BC-system, absolute cyclotomy and the quantized calculus

We give a short survey on several developments on the BC-system, the adele class space of the rationals, and on the understanding of the"zeta sector"of the latter space as the Scaling Site. The new result that we present concerns the description of the BC-system as the universal Witt ring (i.e. K-theory of endomorphisms) of the"algebraic closure"of the absolute base S. In this way we attain a conceptual meaning of the BC dynamical system at the most basic algebraic level. Furthermore, we define an invariant of Schwartz kernels in 1 dimension and relate the Fourier transform (in 1 dimension) to its role over the algebraic closure of S. We implement this invariant to prove that, when applied to the quantized differential of a function, it provides its Schwarzian derivative. Finally, we survey the roles of the quantized calculus in relation to Weil's positivity, and that of spectral triples in relation to the zeros of the Riemann zeta function.


Introduction
We dedicate this paper to Dennis Sullivan, whose genuine love for understanding mathematics and his generosity in communicating new ideas has always been an inspirational example to us.We take this opportunity to write an overview on the actual state of our enduring interest in the Riemann zeta function.The Riemann zeta function has many intriguing manifestations in science: our own interest was triggered with the discovery of the surprising relation that this function has with noncommutative geometry.In quantum statistical mechanics in the first place, the zeta function appears as the partition function of a dynamical system determined by the analysis of the Hecke algebra of the affine group of rational numbers [3].This result leads to the noncommutative space of adele classes of the rationals [6].The impact in number theory is the spectral realization of the zeros of the Riemann zeta function and the geometric understanding of the Riemann-Weil explicit formula as a trace formula [6,34].Moreover, the quantized calculus in noncommutative geometry, jointly with the theory of prolate spheroidal functions in analysis provides, by means of a semilocal trace formula computation, a conceptual reason for the positivity of Weil's functional [16].The understanding of the adele class space of the rationals from a more classical geometric standpoint is provided by the theory of Grothendieck toposes: indeed, in many cases the space of the points of these toposes is noncommutative.The "zeta sector" of the adele class space of the rationals is described precisely by the set of points of the Scaling Site [14].This result led us to a parallel and independent investigation of the algebraic landscape of semirings of characteristic one, where each integer acts by an endomorphism, thus generalizing the Frobenius operator on geometries in finite characteristic, and where categorically, one only adds one more (prime) "field": the Boolean semifield B [19].However, and in spite of its elementary definition, the Boolean B cannot qualify as realizing the original dream of J. Tits in his search for a basic algebraic structure rooting fundamental examples of combinatorial geometries, neither as an algebraic incarnation of Waldhausen "initial ring".The weakness of the classical algebraic approach is in part due to the inherent deficiency of set-theory when compared to the more flexible categorical counterpart.By following the simple idea that an abelian group A is entirely encoded by the covariant functor HA that assigns to a pointed set X the pointed set of A-valued "divisors" on X, one quickly realizes that the functorial viewpoint is a very natural and versatile generalization of the original set-theoretical notion of abelian group.This idea has led us to develop algebraic geometry over a "base" S that is the spherical counterpart of the multiplicative monoid F 1 " t0, 1u and the categorical backbone of the sphere spectrum in homotopy theory [28].Rings over this base are the Γ-rings of G. Segal and the simplest of them is S, namely the identity endofunctor on the category of pointed sets.Ordinary rings become Γ-rings through the Eilenberg-MacLane functor H.In [20] we gave an arithmetic application of these ideas by extending, at archimedean infinity, the structure sheaf of the algebraic spectrum of the ring of integers as a sheaf of Γ-rings, and more precisely as a subsheaf of the constant sheaf HQ.One instance of the relevance of the stalk at the archimedean place of this compactification (the Arakelov one point compactification) is exemplified through its relation with the Gromov norm [21].The choice of this new base has also the further advantage to provide the right framework for Hochschild and cyclic homologies, since simplicial Γ-sets (i.e.Γ-spaces) are well understood and homotopy theory over them becomes, by means of the Dold-Kan correspondence, homological algebra taking place over S [22].In Section 1 we give a short survey of all these developments centered on the key role played by the BC-system and the adele class space, and on the understanding of the "zeta sector" of the latter space as the Scaling Site.The new result that we present in this context is introduced in Section 2, where we describe the BC-system as the Witt ring (i.e.K-theory of endomorphisms) of the "algebraic closure" of S. In this way we obtain a conceptual meaning of the BC dynamical system no longer from analysis (in terms of quantum statistical mechanics) but at the most basic algebraic level.The algebraic closure S is defined by adjoining to S all (abstract) roots of unity, and the relation between its Witt ring and the BC-system suggests to perform the following two steps (i) Determine the extension of scalars Spec Z ˆS S.
(ii) Define an appropriate De Rham-Witt complex for Spec Z ˆS S.
An educated guess on Spec Z ˆS S suggests that this space ought to involve algebraically the cyclotomic extension of the field of rational numbers.The De Rham-Witt complex of Spec Z ˆS S should mainly provide a strengthening of the link between two worlds: on one side (say on the left) the classical world of Arakelov geometry now enriched over Γ-rings, while on the other side (the right) the analytic framework of noncommutative geometry stemming from the BC-system and directly related to the understanding of the zeros of the Riemann zeta function.These two worlds are, a priori, quite different in nature.Homological algebra over Γ-rings is, through the Dold-Kan correspondence, naturally encoded by the homotopy theory of Γ-spaces, so that the world on the left is that of homotopy theory, spectra, animas....The world on the right side instead, is that of analysis, Hilbert space operators, the quantum...One fundamental relation between these two worlds is the assembly map [1] which associates to K-homology classes of the universal proper quotient, classes in the K-theory of the reduced C ˚-algebra of a group.This creates a bridge, of index theoretic nature, between the world of homotopy theory and the world of analysis where K-theory of C ˚-algebras plays a key role.More precisely, the assembly map relates together two ways of effecting the quotient of a space by a group action.On the left world one sees an homotopy quotient as a special case of a homotopy colimit, while on the right world one effects a cross product which is a special case of a general principle in noncommutative geometry of encoding difficult quotients (such as leaf spaces of foliations) by noncommutative algebras.It is also worth noticing that aside from quotient spaces, these tricky spaces also appear naturally as sets of points of a topos.For example, to a small category C one may associate the presheaf topos p C of contravariant functors from C to the category of sets.In general, the nature of the space of points of the topos p C is as delicate as that of a quotient space, and one may either use as a substitute the classifying space BC in the left world or view such spaces as noncommutative spaces, if one prefers to work in the right world.In Section 3 we describe the role of the 1-dimensional quantized calculus in relation to Weil's positivity.A central role is played both by the unitary obtained by composing Fourier transform with inversion, and by its quantized logarithmic derivative.An elementary lemma only meaningful in quantized calculus (called the Main Lemma in this paper) gives the conceptual reason to expect Weil's positivity.The fact that the hypothesis of this lemma is only verified up to an infinitesimal prevents one to conclude immediately that positivity holds.In [24] we showed that positivity can still be obtained, for a single archimedean place, by treating separately this infinitesimal.Moreover, in the semilocal case, (i.e. when finitely many places, including the archimedean one, are involved), the same infinitesimal property continues to hold [16], and this fact opens the way for a strategy toward RH.In §3.1 we define an invariant of Schwartz kernels in 1 dimension and relate the Fourier transform (in 1 dimension) to its role over S (see §2.3).Then, we implement this invariant to prove that, when applied to the quantized differential of a function, it delivers its Schwarzian derivative.This shows in particular that, as emphasized in the fax of D. Sullivan reported in Figure 1 (beginning of Section 3), the quantized differential calculus encodes in a subtle manner the conformal structure also in dimension 1, where the Riemannian point of view gives no clue.In §3.2 we state the Main Lemma in quantized calculus that yields Weil's positivity as a consequence of the triangular property of the quantized differential, and in §3.3 we discuss the triptych formed by Fourier, Zeta and Poisson.The quantized calculus is then applied in the semilocal framework ( §3.4) and provides, through the semilocal trace formula, both the operator theoretic formalism for the explicit formulas of Riemann-Weil and a conceptual reason for Weil's positivity.We discuss the radical of Weil's quadratic form in §3.5 and the "almost radical" of its restriction to an interval rλ ´1, λs in §3.6.We then use spectral triples (through Dirac operators) to detect the zeros of the Riemann zeta function up to imaginary part 2πλ 2 .This provides the operator theoretic replacement for the Riemann-Siegel formula in analytic number theory and the approximation to the sought for cohomology discovered in [17].

