RIEMANN-ROCH FOR ${\overline{{\rm Spec\,}{\mathbb{Z}}}}$^††thanks: Research supported by The Simons Foudation

The development of the theory of curves from complex Riemann surfaces to algebraic curves over arbitrary fields, led F. K. Schmidt, in the 1930s, to the first proof of the Riemann-Roch theorem for function fields over finite fields. In nowadays terminology, that result involves the integer dimensions of the two cohomologies $H^{0}(D)$ and $H^{1}(D)$ of a divisor $D$ on the algebraic curve associated to the function field, as vector spaces over the (finite) field of constants. The analogy between function fields in one variable over fields of positive characteristic and number fields suggests, as pointed out by A. Weil [13], to investigate the existence of a Riemann-Roch formula for number fields. Both the notions of a divisor $D$ with its degree, and the finite set underlying $H^{0}(D)$ are easy to define. These ideas led S. Lang [9] to an asymptotic formula that relates for ${\mathbb{Q}}$, $\log\#H^{0}(D)$ and $\deg D+\log 2$, when $\deg D\to\infty$. Another attempt to a Riemann-Roch formula for a number field was promoted two decades ago by G. van der Geer and R. Shoof [11] and more recently, in the work of J.B. Bost [1]. In this approach one starts from the functional equation in terms of the theta function and rewrites that equation as a Riemann-Roch formula for the log-theta number, which is thus promoted to the status of a dimension. This view of the log-theta number as a dimension remains virtual for the obvious reason that it outputs real numbers rather than integers. An interpretation of these numbers as Murray-von Neumann dimensions could greatly improve their status as dimensions, but this step has not been achieved so far.
The most fundamental number field is the field ${\mathbb{Q}}$ of rational numbers that governs elementary arithmetics through its ring ${\mathbb{Z}}$ of integers. In this paper we prove a Riemann-Roch theorem of an entirely novel nature for this basic arithmetic structure. This result relies on the introduction of three key concepts: the two cohomologies $H^{\bullet}(D)$ (attached to a divisor $D$), their (integer) dimension, and Serre duality. These three notions directly extend their classical analogues for function fields. Our main result is the following theorem.

