孔涅 非交换几何与黎曼δ函数

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 Andre´ 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 Gro¨ssencharakter. 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∗ = GL1(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∗+ , 1 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 Fq (with q = p ` a prime power). One has, (5) Mod(K) = qZ ⊂ 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 ∈ AutK(K ′) such that, (6) θ(µ) = µq , 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 6= {1} . Let then K be an algebraic closure of K, Ksep ⊂ K the separable algebraic closure, Kab ⊂ Ksep the maximal abelian extension of K and Kun ⊂ Kab the maximal unramified extension of K, i.e. the union of all unramified extensions of finite degree. One has, (8) K ⊂ Kun ⊂ Kab ⊂ Ksep ⊂ K , and the Galois group Gal(Kun : K) is topologically generated by θ called the Frobenius automorphism. 2 The correspondence (7) is given by, (9) K ′ = {x ∈ Kun ; θλ(x) = x ∀λ ∈ Γ} , with rather obvious notations so that θq is the θ of (6). Let then WK be the subgroup of Gal(Kab : K) whose elements induce on Kun an integral power of the Frobenius automorphism. One endows WK with the locally compact topology dictated by the exact sequence of groups, (10) 1 → Gal(Kab : Kun) → WK → Mod(K) → 1 , and the main result of local class field theory asserts the existence of a canonical isomorphism, (11) WK ∼ → K∗ , compatible with the module. The basic step in the construction of the isomorphism (11) is the clas- sification of finite dimensional central simple algebras A over K. Any such algebra is of the form, (12) A = Mn(D) , where D is a (central) division algebra over K and the symbol Mn 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 Archi- median. For Archimedian fields R and C the same questions have an idioti- cally simple answer. Since C is algebraically closed one has K = 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 ∗. 3 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 kv , where the product is the restricted product of the local fields kv labelled by the places of k. When the characteristic of k is p > 1 so that k is a function field over Fq, one has (16) k ⊂ kun ⊂ kab ⊂ ksep ⊂ k , where, as above k is an algebraic closure of k, ksep the separable algebraic closure, kab the maximal abelian extension and kun is obtained by adjoining to k all roots of unity of order prime to p. One defines the Weil group Wk as above as the subgroup of Gal(kab : k) of those automorphisms which induce on kun an integral power of θ, (17) θ(µ) = µq ∀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) Wk ' Ck = GL1(A)/GL1(k) , of locally compact groups. When k is of characteristic 0, i.e. is a number field, one has a canonical isomorphism, (19) Gal(kab : k) ' Ck/Dk , where Dk is the connected component of identity in the Idele class group Ck = GL1(A)/GL1(k), but because of the Archimedian places of k there is no interpretation of Ck analogous to the Galois group interpretation for function fields. According to A. Weil [28], “La recherche d’une interpre´tation pour Ck si k est un corps de nombres, analogue en quelque manie`re a` l’interpre´tation par un groupe de Galois quand k est un corps de fonctions, me semble constituer l’un des proble`mes fondamentaux de la the´orie des 4 nombres a` l’heure actuelle ; il se peut qu’une telle interpre´tation renferme la clef de l’hypothe`se 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 Dk 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 = Mn(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 M1 is isomorphic to M2 iff Mod(M1) ∼= Mod(M2). 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 dimen- sionality. (C) The proof that Mod is a complete invariant and that all virtual sub- groups 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(∆ϕ) ⊂ R+ 5 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 ∆ϕ is the modular operator ([24]) (23) ∆ϕ = S ∗ ϕ Sϕ , 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) → λit ∀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. 6 The condition (29) is obviously what one would expect for an approximately finite dimensional algebra. Condition (30) is similar to amenability for dis- crete 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 construc- tion 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 approxi- mately finite dimensional factor of type II∞. It is (33) R0,1 = R⊗M∞(C) , where R is the unique approximately finite dimensional factor with a finite trace τ0, i.e. a state such that, (34) τ0(xy) = τ0(yx) ∀x, y ∈ R . The tensor product of τ0 by the standard semifinite trace on M∞(C) yields a semi-finite trace τ on R0,1. There exists, up to conjugacy, a unique one parameter group of automorphisms θλ ∈ Aut(R0,1), λ ∈ R ∗ + such that, (35) τ(θλ(a)) = λτ(a) ∀ a ∈ Domain τ , λ ∈ R ∗ + . Let first Γ ⊂ R∗+ be an ordinary closed subgroup of R ∗ +. Then the corre- sponding factor RΓ with modulo Γ is given by the equality: (36) RΓ = {x ∈ R0,1 ; θλ(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) RΓ = {x ∈ R0,1 ⊗A ; (θλ ⊗ αλ)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.) 