The BC-system and its role
The origin of the relation between noncommutative geometry and the Riemann zeta function is a fundamental interplay between the mechanism of symmetry breaking in physics and the theory of ambiguity of E. Galois.In physics, the choice of an extremal equilibrium state at zero temperature breaks the symmetry of a system.On the Galois side the choice of such a state selects a group isomorphism of the abstract group Q{Z with the group of roots of unity in C. The link is established explicitly by implementing the formalism of quantum statistical mechanics [4] that encodes a quantum statistical system by a pair pA, σ t q of a C ˚-algebra A and a 1-parameter group of automorphisms σ : R Ñ AutpAq.The main tool is the KMS condition that analytically encapsulates the relation existing in quantum mechanics between the Heisenberg time evolution of observables σ t pAq " exppitHqA expp´itHq (H is the Hamiltonian of the system) and an equilibrium state φ at inverse temperature β " 1 kT , whose evaluation on an observable A is φpAq :" Z ´1TrpA expp´βHq, where Z " Trpexpp´βHq.The precise mathematical encoding of this relation was obtained by Haag, Hugenholtz and Winnink [30], starting from earlier work of Kubo, Martin and Schwinger.A way to understand the KMS β condition is provided by the equality pϕpxσ t pyqqq t"iβ " ϕpyxq whose heuristic meaning is that σ t at t " iβ compensates for the lack of tracial property of the state ϕ by allowing one to replace ϕpyxq with ϕpxσ t pyqq at t " iβ.The states fulfilling the KMS β condition form a (possibly empty) convex compact simplex.The specific system that exhibits the interplay between the phenomenon of symmetry breaking in physics and the theory of ambiguity of E. Galois is the BC-system [3].It is defined using the affine group The subgroup P `pZq of integral translations obtained by requiring that a, b P Z is almost normal in P `pQq, and this fact allows one to define a Hecke algebra A in place of the convolution algebra of the quotient P `pQq{P `pZq.The action of A in the Hilbert space 2 pP `pQq{P `pZqq plays the role of the regular representation.The significant fact here is that this representation determines a factor of type III, thus naturally endowed with a one parameter group of automorphisms σ t of A (the time evolution).The pair pA, σ t q constitutes the BC-system.Its first properties are the following: § The system exhibits a phase transition with spontaneous symmetry breaking.The KMS β state is unique for β ď 1.For β ą 1 the extremal KMS β states are parameterized by the points of the zero-dimensional Shimura variety ShpGL 1 , t˘1uq.§ The symmetries of the system are given by the group GL 1 p Ẑq " Ẑ˚o f invertible elements of the profinite completion of the integers.The zero-temperature KMS states evaluated on a natural arithmetic subalgebra of the algebra of observables of the system take values that are algebraic numbers and generate the maximal abelian extension Q cycl of Q.
§ The class field theory isomorphism intertwines the action of the symmetries and the Galois action on the values of states, thus providing a quantum statistical mechanical reinterpretation of the explicit class field theory of Q. § The partition function Zpβq of the system is the Riemann zeta function evaluated at β P R.This last property establishes the link between noncommutative geometry and the Riemann zeta function.The algebra of the BC-system describes the quotient space Q ˆzA f of finite adeles of Q acted upon by the multiplicative group Q ˆ.When passing to the dual system using the dynamics, and combining the dual action of R ˚together with the symmetries GL 1 p Ẑq " Ẑ˚o f the system, one obtains the action of the idele class group on the adele class space Q ˆzA Q .This is the space that provides a geometric interpretation of the Riemann-Weil explicit formulas [6].This latter result was the starting point of a "longue marche" pursuing the study of the geometry of the adele class space.This space provides the spectral realization of the zeros of L-functions with Grössencharacter, where the Riemann zeta function is associated to the trivial character and whose related space is the "zeta-sector" X " Q ˆzA Q { Ẑ˚.This zeta-sector provides a Hasse-Weil formula for the Riemann zeta function using the action of R ˚on X [10,11].In view of this result it is clear that X may play the role of the space of the points of the curve for function fields.The geometric structure of X came with the discovery of the "Arithmetic Site" [12,13]: this is the presheaf topos x N ˆdual to the multiplicative monoid of positive integers, endowed with the structure sheaf provided by the only semifield F whose multiplicative group is infinite cyclic.The geometry of the Arithmetic Site is tropical and of characteristic one (the addition is unipotent: 1 `1 " 1).The structure sheaf of this topos is obtained by implementing the action of the semigroup N ˆon the semifield F by power maps x Þ Ñ x n .It is a general fact that in characteristic one the power maps define injective endomorphisms of a semifield and that there exists only one semifield which is finite and not a field, namely the Boolean semifield B :" t0, 1u.The Arithmetic Site is defined over B (because F is of characteristic one) and a key result is that the "zeta-sector" X gets canonically identified with the set of points of the Arithmetic Site defined over the semifield R max `of tropical real numbers.This semifield appears both in tropical geometry and also in semiclassical analysis as a limit of deformations of real numbers.One extremely convincing result of the dequantization program [33] is that the Fourier transform becomes the Legendre transform when taken to the classical limit.The semifield R max ìs an infinite extension of B and its absolute Galois group is determined by the power maps Aut B pR max `q " tFr λ | λ P R ˚u, Fr λ pxq :" This group acts on the points of the Arithmetic Site defined over R max `and, under the canonical identification of these points with the "zeta-sector" X " Q ˆzA Q { Ẑ˚, this action corresponds to the action of the idele class group.In spite of the fact that the Arithmetic Site is an object of countable nature (the semigroup N ˆand the semifield F are countable) and hence there is no non-trivial action of R ˚on the topos, R ˚acts meaningfully using the theory of correspondences [12,13].The extension of scalars of the Arithmetic Site to R max `determines the Scaling Site [14], namely the Grothendieck topos r0, 8q ¸Nˆ( N ˆacts by multiplication) endowed with the structure sheaf of continuous convex functions with integral slopes.The set of points of the topos r0, 8q ¸Nˆi dentifies canonically with the "zeta sector" X.The restriction of the structure sheaf of the Scaling Site to the periodic orbits in X determines, for each prime p, the quotient R ˚{p Z which appears in X as the counterpart of the prime-point p of Spec Z.The emerging tropical structure describes an analogue of an elliptic curve and it also exhibits a few totally new features.For instance, the divisor degree on these curves is a real number and the Riemann-Roch formula is real valued.Such real valued indices are ubiquitous in the noncommutative geometry of foliations and the tropical geometry of the Scaling Site can be lifted in complex geometry [18].In order to extend the geometric positivity argument used by Mattuck-Tate and Grothendieck for function fields, to the field of rational numbers and on the above geometric space one needs to show a Riemann-Roch formula holding on the square of the Scaling Site.In this respect, the case of periodic orbits is far too simplified since for curves one can bypass the construction of a cohomology theory for divisors beyond H 0 using Serre duality as a definition of H 1 .For surfaces, and in particular for the square of the Scaling Site, this trick handles only H 2 leaving H 1 still out of reach.One is thus faced with the problem of developing a good cohomology theory in characteristic one.Motivated by this application we developed a general theory of homological algebra for the (non abelian) category of B-modules [9], however the lack of the additive inverse makes the elimination of certain technical difficulties apparently quite hard.While trying to bypass this issue, we were lead to investigate a more fundamental base for algebraic manipulations, which is, as explained in the introduction, independent of the choice of a characteristic.The main reason for our turn of interests toward this new base is that it is the most natural one for Hochschild and cyclic homology theories.In the next section we show that the fundamental basis S provides the conceptual interpretation of the BC-system as the Witt construction over the algebraic closure of S.

