} are non trivial and have the same structure as K_{∞}, which we now describe. First some terminology, a KMS_{β} state for β = ∞ is called a ground state and the KMS_{∞} condition is equivalent to positivity of energy in the corresponding Hilbert space representation.

Remember that H_{ℂ}(Γ, Γ_{0}) contains H_{ℚ}(Γ, Γ_{0}) so,

(12)  H_{ℚ}(Γ, Γ_{0}) ⊂ A .

By [3] theorem 5 and proposition 24 one has,

Theorem. Let E(K_{∞}) be the set of extremal KMS_{∞} states. a) The group G acts freely and transitively on E(K_{∞}) by composition, φ → φ ∘ g^{−1}, ∀ g ∈ G.

b) For any φ ∈ E(K_{∞}) one has,

φ(H_{ℚ}) = ℚ_{ab} ,

and for any element α ∈ Gal(ℚ_{ab} : ℚ) there exists a unique extension of α ∘ φ, by continuity, as a state of A. One has α ∘ φ ∈ E(K_{∞}).

c) The map α → (α ∘ φ)φ^{−1} ∈ G = C_{k}/D_{k} defined for α ∈ Gal(ℚ_{ab} : ℚ) is the isomorphism of global class field theory (I.19).

This last map is independent of the choice of φ. What is quite remarkable in this result is that the existence of the subalgebra H_{ℚ} ⊂ H_{ℂ} allows to bring into action the Galois group of ℂ on the values of states. Since the Galois group of ℂ : ℚ is (except for z → \overline{z}) formed of discontinuous automorphisms it is quite surprising that its action can actually be compatible with the characteristic 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_{∞} be the weak closure of H_{ℂ}(Γ, Γ_{0}) in ℓ^{2}(Γ_{0}\Γ), we can compute the associated pair (R_{0,1}, θ_{λ}) of section I.

The C^{∗} algebra closure of H_{ℂ} is Morita equivalent (cf. M. Laca) to the crossed product C^{∗} algebra,

(13)  C_{0}(A) >◁ ℚ_{+}^{∗} ,

where A is the locally compact space of finite Adeles. It follows immediately that,

(14)  R_{0,1} = L^{∞}(ℚ_{A}) >◁ ℚ^{∗} ,

i.e. the von Neumann algebra crossed product of the L^{∞} functions on Adeles of ℚ by the action of ℚ^{∗} by multiplication.

The one parameter group of automorphisms, θ_{λ} ∈ Aut(R_{0,1}), is obtained as the restriction to,

(15)  D_{ℚ} = ℝ_{+}^{∗} ,

of the obvious action of the Idele class group C_{ℚ},

(16)  (g, x) → g x  ∀ g ∈ C_{ℚ} , x ∈ A_{ℚ}/ℚ^{∗} ,

on the space X = A_{ℚ}/ℚ^{∗} of Adele classes.

Our next goal will be to show that the latter space is intimately related to the zeros of the Hecke L-functions with Grossencharakter.

(We showed in [3] that the partition function of the above system is the Riemann zeta function.)

3 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 Σ defined over 𝔽_{q},

Algebraic Geometry  Function Theory
Eigenvalues of action of  Zeros of ζ

Frobenius on H_{et}^{1}(\overline{Σ}, ℚ_{ℓ})

Poincaré duality in  Functional equation

ℓ-adic cohomology

Lefschetz formula for  Explicit formulas

the Frobenius

Castelnuovo positivity  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,

(1)  X = A/k^{∗} , A = Adeles of k .

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, (A) Let N(E) be the number of zeros of the Riemann zeta function satisfying 0 < Imρ < E, then ([22])

(2)  N(E) = ⟨N(E)⟩ + N_{osc}(E) ,

where the smooth function ⟨N(E)⟩ is given by

(3)  ⟨N(E)⟩ = \frac{E}{2π} (log \frac{E}{2π} − 1) + \frac{7}{8} + o(1) ,

while the oscillatory part is

(4)  N_{osc}(E) = \frac{1}{π} Im log ζ (\frac{1}{2} + i E) .

