# The Arithmetic Site and the Scaling Site

##### Introduction

In The Riemann Hypothesis for Curves over Finite Fields, we gave a rough outline of Andre Weil’s strategy to prove the analogue of the famous Riemann hypothesis for curves over finite fields. A rather natural question to ask would be, does this strategy give us any suggestions on how to take on the original Riemann hypothesis? We mentioned briefly in The Field with One Element that some mathematicians hope to find in the theory of the so-called “field with one element” something that will allow them to apply the ideas of Weil’s proof to the original Riemann hypothesis, by viewing the scheme $\text{Spec}(\mathbb{Z})$  as some kind of “curve” over the “field with one element”.

In this post we will consider something along similar lines, examining a kind of “space” to which we can apply an analogue of Weil’s strategy. This approach is due to the mathematicians Alain Connes and Caterina Consani, and makes use of the concepts of sites and toposes (see More Category Theory: The Grothendieck Topos and Even More Category Theory: The Elementary Topos). This is perhaps appropriate, since sites or toposes are often referred to as “generalized spaces”.

We recall from The Riemann Hypothesis for Curves over Finite Fields some aspects of Weil’s strategy. The object in consideration is a curve $C$ over a finite field $\mathbb{F}_{q}$. In order to write down the zeta function for $C$, we need to count the number of points over $\mathbb{F}_{q^{n}}$, for every $n$ from $1$ to infinity. We can do this by counting the fixed points of powers of the Frobenius morphism. Explicitly this means taking intersection numbers of the diagonal and the divisor formed by integral linear combinations of powers of the Frobenius morphism on $\bar{C}\times_{\bar{\mathbb{F}}_{q}}\bar{C}$, where $\bar{\mathbb{F}}_{q}$ is an algebraic closure of $\mathbb{F}_{q}$ (it is the direct limit of the directed system formed by all the $\mathbb{F}_{q^{n}}$) and $\bar{C}=C\otimes_{\mathbb{F}_{q}}\bar{\mathbb{F}}_{q}$. The number of points of $\bar{\mathbb{F}}_{q}$ will be the same as the number of points of $C$ over $\mathbb{F}_{q^{n}}$. Throughout this post we should keep these steps of Weil’s strategy in mind.

In order to transfer this strategy of Weil to the original Riemann hypothesis, Connes and Consani construct the arithmetic site, meant to be the analogue of $C$, and the scaling site, meant to be the analogue of $\bar{C}$. The intuition behind these constructions is that the points of the scaling site, which is the same as the points of the arithmetic site “over $\mathbb{R}_{+}^{\text{max}}$“, is the same as the points of the “adele class space$\mathbb{Q}^{\times}\backslash\mathbb{A}_{\mathbb{Q}}/\hat{\mathbb{Z}}^{\times}$, which originally came up in earlier work of Connes where he constructed a quantum-mechanical system which gives Riemann’s prime-counting function (whose study provided the historical origin of the Riemann hypothesis), in the form of Weil’s “explicit formula”, as a quantum-mechanical trace formula! In essence this work restates the Riemann hypothesis in terms of mathematical language more commonly associated to physics, and is part of Connes’ pioneering work in noncommutative geometry, a new area of mathematics also closely related to physics, in particular quantum mechanics and quantum field theory. In the definition of the adele class space, $\mathbb{A}_{\mathbb{Q}}$ refers to the ring of adeles of $\mathbb{Q}$ (see Adeles and Ideles), while $\hat{\mathbb{Z}}$ refers to $\prod_{p}\mathbb{Z}_{p}$, where $\mathbb{Z}_{p}$ are the $p$-adic integers, which can be defined as the inverse limit of the inverse system formed by $\mathbb{Z}/p^{n}\mathbb{Z}$.

##### The Arithmetic Site

We now proceed to discuss the arithmetic site. It is described as the pair $(\widehat{\mathbb{N}^{\times}},\mathbb{Z}_{\text{max}})$, where $\widehat{\mathbb{N}^{\times}}$ a Grothendieck topos, which, as we may recall from More Category Theory: The Grothendieck Topos, is defined as a category equivalent to the category $\text{Sh}(\mathbf{C},J)$ of sheaves on a site $(\mathbf{C},J)$. In the case of $\widehat{\mathbb{N}^{\times}}$, $\mathbf{C}$ is the category with only one object, and whose morphisms correspond to the multiplicative monoid of nonzero natural numbers $\mathbb{N}^{\times}$ (we also use $\mathbb{N}^{\times}$ to denote this category, and $\mathbb{N}_{0}^{\times}$ to denote the category with one object and whose morphisms correspond to $\mathbb{N}^{\times}\cup\{0\}$), while $J$ is the indiscrete, or chaotic, Grothendieck topology, where all presheaves are also sheaves.

As part of the definition of the arithmetic site, we must also specify a structure sheaf. In this case is provided by $\mathbb{Z}_{\text{max}}$, the semiring (a semiring is like a ring, but is only a monoid, and not a group, under the “addition” operation – a semiring is also sometimes called a “rig“, because it is a ring without the “n” – the negative elements, and the most common example is the natural numbers $\mathbb{N}$ with the usual addition and multiplication) whose elements are just the integers, together with $-\infty$, but where the “addition” is provided by the “maximum” operation, and the “multiplication” is provided by the ordinary addition! With the arithmetic site thus defined, we denote it by $\mathcal{A}$.

We digress for a while to discuss the semiring $\mathbb{Z}_{\text{max}}$, as well as the closely related semirings $\mathbb{R}_{\text{max}}$ (defined similarly to $\mathbb{Z}_{\text{max}}$, but with the real numbers instead of the integers), $\mathbb{R}_{+}^{\text{max}}$ (whose elements are the positive real numbers, with the addition given by the maximum operation, and the multiplication given by the ordinary multiplication), and the so-called Boolean semifield $\mathbb{B}$ (whose elements are $0$ and $1$, with the addition again given by the maximum operation, and the multiplication again given by the ordinary multiplication). These semirings have origins in the area of mathematics known as tropical geometry, so named because one of its pioneers, Imre Simon, comes from Brazil, which is a tropical country. However, another source of inspiration is the work of the mathematical physicist Viktor Pavlovich Maslov in “semiclassical” quantum mechanics, where certain approximations could be made as the quantum mechanical systems being studied approached the classical limit. Maslov considered a “conjugated” addition

$\displaystyle \lim_{\epsilon\to 0}(x^{\frac{1}{\epsilon}}+y^{\frac{1}{\epsilon}})^{\epsilon}$

and this just happened to be the same as $\text{max}(x,y)$.

Going back to the arithmetic site, we now discuss its points. Recall from Even More Category Theory: The Elementary Topos that a point of a topos (we discussed elementary toposes in that post, but this also applies to Grothendieck toposes) is defined by a geometric morphism from the topos $\mathfrak{P}$ of sheaves of sets on the singleton set (the set with a single element) to the topos. This refers to a pair of adjoint functors such that the left-adjoint is left-exact (preserves finite limits). Therefore, for the arithmetic site, a point $p$ is given by such a pair $p^{*}$ and $p_{*}$ such that $p^{*}:\widehat{\mathbb{N}^{\times}}\rightarrow\textbf{Sets}$ is left-exact. The point $p$ is also uniquely determined by the covariant functor $\mathscr{P}=p^{*}\circ\epsilon:\mathbb{N}^{\times}\rightarrow\textbf{Sets}$ where $\epsilon:\mathbb{N}^{\times}\rightarrow\widehat{\mathbb{N}^{\times}}$ is the Yoneda embedding.

There is an equivalence of categories between the category of points of the arithmetic site and the category of totally ordered groups which are isomorphic to the nontrivial subgroups of $(\mathbb{Q},\mathbb{Q}_{+})$ and injective morphisms of ordered groups. For such an ordered group $\textbf{H}$ we therefore have a point $\mathscr{P}_{\textbf{H}}$. This gives us a correspondence with $\mathbb{Q}_{+}^{\times}\backslash\mathbb{A}_{\mathbb{Q}}^{f}/\hat{\mathbb{Z}}^{\times}$ (where $\mathbb{A}_{\mathbb{Q}}^{f}$ refers to the ring of finite adeles of $\mathbb{Q}$, which is defined similarly to the ring of adeles of $\mathbb{Q}$ except that the infinite prime is not considered) because any such ordered group $\textbf{H}$ is of the form $\textbf{H}_{a}$, the ordered group of all rational numbers $q$ such that $aq\in\hat{\mathbb{Z}}$, for some unique $a\in \mathbb{A}_{\mathbb{Q}}^{f}/\hat{\mathbb{Z}}$. We can also now describe the stalks of the structure sheaf $\mathbb{Z}_{\text{max}}$ at the point $\mathscr{P}_{\textbf{H}}$; it is isomorphic to the semiring $H_{\text{max}}$, with elements given by the set $(\textbf{H}\cup\{-\infty\})$, addition given by the maximum operation, and multiplication given by the ordinary addition.