The conceptual meaning of the BC-system
The convolution algebra of the quotient P `pQq{P `pZq has an integral model [8] §3, given by the Hecke algebra H Z " ZrQ{Zs ¸N.The ring endomorphisms σ n peprqq " epnrq, n P N act on the canonical generators of the group ring eprq P ZrQ{Zs, r P Q{Z.There are natural quasi-inverse linear maps ρn : ZrQ{Zs Ñ ZrQ{Zs , ρn pepγqq " These two operators are used both in the definition of the crossed product ZrQ{Zs ¸N and in the presentation of the algebra.
There is a striking analogy between the algebraic rules fulfilled by the pair tσ n , ρn u and the relations fulfilled, in the global Witt construction, by the Frobenius and Verschiebung operators.
The invariant part of the group ring ZrQ{Zs for the action of the group AutpQ{Zq " p Z ˚is described in terms of Almkvist's ring of endomorphisms W 0 pSq as follows Theorem 2.1.( [22] Theorem 2.3) The ring W 0 pSq is canonically isomorphic to the invariant part of the group ring ZrQ{Zs for the action of the group AutpQ{Zq " p Z ˚.
In this section we extend Theorem 2.1 by showing a natural isomorphism of the group ring ZrQ{Zs with the ring W 0 pSq, where S denotes the monoid S-algebra SrM s of the multiplicative pointed monoid M " pQ{Zq `, with elements the base point ˚" 0 and the eprq's for r P Q{Z.The multiplication in M is defined by: eprqepsq " epr `sq @ r, s P Q{Z.The functor S : Γ op ÝÑ Sets ˚, is defined by SrXs " X ^M , where the monoid structure in M yields the algebra structure SrXs ^SrY s Ñ SrX ^Y s.

