We show that the non-commutative geometric approach to the Riemann zeta
function has an algebraic geometric incarnation: the "Arithmetic Site". This
site involves the tropical semiring viewed as a sheaf on the topos which is the
dual of the multiplicative semigroup of positive integers. We prove that the
set of points of the arithmetic site over the maximal compact subring of the
tropical semifield is the non-commutative space quotient of the adele class
space of Q by the action of the maximal compact subgroup of the idele class
group. We realize the Frobenius correspondences in the square of the
"Arithmetic Site" and compute their composition. This note provides the
algebraic geometric space underlying the non-commutative approach to RH.Comment: 1 Figure, submitted to Comptes Rendu

Connes, Alain

Consani, Caterina

English

arXiv

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 352 (2014) 971–975Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comNumber theory/Algebraic geometryThe Arithmetic Site ✩Le Site arithmétiqueAlain Connes a,b,c, Caterina Consani d,1a Collège de France, 3, rue d’Ulm, 75005 Paris, Franceb I.H.E.S., Francec Ohio State University, USAd The Johns Hopkins University, Baltimore, MD 21218, USAa r t i c l e i n f o a b s t r a c tArticle history:Received 18 May 2014Accepted 7 July 2014Available online 8 October 2014Presented by the Editorial BoardWe show that the non-commutative geometric approach to the Riemann zeta function has an algebraic geometric incarnation: the “Arithmetic Site”. This site involves the tropical semiring N̄ viewed as a sheaf on the topos N̂× dual to the multiplicative semigroup of positive integers. We realize the Frobenius correspondences in the square of the “Arithmetic Site”.© 2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éLe « Site arithmétique » est l’incarnation en géométrie algébrique de l’espace non commu-tatif, de nature adélique, qui permet d’obtenir la fonction zêta de Riemann comme fonction de dénombrement de Hasse–Weil. Ce site est construit à partir du semi-anneau tropical N̄ vu comme un faisceau sur le topos N̂× dual du semigroupe multiplicatif des entiers positifs. Nous réalisons les correspondances de Frobenius dans le carré du « Site arithmétique ».© 2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionWe unveil the “Arithmetic Site” as a ringed topos deeply related to the non-commutative geometric approach to RH. The topos is the presheaf topos N̂× of functors from the multiplicative semigroup N× of positive integers to the category of sets. The structure sheaf is a sheaf of semirings of characteristic 1 and (as an object of the topos) is the tropical semiring N̄ := (N ∪ ∞, inf, +), N = Z≥0, on which the semigroup N× acts by multiplication. We prove that the set of points of the arithmetic site (N̂×, N̄) over the maximal compact subring [0,1]max ⊂ Rmax+ of the tropical semifield is the non-commutative space Q×\AQ/Ẑ∗ , quotient of the adèle class space of Q by the action of the maximal compact subgroup Ẑ∗ of the idèle class group. In [5,6] it was shown that the action of R∗+ on Q×\AQ/Ẑ∗ yields the counting distribution whose Hasse–Weil ✩ Both authors thank Ohio State University where this paper was written.E-mail addresses: alain@connes.org (A. Connes), kc@math.jhu.edu (C. Consani).1 Partially supported by the NSF grant DMS 1069218.http://dx.doi.org/10.1016/j.crma.2014.07.0091631-073X/© 