The arithmetic site is analogous to the curve $C$ over the finite field $\mathbb{F}_{q}$. As for the finite field $\mathbb{F}_{q}$, its analogue is given by the Boolean semifield $\mathbb{B}$ mentioned earlier, which has “characteristic $1$“, reminiscent of the field with one element (see The Field with One Element). Next we want to find the analogues of the algebraic closure $\bar{\mathbb{F}}_{q}$, as well as the Frobenius morphism. The former is given by the semiring $\mathbb{R}_{+}^{\text{max}}$, which contains $\mathbb{B}$, while the latter is given by multiplicative group of the positive real numbers $\mathbb{R}_{+}^{\times}$, as it is isomorphic to the group of automorphisms of $\mathbb{R}_{+}^{\text{max}}$ that keep $\mathbb{B}$ fixed.

But while we do know that the points of the arithmetic topos are given by geometric morphisms $p:\mathfrak{P}\rightarrow \widehat{\mathbb{N}^{\times}}$ and determined by contravariant functors $\mathscr{P}_{\textbf{H}}:\mathbb{N}^{\times}\rightarrow\textbf{Sets}$, what do we mean by its “points over $\mathbb{R}_{+}^{\text{max}}$“? A point of the arithmetic site “over $\mathbb{R}_{+}^{\text{max}}$” refers to the pair $(\mathscr{P}_{\textbf{H}},f_{\mathscr{P}}^{\#})$, where $\mathscr{P}_{\textbf{H}}:\mathbb{N}^{\times}\rightarrow\textbf{Sets}$ as earlier, and $f_{\mathscr{P}_{\textbf{H}}}^{\#}:H_{\text{max}}\rightarrow\mathbb{R}_{+}^{\text{max}}$ (we recall that $H_{\text{max}}$ are the stalks of the structure sheaf $\mathbb{Z}_{\text{max}}$). The points of the arithmetic site over $\mathbb{R}_{+}^{\text{max}}$ include its points “over $\mathbb{B}$“, which are what we discussed earlier, and mentioned to be in correspondence with $\mathbb{Q}_{+}^{\times}\backslash\mathbb{A}_{\mathbb{Q}}^{f}/\hat{\mathbb{Z}}^{\times}$. But in addition, there are also other points of the arithmetic site over $\mathbb{R}_{+}^{\text{max}}$ which are in correspondence with $\mathbb{Q}_{+}^{\times}\backslash((\mathbb{A}_{\mathbb{Q}}^{f}/\hat{\mathbb{Z}}^{\times})\times\mathbb{R}_{+}^{\times})$, just as $\mathbb{R}_{+}^{\text{max}}$ contains all of $\mathbb{B}$ but also other elements. Altogether, the points of the arithmetic site over $\mathbb{R}_{+}^{\text{max}}$ correspond to the disjoint union of $\mathbb{Q}_{+}^{\times}\backslash\mathbb{A}_{\mathbb{Q}}^{f}/\hat{\mathbb{Z}}^{\times}$ and $\mathbb{Q}_{+}^{\times}\backslash((\mathbb{A}_{\mathbb{Q}}^{f}/\hat{\mathbb{Z}}^{\times})\times\mathbb{R}_{+}^{\times})$, which is $\mathbb{Q}^{\times}\backslash\mathbb{A}_{\mathbb{Q}}/\hat{\mathbb{Z}}^{\times}$, the adele class space as mentioned earlier.

There is a geometric morphism $\Theta:\text{Spec}(\mathbb{Z})\rightarrow \widehat{\mathbb{N}_{0}^{\times}}$ (here $\widehat{\mathbb{N}_{0}^{\times}}$ is defined similarly to $\widehat{\mathbb{N}^{\times}}$, but with $\mathbb{N}_{0}^{\times}$ in place of $\mathbb{N}^{\times}$) uniquely determined by

$\displaystyle \Theta^{*}:\mathbb{N}_{0}^{\times}\rightarrow \text{Sh}(\text{Spec}(\mathbb{Z}))$

which sends the single object of $\mathbb{N}_{0}^{\times}$ to the sheaf $\mathcal{S}$ on $\text{Spec}(\mathbb{Z})$, which we now describe. Let $H_{p}$ denote the set of all rational numbers $q$ such that $a_{p}q$ is an element of $\hat{Z}$, where $a_{p}$ is the adele with a $0$ for the $p$-th component and $1$ for all other components. Then the sheaf $\mathcal{S}$ can be described in terms of its stalks $\mathcal{S}_{\mathscr{P}}$, which are given by $H_{p}^{+}$, the positive part of $H_{p}$, and $\mathcal{S}_{0}$, given by $\{0\}$. The sections $\Gamma(U,\mathcal{S})$ are given by the maps $\xi:U\rightarrow \coprod_{p}H_{p}^{+}$ such that $\xi_{p}\neq 0$ for finitely many $p\in U$.

##### The Scaling Site

Now that we have defined the arithmetic site, which is the analogue of $C$, and the points of the arithmetic site over $\mathbb{R}_{+}^{\text{max}}$, which is the analogue of the points of $C$ over the algebraic closure $\bar{\mathbb{F}}_{q}$, we now proceed to define the scaling site, which is the analogue of $\bar{C}=C\otimes_{\mathbb{F}_{q}}\bar{\mathbb{F}}_{q}$. The points of the scaling site are the same as the points of the arithmetic site over $\mathbb{R}_{+}^{\text{max}}$, which is analogous to the points of $\bar{C}$ being the same as the points of $C$ over $\bar{\mathbb{F}}_{q}$. But the importance of the scaling site lies in the fact that we can construct the analogue of a sheaf of rational functions on it, and a Riemann-Roch theorem, which, as we may recall from The Riemann Hypothesis for Curves over Finite Fields, it is also an important part of Weil’s proof.

The scaling site is once again given by a pair $([0,\infty)\rtimes\mathbb{N}^{\times},\mathcal{O})$, where $[0,\infty)\rtimes\mathbb{N}^{\times}$ is a Grothendieck topos and $\mathcal{O}$ is a structure sheaf, but both are quite sophisticated constructions compared to the arithmetic site. To describe the Grothendieck topos $[0,\infty)\rtimes\mathbb{N}^{\times}$ we recall that it must be a category equivalent to the category $\text{Sh}(\mathbf{C},J)$ of sheaves on some site $(\mathbf{C},J)$. Here $\mathbf{C}$ is the category whose objects are given by bounded open intervals $\Omega\subset [0,\infty)$, including the empty interval $\null$, and whose morphisms are given by

$\displaystyle \text{Hom}(\Omega,\Omega')=\{n\in\mathbb{N}^{\times}|n\Omega\subset\Omega'\}$

and in the special case that $\Omega$ is the empty interval $\null$, we have

$\displaystyle \text{Hom}(\Omega,\Omega')=\{*\}$.

The Grothendieck topology $J$ here is defined by the collection $K(\Omega)$ of all ordinary covers of $\Omega$ for any object $\Omega$ of the category $\mathbf{C}$:

$\displaystyle \{\Omega_{i}\}_{i\in I}=\{\Omega_{i}\subset\Omega|\cup_{i}\Omega_{i}=\Omega\}$

Now we have to describe the structure sheaf $\mathcal{O}$. We start by considering $\mathbb{Z}_{\text{max}}$, the structure sheaf of the arithmetic site. By “extension of scalars” from $\mathbb{B}$ to $\mathbb{R}_{+}^{\text{max}}$ we obtain the reduced semiring $\mathbb{Z}_{\text{max}}\hat{\otimes}_{\mathbb{B}}\mathbb{R}_{+}^{\text{max}}$. This is not yet the structure sheaf $\mathcal{O}$, because the underlying category and Grothendieck topology for the scaling site is more complicated than the arithmetic site, and unlike the case for the arithmetic site, for the scaling site not every presheaf is a sheaf. So we must first “localize” $\mathbb{Z}_{\text{max}}\hat{\otimes}_{\mathbb{B}}\mathbb{R}_{+}^{\text{max}}$, and this gives us the structure sheaf $\mathcal{O}$.

Let us describe $\mathbb{Z}_{\text{max}}\hat{\otimes}_{\mathbb{B}}\mathbb{R}_{+}^{\text{max}}$ in more detail. Let $H$ be a rank $1$ subgroup of $\mathbb{R}$. Then an element of $H_{\text{max}}\hat{\otimes}_{\mathbb{B}}\mathbb{R}_{+}^{\text{max}}$ is given by a Newton polygon $N\subset\mathbb{R}^{2}$, which is the convex hull of the union of finitely many quadrants $(x_{j},y_{j}-Q)$, where $Q=H\times\mathbb{R}_{+}$ and $(x_{j},y_{j})\in H\times R$ (a set is a convex set if it contains the line segment connecting any two of its points; the convex hull of a set is the smallest convex set that contains it). The Newton polygon $N$ is uniquely determined by the function

$\displaystyle \ell_{N}(\lambda)=\text{max}(\lambda x_{j}+y_{j})$

for $\lambda\in\mathbb{R}_{+}$. This correspondence gives us an isomorphism between $H\hat{\otimes}_{\mathbb{B}}\mathbb{R}_{+}^{\text{max}}$ and $\mathcal{R}(H)$, the semiring of convex, piecewise affine, continuous functions on $[0,\infty)$ with slopes in $H\subset\mathbb{R}$ and finitely many singularities, with the pointwise operations (function is a convex function if the points on and above its graph form a convex set).