Endomorphisms and matrices
In [22] we considered the class of S-modules of the form SrF s " S ^F , where F is a finite object of the category Sets ˚of pointed sets.As a functor SrF s : Γ op ÝÑ Sets ˚associates to a finite pointed set X the smash product SrF spXq :" F ^X and to a map of finite pointed sets g : X Ñ Y the map SrF spgq :" Id ^g.An endomorphism of SrF s is a natural transformation.Lemma 2.2.Let F, F 1 be two finite objects in Sets ˚.The map where φp1 `q denotes the restriction of φ to 1 `" t0, 1u is a bijection of sets.The inverse map is where ψpXq " Id X ^ψ : X ^F Ñ X ^F 1 .
Proof.Let φ P Hom S pSrF s, SrF 1 sq and X a finite pointed set.An element y P SrF spXq " F ^X, y ‰ ˚, is determined by a pair y " pf, xq P F ˆX, and there exists a (unique) map of pointed sets g : 1 `Ñ X with gp1q " x.By the naturality of the transformation φ one has: φ ˝SrF spgq " SrF spgq ˝φ.This shows that φ is uniquely determined by its restriction φp1 `q on SrF sp1 `q " F , with φp1 `q P Hom Sets˚p F, F 1 q.Conversely, given ψ P Hom Sets˚p F, F 1 q one associates to it the natural transformation ψ : SrF s Ñ SrF 1 s that maps a finite pointed set X to the map Id X ^ψ : SrF s Ñ SrF 1 s.It is immediate to verify that the two maps are inverse of each other.
In the following part we shall consider endomorphisms of S-modules of the form SrF s " S^F , with F a finite pointed set.For n P N, n `:" t0, 1, . . ., n ´1, nu.
n pSq be the multiplicative pointed monoid of nˆn matrices with entries in the multiplicative monoid Sp1 `q " M " pQ{Zq `, which have only one non-zero (i.e.not equal to the base point ˚) entry in each column.
Given µ " pµ ij q P Mat R n pSq one defines a map of pointed sets by setting ρpµq : M ^n`Ñ M ^n`ρ pµqpα, jq :" Note that for µ P Mat R n pSq, there exists, for a given j, at most one i P t1, . . ., nu with µ ij ‰ ˚.
Proof.With µ " pµ ij q P Mat R n pSq, Ć ρpµq defines a natural transformation hat commutes with the action of M .Thus it determines an endomorphism Ć ρpµq P End S pSrF sq.Let µ, µ 1 P Mat R n pSq: their product is given by By applying (2) one gets: ρpµµ 1 q " ρpµq ˝ρpµ 1 q, since ρpµq ˝ρpµ 1 qpα, kq ‰ ˚if and only if there exist j with µ 1 jk ‰ ˚and i with µ ij ‰ ˚.In that case, one has: ρpµq ˝ρpµ 1 qpα, kq " pµ ij µ 1 jk α, iq " ppµµ 1 q ik α, iq.This shows that ρ is a multiplicative map.It is injective by construction.Next we show that it is also surjective.Let φ P End S pSrF sq.Then by Lemma 2.2 φ " ψ where ψ is the restriction φp1 `q.This restriction commutes with the action of M on pSrF sqp1 `q " F ^M and thus it is given by a matrix ρpµq acting as in (2).
For a given S-algebra A, we denote by Mat R n pAq the S-algebra of matrices over A defined in [28] ( §2.1.4.1, example 2.1.4.3, 6).Note that, up to transposition, there are two equivalent definitions for such matrices: we let Mat L n pAq be the functor (S-algebra) from finite pointed sets to pointed sets that maps a finite pointed set X to the set of n ˆn matrices of elements of ApXq with only one non-zero entry in each row.Similarly, Mat R n pAq is the functor mapping a finite set X to the set of n ˆn matrices of elements of ApXq with only one non-zero entry in each column.
Next proposition shows that one can define a bimodule Mat n pAq over these two S-algebras as the functor from finite pointed sets to pointed sets mapping X to the set of n ˆn matrices of elements of ApXq with no restriction on the matrix entries.The proposition is in fact a special case of the composition law for S-algebras viewed as endofunctors.Mat n pAqpXq ˆMat n pAq R pY q Ñ Mat n pAqpX ^Y q turns Mat n pAq into a right module over Mat R n pAq.
Proof.The proof is the same as the one in [28]: one simply needs to check that the product in the S-algebra A determines a well defined product of matrices.To this end, the point is that the sum involved in determining the matrix element at position pi, jq is obtained from a row by column product of two matrices that only contain one non-zero term.This fact holds as long as either the rows of one matrix or the columns of the other one contain only one non-zero element: this is the case in (i) and (ii).
For A " S and X " 1 `there is an isomorphism of pointed monoids1 Mat R n pSqp1 `q " Mat R n pSq.Moreover, the set of matrices Mat n pSq " M n pSrM sq (M " pQ{Zq `) coincide with Mat n pSqp1 `q.By Proposition 2.5, they form a bimodule with right and left actions provided by Mat R n pSq acting on the right of Mat n pSq by matrix multiplication and by Mat L n pSq " Mat L n pSqp1 `q acting similarly on the left.The role of the bimodule Mat n pSq " Mat n pSqp1 `q is to encode similarities.Given a field k and the associated S-algebra Hk, morphisms of S-algebras S " SrM s Ñ Hk correspond bijectively to (multiplicative) monoid homomorphisms M Ñ k ([23] Proposition 2.2 piq).In particular, to an injective morphism M Ñ k corresponds an extension of S by the field k.In view of this fact, we introduce the following Definition 2.6.An element α P Mat n pSq " M n pSrM sq is invertible if and only if the matrix α P M n pkq is invertible in all field extensions k of S.
Matrix similarity in Mat n pSq is stable by taking powers, as illustrated by the following Lemma 2.7.Let α P Mat n pSq, µ P Mat R n pSq, γ P Mat L n pSq, such that γα " αµ.Then one has γ k α " αµ k for all k P N.
In view of Proposition 2.4, it is equivalent to consider endomorphisms T P End S pSrF sq (F finite pointed set) of S-modules E " SrF s, or matrices µ P Mat R ˚pSq, where ˚is the integer recording the cardinality of the complement of the base point in F .One defines the notion of invariant (of endomorphisms) as follows Definition 2.8.An invariant is a map χ : Mat R ˚pSq Ñ R to a commutative ring R that satisfies the following conditions: where the smash product is taken over S (iii) χ is invariant under similarity, i.e. χpγq " χpµq if γα " αµ for an invertible matrix α P Mat ˚pSq.
Condition (i) is the same as in Definition 2.2 of [22], and has the role to mod out the zero endomorphisms.The second condition implements the ring structure.Finally, (iii) realizes invariance under similarity.

