# p-adic Hodge Theory: An Overview

In The Theory of Motives we discussed the notion of a Weil cohomology, and mentioned four “classical” examples, the singular (also known as Betti) cohomology, the de Rham cohomology, the $\ell$-adic cohomology, and the crystalline cohomology.

Cohomology theories may be thought of as a way to study geometric objects using linear algebra, by associating vector spaces (or more generally, modules or abelian groups) to such a geometric object. But the four Weil cohomology theories above actually give more than just a vector space:

• The singular cohomology has an action of complex conjugation.
• The de Rham cohomology has a Hodge filtration.
• The $\ell$-adic cohomology has an action of the Galois group.
• The crystalline cohomology has an action of Frobenius (and a Hodge filtration as well).

There are relations between these different cohomologies. For example, for a smooth projective variety $X$ over the complex numbers $\mathbb{C}$, the singular cohomology of the corresponding complex analytic manifold $X(\mathbb{C})$, with complex coefficients (this can be obtained from singular cohomology with integral coefficients by tensoring with $\mathbb{C}$) and the de Rham cohomology are isomorphic:

$\displaystyle H_{\mathrm{sing}}^{k}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{C}=H_{\mathrm{dR}}^{k}(X)$

The roots of this idea go back to de Rham’s work on complex manifolds, where chains in singular homology (which is dual to singular cohomology, see also Homology and Cohomology) can be paired with the differential forms of de Rham cohomology (see also Differential Forms), simply by integrating the differential forms along these chains. By the machinery developed by Alexander Grothendieck, this can be ported over into the world of algebraic geometry.

Again borrowing from the world of complex manifolds, the machinery of Hodge theory gives us the following Hodge decomposition (see also Shimura Varieties):

$\displaystyle H_{\mathrm{sing}}^{k}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{C}=\bigoplus_{i+j=k} H^{i}(X,\Omega_{X/\mathbb{C}}^{j})$

Now again for the case of smooth projective varieties over the complex numbers , $\ell$-adic cohomology also has such an isomorphism with singular cohomology – but this time if it has $\ell$-adic coefficients (i.e. in $\mathbb{Q}_{\ell}$).

$\displaystyle H_{\mathrm{sing}}^{k}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{Q}_{\ell}\simeq H_{\mathrm{et}}^{k}(X,\mathbb{Q}_{\ell})$

Such isomorphisms are also known as comparison isomorphisms (or comparison theorems).

More generally, if we have a field $B$ into which we can embed both $\mathbb{Q}_{\ell}$ and $\mathbb{C}$ (for instance $\mathbb{C})$, we obtain the following comparison theorem:

$\displaystyle H_{\mathrm{et}}^{k}(X,\mathbb{Q}_{\ell}) \otimes_{\mathbb{Q_{\ell}}} B\simeq H_{\mathrm{dR}}^{k}(X) \otimes_{\mathbb{C}} B$

Here is a very interesting thing that these comparison theorems can give us. Let $X$ be a modular curve. Then the Hodge decomposition for the first cohomology gives us

$\displaystyle H_{\mathrm{sing}}^{1}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{C}=H^{1}(X,\Omega_{X/\mathbb{C}}^{0})\oplus H^{0}(X,\Omega_{X/\mathbb{C}}^{1})$

But the $H^{0}(X,\Omega_{X/\mathbb{C}}^{1})$ is the cusp forms of weight $2$ as per the discussion in Modular Forms (see also Galois Representations Coming From Weight 2 Eigenforms). By the results of Hodge theory, the other summand $H^{1}(X,\Omega_{X/\mathbb{C}}^{0})$ is just the complex conjugate of $H^{0}(X,\Omega_{X/\mathbb{C}}^{1})$. But we now also have a comparison with etale cohomology, which has a Galois representation! For this the modular form must lie in the cohomology with $\mathbb{Q}$ coefficients, which happens if it is a Hecke eigenform whose Hecke eigenvalues are in $\mathbb{Q}$. So one of the great things that these comparison theorems gives us is this way of relating modular forms and Galois representations.