Therefore, we can describe the sections $\Gamma(\Omega,\mathcal{O})$ of the structure sheaf $\mathcal{O}$, for any bounded open interval $\Omega$, as the set of all convex, piecewise affine, continuous functions from $\Omega$ to $\mathbb{R}_{\text{max}}$ with slopes in $\mathbb{Z}$. We can also likewise describe the stalks of the structure sheaf $\mathcal{O}$ – for a point $\mathfrak{p}_{H}:[0,\infty)\rtimes\mathbb{N}^{\times}\rightarrow\textbf{Sets}$ associated to a rank 1 subgroup $H\subset\mathbb{R}$, the stalk $\mathcal{O}_{\mathfrak{p}_{H}}$ is given by the semiring $\mathcal{R}_{H}$ of germs of $\mathbb{R}_{+}^{\text{max}}$-valued, convex, piecewise affine, continuous functions with slope in $H$. We also have points $\mathfrak{p}_{H}^{0}:[0,\infty)\rtimes\mathbb{N}^{\times}\rightarrow\textbf{Sets}$ with “support $\{0\}$“, corresponding to the points of the arithmetic site over $\mathbb{B}$. For such a point, the stalk $\mathcal{O}_{\mathfrak{p}_{H}^{0}}$ is given by the semiring $(H\times\mathbb{R})_{\text{max}}$ associated to the totally ordered group $H\times\mathbb{R}$.

Now that we have decribed the Grothendieck topos $[0,\infty)\rtimes\mathbb{N}^{\times}$ and the structure sheaf $\mathcal{O}$, we describe the scaling site as being given by the pair $([0,\infty)\rtimes\mathbb{N}^{\times},\mathcal{O})$, and we denote it by $\hat{\mathcal{A}}$.

Our next task, now that we have described the arithmetic site and the scaling site, is to find the analogue of the Riemann-Roch theorem. We start by noting that we have a sheaf of semifields $\mathcal{K}$, defined by letting $\mathcal{K}(\Omega)$ be the semifield of fractions of $\mathcal{O}(\Omega)$. For an element $f_{H}$ in the stalk $\mathcal{K}_{\mathfrak{p}_{H}}$ of $\mathcal{K}$, we define its order as

$\displaystyle \text{Order}_{H}(f):=h_{+}-h_{-}$

where

$\displaystyle h_{\pm}:=\lim_{\epsilon\to 0_{\pm}}(f((1+\epsilon)H)-f(H))/\epsilon$

for $\epsilon\in\mathbb{R}_{+}$.

We let $C_{p}$ be the set of all points $\mathfrak{p}_{H}:[0,\infty)\rtimes\mathbb{N}^{\times}\rightarrow\textbf{Sets}$ of the scaling site $\hat{\mathcal{A}}$ such that $H$ is isomorphic to $H_{p}$. The $C_{p}$ are the analogues of the orbits of Frobenius. There is a topological isomorphism $\eta_{p}:\mathbb{R}_{+}^{\times}/p^{\mathbb{Z}}\rightarrow C_{p}$. It is worth noting that the expression $\mathbb{R}_{+}^{\times}/p^{\mathbb{Z}}$ is reminiscent of the Tate uniformization of an elliptic curve (which generalizes the idea that an elliptic curve over the complex numbers forms a lattice in the complex plane to other complete fields besides the complex numbers –  see The Moduli Space of Elliptic Curves).

We have a pullback sheaf $\eta_{p}^{*}(\mathcal{O}|_{C_{p}})$, which we denote suggestively by $\mathcal{O}_{p}$. It is the sheaf on $\mathbb{R}_{+}^{\times}/p^{\mathbb{Z}}$ whose sections are convex, piecewise affine, continuous functions with slopes in $H_{p}$. We can consider the sheaf of quotients $\mathcal{K}_{p}$ of $\mathcal{O}_{p}$ and its global sections $f:\mathbb{R}_{+}^{\times}\rightarrow\mathbb{R}$, which are piecewise affine, continuous functions with slopes in $H_{p}$ such that $f(p\lambda)=f(\lambda)$ for all $\lambda\in\mathbb{R}_{+}^{\times}$. Defining

$\displaystyle \text{Order}_{\lambda}(f):=\text{Order}_{\lambda H_{p}}(f\circ\eta_{p}^{-1})$

we have the following property for any $f\in H^{0}(\mathbb{R}_{+}^{\times}/p^{\mathbb{Z}},\mathcal{K}_{p})$ (recall that the zeroth cohomology group $H^{0}(\mathbb{R}_{+}^{\times}/p^{\mathbb{Z}},\mathcal{K}_{p})$ is defined as the space of global sections of $\mathcal{K}_{p}$):

$\displaystyle \sum_{\lambda\in\mathbb{R}_{+}^{\times}/p^{\mathbb{Z}}}\text{Order}_{\lambda}(f)=0$

We now want to define the analogue of divisors on $C_{p}$ (see Divisors and the Picard Group). A divisor $D$ on $C_{p}$ is a section $C_{p}\rightarrow H$, mapping $\mathfrak{p}_{H}\in C_{p}$ to $D(H)\in H$, of the bundle of pairs $(H,h)$, where $H\subset\mathbb{R}$ is isomorphic to $H_{p}$, and $h\in H$. We define the degree of a divisor $D$ as follows:

$\displaystyle \text{deg}(D)=\sum_{\mathfrak{p}\in C_{p}}D(H)$

Given a point $\mathfrak{p}_{H}\in C_{p}$ such that $H=\lambda H_{p}$ for some $\lambda\in\mathbb{R}_{+}^{*}$, we have a map $\lambda^{-1}:H\rightarrow H_{p}$. This gives us a canonical mapping

$\displaystyle \chi: H\rightarrow H_{p}/(p-1)H_{p}\simeq\mathbb{Z}/(p-1)\mathbb{Z}$

Given a divisor $D$ on $C_{p}$, we define

$\displaystyle \chi(D):=\sum_{\frak{p}_{H}\in C_{p}}\chi(D(H))\in\mathbb{Z}/(p-1)\mathbb{Z}$

We have $\text{deg}(D)=0$ and $\chi(D)=0$ if and only if $D=(f)$, for $f\in H^{0}(\mathbb{R}_{+}^{\times}/p^{\mathbb{Z}}\mathcal{K}_{p})$ i.e. $D$ is a principal divisor.

We define the group $J(C_{p})$ as the quotient $\text{Div}^{0}(C_{p})/\mathcal{P}$ of the group $\text{Div}^{0}(C_{p})$ of divisors of degree $0$ on $C_{p}$ by the group $\mathcal{P}$ of principal divisors on $C_{p}$. The group $J(C_{p})$ is isomorphic to $\mathbb{Z}/(p-1)\mathbb{Z}$, while the group $\text{Div}(C_{p})/\mathcal{P}$ of divisors on $C_{p}$ modulo the principal divisors is isomorphic to $\mathbb{R}\times(\mathbb{Z}/(p-1)\mathbb{Z})$.

In order to state the analogue of Riemann-Roch theorem we need to define the following module over $\mathbb{R}_{+}^{\text{max}}$:

$\displaystyle H^{0}(D):=\{f\in\mathcal{K}_{p}|D+(f)\geq 0\}$

Given $f\in H^{0}(C_{p},\mathcal{K}_{p})$, we define

$\displaystyle \|f\|_{p}:=\text{max}\{h(\lambda)|_{p}/\lambda,\lambda\in C_{p}\}$

where $h(\lambda)$ is the slope of $f$ at $\lambda$. Then we have the following increasing filtration on $H^{0}$:

$\displaystyle H^{0}(D)^{\rho}:=\{f\in H^{0}(D)|\|f\|_{p}\leq\rho\}$

This allows us to define the following notion of dimension for $H^{0}(D)$ (here $\text{dim}_{\text{top}}$ refers to what is known as the topological dimension or Lebesgue covering dimension, a notion of dimension defined in terms of refinements of open covers):

$\displaystyle \text{Dim}_{\mathbb{R}}(H^{0}(D))=\lim_{n\to\infty}p^{-n}\text{dim}_{\text{top}}(H^{0}(D)^{p^{n}})$

The analogue of the Riemann-Roch theorem is now given by the following:

$\displaystyle \text{Dim}_{\mathbb{R}}(H^{0}(D))+\text{Dim}_{\mathbb{R}}(H^{0}(-D))=\text{deg}(D)$

##### S-Algebras

This concludes our discussion of the arithmetic site and the scaling site, but I would like to discuss one more related topic also being explored by Connes and Consani – the use of $\mathbb{S}$-algebras, which is closely related to the $\Gamma$-sets we have already introduced in The Field with One Element. Both of these concepts have their origins in homotopy theory.