Construction of the universal invariant
We shall define the universal invariant of endomorphisms after applying the extension of scalars from S to the maximal cyclotomic extension of Q.In that set-up Almkivst's original result applies and associates to (square) matrices a divisor with coefficients in the multiplicative group of the field.Our result states that the divisor has coefficients in the group of roots of unity.Next proposition gives the construction of the invariant of endomorphisms.We keep the same notations of §2.1, in particular M " pQ{Zq `denotes the multiplicative, pointed monoid of abstract roots of unity.Proposition 2.9.Let T P Mat R n pSq, and κ : M ãÑ k be an injective morphism into an algebraically closed field extension of S of characteristic zero.
(i) The divisor D defined by Almkvist's invariant of κpT q P Mat n pkq has coefficients in κpM ˆq.
Proof.(i) Let t " pt ij q P M n pkq be a matrix whose non-zero entries are roots of unity and with at most one non-zero element t ij in each column.We claim that the eigenvalues of t are either 0 or roots of unity.Let E " k n be the k-vector space on which t acts.The subspaces E j :" t j pEq form a decreasing filtration of E for which there exists a finite index such that E `1 " E .The non-zero eigenvalues of t are the same as the eigenvalues of the restriction t of t on E .We verify that the endomorphism t has finite order.Indeed, let The range of φ labels a basis of E : in this basis the matrix of t describes the permutation obtained by restricting φ, whose entries are in roots of unity.Such a matrix is periodic thus all of its eigenvalues are roots of unity.This shows that Almkivst's invariant of κpT q P Mat n pkq, i.e. a divisor D with coefficients in k ˆ, has in fact coefficients in κpM ˆq.Moreover one also derives that the divisor τ pT q :" κ ´1pDq with coefficients in M ˆis independent of the choice of κ.
(ii) The map τ fulfills the three conditions of Definition 2.8 since they hold true for Almkivst's invariant, in particular the operations in condition (ii) correspond to direct sum and tensor product of modules.

Completeness of the invariant τ
To prove that the above construction defines a universal invariant, one applies the same proof as in Theorem 3.3 of [22] (in the case of endomorphisms of finite S-modules), to show the injectivity of τ .The main fact to verify is that by implementing the (algebraic) Fourier transform one can diagonalize any matrix in Mat n pSq corresponding to a permutation at the set level.We shall see (in the proof of Theorem 2.11) that for a cycle of such permutation one can choose a basis so that the matrix of such permutation is equivalent to the cyclic permutation matrix multiplied by a root of unity.Next proposition gives the algebraic relation between the endomorphisms determined by the cyclic permutation matrix Cpnq of order n Cpnq ij :" # 1 if i " j `1 pnq 0 otherwise , @i, j P t0, . . ., n ´1u and the diagonal matrix ∆pnq whose entries are the full set of n-th roots of unity, i.e. ∆pnq jj " epj{nq for j P t0, . . ., n ´1u.The proposition shows that there is a non-trivial algebraic relation between ∆pnq and Cpnq, and by Lemma 2.7 the same relation holds true when arbitrary powers of these two matrices are involved.
Proposition 2.10.For n P N, let µ n :" tepa{nq | a P Z{nZu be the group of n-th roots of unity.
(i) The matrix V " pV ij q P Mat n pSrµ n sq: V ij " epij{nq is the matrix of the Fourier transform on the cyclic group Z{nZ.
(ii) In any field extension of Srµ n s one has n ‰ 0, and the inverse of V is, up to the overall factor n, the matrix W " pW ij q: W ij " ep´ij{nq.
(iii) The following relations hold Proof.(i) It suffices to recall that the Fourier transform on the cyclic group Z{nZ is the transformation F of functions f : Z{nZ Ñ C defined by F pf qpaq " ÿ pab{nqf pbq, pxq :" expp´2πixq , @x P R.
(ii) Let k be a field extension of Srµ n s, then k contains n distinct roots of unity of order n, thus the characteristic of k is prime to n.It follows that n ‰ 0 in k and the inverse of V is 1 n W . (iii) One checks (4) by direct computation using the equality epxqepyq " epx `yq.
We can now state and prove the main result of this section Theorem 2.11.The ring W 0 pSq is canonically isomorphic to the group ring ZrQ{Zs.The invariant τ : Mat R ˚pSq Ñ ZrQ{Zs is universal and it extends the additive invariant of Theorem 2.1.
Proof.Let χ : Mat R ˚pSq Ñ R be an invariant (thus fulfilling the conditions of Definition 2.8), then consider the map β : Q{Z Ñ R, βprq :" χpreprqsq, @r P Q{Z, where reprqs is the endomorphism of the one dimensional module S given by multiplication by eprq.By Definition 2.8 (ii), β : Q{Z Ñ R ˆis a group homomorphism and hence it extends to a ring homomorphism β : ZrQ{Zs Ñ R. Next, we show that χ " β ˝τ .Let T P End S pSrF sq, we prove that χpT q " βpτ pT qq.We identify F " n `and let µ " pµ ij q be the matrix with ρpµq " T (see Proposition 2.4).Let φ : n `Ñ n be the map as in (3).The ranges X of the powers φ form a decreasing sequence of subsets and we let be such that X " X `1.The matrix of the restriction of T to X is the matrix of the permutation obtained by restricting φ, thus using Definition 2.8 (i) we can just consider the case where µ is the matrix of a permutation with entries in roots of unity.The required additivity in (ii) of Definition 2.8, allows one to assume that the permutation is a cyclic permutation.Then we observe that the matrix of a cyclic permutation ν with entries roots of unity is equivalent, using a diagonal matrix whose entries are ratios of entries of ν, to the matrix of a cyclic permutation of type epsqCpmq, whose entries are all the same root of unity epsq.It then follows from Proposition 2.10 and Definition 2.8 (iii) that χpT q " χpepsq∆pmqq, and finally, using again (ii) of Definition 2.8, one obtains χpT q " βpτ pT qq.

