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Comptes Rendus Mathématique
Volume 354, n° 1
pages 1-6 (janvier 2016)
Doi : 10.1016/j.crma.2015.09.027
Received : 24 September 2015 ;  accepted : 29 September 2015
The scaling site
Le site des fréquences
 

Alain Connes a, b, c , Caterina Consani d, 1
a Collège de France, 3 rue d'Ulm, 75005 Paris, France 
b I.H.E.S., France 
c Ohio State University, USA 
d The Johns Hopkins University, Baltimore, MD 21218, USA 

Abstract

We investigate the semi-ringed topos obtained from the arithmetic site   of [[3], [4]], by extension of scalars from the smallest Boolean semifield   to the tropical semifield  . The obtained site   is the semi-direct product of the Euclidean half-line and the monoid   of positive integers acting by multiplication. Its points are the same as the points   of   over   and form the quotient of the adele class space of   by the action of the maximal compact subgroup   of the idèle class group. The structure sheaf of the scaling topos endows it with a natural structure of tropical curve over the topos  . The restriction of this structure to the periodic orbits of the scaling flow gives, for each prime p , an analogue   of an elliptic curve whose Jacobian is  . The Riemann–Roch formula holds on   and involves real-valued dimensions and real degrees for divisors.

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Résumé

Le site des fréquences   est obtenu à partir du site arithmétique   de [[3], [4]] par extension des scalaires du semicorps booléen   au semicorps tropical  . C'est le produit semi-direct de la demi-droite euclidienne   par l'action du semi-groupe   des entiers positifs par multiplication. Ses points sont les mêmes que ceux du site arithmétique définis sur   et forment le quotient de l'espace des classes d'adèles de   par l'action du sous-groupe compact maximal du groupe des classes d'idèles. Le faisceau structural du site des fréquences en fait une courbe tropicale dans le topos  . La restriction de cette structure aux orbites périodiques donne, pour chaque nombre premier p , un analogue   d'une courbe elliptique dont la jacobienne est  . La formule de Riemann–Roch pour   fait apparaître des dimensions à valeurs réelles et les degrés des diviseurs sont des nombres réels.

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Keywords : Arithmetic site, Scaling site, Adele class space, Topos, Characteristic 1

Mots-clés : Site arithmétique, Site des fréquences, Classes d'adèles, Topos, Caractéristique 1


Introduction

This note describes the Scaling Site as the algebraic geometric space obtained from the arithmetic site   of [[3], [4]] by extension of scalars from the Boolean semifield   to the tropical semifield  . The underlying site   inherits, from its sheaf of regular functions, a natural structure of tropical curve allowing one to define the sheaf of rational functions and to investigate an adequate version of the Riemann–Roch theorem in characteristic 1. We test this structure by restricting it to the periodic orbits of the scaling flow, i.e. to the points over the image of   (cf. [[4]], §5.1). We find that for each prime p the corresponding circle of length   is endowed with a quasi-tropical structure which turns this orbit into the analogue   of a classical elliptic curve  . In particular the notions of rational functions, divisors, etc. are all meaningful. A new feature is that the degree of a divisor can now be any real number. We determine the Jacobian of the curve  , i.e. the quotient   of the group of divisors of degree 0 by principal divisors and show in Theorem 6.4 that it is a cyclic group of order  . For each divisor D on   we define the corresponding Riemann–Roch problem with solution space  . We introduce the continuous dimension   of this  -module using a limit of normalized topological dimensions and find that   is a real number. Finally, in Theorem 6.6 we prove that the Riemann–Roch formula holds for  . The appearance of arbitrary positive real numbers as continuous dimensions in this formula is due to the density in   of the subgroup   of fractions with denominators a power of p and the fact that continuous dimensions are obtained as limits of normalized dimensions  . We view this outcome as the analogue in characteristic 1 of what happens for matroid  -algebras and the type-II normalized traces as in [[5]].

Notations

For any abelian ordered group H we let   be the semifield obtained from H by applying the max-plus construction, i.e. the addition is given by the max, and the multiplication by the addition in H . In particular   is isomorphic to   by the exponential map (cf. [[7]]).

The scaling site

The scaling site   is, as a site, given by a small category C endowed with a Grothendieck topology J [[1]]. The objects of C are the (possibly empty) bounded open intervals  . The morphisms between two objects are defined by  , if   and by  , i.e. the one point set, for any object of C . Thus the empty set is the initial object of C . The category C admits pullbacks. Indeed, let   ( ) and consider two morphisms   given by integers  . Let   be their lowest common multiple, write   and let  . If   the initial object is the pullback. Otherwise this gives an object   of C and morphisms   such that  . One sees that   is the pullback of the pair  . Since the category C has pullbacks we can describe a Grothendieck topology J on C by providing a basis (cf. [[8]], Definition III.2).

Proposition 2.1

(i ) For each object Ω of C, let   be the collection of all ordinary covers   of Ω. Then K defines a Grothendieck topology J on C.