In this formula, the neperian logarithm that is traditionally used to define the degree of a divisor $D=\sum_{j}a_{j}\{p_{j}\}+a\{\infty\}$, is replaced by the logarithm in base $3$.¹¹1All throughout the paper, to avoid confusion, we shall nevertheless use the neperian logarithm This alteration is equivalent to the division by $\log 3$ i.e. $\deg^{\prime}(D):=\deg(D)/\log 3$. Let us now comment on the key role played by the number $3$. In the proof of Theorem 1.1, the ring ${\mathbb{Z}}$ appears in a new light as a ring of polynomials $\sum a_{j}3^{j}$ with coefficients in the absolute base ${{{\mathbb{S}}}[\pm 1]}=\{0,\pm 1\}$. At an elementary level, it is not difficult to check that any integer $n\in{\mathbb{Z}}$ can be written uniquely as a finite sum of powers of $3$ with coefficients in $\{0,\pm 1\}$. The deep reason behind this fact is that, for $p=3$ (and only for this rational prime) the Witt vectors with only finitely many non-zero components form a subring inside the ring of $p$-adic integers, which turns out to be isomorphic to ${\mathbb{Z}}$. In particular, the formula for the addition of integers written as polynomials $\sum a_{j}3^{j}$ coincides with the formula for the addition of Witt vectors over ${\mathbb{F}}_{3}$.
Next, we describe the general formalism that allows us to give a precise mathematical meaning both to the base ${{{\mathbb{S}}}[\pm 1]}=\{0,\pm 1\}$ and (to) the two cohomologies $H^{\bullet}(D)$ with their integer-valued dimension. The basic step underlying this formalism is taken by going beyond the traditional use of abelian groups and rings in algebra using Segal’s $\Gamma$-rings. In [2, 5] we developed the fundamentals of a geometric theory where the initial ring ${\mathbb{Z}}$ of the category of rings is replaced by the sphere spectrum ${{\mathbb{S}}}$. A similar change of structures is familiar in homotopy theory, but in our work we insist on staying at a concrete computational level in the definition of the basic objects, while the link with the $\Gamma$-spaces of homotopy theory [6] enters only when doing homological algebra. The extension of the category of abelian groups is obtained by embedding faithfully and fully this category in the category ${{{\mathbb{S}}}-{\rm{Mod}}}$ of pointed covariant functors from finite pointed sets to pointed sets. Given an abelian group $H$ one assigns to a finite pointed set $X$ the pointed set of $H$-valued divisors on $X$. These divisors push forward by summing over the pre-image of a point. In ${{{\mathbb{S}}}-{\rm{Mod}}}$ the traditional notion of ring becomes that of Segal’s $\Gamma$-ring, and the (classical) base ${\mathbb{Z}}$ is replaced by the $\Gamma$-ring ${{\mathbb{S}}}$ here understood in its most elementary form of identity functor (from finite pointed sets to pointed sets). The base ${{\mathbb{S}}}$ is in fact the most elementary categorical form of the sphere spectrum in homotopy theory [6]. In analogy with the theory of algebraic curves over finite fields, we have (previously) defined the structure sheaf of the one-point compactification ${\overline{{\rm Spec\,}{\mathbb{Z}}}}$ of the algebraic spectrum of the integers by suitably extending, at the archimedean place, the structure sheaf of ${{\rm Spec\,}{\mathbb{Z}}}$ as a subsheaf of the constant sheaf ${\mathbb{Q}}$. The global sections of this sheaf determine the ${{\mathbb{S}}}$-algebra ${{{\mathbb{S}}}[\pm 1]}$. In [3] we have generalized the notion of homology for simplicial complexes, when group coefficients are replaced by ${{\mathbb{S}}}$-modules. In [4] we implemented homological algebra in this context by showing an extension of the Dold-Kan correspondence which associates a $\Gamma$-space to a short complex of ${{\mathbb{S}}}$-modules.
Let us now introduce the cohomologies and their dimensions. An Arakelov divisor $D=\sum_{j}a_{j}\{p_{j}\}+a\{\infty\}$ on ${\overline{{\rm Spec\,}{\mathbb{Z}}}}$ determines a compact subset ${\mathcal{O}}(D)\subset{\mathbb{A}}_{\mathbb{Q}}$ of the adeles of ${\mathbb{Q}}$. This is the product, indexed by the places of ${\mathbb{Q}}$, of the abelian group ${\mathbb{Z}}_{p}\subset{\mathbb{Q}}_{p}$ for each finite prime $p\notin\{p_{j}\}_{j}$, the abelian groups $p_{j}^{-a_{j}}{\mathbb{Z}}_{p_{j}}$, and the interval $[-e^{a},e^{a}]\subset{\mathbb{R}}$. This last component is implemented by a sub-module of the Eilenberg-MacLane ${{{\mathbb{S}}}[\pm 1]}$-module $H{\mathbb{R}}$ which functorially encodes the additive structure of the group ${\mathbb{R}}$. The morphisms of abelian groups used by Weil in the development of the adelic geometry of function fields still retain a meaning in this absolute set-up and they are viewed as morphisms of ${{{\mathbb{S}}}[\pm 1]}$-modules. We focus, in particular, on the morphism

where ${\mathbb{Q}}$ is embedded diagonally in the adeles. The $\Gamma$-space ${\bf H}(D)$ associated by the Dold-Kan correspondence to (1.2) ($\psi$ is viewed here as a short complex of ${{{\mathbb{S}}}[\pm 1]}$-modules: see appendix B) provides the absolute incarnation of the Riemann-Roch problem for the divisor $D$. By implementing linear equivalence i.e. the multiplicative action of ${\mathbb{Q}}^{\times}$ on ${\mathbb{A}}_{\mathbb{Q}}$, one is reduced to consider only the case $D=a\{\infty\}$, for which $\ker(\psi)$ is the ${{{\mathbb{S}}}[\pm 1]}$-module $H^{0}(D)=\|H{\mathbb{Z}}\|_{e^{a}}$ that associates to a finite pointed set $X$ the ${\mathbb{Z}}$-valued divisors $\sum_{i}n_{i}x_{i}$, $x_{i}\in X$, fulfilling the condition $\sum_{i}|n_{i}|\leq e^{a}$ ($|\cdot|=$ euclidean absolute value). By definition, $H^{0}(D)$ is a covariant functor $\Gamma^{o}\longrightarrow\mathfrak{Sets}_{*}$ from the small category $\Gamma^{o}$ (a skeleton of the category of finite pointed sets) to pointed sets. It keeps track of the partially defined addition in $I=\|H{\mathbb{Z}}\|_{e^{a}}(1_{+})=[-e^{a},e^{a}]\cap{\mathbb{Z}}$, so that for $n_{i}\in I$, the sum $\sum n_{i}$ is meaningful provided $\sum_{i}|n_{i}|\leq e^{a}$. The dimension $\dim_{{{{\mathbb{S}}}[\pm 1]}}H^{0}(D)$ is defined to be the smallest number of linear generators of $I$. More precisely, a subset $F\subset I$ linearly generates if and only if for every $m\in I$ there exist coefficients $\alpha(f)\in\{-1,0,1\}={{{\mathbb{S}}}[\pm 1]}(1_{+})$ such that $m=\sum\alpha(f)f$, $f\in F$, and $\sum|\alpha(f)f|\leq e^{a}$ (see section 3). The ${{{\mathbb{S}}}[\pm 1]}$-module $H^{0}(D)$ only depends upon the integer part $n=\lfloor e^{a}\rfloor$ of $e^{a}$ and Proposition 3.3 determines its dimension to be²²2For the values $n=4,13,40,\ldots,(3^{k}-1)/2$, every element of $I$ is uniquely written in terms of the generating set $F=\{3^{i}\mid 0\leq i<k\}$ (Proposition 3.3 $(iii)$).