2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.972 A. Connes, C. Consani / C. R. Acad. Sci. Paris, Ser. I 352 (2014) 971–975zeta function is the complete Riemann zeta function. This result is now applied to the arithmetic site to show that its Hasse–Weil zeta function is the complete Riemann zeta function. The action of R∗+ on Q×\AQ/Ẑ∗ indeed corresponds to the action of the Frobenius automorphisms Frλ ∈ Aut(Rmax+ ), λ ∈ R∗+ , on the points of (N̂×, N̄) over [0,1]max ⊂ Rmax+ . The square of the arithmetic site over the semifield B = ({0, 1}, max, ×) has an unreduced and a reduced version. In both cases the underlying topos is N̂×2. The structure sheaf in the unreduced case is N̄ ⊗B N̄ and in the reduced case it is the multiplicatively cancellative semiring canonically associated with N̄⊗B N̄. We determine this latter semiring as the semiring Conv≥(N × N) of Newton polygons with the operations of convex hull of the union and sum. On both versions there is a canonical action of N×2 by endomorphisms Frn,m . By composing this action with the diagonal (given by the product μ), one obtains the Frobenius correspondences Ψ (λ) = μ ◦ Frn,m for rational values λ = n/m. The Frobenius correspondences Ψ (λ)for arbitrary positive real numbers λ are realized as curves in the square obtained from the rational case using diophantine approximation. Finally we determine the composition law of these correspondences and show that it is given by the product law in R∗+ with a subtle nuance in the case of two irrational numbers whose product is rational.This note provides the algebraic geometric space underlying the non-commutative approach to RH. It gives a geometric framework reasonably suitable to transpose the conceptual understanding of the Weil proof in finite characteristic as in [7]. This translation would require in particular an adequate version of the Riemann–Roch theorem in characteristic 1.2. The arithmetic siteGiven a small category C , we denote by Ĉ the topos of contravariant functors from C to the category of sets. We let N×be the category with a single object ∗, End(∗) = N× .Definition 2.1. We define the arithmetic site (N̂×, N̄) as the topos N̂× endowed with the structure sheaf N̄ := (N ∪ ∞, inf, +)viewed as a semiring in the topos.Notice that N̂×  Sh(N×, J), where J is the chaotic topology on N× ([1] Exposé IV, 2.6).2.1. The points of the topos N̂×A point of a topos T is defined as a geometric morphism from the topos of sets to T [1,8].Theorem 2.2.(i) The category of points of the topos N̂× is canonically equivalent to the category of totally ordered groups isomorphic to non-trivial subgroups of (Q, Q+), and injective morphisms of ordered groups.(ii) Let A f be the ring of finite adèles of Q. The space of isomorphism classes of points of N̂× is canonically isomorphic to the double quotient Q×+\A f /Ẑ∗ where Q×+ acts by multiplication on A f .We denote by F = Zmax the semifield of fractions of the semiring N̄.Corollary 2.3. The category of points of the topos N̂× is equivalent to the category of algebraic extensions of the semifield F = Zmaxi.e. of extensions: F ⊂ K ⊂ Qmax . The morphisms are the injective morphisms of semifields.2.2. The structure sheaf N̄The next result provides an explicit description of the semiring structure inherited automatically by the stalks of the sheaf N̄ on the topos N̂× .Theorem 2.4. At the point of the topos N̂× associated with