We recall from the short discussion at the end of The Riemann Hypothesis for Curves over Finite Fields that the Weil conjectures, which are Weil’s generalization of the Riemann hypothesis for curves over finite fields to varieties of higher dimension, were proven by making use of cohomology (in particular etale cohomology) to find the fixed points of the powers of the Frobenius morphism (the formula that gives us the fixed points of a map using cohomology is called the Lefschetz fixed point formula). Now, concepts such as monoids, semirings, and many others (including the mathematician Nikolai Durov’s approach to the field with one element, which he also uses to develop a new version of Arakelov geometry) are all subsumed under the concept of $\mathbb{S}$-algebras, and doing so allows us to make use of a cohomology theory called topological cyclic cohomology. Connes and Consani hope that topological cyclic cohomology will help prove the original Riemann hypothesis the way that etale cohomology helped prove the Weil conjectures. Let us discuss briefly the work of Connes and Consani on this topic.

We recall from The Field with One Element the definition of a $\Gamma$-set (there also referred to as a $\Gamma$-space). A $\Gamma$-set is defined to be a covariant functor from the category $\Gamma^{\text{op}}$, whose objects are pointed finite sets and whose morphisms are basepoint-preserving maps of finite sets, to the category $\textbf{Sets}_{*}$ of pointed sets. An $\mathbb{S}$-algebra is defined to be a $\Gamma$-set $\mathscr{A}:\Gamma^{\text{op}}\rightarrow \textbf{Sets}_{*}$ together with an associative multiplication $\mu:\mathscr{A}\wedge \mathscr{A}\rightarrow\mathscr{A}$ and a unit $1:\mathbb{S}\rightarrow\mathscr{A}$, where $\mathbb{S}:\Gamma^{\text{op}}\rightarrow\textbf{Sets}_{*}$ is the inclusion functor (also known as the sphere spectrum). An $\mathbb{S}$-algebra is a monoid in the symmetric monoidal category of $\Gamma$-sets with the wedge product and the sphere spectrum.

Any monoid $M$ defines an $\mathbb{S}$-algebra $\mathbb{S}M$ via the following definition:

$\displaystyle \mathbb{S}M(X):=M\wedge X$

for any pointed finite set $X$. Here $M\wedge X$ is the smash product of $M$ and $X$ as pointed sets, with the basepoint for $M$ given by its zero element element. The maps are given by $\text{Id}_{M}\times f$, for $f:X\rightarrow Y$.

Similarly, any semiring $R$ defines an $\mathbb{S}$-algebra $HR$ via the following definition:

$\displaystyle HR(X):=X^{R/*}$

for any pointed finite set $X$. Here $X^{R/*}$ refers to the set of basepoint preserving maps from $R$ to $X$. The maps $HR(f)$ are given by $HR(f)(\phi)(y):=\sum_{x\in f^{-1}(y)}\phi(x)$ for $f:X\rightarrow Y$, $x\in X$, and $y\in Y$. The multiplication $HR(X)\wedge HR(Y)\rightarrow HR(X\wedge Y)$ is given by $\phi\psi(x,y)=\phi(x)\psi(y)$ for any $x\in X\setminus *$ and $y\in Y\setminus *$. The unit $1_{X}:X\rightarrow HR(X)$ is given by $1_{X}(x)=\delta_{x}$ for all $x$ in $X$, where $\delta_{x}(y)=1$ if $x=y$, and $0$ otherwise.

Therefore we can see that the notion of $\mathbb{S}$-algebra subsumes the notions of monoids and semirings, and other notions such as that of “hyperrings“, which we leave to the references for the moment. Instead, we will discuss how $\mathbb{S}$-algebras are related to the approach of Durov to the field with one element and Arakelov geometry. As we mentioned in Arakelov Geometry, the main idea of the theory is to consider the “infinite prime” along with the other points of $\text{Spec}(\mathbb{Z})$. We therefore define $\overline{\text{Spec}(\mathbb{Z})}$ as $\text{Spec}(\mathbb{Z})\cup \{\infty\}$. Let $\mathcal{O}_{\text{Spec}(\mathbb{Z})}$ be the structure sheaf of $\text{Spec}(\mathbb{Z})$. We want to extend this to a structure sheaf on $\overline{\text{Spec}(\mathbb{Z})}$, and to accomplish this we will use the functor $H$ from semirings to $\mathbb{S}$-algebras defined earlier. For any open set $U$ containing $\infty$, we define

$\displaystyle \mathcal{O}_{\overline{\text{Spec}(\mathbb{Z})}}(U):=\|H\mathcal{O}_{\text{Spec}(\mathbb{Z})}(U\cup\text{Spec}(\mathbb{Z}))\|_{1}$.

The notation $\|\|_{1}$ is defined for the $\mathbb{S}$-algebra $HR$ associated to the semiring $R$ as follows:

$\displaystyle \|HR(X)\|_{1}:=\{\phi\in HR(X)|\sum_{X\*}\|\phi(x)\|\leq 1\}$

where $\|\|$ in this particular case comes from the usual absolute value on $\mathbb{Q}$. This becomes available to us because the sheaf $\mathcal{O}_{\overline{\text{Spec}(\mathbb{Z})}}$ is a subsheaf of the constant sheaf $\mathbb{Q}$.

Given an Arakelov divisor on $\overline{\text{Spec}(\mathbb{Z})}$ (in this context an Arakelov divisor is given by a pair $(D_{\text{finite}},D_{\infty})$, where $D_{\text{finite}}$ is an ordinary divisor on $\text{Spec}(\mathbb{Z})$ and $D_{\infty}$ is a real number) we can define the following sheaf of $\mathcal{O}_{\overline{\text{Spec}(\mathbb{Z})}}$-modules over $\overline{\text{Spec}(\mathbb{Z})}$:

$\displaystyle \mathcal{O}_{\overline{\text{Spec}(\mathbb{Z})}}(D)(U):=\|H\mathcal{O}(D_{\text{finite}})(U\cup\text{Spec}(\mathbb{Z}))\|_{e^{a}}$

where $a$ is the real number “coefficient” of $D_{\infty}$, and $\|\|_{\lambda}$ means, for an $R$-module $E$ (here the $\mathbb{S}$-algebra $HE$ is constructed the same as $HR$, except there is no multiplication or unit) with seminorm $\|\|^{E}$ such that $\|a\xi\|^{E}\leq\|a\|\|\xi\|^{E}$ for $a\in R$ and $\xi\in E$,

$\displaystyle \|HE(X)\|_{\lambda}:=\{\phi\in HE(X)|^{E}\sum_{X\*}\|\phi(x)\|^{E}\leq \lambda\}$

With such sheaves of $\mathbb{S}$-algebras on $\overline{\text{Spec}(\mathbb{Z})}$ now constructed, the tools of topological cyclic cohomology can be applied to it. The theory of topological cyclic cohomology is left to the references for now, but will hopefully be discussed in future posts on this blog.

##### Conclusion

The approach of Connes and Consani, whether making use of the arithmetic site and the scaling site to apply Weil’s strategy to the original Riemann hypothesis, or making use of $\mathbb{S}$-algebras and topological cyclic cohomology in analogy with the proof of the Weil conjectures, is still currently facing several technical obstacles. In the former case, an intersection theory and a Riemann-Roch theorem on the square of the scaling site is yet to be constructed. In the latter, there is the problem of appropriate coefficients for the cohomology theory. There are already several proposed strategies for dealing with these obstacles. Such efforts, aside from aiming to prove the Riemann hypothesis, widens the scope of the mathematics that we have today, and, perhaps more importantly, uncovers more and more the mysterious geometry underlying the familiar everyday concept of numbers.

References:

On the Geometry of the Adele Class Space of Q by Caterina Consani

An Essay on the Riemann Hypothesis by Alain Connes

The Arithmetic Site by Alain Connes and Caterina Consani

Geometry of the Arithmetic Site by Alain Connes and Caterina Consani

The Scaling Site by Alain Connes and Caterina Consani

Geometry of the Scaling Site by Alain Connes and Caterina Consani

Absolute Algebra and Segal’s Gamma Sets by Alain Connes and Caterina Consani

New Approach to Arakelov Geometry by Nikolai Durov

# Arakelov Geometry

In many posts on this blog, such as Basics of Arithmetic Geometry and Elliptic Curves, we have discussed how the geometry of shapes described by polynomial equations is closely related to number theory. This is especially true when it comes to the thousands-of-years-old subject of Diophantine equations, polynomial equations whose coefficients are whole numbers, and whose solutions of interest are also whole numbers (or, equivalently, rational numbers, since we can multiply or divide both sides of the polynomial equation by a whole number). We might therefore expect that the more modern and more sophisticated tools of algebraic geometry (which is a subject that started out as just the geometry of shapes described by polynomial equations) might be extremely useful in answering questions and problems in number theory.

One of the tools we can use for this purpose is the concept of an arithmetic scheme, which makes use of the ideas we discussed in Grothendieck’s Relative Point of View. An arithmetic variety is defined to be a a regular scheme that is projective and flat over the scheme $\text{Spec}(\mathbb{Z})$. An example of this is the scheme $\text{Spec}(\mathbb{Z}[x])$, which is two-dimensional, and hence also referred to as an arithmetic surface.