The cokernel of $\psi$ in (1.2), that is $H^{1}(D)$, is defined in terms of the $\Gamma$-space ${\bf H}(D)$. The lack of transitivity of the homotopy relation in non-Kan complexes is dealt with using the classical notion of tolerance relation [12]. Thus $H^{1}(D)$ is the couple of the ${{{\mathbb{S}}}[\pm 1]}$-module $H{\mathbb{A}}_{\mathbb{Q}}$ and a suitable tolerance relation ${\mathcal{R}}$ on it (see Proposition 2.2 and appendix A). This relation is defined by pairs whose difference belongs to the image of $\psi$. More specifically, $H^{1}(D)$ defines an object of the category $\Gamma{\mathcal{T}}_{*}$ of tolerance ${{\mathbb{S}}}$-modules (see appendix A). Its dimension $\dim_{{{{\mathbb{S}}}[\pm 1]}}H^{1}(D)$ is defined to be the smallest number of linear generators where a subset $F\subset H{\mathbb{A}}_{\mathbb{Q}}(1_{+})={\mathbb{A}}_{\mathbb{Q}}$ linearly generates if and only if for every $x\in H{\mathbb{A}}_{\mathbb{Q}}(1_{+})$ there exists coefficients $\alpha(f)\in\{-1,0,1\}$ such that the pair $(x,\sum\alpha(f)f)$ belongs to the relation ${\mathcal{R}}$ (see section 4).
An attentive reader should readily recognize that our construction strictly parallels Weil’s adelic proof of the Riemann-Roch formula for function fields and, in particular, that our notion of dimension is a straigthforward generalization of the classical dimension for vector spaces. For an archimedean divisor $D=a\{\infty\}$, the dimension of $H^{1}(D)$ is the same as the dimension of the pair $(H({\mathbb{R}}/{\mathbb{Z}}),{\mathcal{R}})$, where the relation ${\mathcal{R}}$ on $H({\mathbb{R}}/{\mathbb{Z}})(1_{+})={\mathbb{R}}/{\mathbb{Z}}$ is given by

for $d$ the translation invariant Riemannian metric of length $1$ on ${\mathbb{R}}/{\mathbb{Z}}$. In Proposition 4.1 we show that this dimension is the integer

Theorem 1.1 follows from (1.3) when $\deg D\geq-\log 2$, since in this case $\dim_{{{{\mathbb{S}}}[\pm 1]}}H^{1}(D)=0$. The case $\deg D<-\log 2$ is proved using (1.5). The symmetry of the term $\lceil\frac{\deg D+\log 2}{\log 3}\rceil^{\prime}$ under the replacement of $D$ by the divisor $K-D$, with $K:=-2\{2\}$, is the numerical evidence of a geometric duality holding on the curve ${\overline{{\rm Spec\,}{\mathbb{Z}}}}$ over ${{{\mathbb{S}}}[\pm 1]}$ (see section 5). This is precisely stated by the following (see Theorem 5.3)

The paper is organized as follows, in section 2 we adapt to the adelic framework the construction of the $\Gamma$-space ${\bf H}(D)$ naturally associated to an Arakelov divisor $D$. This gives, in Proposition 2.2, the cohomologies $H^{0}(D)$ and $H^{1}(D)$. We compute the dimension of $H^{0}(D)$ in section 3 (Theorem 3.4), and of $H^{1}(D)$ in section 4 which concludes with the proof of the main Theorem 4.3. Section 5 is devoted to Serre and Pontrjagin dualities. In appendix A we develop the needed generalities on tolerance ${{\mathbb{S}}}$-modules and in appendix B the technical details of the construction, through the Dold-Kan correspondence, of the $\Gamma$-space ${\bf H}(D)$.