Frobenius and Verschiebung
The Frobenius endomorphisms and the Verschiebung maps are operators in W 0 pSq [22].The Frobenius ring endomorphisms F n , n P N, are defined by the equality F n ppE, T qq :" pE, T n q (we refer to op.cit.for the notations).One easily checks that τ pF n pxqq " σ n pτ pxqq , @n P N, x P W 0 pSq, where the group ring endomorphism σ n is defined, for each n P N, by The Verschiebung maps V n replace a pair pE, T q by the endomorphism of the sum _ n E that cyclicly permutes the terms and uses T : E Ñ E to turn back from the last term to the first.The map V n is additive by construction and when applied to a one dimensional S-module repaqs it gives the sum of the repbqs's where nb " a.One thus obtains τ pV n pxqq " ρn pτ pxqq , @n P N, x P W 0 pSq, where ρn , n P N, is defined by ρn : ZrQ{Zs Ñ ZrQ{Zs, ρn pepγqq " Then, in analogy to and generalizing what stated in [22] §2.2, one derives the following Theorem 2.12.The correspondences σ n Ñ F n , ρn Ñ V n determine a canonical isomorphism of the integral BC-system (i.e. the Hecke algebra H Z " ZrQ{Zs ¸N) with the Witt ring W 0 pSq endowed with the Frobenius and Verschiebung maps.

Analytic approach and noncommutative geometry
On September 27 1993, Dennis Sullivan sent to the first author a fax, reproduced in Figure 1, that sets the scene of the interactions between noncommutative geometry and geometry of manifolds 2 .In [7], the account of the analytic approach based on [6] was reduced to the minimum.This section is dedicated to explain our recent results on this analytic approach.Two main tools in noncommutative geometry play a key role here, they are: -The quantized calculus -The notion of spectral triple The quantized calculus is applied in the semilocal framework and it provides, through the semilocal trace formula, both the operator theoretic formalism for the explicit formulas of Riemann-Weil and a conceptual reason for Weil's positivity ( § §3.2, 3.3, 3.4).Spectral triples (through Dirac operators) together with the understanding of the radical of the Weil quadratic form restricted to an interval rλ ´1, λs using prolate functions, and the implementation of the map Epf qpxq :" x 1{2 ř λ 1 f pnxq, allow one to detect the zeros of the Riemann zeta function up to imaginary part 2πλ 2 , thus providing the operator theoretic replacement for the Riemann-Siegel formula in analytic number theory ( § §3.5, 3.6).Both notions make essential use of operators in Hilbert space and of the following dictionary Conformal Geometry Fredholm module pA, H, F q, F 2 " 1. Perturbation by Beltrami differential F Þ Ñ paF `bqpbF `aq ´1, a " p1 ´µ2 q ´1{2 , b " µa Distributional derivative Quantized differential dZ :" rF, Zs

Schwartz kernels and Schwarzian derivative
Let V be a 1-dimensional manifold and let H " L 2 pV q be the Hilbert space of square integrable half densities: ξpxq " f pxqdx 1{2 P H.The Schwartz kernel of an operator T : H Ñ H is of the form: k T px, yqdx 1{2 dy 1{2 .This means that, as a function of two variables, the kernel depends on choices of positive sections of the 1-dimensional bundle of 1{2-densities.By varying the choice of a section, i.e. by dividing it with a positive function ρpxq, the kernel k T px, yq gets modified accordingly to ρpxqρpyqk T px, yq.Next lemma detects an invariant of the above change Lemma 3.1.The differential form is independent of the choice of sections of the bundle of 1/2-densities and defines an invariant ωpT q of the operator T .
Proof.One sees that by taking the log of the variation of the Schwartz kernel logpρpxqρpyqkpx, yqq " logpkpx, yqq `log ρpxq `log ρpyq and then applying B x B y , the output is independent of ρ.
Let now V " R and T " F the Fourier transform, then k F px, yq " expp´2πixyq and the differential form becomes ωpFq " ´2πidxdy.Note that the Schwartz kernel of the Fourier transform already appeared in §2.3 (in matrix form) and there it played a crucial role in the determination of the K-theory of endomorphisms of S.
Next, we compute ωp df q for the quantized differential of a function f on R. Lemma 3.2.Let f be a smooth, complex valued function on R. Then ωp df q " ˆf 1 pxqf 1 pyq pf pxq ´f pyqq 2 ´1 px ´yq 2 ˙dxdy.
The restriction of ωp df q to the diagonal is 1  6 Spf qdx 2 , where Spf q " f p3q pxq Proof.The Schwartz kernel of the quantized differential is kpx, yq " i π f pxq´f pyq x´y . By applying (9) one obtains the stated equality (10).
Note, in particular, that the second statement of the above lemma shows that the quantized differential determines the Schwarzian derivative.

The Main Lemma
The conceptual reason for the link between Weil's explicit formula and the trace of the compression of the scaling action on Sonin's space [24,16] is rooted in the following two general facts.
Let H be a Hilbert space, and let F " 2P ´1 be the operator defining the quantized calculus.
An operator T in H is encoded by a matrix T "  with T 11 " p1 ´P qT p1 ´P q, T 12 " p1 ´P qT P, T 21 " P T p1 ´P q, T 22 " P T P be the upper-triangular matrix of an operator in H. Then U is unitary if and only if the following conditions hold (i) U 11 is an isometry (ii) U 22 is a coisometry (iii) U 12 is a partial isometry from kerpU 22 q to the cokerpU 11 q Next Lemma (see also [15] Lemma 3.4) relates the sign of the quantized differential of a triangular unitary operator U to the kernel of the compression P U P of U on P (corresponding to Sonin's space in the application related to Weil's explicit formula).Lemma 3.4.With the notation of Proposition 3.3, let f be a positive operator and S the orthogonal projection to kerpU 22 q.Let S " Proof.We first show that ´1 2 U ˚dU " S. We have Then it follows that U ˚P U ´P " . One also has U 22 U 22 ´1 " ´S since U 22 is a coisometry.Then the claim follows from the equality U ˚dU " 2pU ˚P U ´P q together with the fact that the trace of a product of two positive operators is nonnegative.