the intermediate semifield F ⊂ K ⊂ Qmax , the stalk of the structure sheaf O := N̄ is the semiring OK := {r ∈ K | r ∨ 1 = 1} where ∨ denotes addition.2.3. The points of the arithmetic site (N̂×, N̄) over [0,1]maxThe following definition provides the notion of point of the arithmetic site over a local semiring.Definition 2.5. Let R be a local semiring. Then a point of (N̂×, N̄) over R is a pair of a point p of N̂× and a local morphism of semirings f #p :Op → R .We let [0,1]max ⊂ Rmax+ be the maximal compact sub-semiring of the tropical semifield Rmax+ . The next crucial statement determines the interpretation of the space underlying the non-commutative geometric approach to RH in terms of algebraic geometry.A. Connes, C. Consani / C. R. Acad. Sci. Paris, Ser. I 352 (2014) 971–975 973Theorem 2.6. The points of the arithmetic site (N̂×, N̄) over the local semiring [0,1]max form the quotient Q×\AQ/Ẑ∗ of the adèle class space of Q by the action of Ẑ∗. The action of the Frobenius automorphisms Frλ ∈ Aut([0,1]max) on these points corresponds to the action of the idèle class group (mod Ẑ∗) on the above quotient of the adèle class space.Notice that the quotient Q×\AQ/Ẑ∗ is the disjoint union of the following two spaces:(i) Q×+\A f /Ẑ∗ is the space of adèle classes whose Archimedean component vanishes. The corresponding points of the arithmetic site (N̂×, N̄) are those which are defined over B; they are given by the points of N̂× (Theorem 2.2).(ii) Q×+\((A f /Ẑ∗) ×R∗+) is the space of adèle classes whose Archimedean component does not vanish. It is in canonical bijection with rank-one subgroups of R through the map(a, λ) → λHa, ∀a ∈A f /Ẑ∗, λ ∈R∗+, Ha := {q ∈ Q | qa ∈ Ẑ}.2.4. Hasse–Weil formula for the Riemann zeta functionIn order to count the number of fixed points of the Frobenius action on points of (N̂×, N̄) over [0,1]max, we let ϑuξ(x) =ξ(u−1x) be the scaling action of the idèle class group G = GL1(AQ)/GL1(Q) on functions on the adèle class space AQ/Q∗and use the trace formula [3,4,9] in the form (ΣQ = places of Q, d∗u multiplicative Haar measure)Trdistr(∫Gh(u)ϑud∗u)=∑v∈ΣQ∫Q∗vh(u−1)|1 − u| d∗u. (1)We apply (1) to test functions of the form h(u) = g(|u|) where the support of g is contained in (1, ∞) and |u| is the module. The invariance of h under the kernel Ẑ∗ of the module G → R∗+ corresponds at the geometric level to taking the quotient of the adèle class space by the action of Ẑ∗ . Using Theorem 2.6 and [6], §2, one obtains the counting distribution N(u), u ∈ [1, ∞) associated with the Frobenius action on points of (N̂×, N̄) over [0,1]max.Theorem 2.7. The zeta function ζN associated by the equation∂sζN(s)ζN(s)= −∞∫1N(u) u−sd∗u, s ∈C, (s) > 1 (2)with the counting distribution N(u) is the complete Riemann zeta function ζQ(s) = π−s/2Γ (s/2)ζ(s).In [5], Eq. (2) was shown (following a suggestion made in [11]) to be the limit, when q → 1, of the Hasse–Weil formula for counting functions over finite fields Fq .3. The square of the arithmetic site3.1. The unreduced square (N̂×2,N⊗B N)Given a partially ordered set J , we let Sub≥( J ) be the set of subsets E ⊂ J , which are hereditary, i.e. such that x ∈ E ⇒y ∈ E , ∀y ≥ x. Then Sub≥( J ), endowed with the operation E ⊕ E ′ := E ∪ E ′ , is a B-module. We refer to [10] for the general treatment of tensor products of semi-modules.Proposition 3.1.(i) Let N × N be endowed with the partial order (a, b) ≤ (c, d) ⇐⇒ a ≤ c & b ≤ d. Then one has a canonical isomorphism of B-modules N⊗B N Sub≥(N ×N).