The numbers x_{j} = ⟨N(ρ_{j})⟩ where ρ_{n} is the imaginary part of the n^{th} zero are of average density one and behave like the eigenvalues of a random Hermitian matrix. This was discovered by H. Montgomery [18] who conjectured (and proved for suitable test functions) that when M → ∞, with α, β > 0,

(5) # \{(i, j) ∈ \{1, …, M\}^{2} ; x_{i} − x_{j} ∈ [α, β]\} ∼ M ∫_{α}^{β} 1 − (\frac{sin πu}{πu})^{2} du

which is exactly what happens in the Gaussian Unitary Ensemble. Numerical tests by A. Odlyzko [20] and further theoretical work by Katz-Sarnak [17] and J. Keating give overwhelming evidence that zeros of zeta should be the eigenvalues of a hermitian matrix. (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. (C) The deep arithmetic significance of the work of A. Selberg on the spectral analysis of the Laplacian on L^{2}(G/Γ) where Γ is an arithmetic subgroup of a semi simple Lie group G.

Direct atempts (cf. [2]) to construct the Polya-Hilbert space giving a spectral realization of the zeros of ζ using quantum mechanics, meet the following − sign problem: Let H be the Hamiltonian of the quantum mechanical system obtained by quantizing the classical system,

(6)  (X, F_{t})

where X is phase space and t ∈ ℝ → F_{t} the Hamiltonian flow. Let N(E) be the number of eigenvalues λ of H such that 0 ≤ λ ≤ E. Then, as for ζ,

(7)  N(E) = ⟨N(E)⟩ + N_{osc}(E) ,

where ⟨N(E)⟩ is essentially a volume in phase space, while the oscillatory part admits a heuristic asymptotic expansion (cf. [2]) of the form,

(8)  N_{osc}(E) ∼ \frac{1}{π} ∑_{γ} ∑_{m = 1}^{∞} \frac{1}{m} \frac{1}{2sh (\frac{mλ_{γ}}{2} )} sin(T_{γ}^{#} m E)

where the γ are the periodic orbits of the flow F, the T_{γ}^{#} are their periods and the λ_{γ} the unstability exponents of these orbits.

One can compare ([2]) (8) with the equally heuristic asymptotic expansion of (4) using the Euler product of ζ which gives, using − log(1 − x) =

∑_{m=1}^{∞} \frac{x^{m}}{m},

(9)  N_{osc}(E) ≃ −\frac{1}{π} ∑_{p} ∑_{m=1}^{∞} \frac{1}{m} \frac{1}{p^{m/2}} sin((log p) m E) .

Comparing (8) and (9) one gets precious information on the hypothetical “Riemann flow” of M. Berry. The periodic orbits γ should be labelled by the primes p, the periods should be the log p as well as the unstability exponents λ_{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:

(10)  (X, F_{−t}) .

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,

(11)  ∑_{L(χ,ρ) = 0} ĥ(χ, ρ) − ĥ(0) − ĥ(1) = − ∑_{v} ∫′_{k_{v}^{∗}} \frac{h(u^{−1})}{|1 − u|} d^{∗}u ,

where h is a test function on the Idele class group C_{k}, ĥ is its Fourier transform,

(12)  ĥ(x, z) = ∫_{C_{k}} h(u) χ(u) |u|^{z} d^{∗}u ,

and the finite values ∫′ are suitably normalized. If we use the above dictionary when char(k) ≠ 0, the geometric origin of this − sign becomes clear, the formula (11) is the Lefschetz formula,

(13) # of fixed points of φ = Trace φ/H^{0} − Trace φ/H^{1} + Trace φ/H^{2}

in which the space H_{et}^{1}(\overline{Σ}, ℚ_{ℓ}) 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,

(14)  0 → H_{0} →^{T} H_{1} → H → 0 .

The example we have in mind for (14) is the assembled Euler complex for a Riemann surface, where H_{0} is the codimension 2 subspace of differential forms of even degree orthogonal to harmonic forms, where 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 Grössencharakter we do not expect to have only an action of ℤ for char(k) > 0 corresponding to the Frobenius, or of the group ℝ_{+}^{∗} if 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 H_{0} a suitable completion of the codimension 2 subspace of functions on X such that,

(15)  f(0) = 0 , ∫ f dx = 0 ,

while H_{1} = L^{2}(C_{k}) and T is the restriction map coming from the inclusion C_{k} → X, multiplied by |a|^{1/2},

(16)  (T f)(a) = |a|^{1/2} f(a) .

The action of C_{k} is then the obvious one, for H_{0}

(17)  (U(g)f)(x) = f(g^{−1} x)  ∀ g ∈ C_{k}

using the action II.15 of C_{k} on X, and similarly the regular representation V for 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”

(18)  f → f_{q} , f_{q}(x) = f(xq) , ∀ q ∈ k^{∗} .

Here the natural function space is the Bruhat-Schwarz space S(A) and by (15) the codimension 2 subspace,

(19)  S(A)_{0} = {f ∈ S(A) ; f(0) = 0 , ∫ f dx = 0} .