(ii ) The category   of sheaves on   is canonically isomorphic to the category of  -equivariant sheaves on  .

Definition 2.2

The scaling site   is the small category C endowed with the Grothendieck topology J . The scaling topos is the category  .

The points of the scaling topos

We recall from [[4]] that the space   of points of the arithmetic site   over   is the disjoint union of the following two spaces:

(i ) The points which are defined over  : they correspond to the points of   and are in canonical bijection with the space   of adele classes whose Archimedean component vanishes.

(ii ) The points of   are in canonical bijection with the space   of adele classes whose Archimedean component does not vanish. Equivalently, these points correspond to the space   of rank-one subgroups of   through the map
(a,λ)↦λHa,∀a∈Af/Zˆ⁎,λ∈R+⁎,Ha:={q∈Q|qa∈Zˆ}. The next statement shows that the points of the scaling topos   are in canonical bijection with  . We recall that the points of a topos of the form   are equivalently described as flat, continuous functors   (cf. [[8]]). In our context, we define the support of such a functor as the complement of the union of the open intervals I such that  .

Theorem 3.1

(i ) The category of points of the scaling topos with support {0} is the same as the category of points of  .

(ii ) The category of points of the scaling topos with support different from {0} is canonically equivalent to the category of rank-one subgroups of  .

The proof of the above theorem follows from the next four lemmas.

Lemma 3.2

(i ) Let   be a rank-one subgroup. Then   defines a flat, continuous functor  .

(ii ) The map  , which associates with a rank-one subgroup of   the point of   represented by the flat continuous functor   is an injection of   in the space of points of the scaling topos up to isomorphism.

The next lemma shows that the category of points of the scaling topos with support {0} is the same as the category of points of  .

Lemma 3.3

Let   be a flat continuous functor. Assume that   when  . Then there exists a unique flat functor   such that   for any object V of C containing 0.

The next two lemmas show that the category of points of the scaling topos with support   is equivalent to the category of rank-one subgroups of  .

Lemma 3.4

Let   be a flat continuous functor. Let   and   be the co-stalk of F at λ. Then there exists at most one element in the set   and for any bounded open interval  ,   is the disjoint union  .

Lemma 3.5

Let   be a flat continuous functor. Assume that   for some open interval V not containing 0. Then the set   is the positive part of a rank-one subgroup   of  .

The structure sheaf of the scaling site

The Legendre transform allows one to describe the reduced semiring   involved in the extension of scalars of the arithmetic site   from   to   in terms of  -valued functions on   which are convex, piecewise affine functions with integral slopes. We first discuss an analogous result that holds when   is replaced by the semiring   associated by the max-plus construction with a rank-one subgroup  .

The Legendre transform

Let us fix a rank-one subgroup   and consider the tensor product   and the associated multiplicatively cancellative semiring   whose elements are viewed as Newton polygons with vertices pairs   [[4]]. Let  . Any element of R is given by the convex hull N in   of the union of finitely many quadrants  . This convex hull N is the intersection of half planes   of the form  , where   and  . This description shows that N is uniquely determined by the function   and that this function is given in terms of the finitely many vertices   of the Newton polygon N by the formula
(1)ℓN(λ)=maxj⁡λxj+yj.

Proposition 4.1

Let   be a subgroup of rank one. The map   is an isomorphism of the multiplicatively cancellative semiring   with the semiring   of convex, piecewise affine continuous functions on   with slopes in   and only finitely many singularities. The operations are the pointwise operations of  -valued functions.

The stalks of  

Proposition 4.1 gives the relation between the reduced semiring   involved in the extension of scalars of the arithmetic site from   to  , and the semiring  . The structure sheaf   of   is defined by localizing the semiring  . The sections   on an open set   are convex, piecewise affine continuous functions on Ω with slopes in  . The action of   on   is given by the morphisms
(2)γn:O(Ω)→O(1nΩ),γn(ξ)(λ):=ξ(nλ),∀λ∈[0,∞),n∈N×. For   as in ((1)) one has   so that  . Note that these maps are not invertible.

Theorem 4.2

(i ) Let   be a rank-one subgroup of   and   be the associated point of the scaling topos. The stalk of the structure sheaf   at   is the semiring   of germs of  -valued, piecewise affine, convex continuous functions with slope in H.

(ii ) Let H be an abstract rank-one ordered group and   the point of the scaling topos with support {0}, associated with H. The stalk of the structure sheaf   at   is the semiring   associated by the max-plus construction with the totally ordered group   endowed with the lexicographic order.

The germs at   of  -valued, piecewise affine, convex continuous functions   with slopes in H are characterized by a triple  , such that   for   small enough. Here, one has  ,  ,  . The only additional element of this semiring   corresponds to the germ of the constant function −∞. This function is the zero element of the semiring. The algebraic rules for non-zero elements in   are as follows. The addition ∨ in   is given by the max of the two germs:
(x,h+,h−)∨(x′,h+′,h−′):={(x,h+,h−)if x>x′(x′,h+′,h−′)ifx′>x(x,h+∨h+′,h−∧h−′)ifx=x′ The product in   is given by the sum of the two germs  .