(ii) There exists on the B-module S = N ⊗B N a unique bilinear multiplication such that, using multiplicative notation where q is a formal variable, one has(qa ⊗B qb)(qc ⊗B qd) = qa+c ⊗B qb+d. (3)(iii) The multiplication (3) turns N⊗B N into a semiring of characteristic 1.(iv) The following formula defines an action of N× ×N× by endomorphisms on N⊗B NFrn,m(∑qa ⊗B qb):=∑qna ⊗B qmb.Definition 3.2. The unreduced square (N̂×2,N⊗B N) of the arithmetic site (N̂×, N̄) is the topos N̂×2 with the structure sheafN⊗B N, viewed as a semiring in the topos.974 A. Connes, C. Consani / C. R. Acad. Sci. Paris, Ser. I 352 (2014) 971–975Fig. 1. Typical element E ∈ Sub≥(N ×N) (in yellow); its image under μ (red), under the Frobenius correspondence for λ = 13 (blue) and under γ (green or yellow). Removing the element q5 ⊗B q3 does not alter the convex hull γ (E). (For interpretation of references to color in this figure, the reader is referred to the web version of this article.)3.2. The Frobenius correspondencesThe product in the semiring N yields a morphism of semirings μ : (N⊗B N) → N, given on simple tensors by μ(qa ⊗Bqb) = qa+b (Fig. 1).Proposition 3.3.(i) The range of the morphism μ ◦ Frn,m : N ⊗B N → N only depends, up to the canonical isomorphism, on the ratio r = n/m. Assuming that n, m are relatively prime, this range contains the ideal{qa∣∣ a ≥ (n − 1)(m − 1)} ⊂ N̄.(ii) Let r = n/m, q ∈ (0, 1) and let mr : N⊗B N→ Rmax+ be given bymr(∑(qni ⊗B qmi)) = qα, α = inf(rni + mi).Up to the canonical isomorphism of their ranges, the morphisms μ ◦ Frn,m and mr are equal.Proposition 3.3(ii) allows one to extend the definition of the Frobenius correspondence to arbitrary positive real numbers.Proposition 3.4.(i) Let λ ∈R∗+ and q ∈ (0, 1) then the following formula defines a homomorphismF(λ,q) : N⊗B N→Rmax+ , F(λ,q)(∑(qni ⊗B qmi)) = qα, α = inf(λni + mi). (4)(ii) The semiring R(λ) := Im(F(λ, q)) is independent, up to the canonical isomorphism, of q ∈ (0, 1).(iii) The semirings R(λ) and R(λ′) are isomorphic if and only if λ′ = λ or λ′ = 1/λ.3.3. The reduced square (N̂×2,Conv≥(N×N))Let R be a semiring without zero divisors and ι : R → FracR the canonical morphism to the semifield of fractions. It is not true in general that ι is injective (cf. [2]). We shall refer to ι(R) as the reduced semiring of R .Definition 3.5. We let Conv≥(N × N) be the set of closed convex subsets C of the quadrant Q := R+ × R+ such that (i) C + Q = C and (ii) the extreme points ∂C belong to N ×N ⊂ Q .The set Conv≥(N ×N) is a semiring for the operations of convex hull of the union and sum.Proposition 3.6.(i) The semiring Conv≥(N ×N) is multiplicatively cancellative.A. Connes, C. Consani / C. R. Acad. Sci. Paris, Ser. I 352 (2014) 971–975 975(ii) The homomorphism γ : N⊗B N  Sub≥(N × N) → Conv≥(N × N) given by convex hull is the same as the homomorphism ι : N⊗B N→ ι(N⊗B N).