The restriction map T is then given by,

(20)  T (f)(a) = |a|^{1/2} ∑_{q ∈ k^{∗}} f(aq)  ∀ a ∈ C_{k} .

The corresponding function T (f) belongs to 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_{δ}^{2}(C_{k}) using the polynomial weight (log^{2} |a|)^{δ/2}, i.e. the norm,

(21)  ∥ξ∥_{δ}^{2} = ∫_{C_{k}} |ξ(a)|^{2} (1 + log^{2} |a|)^{δ/2} d^{∗}a .

Let char(k) = 0 so that Mod k = ℝ_{+}^{∗} and C_{k} = K × ℝ_{+}^{∗} where K is the compact group C_{k,1} = {a ∈ C_{k}; |a| = 1}.

Theorem. Let δ > 1, H be the cokernel of T in L_{δ}(C_{k}) and W the quotient representation of C_{k}. Let χ be a character of K, χ̃ = χ × 1 the corresponding character of C_{k}. Let H_{χ} = {ξ ∈ H; W (g) ξ = χ(g) ξ ∀ g ∈ K} and D_{χ} = lim_{ϵ→0} \frac{1}{ϵ} (W (e^{ϵ}) − 1). Then D_{χ} is an unbounded closed operator with discrete spectrum, Sp D_{χ} ⊂ i ℝ is the set of imaginary parts of zeros of the L function with Grössencharakter χ̃ which have real part 1/2. Moreover the spectral multiplicity of ρ is the largest integer n < \frac{1+δ}{2} in {1, …, multiplicity as a zero of L}.

A similar result holds for char(k) > 0. This allows to compute the character of the representation W as,

(22)  Trace(W (h)) = ∑_{{L(χ, \frac{1}{2} +ρ) = 0 \atop ρ ∈ i ℝ/N^{⊥}}} ĥ(χ, ρ)

where N = Mod(k), W (h) = ∫ W (g) h(g) d^{∗}g, h ∈ S(C_{k}), ĥ is defined in (12) and the multiplicity is counted as in the theorem.

This result is only preliminary because of the unwanted parameter δ which artificially restricts the multiplicities. The restriction Re ρ = \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 ∥ξ∥^{2} = ∫_{X} |ξ(x)|^{2} dx is naturally given upstairs on S(A)_{0} by:

(23)  ∥f∥^{2} = ∫_{D} |Σ f(xq)|^{2} |x| d^{∗}x , ∀ f ∈ S(A)_{0} ,

where D is a fundamental domain for k^{∗} acting on Ideles. For a local field one has indeed the equality

(24)  dx = |x| d^{∗}x ,

(up to normalization) between the additive Haar measure and the multiplicative one. In the global case one has,

(25)  dx = lim_{ϵ→0} ϵ |x|^{1+ϵ} d^{∗}x ,

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,

(26)  T : L^{2}(X)_{0} → L^{2}(C_{k}) .

The analogue of the Hodge ∗ operation is given on H_{0} by the Fourier transform,

(27)  (F f)(x) = ∫_{A} f(y) α(xy) dy  ∀ f ∈ S(A)_{0}

which, because we take the quotient by (18), is independent of the choice of additive character α of A such that α ≠ 1 and α(q) = 1 ∀ q ∈ k. Note also that F^{2} = 1 on the quotient. On H_{1} the Hodge ∗ is given by,

(28)  (∗ ξ)(a) = ξ(a^{−1})  ∀ a ∈ C_{k} .

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 ([1]) of the Lefschetz formula we do obtain a clear geometric meaning for the Weil distribution. One can of course as in [10] 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 φ be a diffeomorphism of a smooth manifold Σ and assume that the graph of φ is transverse to the diagonal, one can then easily define and compute (cf. [1]) the distribution theoretic trace of the permutation U of functions on Σ associated to φ,

(29)  (Uξ)(x) = ξ(φ(x))  ∀ x ∈ Σ .

One has “Trace” (U) = ∫ k(x, x) dx, where k(x, y) dy is the Schwarz kernel associated to U, i.e. the distribution on Σ × Σ such that,

(30)  (Uξ)(x) = ∫ k(x, y) ξ(y) dy .

Now near the diagonal and in local coordinates one has,

(31)  k(x, y) = δ(y − φ(x)) ,

where δ is the Dirac distribution. One then obtains,

(32)  “Trace” (U) = ∑_{φ(x) = x} \frac{1}{|1 − φ′(x)|} ,

where φ′ is the Jacobian of φ and | | stands for the absolute value of the determinant.