We shall denote by   the semi-ringed topos  . We view it as a relative topos over   in the sense that the structure semirings are over  . Likewise, for the arithmetic site, the structure sheaf has no non-constant global sections.

The points of   over  

The next theorem states that extension of scalars from   to   does not affect the points over  .

Theorem 5.1

The canonical projection from points of   defined over   to points of the scaling topos is bijective.

The real valued Riemann–Roch theorem on periodic orbits

To realize the notion of rational functions in our context we proceed as in the definition of Cartier divisors and consider the sheaf obtained from the structure sheaf   by passing to the semifield of fractions.

Proposition 6.1

For any object Ω of C the semiring   is multiplicatively cancellative and the canonical morphism to its semifield of fractions   is the inclusion of convex, piecewise affine, continuous functions among continuous, piecewise affine functions, endowed with the two operations of max and plus.

The natural action of   on   defines a sheaf of semifields in the scaling topos. One determines its stalks in the same way as for the structure sheaf  . The local convexity no longer holds, i.e. the difference   is no longer required to be positive.

Definition 6.2

Let   be the point of the scaling topos associated with the rank-one subgroup   and let f be an element of the stalk of   at  . The order of f at H is defined as   where  .

Let p be a prime, consider the subspace   of points   of   corresponding to subgroups   which are abstractly isomorphic to the subgroup   of fractions with denominator a power of p .

Lemma 6.3

(i ) The map  ,   induces a topological isomorphism  . The pullback by   of the structure sheaf   is the sheaf   on   of piecewise affine, continuous convex functions, with slopes in  .

(ii ) The sheaf of quotients   of the sheaf of semirings   is the sheaf (on  ) of piecewise affine, continuous functions with slopes in  , endowed with the operations of max and plus.

(iii ) The sheaf   admits global sections and for any   one has:
∑λ∈R+⁎/pZOrderλ(f)=0, where Orderλ(f):=OrderλHp(f∘ηp−1)∈λHp,∀λ∈R+⁎/pZ.

A divisor D is a section  , vanishing except on a finite subset, of the projection on the base from the total space of the bundle formed by pairs   where   is a subgroup abstractly isomorphic to the subgroup   and where  . The degree of a divisor D is the finite sum  . Next, we define an invariant of divisors with values in the group  . Note that given  , the elements   such that   determine maps   differing from each other by multiplication by a power of p , thus the corresponding map   is canonical. For any divisor D on   we define  .

Theorem 6.4

The map χ vanishes on principal divisors and it induces an isomorphism of groups   of the quotient   of the group of divisors of degree 0 by the subgroup   of principal divisors.

Since the group law on divisors is given by pointwise addition of sections, both the maps   and   are group homomorphisms and the pair   is an isomorphism of groups
(3)(deg⁡,χ):Div(Cp)/P→R×(Z/(p−1)Z). Given a divisor   one defines the following module over  :H0(D)=Γ(Cp,Op(D))={f∈Kp|D+(f)≥0}.

Definition 6.5

Let  . One sets  , and   is the slope2 of f at λ , where   is the p -adic norm.

Let   be a divisor. We introduce the following increasing filtration of   by  -submodules:  . We denote by   the topological covering dimension of an  -module   (cf. [[10]]) and define
(4)DimR(H0(D)):=limn→∞⁡p−ndimtop(H0(D)pn). One shows that the above limit exists: indeed, one has the following

Theorem 6.6

(i ) Let   be a divisor with  . Then the limit in ((4)) converges and one has  .

(ii ) The following Riemann–Roch formula holds
DimR(H0(D))−DimR(H0(−D))=deg⁡(D),∀D∈Div(Cp).

One can compare the above Riemann–Roch theorem with the tropical Riemann–Roch theorem of [[2], [6], [9]] and its variants. More precisely, let us consider an elliptic tropical curve C , given by a circle of length L . In this case, the structure of the group   of divisor classes is inserted into an exact sequence of the form   (cf. [[9]]). This sequence is very different from the split exact sequence associated with   and deduced from ((3)), i.e.  . The reason for this difference is due to the nature of the structure sheaf of  , when this sheaf is written in terms of the variable  . This choice is dictated by the requirement that the periodicity condition   becomes translation invariance by  . The condition for f of being piecewise affine in the parameter λ is expressed in the variable u by the piecewise vanishing of  , where   is the elliptic translation invariant operator  . In terms of the variable  , this operator takes the form   and this fact explains why the structure sheaf of   is considered as tropical (in terms of the variable λ ).

References

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1  The second author thanks the Collège de France for hospitality and financial support.
2  At a point of discontinuity of the slopes one takes the max of the two values  .


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