(iii) Let R be a multiplicatively cancellative semiring and ρ : N⊗B N → R a homomorphism of semirings such that ρ−1({0}) = {0}. Then there exists a unique semiring homomorphism ρ ′ : Conv≥(N ×N) → R such that ρ = ρ ′ ◦ γ .Definition 3.7. The reduced square (N̂×2,Conv≥(N×N)) of the arithmetic site (N̂×, N̄) is the topos N̂×2 with the structure sheaf Conv≥(N ×N), viewed as a semiring in the topos.4. Composition of Frobenius correspondences4.1. Reduced correspondencesDefinition 4.1. A reduced correspondence over the arithmetic site (N̂×, N̄) is given by a triple (R, , r) where R is a multi-plicatively cancellative semiring, , r : N̄ → R are semiring morphisms such that −1({0}) = {0}, r−1({0}) = {0} and that R is generated by (N̄)r(N̄).By construction, cf. Proposition 3.4, the Frobenius correspondence gives a reduced correspondence:Ψ (λ) := (R, (λ), r(λ)), R := R(λ), (λ)(qn) := F(λ,q)(qn ⊗ 1), r(λ)(qn) := F(λ,q)(1 ⊗ qn). (5)By (4) one gets that the elements of R(λ) are powers qα where α ∈ N + λN and that the morphisms (λ) and r(λ) are described as follows:(λ)(qn)qα = qα+nλ, r(λ)(qn)qα = qα+n.4.2. The composition of the correspondences Ψ(λ) ◦ Ψ (λ′)The composition Ψ (λ) ◦ Ψ (λ′) of the Frobenius correspondences is obtained as the left and right actions of N̄ on the reduced semiring of the tensor product R(λ) ⊗N̄ R(λ′). In order to state the general result, we introduce a variant Idε of the identity correspondence. We let Germε=0(Rmax+ ) be the semiring of germs of continuous functions from a neighborhood of 0 ∈ R to Rmax+ , endowed with the pointwise operations. Let N̄ε be the sub-semiring of Germε=0(Rmax+ ) generated, for fixed q ∈ (0, 1), by q, and Fr1+ε(q) = q1+ε . N̄ε is independent, up to canonical isomorphism, of the choice of q ∈ (0, 1).Definition 4.2. The tangential deformation of the identity correspondence is given by the triple (N̄ε , ε, rε) where ε(qn) :=Fr1+ε(qn) and rε(qn) := qn , ∀n ∈N.Theorem 4.3. Let λ, λ′ ∈R∗+ such that λλ′ /∈ Q. The composition of the Frobenius correspondences is then given byΨ (λ) ◦ Ψ (λ′) = Ψ (λλ′).The same equality holds if λ and λ′ are rational. When λ, λ′ are irrational and λλ′ ∈Q,Ψ (λ) ◦ Ψ (λ′) = Ψ (λλ′) ◦ Idε = Idε ◦ Ψ (λλ′),where Idε is the tangential deformation of the identity correspondence.References[1] M. Artin, A. Grothendieck, J.-L. Verdier (Eds.), SGA4, LNM, vols. 269, 270, 305, Springer-Verlag, Berlin, New York, 1972.[2] D. Castella, Algèbres de polynômes tropicaux, Ann. Math. Blaise Pascal 20 (2013) 301–330.[3] A. Connes, Trace formula in noncommutative geometry and the zeros of the Riemann zeta function, Sel. Math. New Ser. 5 (1) (1999) 29–106.[4] A. Connes, M. Marcolli, Noncommutative Geometry, Quantum Fields, and Motives, Colloquium Publications, vol. 55, American Mathematical Society, 2008.[5] A. Connes, C. Consani, Schemes over F1 and zeta functions, Compos. Math. 146 (6) (2010) 1383–1415.[6] A. Connes, C. Consani, From monoïds to hyperstructures: in search of an absolute arithmetic, in: Casimir Force, Casimir Operators and the Riemann Hypothesis, de Gruyter, 2010, pp. 147–198.[7] A. Grothendieck, Sur une note de Mattuck–Tate, J. Reine Angew. Math. 200 (1958) 208–215.[8] S. Mac Lane, I. Moerdijk, Sheaves in Geometry and Logic. A First Introduction to Topos Theory, Universitext, Springer-Verlag, New York, 1994, corrected reprint of the 1992 edition.[9] R. Meyer, On a representation of the idèle class group related to primes and zeros of L-functions, Duke Math. J. 127 (3) (2005) 519–595.[10] B. Pareigis, H. Rohrl, Remarks on semimodules, arXiv:1305.5531v2 [mathRA], 2013.[11] C. Soulé, Les variétés sur le corps à un élément, Mosc. Math. J. 4 (1) (2004) 217–244.

The Arithmetic Site

Alain Connes

Caterina Consani

Comptes Rendus Mathématique

Comptes Rendus Mathematique

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The Arithmetic Site

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