We recall that the points of an affine scheme $\text{Spec}(R)$, for some ring $R$, are given by the prime ideals of $R$. Therefore the scheme $\text{Spec}(\mathbb{Z})$ has one point for every prime ideal – one “closed point” for every prime number $p$, and a “generic point” given by the prime ideal $(0)$.

However, we also recall from Adeles and Ideles the concept of the “infinite primes” – which correspond to the archimedean valuations of a number field, just as the finite primes (primes in the classical sense) correspond to the nonarchimedean valuations. It is important to consider the infinite primes when dealing with questions and problems in number theory, and therefore we need to modify some aspects of algebraic geometry if we are to use it in helping us with number theory.

We now go back to arithmetic schemes, taking into consideration the infinite prime. Since we are dealing with the ordinary integers $\mathbb{Z}$, there is only one infinite prime, corresponding to the embedding of the rational numbers into the real numbers. More generally we can also consider an arithmetic variety over $\text{Spec}(\mathcal{O_{K}})$ instead of $\text{Spec}(\mathbb{Z})$, where $\mathcal{O}_{K}$ is the ring of integers of a number field $K$. In this case we may have several infinite primes, corresponding to the embediings of $K$ into the real and complex numbers. In this post, however, we will consider only $\text{Spec}(\mathbb{Z})$ and one infinite prime.

How do we describe an arithmetic scheme when the scheme $\text{Spec}(\mathbb{Z})$ has been “compactified” with the infinite prime? Let us look at the fibers of the arithmetic scheme. The fiber of an arithmetic scheme $X$ at a finite prime $p$ is given by the scheme defined by the same homogeneous polynomials as $X$, but with the coefficients taken modulo $p$, so that they are elements of the finite field $\mathbb{F}_{p}$. The fiber over the generic point $(0)$ is given by taking the tensor product of the coordinate ring of $X$ with the rational numbers. But how should we describe the fiber over the infinite prime?

It was the idea of the mathematician Suren Arakelov that the fiber over the infinite prime should be given by a complex variety – in the case of an arithmetic surface, which Arakelov originally considered, the fiber should be given by a Riemann surface.  The ultimate goal of all this machinery, at least when Arakelov was constructing it, was to prove the famous Mordell conjecture, which states that the number of rational solutions to a curve of genus greater than or equal to $2$ was finite. These rational solutions correspond to sections of the arithmetic surface, and Arakelov’s strategy was to “bound” the number of these solutions by constructing a “height function” using intersection theory (see Algebraic Cycles and Intersection Theory) on the arithmetic surface. Arakelov unfortunately was not able to carry out his proof. He had only a very short career, being forced to retire after being diagnosed with schizophrenia. The Mordell conjecture was eventually proved by another mathematician, Gerd Faltings, who continues to develop Arakelov’s ideas.

Since we will be dealing with a complex variety, we must first discuss a little bit of differential geometry, in particular complex geometry (see An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry). Let $X$ be a smooth projective complex equidimensional variety with complex dimension $d$. The space $A^{n}(X)$ of differential forms (see Differential Forms) of degree $n$ on $X$ has the following decomposition:

$\displaystyle A^{n}(X)=\bigoplus_{p+q=n}A^{p,q}(X)$

We say that $A^{p,q}(X)$ is the vector space of complex-valued differential forms of type $(p,q)$. We have differential operators

$\displaystyle \partial:A^{p,q}(X)\rightarrow A^{p+1,q}(X)$

$\displaystyle \bar{\partial}:A^{p,q}(X)\rightarrow A^{p,q+1}(X)$.

$\displaystyle d=\partial+\bar{\partial}:A^{n}\rightarrow A^{n+1}$.

We let $D_{p,q}(X)$ be the dual to the vector space $A^{p,q}(X)$, and we write $D^{p,q}(X)$ to denote $D_{d-p,d-q}(X)$. We refer to an element of $D^{p,q}$ as a current of type $(p,q)$. We have an inclusion map

$\displaystyle A^{p,q}\rightarrow D^{p,q}$

mapping a differential form $\omega$ of type $(p,q)$ to a current $[\omega]$ of type $(p,q)$, given by

$\displaystyle [\omega](\alpha)=\int_{X}\omega\wedge\alpha$

for all $\alpha\in A^{d-p,d-q}(X)$.

The differential operators $\partial$, $\bar{\partial}$, $d$, and induce maps $\partial'$, $\bar{\partial}'$, and $d'$ on $D^{p,q}$. We define the maps $\partial$, $\bar{\partial}$, and $d$ on $D^{p,q}$ by

$\displaystyle \partial=(-1)^{n+1}\partial'$

$\displaystyle \bar{\partial}=(-1)^{n+1}\bar{\partial}'$

$\displaystyle d=(-1)^{n+1}d'$

We also define

$\displaystyle d^{c}=(4\pi i)^{-1}(\partial-\bar{\partial})$.

For every irreducible analytic subvariety $i:Y\hookrightarrow X$ of codimension $p$, we define the current $\delta_{Y}\in D^{p,p}$ by

$\displaystyle \delta_{Y}(\alpha):=\int_{Y^{ns}}i^{*}\alpha$

for all $\alpha\in A^{d-p,d-q}$, where $Y^{ns}$ is the nonsingular locus of $Y$.

A Green current $g$ for a codimension $p$ analytic subvariety $Y$ is defined to be an element of $D^{p-1,p-1}(X)$ such that

$\displaystyle dd^{c}g+\delta_{Y}=[\omega]$

for some $\omega\in A^{p,p}(X)$.

Let $\tilde{X}$ be the resolution of singularities of $X$. This means that there exists a proper map $\pi: \tilde{X}\rightarrow X$ such that $\tilde X$ is smooth, $E:=\pi^{-1}(Y)$ is a divisor with normal crossings (this means that each irreducible component of $E$ is nonsingular, and whenever they meet at a point their local equations  are linearly independent) whenever $Y\subset X$ contains the singular locus of $X$, and $\pi: \tilde{X}\setminus E\rightarrow X\setminus Y$ is an isomorphism.

A smooth form $\alpha$ on $X\setminus Y$ is said to be of logarithmic type along $Y$ if there exists a projective map $\pi:\tilde{X}\rightarrow X$ such that $E:= \pi^{-1}(Y)$ is a divisor with normal crossings, $\pi:\tilde{X}\setminus E\rightarrow X\setminus Y$ is smooth, and $\alpha$ is the direct image by $\pi$ of a form $\beta$ on $X\setminus E$ satisfying the following equation

$\displaystyle \beta=\sum_{i=1}^{k}\alpha_{i}\text{log}|z_{i}|^{2}+\gamma$

where $z_{1}z_{2} ... z_{k}=0$ is a local equation of $E$ for every $x$ in $X$, $\alpha_{i}$ are $\partial$ and $\bar{\partial}$ closed smooth forms, and $\gamma$ is a smooth form.

For every irreducible subvariety $Y\subset X$ there exists a smooth form $g_{Y}$ on $X\setminus Y$ of logarithmic type along $Y$ such that $[g_{Y}]$ is a Green current for $Y$:

$\displaystyle dd^{c}[g_{Y}]+\delta_{Y}=[\omega]$

where w is smooth on X. We say that $[g_{Y}]$ is a Green current of logarithmic type.

We now proceed to discuss this intersection theory on the arithmetic scheme. We consider a vector bundle $E$ on the arithmetic scheme $X$, a holomorphic vector bundle (a complex vector bundle $E_{\infty}$ such that the projection map is holomorphic) on the fibers $X_{\infty}$ at the infinite prime, and a smooth hermitian metric (a sesquilinear form $h$ with the property that $h(u,v)=\overline{h(v,u)}$) on $E_{\infty}$ which is invariant under the complex conjugation on $X_{\infty}$. We refer to this collection of data as a hermitian vector bundle $\bar{E}$ on $X$.

Given an arithmetic scheme $X$ and a hermitian vector bundle $\bar{E}$ on $X$, we can define associated “arithmetic”, or “Arakelov-theoretic” (i.e. taking into account the infinite prime) analogues of the algebraic cycles and Chow groups that we discussed in Algebraic Cycles and Intersection Theory.

An arithmetic cycle on $X$ is a pair $(Z,g)$ where $Z$ is an algebraic cycle on $X$, i.e. a linear combination $\displaystyle \sum_{i}n_{i}Z_{i}$ of closed irreducible subschemes $Z_{i}$ of $X$, of some fixed codimension $p$, with integer coefficients $n_{i}$, and $g$ is a Green current for $Z$, i.e. $g$ satisfies the equation

$\displaystyle dd^{c}g+\delta_{Z}=[\omega]$

where

$\displaystyle \delta_{Z}(\eta)=\sum_{i}n_{i}\int_{Z_{i}}\eta$

for differential forms $\omega$ and $\eta$ of appropriate degree.

We define the arithmetic Chow group $\widehat{CH}^{p}(X)$ as the group of arithmetic cycles $\widehat{Z}^{p}(X)$ modulo the subgroup $\widehat{R}^{p}(X)$ generated by the pairs $(0,\partial u+\bar{\partial}v)$ and $(\text{div}(f),-\text{log}(|f|^{2}))$, where $u$ and $v$ are currents of appropriate degree and $f$ is some rational function on some irreducible closed subscheme of codimension $p-1$ in $X$ .