We recall, from [4], the construction of the $\Gamma$-space ${\bf H}(D)$ naturally associated to an Arakelov divisor $D$ on ${\overline{{\rm Spec\,}{\mathbb{Z}}}}$. This space is the absolute homological incarnation of the Riemann-Roch problem for the divisor $D$. We formulate this construction in terms of the adeles of ${\mathbb{Q}}$.

Let $D=\sum_{j}a_{j}\{p_{j}\}+a\{\infty\}$ be an Arakelov divisor on ${\overline{{\rm Spec\,}{\mathbb{Z}}}}$. This is a formal finite sum with $a_{j}\in{\mathbb{Z}}$ and $a\in{\mathbb{R}}$, where $p_{j}$ are rational primes in ${\mathbb{Z}}$ and the symbol $\infty$ stands for the restriction $|\cdot|:{\mathbb{Q}}\stackrel{{\scriptstyle}}{{\to}}{\mathbb{R}}$ of the euclidean absolute value $|\cdot|_{\infty}$. Let $\Sigma_{\mathbb{Q}}$ denote the full set of places $\nu$ of ${\mathbb{Q}}$. To $D$ is naturally associated the following idele $\exp(D)$ of ${\mathbb{Q}}$

The equality $\exp(D)(\nu)=|\exp(D)_{\nu}|_{\nu}$ defines a map $\exp(D):\Sigma_{\mathbb{Q}}\to{\mathbb{R}}_{+}^{*}$ such that $\exp(D)(\nu)\in{\rm mod}({\mathbb{Q}}_{\nu})$, $\forall\nu\in\Sigma_{\mathbb{Q}}$, and $\exp(D)(\nu)=1$, for $\nu\neq\infty,\nu_{j},\forall j$. To the divisor $D$ corresponds the compact subset of the adeles of ${\mathbb{Q}}$

By definition ${\mathcal{O}}(D)={\mathcal{O}}(D)_{f}\times{\mathcal{O}}(D)_{\infty}=\prod_{\nu}{\mathcal{O}}(D)_{\nu}$ with

In particular, the product ${\mathcal{O}}(D)_{f}=\prod_{\nu\neq\infty}{\mathcal{O}}(D)_{\nu}$ is a compact abelian group. The archimedean component, on the other hand, i.e. the real interval $[-e^{a},e^{a}]$, is not, and for this reason we implement the functorial viewpoint to suitably understand ${\mathcal{O}}(D)$. Indeed, to ${\mathcal{O}}(D)_{\infty}$ we associate the covariant functor

from the opposite of the Segal category (see e.g. [6] Chpt. 2 and [2]) to pointed sets, which maps a finite pointed set $F$ to the pointed set of ${\mathbb{R}}$-valued divisors on $F$ vanishing on the base point and whose total mass $\sum_{F\setminus\{*\}}|\phi(x)|$ is bounded by $e^{a}$. Covariant functors $\Gamma^{o}\longrightarrow\mathfrak{Sets}_{*}$ and their natural transformations determine the category $\Gamma{\mathfrak{Sets}_{*}}$ of $\Gamma$-sets (aka ${{\mathbb{S}}}$-modules). In particular, the Eilenberg-MacLane functor $H$ encodes an abelian group $A$ as the covariant functor $HA:\Gamma^{o}\longrightarrow\mathfrak{Sets}_{*}$ that associates to $F$ the pointed set of $A$-valued divisors on $F$ vanishing on the base point. Functoriality holds by taking sums on the inverse images of a point. The functor $HA$ encodes the addition on $A=HA(1_{+})$. In general, let $\sigma\in{\rm{Hom}}_{\Gamma^{o}}(k_{+},1_{+})$ with $\sigma(\ell)=1$ $\forall\ell\neq*$ and

Given an ${{\mathbb{S}}}$-module ${\mathcal{F}}$ and elements $x,x_{j}\in{\mathcal{F}}(1_{+})$, $j=1,\ldots,k$, one writes