With more work ([11]) one obtains a similar formula for the distributional trace of the action of a flow,

(33)  (U_{t} ξ)(x) = ξ(F_{t}(x))  ∀ x ∈ Σ , t ∈ ℝ .

It is given, under suitable transversality hypothesis, by

(34)  “Trace” (U(h)) = ∑_{γ} ∫_{I_{γ}} \frac{h(u)}{|1 − (F_{u})_{∗}|} d^{∗}u ,

where U(h) = ∫ h(t) U(t) dt, h is a test function on ℝ, the γ labels the periodic orbits of the flow, including the fixed points, I_{γ} 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_{γ} which is of covolume 1 in (ℝ, 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,

(35)  “Trace” (U(h)) = ∑_{v} ∫_{k_{v}^{∗}} \frac{h(u^{−1})}{|1 − u|} d^{∗}u .

Thus, the isotropy subgroups I_{γ} are parametrized by the places v of k and coincide with the natural cocompact inclusion k_{v}^{∗} ⊂ C_{k} which relates local to global in class field theory. The denominator |1 − u| 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 [27], by,

(36)  ∫_{1≤|u|≤Λ} d^{∗}u ∼ log Λ  for Λ → ∞ ,

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 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,

(37)  h → ĥ(0) + ĥ(1) .

Thus equating the alternate sum of traces on H_{0}, 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 δ.

The trace of permutation matrices is positive and this explains the Hadamard positivity,

(38) “Trace” (U(h)) ≥ 0  ∀ h , h(1) = 0 , h(u) ≥ 0 ∀ u ∈ C

(not to be confused with Weil postivity).

To eliminate the artificial parameter δ and give rigorous meaning, as a Hilbert space trace, to the distribution “trace”, one proceeds as in the Selberg trace formula [23] 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,

(39)  R_{Λ} = P̂_{Λ} P_{Λ} , Λ ∈ ℝ_{+} ,

where P_{Λ} is the orthogonal projection on the subspace,

(40)  {ξ ∈ L^{2}(K) ; ξ(x) = 0  ∀ x, |x| > Λ} ,

while P̂_{Λ} = F P_{Λ} F^{−1}, F the Fourier transform.

One proves ([9]) in this local case the following analogue of the Selberg trace formula,

(41)  Trace (R_{Λ} U(h)) = 2 h(1) log′(Λ) + ∫ ′ \frac{h(u^{−1})}{|1 − u|} d^{∗}u + o(1)

where h ∈ S(K^{∗}) has compact support, 2 log′(Λ) = ∫_{λ ∈ K^{∗}, |λ| ∈ [Λ^{−1},Λ]} d^{∗}λ, and the principal value ∫′ is uniquely determined by the pairing with the unique distribution on K which agrees with \frac{du}{|1−u|} for u ≠ 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

(42)  C_{S} = ∏_{v∈S} k_{v}^{∗}/O_{S}^{∗} ,

where O_{S}^{∗} ⊂ k^{∗} is the group of S-units. One replaces X by

(43)  X_{S} = ∏_{v∈S} k_{v}/O_{S}^{∗} .

The Hilbert space L^{2}(X_{S}), its Fourier transform F and the orthogonal projection P_{Λ}, P̂_{Λ} = F P_{Λ} F^{−1} continue to make sense, with

(44)  Im P_{Λ} = {ξ ∈ L^{2}(X_{S}) ; ξ(x) = 0 ∀ x , |x| > Λ} .

As soon as S contains more than 3 elements, (e.g. {2, 3, ∞} for k = ℚ) the space X_{S} is an extremely delicate quotient space. It is thus quite remarkable that the trace formula holds, Theorem. For any h ∈ S_{c}(C_{S}) one has, with R_{Λ} = P̂_{Λ} P_{Λ},

Trace (R_{Λ} U(h)) = 2 log′(Λ) h(1) + ∑_{v∈S} ∫′_{k_{v}^{∗}} \frac{h(u^{−1})}{|1 − u|} d^{∗}u + o(1)

where the notations are as above and the finite values ∫′ depend on the additive character of Πk_{v} defining the Fourier transform F. When Char (k) = 0 the projectors P_{Λ}, P̂_{Λ} commute on L_{χ}^{2} for Λ large enough so that one can replace R_{Λ} by the orthogonal projection Q_{Λ} on Im P_{Λ} ∩ Im P̂_{Λ}. The situation for Char (k) = 0 is more delicate since P_{Λ} and P̂_{Λ} do not commute (for Λ large) even in the local Archimedian case. But fortunately [21] these operators commute with a specific second order differential operator, whose eigenfunctions, the Prolate Spheroidal Wave functions provide the right filtration Q_{Λ}. This allows to replace R_{Λ} by Q_{Λ} and to state the global trace formula

(45)  Trace (Q_{Λ} U(h)) = 2 log′(Λ) h(1) + ∑_{v} ∫_{k_{v}^{∗}} \frac{h(u^{−1})}{|1 − u|} d^{∗}u + o(1) .