Next we want to have an intersection product on Chow groups, i.e. a bilinear pairing

$\displaystyle \widehat{CH}^{p}(X)\times\widehat{CH}^{q}(X)\rightarrow\widehat{CH}^{p+q}(X)$

We now define this intersection product. Let $[Y,g_{Y}]\in\widehat{CH}^{p}(X)$ and $[Z,g_{Z}]\in\widehat{CH}^{q}$. Assume that $Y$ and $Z$ are irreducible. Let $Y_{\mathbb{Q}}=Y\otimes_{\text{Spec}(\mathbb{Z})}\text{Spec}(\mathbb{Q})$, and $Z_{\mathbb{Q}}=Z\otimes_{\text{Spec}(\mathbb{Z})}\text{Spec}(\mathbb{Q})$. If $Y_{\mathbb{Q}}$ and $Z_{\mathbb{Q}}$ intersect properly, i.e. $\text{codim}(Y_{\mathbb{Q}}\cap Z_{\mathbb{Q}})=p+q$, then we have

$\displaystyle [(Y,g_{Y})]\cdot [(Z,g_{Z})]:=[[Y]\cdot[Z],g_{Y}*g_{Z}]$

where $[Y]\cdot[Z]$ is just the usual intersection product of algebraic cycles, and $g_{Y}*g_{Z}$ is the $*$-product of Green currents, defined for a Green current of logarithmic type $g_{Y}$ and a Green current $g_{Z}$, where $Y$ and $Z$ are closed irreducible subsets of $X$ with $Z$ not contained in $Y$, as

$\displaystyle g_{Y}*g_{Z}:=[\tilde{g}_{Y}]*g_{Z}\text{ mod }(\text{im}(\partial)+\text{im}(\bar{\partial}))$

where

$\displaystyle [g_{Y}]*g_{Z}:=[g_{Y}]\wedge\delta_{Z}+[\omega_{Y}]\wedge g_{Z}$

and

$[g_{Y}]\wedge\delta_{Z}:=q_{*}[q^{*}g_{Y}]$

for $q:\tilde{Z}\rightarrow X$ is the resolution of singularities of $Z$ composed with the inclusion of $Z$ into $X$.

In the case that $Y_{\mathbb{Q}}$ and $\mathbb{Q}$ do not intersect properly, there is a rational function $f_{y}$ on $y\in X_{\mathbb{Q}}^{p-1}$ such that $\displaystyle Y+\sum_{y}\text{div}(f_{y})$ and $Z$ intersect properly, and if $g_{y}$ is another rational function such that $\displaystyle Y+\sum_{y}\text{div}(f_{y})_{\mathbb{Q}}$ and $Z_{\mathbb{Q}}$ intersect properly, the cycle

$\displaystyle (\sum_{y}\widehat{\text{div}}(f_{y})-\sum_{y}\widehat{\text{div}}(g_{y}))\cdot(Z,g_{Z})$

is in the subgroup $\widehat{R}^{p}(X)$. Here the notation $\widehat{\text{div}}(f_{y})$ refers to the pair $(\text{div}(f),-\text{log}(|f|^{2}))$.

This concludes our little introduction to arithmetic intersection theory. We now give a short discussion what else can be done with such a theory. The mathematicians Henri Gillet and Christophe Soule used this arithmetic intersection theory to construct arithmetic analogues of Chern classes, Chern characters, Todd classes, and the Grothendieck-Riemann-Roch theorem (see Chern Classes and Generalized Riemann-Roch Theorems). These constructions are not so straightforward – for instance, one has to deal with the fact that unlike the classical case, the arithmetic Chern character is not additive on exact sequences. This failure to be additive on exact sequences is measured by the Bott-Chern character. The Bott-Chern character plays a part in defining the arithmetic analogue of the Grothendieck group $\widehat{K}_{0}(X)$.

In order to define the arithmetic analogue of the Grothendieck-Riemann-Roch theorem, one must then define the direct image map $f_{*}:\widehat{K}_{0}(X)\rightarrow\widehat{K}_{0}(Y)$ for a proper flat map $f:X\rightarrow Y$ of arithmetic varieties. This involves constructing a canonical line bundle $\lambda(E)$ on $Y$, whose fiber at $y$ is the determinant of cohomology of $X_{y}=f^{-1}(y)$, i.e.

$\displaystyle \lambda(E)_{y}=\bigotimes_{q\geq 0}(\text{det}(H^{q}(X_{y},E))^{(-1)^{q}}$

as well as a metric $h_{Q}$, called the Quillen metric, on $\lambda(E)$. With such a direct image map we can now give the statement of the arithmetic Grothendieck-Riemann-Roch theorem. It was originally stated by Gillet and Soule in terms of components of degree one in the arithmetic Chow group $\widehat{CH}(Y)\otimes_{\mathbb{Z}}\mathbb{Q}$:

$\widehat{c}_{1}(\lambda(E),h_{Q})=f_{*}(\widehat{\text{ch}}(E,h)\widehat{\text{Td}}(Tf,h_{f})-a(\text{ch}(E)_{\mathbb{C}}\text{Td}(Tf_{\mathbb{C}})R(Tf_{\mathbb{C}})))^{(1)}$

where $\widehat{\text{ch}}$ denotes the arithmetic Chern character, $\widehat{\text{Td}}$ denotes the arithmetic Todd class, $Tf$ is the relative tangent bundle of $f$, $a$ is the map from

$\displaystyle \tilde{A}(X)=\bigoplus_{p\geq 0}A^{p,p}(X)/(\text{im}(\partial)+\text{im}(\bar{\partial}))$

to $\widehat{CH}(X)$ sending the element $\eta$ in $\tilde{A}(X)$ to the class of $(0,\eta)$ in $\widehat{CH}(X)$, and

$\displaystyle R(L)=\sum_{m\text{ odd, }\geq 1}(2\zeta'(-m)+\zeta(m)(1+\frac{1}{2}+...+\frac{1}{m}))\frac{c_{1}(L)^{m}}{m!}$.

Later on Gillet and Soule formulated the arithmetic Grothendieck-Riemann-Roch theorem in higher degree as

$\displaystyle \widehat{\text{ch}}(f_{*}(x))=f_{*}(\widehat{\text{Td(g)}}\cdot(1-a(R(Tf_{\mathbb{C}})))\cdot\widehat{\text{ch}}(x))$

for $x\in\widehat{K}_{0}(X)$.

Aside from the work of Gillet and Soule, there is also the work of the mathematician Amaury Thuillier making use of ideas from $p$-adic geometry, constructing a nonarchimedean potential theory on curves that allows the finite primes and the infinite primes to be treated on a more equal footing, at least for arithmetic surfaces. The work of Thuillier is part of ongoing efforts to construct an adelic geometry, which is hoped to be the next stage in the evolution of Arakelov geometry.

References:

Arakelov Theory on Wikipedia

Arithmetic Intersection Theory by Henri Gillet and Christophe Soule

Theorie de l’Intersection et Theoreme de Riemann-Roch Arithmetiques by Jean-Benoit Bost

An Arithmetic Riemann-Roch Theorem in Higher Degrees by Henri Gillet and Christophe Soule

Theorie du Potentiel sur les Courbes en Geometrie Analytique Non Archimedienne et Applications a la Theorie d’Arakelov by Amaury Thuillier

Explicit Arakelov Geometry by Robin de Jong

Notes on Arakelov Theory by Alberto Camara

Lectures in Arakelov Theory by C. Soule, D. Abramovich, J.-F. Burnol, and J. Kramer

Introduction to Arakelov Theory by Serge Lang

# Chern Classes and Generalized Riemann-Roch Theorems

Chern classes are an ubiquitous concept in mathematics, being part of algebraic geometry, algebraic topology, and differential geometry. In this post we discuss Chern classes in the context of algebraic geometry, where they are part of intersection theory (see Algebraic Cycles and Intersection Theory). Among the applications of the theory of Chern classes is a higher-dimensional generalization of the Riemann-Roch theorem (see More on Sheaves) called the Hirzebruch-Riemann-Roch theorem. There is an even further generalization called the Grothendieck-Riemann-Roch theorem, which concerns a morphism of nonsingular projective varieties $f:X\rightarrow Y$, and for which the Hirzebruch-Riemann-Roch theorem is merely the case where $Y$ is a point.

Let $X$ be a nonsingular projective variety, and let $A(X)$ be the Chow ring of $X$ (see Algebraic Cycles and Intersection Theory). Let $\mathcal{E}$ be a locally free  sheaf of rank $r$ on $X$.

We recall that locally free  sheaves correspond to vector bundles (see Vector Fields, Vector Bundles, and Fiber Bundles and More on Sheaves). Therefore, their fibers are isomorphic to $\mathbb{A}^{r}$. The projective bundle $\mathbb{P}(\mathcal{E})$ associated to the locally free sheaf $\mathcal{E}$ is essentially obtained by replacing the fibers with projective spaces $\mathbb{A}\setminus\{0\}/k^{*}$ (see Projective Geometry).