The semilocal functional equation
In this part we explain the functional equation in the semilocal case, by giving a proof of the local functional equation that extends naturally to the semilocal framework.We first introduce some notations.We denote by I the unitary inversion operator in the subspace L 2 pRq ev of even functions in L 2 pRq defined as Ipξqpxq :" |x| ´1ξpx ´1q.The scaling operator ϑpλq, defined for λ P R ˚, is the unitary operator in L 2 pRq ev given by ϑpλqpξqpxq " λ ´1{2 ξpλ ´1xq.One has I ˝ϑpλq " ϑpλ ´1q ˝I.The Fourier transform F e R is the unitary operator in L 2 pRq ev defined by One has F e R ˝ϑpλq " ϑpλ ´1q ˝Fe R .It follows that I ˝Fe R commutes with the scaling pI ˝Fe R q ˝ϑpλq " I ˝pF e R ˝ϑpλqq " I ˝pϑpλ ´1q ˝Fe R q " pI ˝ϑpλ ´1qq ˝Fe R " ϑpλq ˝pI ˝Fe R q.
The representation ϑ of R ˚by scaling in L 2 pRq ev is unitarily equivalent to the multiplication action with the character λ ´is in L 2 pRq, through the Fourier transform F µ ˝w, where w is the unitary isomorphism w : L 2 pRq ev Ñ L 2 pR ˚, dx{xq, wpξqpλq " λ 1{2 ξpλq @λ ą 0 and F µ denotes the multiplicative Fourier transform F µ pf qpsq :" The von Neumann algebra generated in L 2 pRq by the multiplications operators by λ ´is is equal to L 8 pRq acting by multiplication.The Bicommutant Theorem ensures that a unitary operator commuting with this representation is a multiplication operator by a function of modulus one on R. Then it follows that the composite operator in L 2 pRq pF µ ˝wq ˝pI ˝Fe R q ˝pF µ ˝wq ´1 (11) is the multiplication by a function u P L 8 pRq of modulus one.Next, we shall develop on the following schematic diagram (Figure 2) Proof.Let f P SpRq be an even function with f p0q " 0 " ş f pxqdx.The Poisson formula (x ą 0) then we have: Epf qpxq " EpF e R f qpx ´1q.By applying the multiplicative Fourier F µ on the right hand side of this equality, one has (s P R) F µ pEpf qqpsq " ζp1{2 ´isqF µ pwf qpsq (14) and by the above equality: F µ pEpf qqpsq " F µ pEpF e R f qqp´sq.Thus with (14) one obtains Finally, in view of (11) one writes pF µ ˝wq ˝pI ˝Fe R qpf qpsq " F µ pwF e R f qp´sq " ζp1{2 ´isq ζp1{2 `isq F µ pwf qpsq Thus u " ζp1{2´isq ζp1{2`isq is the ratio of local factors.
An argument similar to the one just developed in the above proof applies in the semilocal case (when finitely many places are involved, inclusive the archimedean) [15].More precisely, one lets S be a finite set of places of Q containing the archimedean place and one considers the semilocal adele class space where A Q,S " ś vPS Q v is the product of the local fields acted upon (by multiplication) by the subgroup Even though X Q,S is a noncommutative space, at the topological level and when S contains at least three places, this space is well behaved at the measure theory level because the additive and multiplicative Haar measures are equivalent on the finite product of local fields [6,25].There is a natural Hilbert space L 2 pX Q,S q of square integrable functions on X Q,S .Moreover, and very importantly, the Fourier transform F α on A Q,S descends to a unitary operator F α in L 2 pX Q,S q.
After passing to the dual of the group C Q,S " GL 1 pA Q,S q{Γ by the Fourier transform F C and using the inversion I, F α reads as the multiplication by a function u of modulus 1 on the dual of C Q,S [25] (Chapter 2) Let π v be the projection from the dual of C Q,S to the dual x Q v , the unitary u is of the form where the u v P L 8 p x Q v q are the functions involved in the local functional equation of Tate [39] Here, F αv denotes the Fourier transform relative to the canonical additive character α v of the local field Q v .One knows that the function u v is the ratio of the local factors of L-functions.When restricting to the "zeta sector" i.e. to the subspace of L 2 pX Q,S q of functions invariant under the action of the maximal compact subgroup of C Q,S , the function u is the product of ratios of local factors of the Riemann zeta function: u " ρ 8 ś ρ p .This can be proved directly using the same argument as in the above proof of Fact 3.5 (see [15]).In [16], we have developed the notion of quasi-inner function as a generalization of Beurling's notion of inner function which we first related to the main Lemma 3.4.In Theorem 4.1 of op.cit.we showed that the product u " ρ 8 ś ρ p of ratios of local factors of the Riemann zeta function is a quasi-inner function Theorem 3.6.The product u " ρ 8 ś ρ p of ratios of local factors over a finite set of places of Q containing the archimedean place is a quasi-inner function relative to C ´" tz P C | pzq ď 1 2 u.
This fact shows that the quantized differential u ˚du fulfills the hypothesis of the main Lemma 3.4 "modulo infinitesimals" i.e. working in the Calkin algebra quotient of the algebra of bounded operators by the ideal of compact ones.

Quantized calculus and the semilocal trace formula
The key fact enabling the development of the semilocal framework of §3.3 is the equivalence of the additive and multiplicative Haar measures on a finite product of local fields.This fact fails in the global adelic case.The reason why one can go around this difficulty in order to understand the location of the zeros of the Riemann zeta function is that Weil's criterion only involves finitely many primes at a time.Indeed, while the sum on the right is extended to all places of Q, for v " p a non archimedean prime, the functional vanishes on test functions with support in the interval pp ´1, pq.Thus W v pg ˚g˚q ‰ 0 for only finitely many v.The functionals W v are given by the Riemann-Weil explicit formula where Z is the multi-set of the non-trivial zeros of the Riemann zeta function.The archimedean distribution W R is defined as The key equality now is the local trace formula of [6] (as revisited in [25]) where the notations for the right hand side are as in §3.3.Lemma 3.4 would imply the negativity criterion (20), if u fulfilled the required hypothesis of that lemma (in §3.3 we pointed out that the failure is only by an infinitesimal).When S is reduced to the single archimedean place, this difficulty can be bypassed by analyzing the effect of the infinitesimal [24], and the expected negativity of the criterion can be derived from Lemma 3.4, where the role of the orthogonal projection to kerpU 22 q is played by the orthogonal projection S on Sonin's space, one has In the next §3.5 we shall see that a difficulty to extend this result in the semilocal case is that the restriction of Weil's quadratic form to test functions with support in the interval rλ ´1, λs Ă R ˚admits (for large values of λ) extremely small eigenvalues.This fact prevents the use of the approximation method developed in [24].