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 Lfunctions with Grössencharakter. Moreover the filtration by Q_{Λ} allows to define the Adelic cohomology and to complete the dictionary between the function theory and the geometry of the Adele class space.

Function Theory  Geometry
Zeros and poles of Zeta  Eigenvalues of action of C_{k}

on Adelic cohomology

Functional Equation  ∗ operation
Explicit formula  Lefschetz formula
RH  Trace formula

References

[1] M.F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes: I, Annals of Math, 86 (1967), 374-407.

[2] M. Berry, Riemann’s zeta function: a model of quantum chaos, Lecture Notes in Physics, 263, Springer (1986).

[3] J.-B. Bost and A. Connes, Hecke Algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Mathematica, New Series 1, n. 3 (1995), 411-457.

[4] A. Connes, Noncommutative Geometry, Academic Press (1994).

[5] A. Connes, Classification of injective factors, Ann. of Math., 104, n. 2 (1976), 73-115.

[6] A. Connes, Une classification des facteurs de type III, Ann. Sci. Ecole Norm. Sup., 6, n. 4 (1973), 133-252.

[7] A. Connes, Formule de trace en Géométrie non commutative et hypothèse de Riemann, C.R. Acad. Sci. Paris Ser. A-B (1996)

[8] A. Connes and M. Takesaki, The flow of weights on factors of type III, Tohoku Math. J., 29 (1977), 473-575.

[9] A. Connes, Trace formula in Noncommutative Geometry and the zeros of the Riemann zeta function. To appear in Selecta Mathematica.

[10] D. Goldfeld, A spectral interpretation of Weil’s explicit formula, Lecture Notes in Math., 1593, Springer Verlag (1994), 135-152.

[11] V. Guillemin, Lectures on spectral theory of elliptic operators, Duke Math. J., 44, n. 3 (1977), 485-517.

[12] U. Haagerup, Connes’ bicentralizer problem and uniqueness of the injective factor of type III_{1}, Acta Math., 158 (1987), 95-148.

[13] S. Haran, Riesz potentials and explicit sums in arithmetic, Invent. Math., 101 (1990), 697-703.

[14] B. Julia, Statistical theory of numbers, Number Theory and Physics, Springer Proceedings in Physics, 47 (1990).

[15] W. Krieger, On ergodic flows and the isomorphism of factors, Math. Ann., 223 (1976), 19-70.

[16] N. Katz and P. Sarnak, Random matrices, Frobenius eigenvalues and Monodromy, (1996) , Book, to appear.

[17] N. Katz and P. Sarnak, Zeros of zeta functions, their spacings and spectral nature, (1997), to appear.

[18] H. Montgomery, The pair correlation of zeros of the zeta function, Analytic Number Theory, AMS (1973).

[19] M.L. Mehta, Random matrices, Academic Press,(1991).

[20] A. Odlyzko, On the distribution of spacings between zeros of zeta functions, Math. Comp. 48 (1987), 273-308.

[21] D. Slepian and H. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty I, Bell Syst. Tech. J. 40 (1961).

[22] B. Riemann, Mathematical Werke, Dover, New York (1953).

[23] A.Selberg, Collected papers, Springer (1989).

[24] M. Takesaki, Tomita’s theory of modular Hilbert algebras and its applications, Lecture Notes in Math. bf 128, Springer (1989).

[25] M. Takesaki, Duality for crossed products and the structure of von Neumann algebras of type III, Acta Math. 131 (1973), 249-310.

[26] A. Weil, Basic Number Theory, Springer, New York (1974).

[27] A. Weil, Sur les formules explicites de la théorie des nombres, Izv. Mat. Nauk., (Ser. Mat.) 36, 3-18.

[28] A. Weil, Sur la théorie du corps de classes, J. Math. Soc. Japan, 3, (1951).

[29] D. Zagier, Eisenstein series and the Riemannian zeta function, Automorphic Forms, Representation Theory and Arithmetic, Tata, Bombay (1979), 275-301.