Let $\xi\in A^{1}(\mathbb{P}(\mathcal{E}))$ be the class of the divisor corresponding to the twisting sheaf (see More on Sheaves) $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$. Let $\pi:\mathbb{P}(\mathcal{E})\rightarrow X$ be the projection of the fiber bundle $\mathbb{P}(\mathcal{E})$ to its “base space” $X$. Then the pullback $\pi^{*}:A^{i}(X)\rightarrow A^{i+r-1}(\mathbb{P}(\mathcal{E}))$ makes $A(\mathbb{P}(\mathcal{E}))$ into a free $A(X)$ module generated by $1$, $\xi$, $\xi^{2},...,\xi^{r-1}$.

We define the $i$-th Chern class $c_{i}(\mathcal{E})\in A^{i}(X)$ by the requirement that $c_{0}(\mathcal{E})=1$ and

$\displaystyle \sum_{i=0}^{r}(-1)^{i}\pi^{*}c_{i}(\mathcal{E}).\xi^{r-i}=0$

where the dot $.$ denotes the intersection product (see Algebraic Cycles and Intersection Theory).

Chern classes are associated to locally free sheaves, which, as we have already mentioned, correspond to vector bundles, and are elements of the Chow ring. We can therefore think of them as generalizing the correspondence between line bundles (vector bundles of dimension $1$) and elements of the Picard group, since, as mentioned in Algebraic Cycles and Intersection Theory, the Chow ring is kind of an analogue of the Picard group for higher dimensions.

We can also define the total Chern class

$\displaystyle c(\mathcal{E})=c_{0}(\mathcal{E})+c_{1}(\mathcal{E})...+c_{r}(\mathcal{E})$

and the Chern polynomial

$\displaystyle c_{t}(\mathcal{E})=c_{0}(\mathcal{E})+c_{1}(\mathcal{E})t+...+c_{r}(\mathcal{E})t^{r}$.

Chern classes satisfy the following important properties:

(i) If $\mathcal{E}$ is the line bundle $\mathcal{L}(D)$ associated to a divisor $D$, then $c_{t}=1+Dt$.

(ii) If $f:X'\rightarrow X$ is a morphism, and $\mathcal{E}$ is a locally free sheaf on $X$, then for each $i$,

$\displaystyle c_{i}(f^{*}\mathcal{E})=f^{*}c_{i}(\mathcal{E})$.

(iii) If $0\rightarrow\mathcal{E}'\rightarrow\mathcal{E}\rightarrow\mathcal{E}''\rightarrow 0$ is an exact sequence (see Exact Sequences) of locally free sheaves, then

$\displaystyle c_{t}(\mathcal{E})=c_{t}(\mathcal{E}')\cdot c_{t}(\mathcal{E}'')$

These three properties can also be considered as a set of axioms which define the Chern classes, instead of the definition that we gave earlier.

Another important property of Chern classes, which comes from the so-called splitting principle, allows us to factor the Chern polynomial into the Chern polynomials of line bundles, and so we have:

$c_{t}(\mathcal{E})=\prod_{i=1}^{r}(1+a_{i}t)$

The $a_{i}$ are called the Chern roots of $\mathcal{E}$.

We define the exponential Chern character (or simply Chern character) as

$\displaystyle \text{ch}(\mathcal{E})=\sum_{i=1}^{r}e^{a_{i}}$

and the Todd class as

$\displaystyle \text{td}(\mathcal{E})=\prod_{i=1}^{r}\frac{(a_{i})}{1-e^{-a_{i}}}$.

Now we can discuss the generalizations of the Riemann-Roch theorem. We first review the statement of the Riemann-Roch theorem for curves, but we restate it slightly in terms of the Euler characteristic.

The Euler characteristic of a coherent sheaf $\mathcal{E}$ on a projective scheme $X$ over a field $k$ is defined to be the alternating sum of the dimensions of the cohomology groups $H^{i}(X,\mathcal{F})$ (see Cohomology in Algebraic Geometry) as vector spaces over $k$.

$\displaystyle \chi(\mathcal{E})=\sum_{i}(-1)^{i}\text{dim}_{k}H^{i}(X,\mathcal{F})$.

Then we can state the Riemann-Roch theorem for curves as

$\chi(\mathcal{L}(D))=\text{deg}(D)+1-g$.

The connection of this formulation with the one we gave in More on Sheaves, where the left-hand side is given by $h^{0}(D)-h^{0}(K_{X}-D)$ is provided by the fact that $h^{0}(D)$ is the same as (and in fact defined as) $\text{dim}_{k}H^{0}(X, \mathcal{L}(D))$, together with the theorem known as Serre duality, which says that $H^{1}(X,\mathcal{L}(D))$ is dual to $H^{0}(X,\omega\otimes\mathcal{L}(D)^{\vee})$, where $\mathcal{L}(D)^{\vee}$ denotes the dual of the line bundle $\mathcal{L}(D)$.

The Hirzebruch-Riemann-Roch theorem says that

$\displaystyle \chi(\mathcal{E})=\text{deg}(\text{ch}(\mathcal{E}).\text{td}(\mathcal{T}_{X}))_{n}$

where $\mathcal{T}_{X}$ is the tangent bundle of $X$ (the dual of the cotangent bundle of $X$, as defined in More on Sheaves) and $(\quad)_{n}$ is the component of degree $n$ in $A(X)\otimes\mathbb{Q}$.

Finally we come to the even more general Grothendieck-Riemann-Roch theorem, but first we must introduce the Grothendieck group $K(X)$ of a scheme $X$, which eventually inspired the area of mathematics known as K-theory.

The Grothendieck group $K(X)$ of a scheme $X$ is defined to be the quotient of the free abelian group generated by the coherent sheaves on $X$ by the subgroup generated by expressions of the form $\mathcal{F}-\mathcal{F}'-\mathcal{F}''$ whenever there is an exact sequence

$\displaystyle 0\rightarrow\mathcal{F'}\rightarrow\mathcal{F}\rightarrow\mathcal{F''}\rightarrow 0$

of coherent sheaves on $X$. Intuitively, we may think of the Grothendieck group as follows. The isomorphism classes of vector bundles on $X$ form a commutative monoid under the operation of taking the direct sum of vector bundles (also called the Whitney sum). There is a way to obtain an abelian group from this monoid, called the group completion, and the abelian group we obtain is the Grothendieck group. The Chern classes and the Chern character are also defined on the Grothendieck group $K(X)$. In K-theory, the Grothendieck group $K(X)$ is also denoted $K_{0}(X)$.

If $f:X\rightarrow Y$ is a proper morphism (a morphism that is separable, of finite type, and universally closed, i.e. for every scheme $Z\rightarrow Y$ , the projection $X\times_{Y}Z\rightarrow Z$ maps closed sets to closed sets), we have a map $f_{!}:K(X)\rightarrow Y$ defined by

$\displaystyle f_{!}(\mathcal{F})=\sum_{i}(-1)^{i}R^{i}f_{*}(\mathcal{F})$

where the $R^{i}f_{*}$ are the higher direct image functors, which are defined as the right derived functors (The Hom and Tensor Functors) of the direct image functor $f_{*}$ (see Direct Images and Inverse Images of Sheaves).

The Grothendieck-Riemann-Roch theorem says that for any $x\in K(X)$, we have

$\displaystyle f_{*}(\text{ch}(x).\text{td}(\mathcal{T}_{X})=\text{ch}(f_{!}(x)).\text{td}(\mathcal{T}_{Y})$.

The Grothendieck-Riemann-Roch theorem is one of the most general versions of the Riemann-Roch theorem, a classic theorem whose origins date back to the 19th century. However, there are also other generalizations, such as the arithmetic Riemann-Roch theorem which is closely related to number theory, and the Atiyah-Singer index theorem which is closely related to physics. We leave these, and the many other details of the topics we have discussed in this post (along with the theory of Chern classes in the context of algebraic topology and differential geometry), to the references for now, until we can discuss them on this blog in the future.

The featured image on this post is a handwritten comment of Alexander Grothendieck, apparently from a lecture in 1971, featuring the Grothendieck-Riemann-Roch theorem.

References:

Chern Class on Wikipedia

Projective Bundle on Wikipedia

Hirzebruch-Riemann-Roch Theorem on Wikipedia

Grothendieck-Riemann-Roch Theorem on Wikipedia

Chern Classes: Part 1 on Rigorous Trivialities

Chern Classes: Part 2 on Rigorous Trivialities

The Chow Ring and Chern Classes on Rigorous Trivialities

The Grothendieck-Riemann-Roch Theorem, Stated on Rigorous Trivialities

The Grothendieck-Riemann-Roch Theorem, a Proof-Sketch on Rigorous Trivialities

Algebraic Geometry by Andreas Gathmann

Algebraic Geometry by Robin Hartshorne

# Functions of Complex Numbers

We have discussed a lot of mathematical topics on this blog, with some of them touching on rather advanced subjects. But aside from a few comments about holomorphic functions and meromorphic functions in The Moduli Space of Elliptic Curves, we have not yet discussed one of the most interesting subjects that every aspiring mathematician has to learn about, complex analysis.