The radical of the Weil quadratic form
Weil's quadratic form QW pf, gq :" admits a non-trivial radical when working with general test functions (i.e. if one drops the compact support condition of ( 20)).This radical contains the range of the map E defined on the codimension two subspace S ev 0 of even Schwartz functions fulfilling the equalities f p0q " 0 " p f p0q grasp the zeros of the Riemann zeta function up to height t " 2πµ, by computing the spectra of the operators Dpλ, kq for λ 2 ď µ.The computation of the involved prolate vectors only requires the use of integers less than the integer part of λ, because all other terms in the sum involved in the definition of the map E vanish due to the support condition.This means that we only use integers n between 1 and λ.In this way we have found a remarkable agreement with the first 31 zeros of zeta only implementing the integers 2, 3, 4!.This fact is all the more remarkable since the above restriction on the involved integers (in the sum defining the map E) coincides exactly with the restriction in the partial sums occurring in the Riemann-Siegel formula (see [2] and [35] §6.1).While these findings are largely depending on computer calculations of spectra of large matrices, we have provided their conceptual explanation by introducing the notion of zeta-cycle [17].With Σ µ denoting the Poincaré series operator, i.e. the linear map defined on functions g : R ˚Ñ C by the formula pΣ µ gqpuq :" We can state the following result, where for µ ą 1 one lets C be the circle C :" R ˚{µ Z Theorem 3.8.piq The spectrum of the action of the multiplicative group R ˚on the orthogonal of Σ µ EpS ev 0 q in L 2 pCq is formed of imaginary parts of zeros of zeta on the critical line.piiq Let s ą 0 be such that ζp 1  2 `isq " 0, then any circle of length an integral multiple of 2π{s is a zeta cycle and its spectrum contains is.
Remark 3.9.The spectral triples Θpλ, kq " pApλq, Hpλq, Dpλ, kqq have the same ultraviolet spectral behavior as the Dirac operator D 0 " ´iuB u on the circle C " R ˚{λ 2Z .In particular the number of eigenvalues with absolute value less than E grows linearly with E. The ultraviolet behavior of the zeros of the Riemann zeta function is given by Riemann's formula for the number N pEq of zeros of imaginary part between 0 and E, N pEq " The problem of finding a Dirac operator with the ultraviolet behavior (30) is solved by the first author and H. Moscovici in [26].Remarkably, the solution involves the prolate spheroidal wave operator W λ whose commutation with the projections P λ and p P λ plays a key role in [37,36,38].

Prolate vectors and the semilocal framework
The construction of the prolate vectors (and of the projection Πpλ, kq) makes use of the map E. In this part we exhibit the relation between this construction and the semi-local framework, showing that the map E appears naturally in the quotient X Q,S for functions with small enough support.

. , uqq
The terms in the sum vanish unless g is an integer since otherwise gp1, 1, . ..qR ś Z v .Moreover in that case the terms are equal to f pguq.Thus the sum becomes wp f qpuq " u 1{2 ÿ gPZXΓ f pguq.
Assume u ą λ ´1.Then for any integer n, with |n| ą µ " λ 2 , one has f pnuq " 0 since the support of f is in r´λ, λs.Moreover the following sets are equal since all prime factors of integers less than µ are in S Y " Z X Γ X r´µ, µs " t˘n | n P N, 0 ă n ď µu and thus for u ą λ ´1 one obtains (since f is even) wp f qpuq " u 1{2 ÿ gPY f pguq " 2Epf qpuq which gives the required equality.
For each prime p the characteristic function 1 Zp is its own Fourier transform on Q p and this implies that the semi-local Fourier transform F α acts as the archimedean Fourier transform F e R on functions f on the quotient X Q,S , associated as above to simple tensors b1 Zp b f .Thus Moreover if the support of f is contained in the ball of radius λ the same holds for f .Together with Proposition (3.10) this fact suggests that the minuscule eigenvalues of the Weil quadratic form of §3.5 can be reinterpreted intrinsically in the semilocal framework, without using the map E, just by analyzing the relative position of the semilocal analogue of the cutoff projections P λ and x P λ .

Proposition 2 . 5 .
Let A be an S-algebra.The following facts hold (i) The action of Mat L n pAq on Mat n pAq by left multiplication Mat L n pAqpXq ˆMat n pAqpY q Ñ Mat n pAqpX ^Y q turns Mat n pAq into a left module over Mat L n pAq.(ii) The action of Mat R n pAq on Mat n pAq by right multiplication

Figure 1 :
Figure1: "...After lecturing it became clear that pA, H, F q is really the conformal structure, as you always say, and the homotopy class pA, H, F 1 q is really the first instance of the quasiconformal structure... "

Figure 2 :Fact 3 . 5 .
Figure 2: The Poisson formula yields a formula for the unitary u as a ratio of the zeta function on the critical line and the functional equation for the complete zeta function expresses this ratio as the ratio of local archimedean factors.

X Q,S associated to the function b vPSz8 1
Zv b f where 1 Zv is the characteristic function of the maximal compact subring Z v Ă Q v .Then the following equality holds wp f qpuq " 2Epf qpuq @u ą λ ´1.(31) Proof.For u P R ˚one has by construction wp f qpuq " u 1{2 ÿ gPΓ pb vPSz8 1 Zv b f qpgp1, 1, . .
[24]rem 3.7.([24])Let g P C 8 c pR ˚q have support in the interval r2 ´1{2 , 2 1{2 s and Fourier transform vanishing at i 2 and 0. Then the following inequality holds ´WR pg ˚g˚q " W 8 pg ˚g˚q ě Trpϑpgq S ϑpgq ˚q.