7 The pair (R0,1, θλ) 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 automor- phisms of the foliated manifold (V, F ) and thus a one parameter group of automorphisms θλ of the von Neumann algebra L ∞(V, F ). This algebra is easy to describe, its elements are random operators T = (Tf ), i.e. bounded measurable families of operators Tf parametrized by the leaves f of the foliation. For each leaf f the operator Tf acts in the Hilbert space L2(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) (T1 + T2)f = (T1)f + (T2)f , (T1 T2)f = (T1)f (T2)f , i.e. are effected pointwise. One proves that, (39) L∞(V, F ) ' R0,1 , and that the geodesic flow θλ 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(Tf ) dΛ(f) , since the geodesic flow satisfies θλ(Λ) = λΛ one obtains (35) from simple ge- ometric considerations. The formula (37) shows that most approximately fi- nite 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”, 8 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 sym- metry 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 Γ0 ⊂ Γ be an almost normal subgroup of a discrete group Γ, i.e. one assumes, (1) Γ0 ∩ sΓ0 s −1 has finite index in Γ0 ∀ s ∈ Γ . Equivalently the orbits of the left action of Γ0 on Γ/Γ0 are all finite. One defines the Hecke algebra, (2) H(Γ,Γ0) , as the convolution algebra of integer valued Γ0 biinvariant functions with finite support. For any field k one lets, (3) Hk(Γ,Γ0) = H(Γ,Γ0)⊗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 = {[ 1 b 0 a ] ; a ∈ Q+ , b ∈ Q } , and Γ0 = P + Z be the subgroup of integral matrices, (5) Γ0 = {[ 1 n 0 1 ] ; n ∈ Z } . One checks that Γ0 is almost normal in Γ. 9 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 Γ0\Γ, i.e. the weak closure of HC(Γ,Γ0) in the Hilbert space, (6) `2(Γ0\Γ) , of Γ0 left invariant function on Γ with norm square, (7) ‖ξ‖2 = ∑ γ ∈Γ0\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 ξ0 ∈ `2(Γ0\Γ) corresponding to the left coset Γ0. The modular automorphism group σϕt leaves the dense subalgebra HC (Γ,Γ0) ⊂ R∞ globally invariant and is given by the formula, (8) σϕt (f)(γ) = L(γ) −it R(γ)it f(γ) ∀ γ ∈ Γ0\Γ/Γ0 for any f ∈ HC(Γ,Γ0). Here we let, (9) L(γ) = Cardinality of the image of Γ0 γ Γ0 in Γ/Γ0 R(γ) = Cardinality of the image of Γ0 γ Γ0 in Γ0\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 (σt)t∈R which is a one parameter group of automor- phisms of A. An equilibrium or KMS (for Kubo-Martin and Schwinger) state, at in- verse 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 (contin- uous on the closed strip), Fx,y(z), 0 ≤ Im z ≤ β such that Fx,y(t) = ϕ(xσt(y)) ∀ t ∈ R Fx,y(t + iβ) = ϕ(σt(y)x) ∀ t ∈ R . 10 For fixed β the KMSβ 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 = 1T up to a conversion factor) the disorder will be predominant so that there will exist only one KMSβ state. For low enough temperatures some order should set in and allow for the coexistence of distinct thermodynamical phases so that the simplex Kβ of KMSβ states should be non trivial. A given symmetry group G of the system will necessarily act trivially on Kβ for large T since Kβ is a point, but acts in general non trivially on Kβ 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 Tc 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 HC(Γ,Γ0) in the algebra of operators in ` 2(Γ0\Γ). We let σt ∈ Aut(A) be the unique extension of the automorphisms σϕt of (8). For β = 1 it is tautological that ϕ is a KMSβ state since we obtained σϕt precisely this way ([24]). One proves ([3]) that for any β ≤ 1 (i.e. for T = 1) there exists one and only one KMSβ state. The compact group G, (11) G = CQ/DQ , quotient of the Idele class group CQ by the connected component of identity DQ ' R ∗ +, acts in a very simple and natural manner as symmetries of the system (A, σt). (To see this one notes that the right action of Γ on Γ0\Γ extends to the action of PA on the restricted product of the trees of SL(2, Qp) 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 Kβ are non trivial and have the same structure as K∞, which we now describe. First some terminology, a KMSβ 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 HC(Γ,Γ0) contains HQ(Γ,Γ0) so, (12) HQ(Γ,Γ0) ⊂ A . By [3] theorem 5 and proposition 24 one has, Theorem. Let E(K∞) be the set of extremal KMS∞ states. 11 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(HQ) = Qab , and for any element α ∈ Gal(Qab : 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 = Ck/Dk defined for α ∈ Gal(Qab : 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 HQ ⊂ HC 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 → 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 HC(Γ,Γ0) in ` 2(Γ0\Γ), we can compute the associated pair (R0,1, θλ) of section I. The C∗ algebra closure of HC is Morita equivalent (cf. M. Laca) to the crossed pr