Complex analysis refers to the study of functions of complex numbers, including properties of these functions related to concepts in calculus such as differentiation and integration (see An Intuitive Introduction to Calculus). Aside from being an interesting subject in itself, complex analysis is also related to many other areas of mathematics such as algebraic geometry and differential geometry.

But before we discuss functions of a complex variable, we will first review the concept of Taylor expansions from basic calculus. Consider a function $f(x)$ where $x$ is a real variable. If $f(x)$ is infinitely differentiable at $x=0$, we can express it as a power series as follows:

$\displaystyle f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^{2}+\frac{f'''(0)}{3!}x^{3}+...$

where $f'(0)$ refers to the first derivative of $f(x)$ evaluated at $x=0$, $f''(0)$ refers to the second derivative of $f(x)$ evaluated at $x=0$, and so on. More generally, the $n$-th coefficient of this power series is given by

$\displaystyle \frac{f^{(n)}(0)}{n!}$

where $f^{(n)}(0)$ refers to the $n$-th derivative of $f(x)$ evaluated at $x=0$. This is called the Taylor expansion (or Taylor series) of the function $f(x)$ around $x=0$. For example, for the sine function, we have

$\displaystyle \text{sin}(x)=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-...$

More generally, if the function $f(x)$ is infinitely differentiable at $x=a$, we can obtain the Taylor expansion of $f(x)$ around $x=a$ using the following formula:

$\displaystyle f(x-a)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^{2}+\frac{f'''(a)}{3!}(x-a)^{3}+...$

If a function $f(x)$ can be expressed as a power series at every point of some interval $U$ in the real line, then we say that $f(x)$ is real analytic on $U$.

Now we bring in complex numbers. Consider now a function $f(z)$ where $z$ is a complex variable. If $f(z)$ can be expressed as a power series at every point of an open disk (the set of all complex numbers $z$ such that the magnitude $|z-z_{0}|$ is less than $\delta$ for some complex number $z_{0}$ and some positive real number $\delta$) $U$ in the complex plane, then we say that $f(z)$ is complex analytic on $U$. Since the rest of this post discusses functions of a complex variable, I will be using “analytic” to refer to “complex analytic”, as opposed to “real analytic”.

Now that we know what an analytic function is, we next discuss the concept of holomorphic functions. If $f(x)$ is a function of a real variable, we define its derivative at $x_{0}$ as follows:

$\displaystyle f'(x_{0})=\lim_{x\to x_{0}}\frac{f(x)-f(x_{0})}{x-x_{0}}$

If we have a complex function $f(z)$, the definition is the same:

$\displaystyle f'(z_{0})=\lim_{z\to z_{0}}\frac{f(z)-f(z_{0})}{z-z_{0}}$

However, note that the value of $z$ can approach $z_{0}$ in many different ways! For example, let $f(z)=\bar{z}$, i.e. $f(z)$ gives the complex conjugate of the complex variable $z$. Let $z_{0}=0$. Since $z=x+iy$, we have $f(z)=\bar{z}=x-iy$.

$\displaystyle f'(0)=\lim_{z\to 0}\frac{x-iy-0}{x+iy-0}$

$\displaystyle f'(0)=\lim_{z\to 0}\frac{x-iy}{x+iy}$

If, for example, $z$ is purely real, i.e. $y=0$, then we have

$\displaystyle f'(0)=\lim_{z\to 0}\frac{x}{x}$

$\displaystyle f'(0)=1$

But if $z$ is purely imaginary, i.e. $x=0$, then we have

$\displaystyle f'(0)=\lim_{z\to 0}\frac{-iy}{iy}$

$\displaystyle f'(0)=-1$

We see that the value of $f'(0)$ is different depending on how we approach the limit $z\to 0$!

A function of complex numbers for which the derivative is the same regardless of how we take the limit $z\to z_{0}$, for all $z_{0}$ in its domain, is called a holomorphic function. The function $f(z)=\bar{z}$ discussed above is not a holomorphic function on the complex plane, since the derivative is different depending on how we take the limit.

Now, it is known that a function of a complex number is holomorphic on a certain domain if and only if it is analytic in that same domain. Hence, the two terms are often used interchangeably, even though the concepts are defined differently. A function that is analytic (or holomorphic) on the entire complex plane is called an entire function.

If a function is analytic, then it must satisfy the Cauchy-Riemann equations (named after two pioneers of complex analysis, Augustin-Louis Cauchy and Bernhard Riemann). Let us elaborate on what these equations are a bit. Just as we can express a complex number $z$ as $x+iy$, we can also express a function $f(z)$ of $z$ as $u(z)+iv(z)$, or, going further and putting together these two expressions, as $u(x,y)+iv(x,y)$. The Cauchy-Riemann equations are then given by

$\displaystyle \frac{\partial{u}}{\partial{x}}=\frac{\partial{v}}{\partial{y}}$

$\displaystyle \frac{\partial{u}}{\partial{y}}=-\frac{\partial{v}}{\partial{x}}$

Once again, if a function $f(z)=u(x,y)+iv(x,y)$ is analytic, then it must satisfy the Cauchy-Riemann equations. Therefore, if it does not satisfy the Cauchy-Riemann equations, we know for sure that it is not analytic. But we should still be careful – just because a function satisfies the Cauchy-Riemann equations does not always mean that it is analytic! We also often say that satisfying the Cauchy-Riemann equations is a “necessary”, but not “sufficient” condition for a function of a complex number to be analytic.

Analytic functions have some very special properties. For instance, since we have already talked about differentiation, we may also now consider integration. Just as differentiation is more complicated in the complex plane than on the real line, because in the former there are different directions in which we may take the limit, integration is also more complicated on the complex plane as opposed to integration on the real line. When we perform integration over the variable $dz$, we will usually specify a “contour”, or a “path” over which we integrate.

We may reasonably expect that the integral of a function will depend not only on the “starting point” and “endpoint”, as in the real case, but also on the choice of contour. However, if we have an analytic function defined on a simply connected (see Homotopy Theory) domain, and the contour is inside this domain, then the integral will not depend on the choice of contour! This has the consequence that if our contour is a loop, the integral of the analytic function will always be zero. This very important theorem in complex analysis is known as the Cauchy integral theorem. In symbols, we write

$\displaystyle \oint_{\gamma}f(z)dz=0$

where the symbol $\oint$ means that the contour of integration is a loop. The symbol $\gamma$ refers to the contour, i.e. it may be a circle, or some other kind of loop – usually whenever one sees this symbol the author will specify the contour that it refers to.

Another important result in complex integration is what is known as the Cauchy integral formula, which relates an analytic function to its values on the boundary of some disk contained in the domain of the function:

$\displaystyle f(z)=\frac{1}{2\pi i}\oint_{\gamma}\frac{f(\zeta)}{\zeta-z}d\zeta$

By taking the derivative of both sides with respect to $z$, we obtain what is also known as the Cauchy differentiation formula:

$\displaystyle f'(z)=\frac{1}{2\pi i}\oint_{\gamma}\frac{f(\zeta)}{(\zeta-z)^{2}}d\zeta$

The reader may notice that on one side of this fascinating formula is a derivative, while on the other side there is an integral – in the words of the Wikipedia article on the Cauchy integral formula, in complex analysis, “differentiation is equivalent to integration”!

These theorems regarding integration lead to the residue theorem, a very powerful tool for calculating the contour integrals of meromorphic functions (see The Moduli Space of Elliptic Curves) – functions which would have been analytic in their domain, except that they have singularities of a certain kind (called poles) at certain points. A more detailed discussion of meromorphic functions, singularities and the residue theorem is left to the references for now.

Aside from these results, analytic functions also have many other interesting properties – for example, analytic functions are always infinitely differentiable. Also, analytic functions defined on a certain domain may possess what is called an analytic continuation – a unique analytic function defined on a larger domain which is equal to the original analytic function on its original domain. Analytic continuation (of the Riemann zeta function) is one of the “tricks” behind such infamous expressions as

$\displaystyle 1+2+3+4+5+....=-\frac{1}{12}$

$\displaystyle 1+1+1+1+1+....=-\frac{1}{2}$

There is so much more to complex analysis than what we have discussed, and some of the subjects that a knowledge of complex analysis might open up include Riemann surfaces and complex manifolds (see An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry), which generalize complex analysis to more general surfaces and manifolds than just the complex plane. For the latter, one has to consider functions of more than one complex variable. Hopefully there will be more posts discussing complex analysis and related subjects on this blog in the future.

References:

Complex Analysis on Wikipedia

Analytic Function on Wikipedia

Holomorphic Function on Wikipedia

Cauchy-Riemann Equations on Wikipedia

Cauchy’s Integral Theorem on Wikipedia

Cauchy’s Integral Formula on Wikipedia

Residue Theorem on Wikipedia

Complex Analysis by Lars Ahlfors

Complex Variables and Applications by James Ward Brown and Ruel V. Churchill