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 quasitropical 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 typeII normalized traces as in [[5]].
For any abelian ordered group H we let be the semifield obtained from H by applying the maxplus 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 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).
(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 .
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 rankone subgroups of through the map
(a,λ)↦λHa,∀a∈Af/Zˆ⁎,λ∈R+⁎,Ha:={q∈Qqa∈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 .
(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 rankone subgroups of .
The proof of the above theorem follows from the next four lemmas.
(i ) Let be a rankone subgroup. Then defines a flat, continuous functor .
(ii ) The map , which associates with a rankone 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 .
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 rankone subgroups of .
Let be a flat continuous functor. Let and be the costalk of F at λ. Then there exists at most one element in the set and for any bounded open interval , is the disjoint union .
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 rankone 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 maxplus construction with a rankone subgroup .
Let us fix a rankone 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)=maxjλxj+yj.
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.
(i ) Let be a rankone 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 rankone 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 maxplus 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 nonzero 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 semiringed 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 nonconstant global sections.


The points of over 
The next theorem states that extension of scalars from to does not affect the points over .
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.
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.
Let be the point of the scaling topos associated with the rankone 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 .
(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 .
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∈KpD+(f)≥0}.
Let . One sets , and is the slope^{2At a point of discontinuity of the slopes one takes the max of the two values h±(λ)p/λ. } 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
(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 ).