For $A={\mathbb{R}}$ the key point is that this general construction respects the bound on the total mass while encoding the addition. One easily sees that $\|H{\mathbb{R}}\|_{e^{a}}$ is an ${{{\mathbb{S}}}[\pm 1]}$-module (an “(${{{\mathbb{S}}}[\pm 1]}\wedge-)$-algebra” as in [6] 2.1.5), where ${{{\mathbb{S}}}[\pm 1]}:\Gamma^{o}\longrightarrow\mathfrak{Sets}_{*}$, ${{{\mathbb{S}}}[\pm 1]}(F):=\{-1,0,1\}\wedge F$ is the spherical monoid algebra of the multiplicative monoid $\{\pm 1\}$. In particular, the inclusion functor $\|H{\mathbb{R}}\|_{e^{a}}\longrightarrow H{\mathbb{R}}$ determines $\|H{\mathbb{R}}\|_{e^{a}}$ as a sub ${{{\mathbb{S}}}[\pm 1]}$-module of $H{\mathbb{R}}$, where ${{\mathbb{S}}}:\Gamma^{o}\longrightarrow\mathfrak{Sets}_{*}$ is the identity functor.

Given two ${{\mathbb{S}}}$-modules ${\mathcal{F}}_{j}$, $j=1,2$, one defines their product as the functor

with the base point of ${\mathcal{F}}_{1}(F)\times{\mathcal{F}}_{2}(F)$ taken to be $(*,*)$. For abelian groups $A,B$ one has a canonical isomorphism $H(A\times B)\simeq HA\times HB$. In particular, the morphism of addition in adeles $\alpha:{\mathbb{Q}}\times{\mathbb{A}}_{\mathbb{Q}}\to{\mathbb{A}}_{\mathbb{Q}}$, $\alpha(q,a)=q+a$ (using the diagonal embedding of ${\mathbb{Q}}$ in ${\mathbb{A}}_{\mathbb{Q}}$) determines a morphism of ${{{\mathbb{S}}}[\pm 1]}$-modules $H\alpha:H{\mathbb{Q}}\times H{\mathbb{A}}_{\mathbb{Q}}\longrightarrow H{\mathbb{A}}_{\mathbb{Q}}$. Next proposition shows how this functorial construction is well combined with the adelic formalism.

The morphism $\psi$ in (2.3) is obtained by restricting $H\alpha$ to the (adelic) divisor $D$, where $\alpha$ is a morphism of abelian groups. The theory developed in [4] associates to $\psi$ a $\Gamma$-space ${\bf H}(D)$, by implementing the Dold-Kan correspondence in the special case of a short complex of abelian groups and its restriction to a sub-${{\mathbb{S}}}$-module. The $\Gamma$-space ${\bf H}(D)$ supplies the absolute cohomology of the divisor $D$. We recall this construction in appendix B. Moreover, we review how the generalized homotopy $\pi^{{\mathcal{T}}}_{\ast}({\bf H}(D))$, where ${\mathcal{T}}$ is the category of tolerance relations defined in A.1, is meaningful in this context supplying, with $\pi^{{\mathcal{T}}}_{1}({\bf H}(D))$ and $\pi^{{\mathcal{T}}}_{0}({\bf H}(D))$, the (tolerant) ${{\mathbb{S}}}$-modules descriptions of resp. the kernel and the cokernel of $\psi$. In other words, one obtains the following result

In this section we consider, for each integer $n>0$, the ${{{\mathbb{S}}}[\pm 1]}$-module $\|H{\mathbb{Z}}\|_{n}$

and evaluate its dimension. It follows from the ${{{\mathbb{S}}}[\pm 1]}$-module structure, that (2.2) holds for a sum of the form

By applying Definition A.2, a subset $F\subset[-n,n]\cap{\mathbb{Z}}=\|H{\mathbb{Z}}\|_{n}(1_{+})$ generates³³3For simplicity we use from now on the term “generates” for “linearly generates” $\|H{\mathbb{Z}}\|_{n}(1_{+})$ if and only if for every element $x\in\|H{\mathbb{Z}}\|_{n}(1_{+})$ there exists coefficients $\alpha_{j}\in\{-1,0,1\}$, $j\in F$, such that $x=\sum_{F}\alpha_{j}j$ and $\sum_{F}|\alpha_{j}j|\leq n$. The dimension $\dim_{{{{\mathbb{S}}}[\pm 1]}}\|H{\mathbb{Z}}\|_{n}$ is the minimal cardinality of a generating set.

The explicit computation of $\dim_{{{{\mathbb{S}}}[\pm 1]}}\|H{\mathbb{Z}}\|_{n}$ follows from the next two statements.

We recall that the ceiling function $\lceil x\rceil$ associates to a positive real number $x$ the smallest integer $n\geq x$.