The comparison isomorphisms above work for smooth projective varieties over the complex numbers, but let us now go to the p-adic world, and let us consider smooth projective varieties over the p-adic numbers.

It was observed by John Tate (and later explored by Gerd Faltings) that the p-adic cohomology (i.e. the etale cohomology of a smooth projective variety over $\mathbb{Q}_{p}$, or more generally some other p-adic field, with p-adic coefficients, distinguishing it from $\ell$-adic cohomology where another prime $\ell$ different from $p$ must be brought in) can have a decomposition akin to the Hodge decomposition, after tensoring it with the p-adic complex numbers (this is the completion of the algebraic closure of the p-adic numbers):

$\displaystyle H^{k}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}\mathbb{C}_{p}=\bigoplus_{i+j=k} H^{i}(X,\Omega_{X/\mathbb{Q}}^{j})\otimes_{\mathbb{Q}}\mathbb{C}_{p}(-j)$

The p-adic complex numbers here play the role of the complex numbers in the singular cohomology case above or the $\ell$-adic numbers in the $\ell$-adic case.

The ideas conjectured by Tate, and later completed by Faltings, was but the prototype of what is now known as p-adic Hodge theory. In its modern form, p-adic Hodge theory concerns comparison isomorphisms between different Weil cohomology theories on smooth projective varieties over the p-adic numbers. However, the role played by the complex numbers, $\ell$-adic numbers (for the complex case), and p-adic complex numbers (for the p-adic case) must now be played by much more complicated objects called period rings, which were developed by Jean-Marc Fontaine. We will discuss the construction of the period rings at the end of this post, but first let us see how they work.

Let $X$ be a smooth projective variety over $\mathbb{Q}_{p}$ (or more generally some other p-adic field). Let $H_{\mathrm{dR}}^{i}(X)$ and $H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})$ be its de Rham cohomology and the p-adic etale cohomology of its base change to the algebraic closure $\overline{\mathbb{Q}}_{p}$ respectively. The comparison isomorphism at the center of p-adic Hodge theory is the following:

$\displaystyle H_{\mathrm{dR}}^{i}(X)\otimes_{\mathbb{Q}_{p}} B_{\mathrm{dR}}=H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}} B_{\mathrm{dR}}$

The object denoted $B_{\mathrm{dR}}$ here is the aforementioned period ring. It is equipped with both a Galois action and a filtration akin to the Hodge filtration. More than just that isomorphism above, we also have a way of obtaining the de Rham cohomology if we are given the p-adic etale cohomology, simply by taking the part that is invariant under the Galois action:

$\displaystyle \displaystyle H_{\mathrm{dR}}^{i}(X)=(H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}} B_{\mathrm{dR}})^{\mathrm{Gal}_{\mathbb{Q}_{p}}}$

To go the other way, i.e. to recover the p-adic etale cohomology from the de Rham cohomology, we will need a different kind of period ring. This period ring is $B_{\mathrm{cris}}$, which aside from having a Galois action and a filtration also has an action of Frobenius. Aside from providing us the same isomorphism between de Rham and p-adic etale cohomology upon tensoring, it also provides us with a solution to our earlier problem (as long as $X$ has a smooth proper integral model) as follows:

$\displaystyle H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})= \mathrm{Fil}^{0}(H_{\mathrm{dR}}^{i}(X)\otimes_{\mathbb{Q}_{p}} B_{\mathrm{cris}})^{\varphi=1}$

This idea can be further abstracted – since etale cohomology provides Galois representations, we can just take some p-adic Galois representation instead, without caring whether it comes from etale cohomology or not, and tensor it with a period ring, then take Galois invariants. For instance let $V$ be some p-adic Galois representation. Then we can take the tensor product

$V_{\mathrm{dR}}=(V\otimes B_{\mathrm{dR}})^{\mathrm{Gal}_{\mathbb{Q}_{p}}}$

If the dimension of $V_{\mathrm{dR}}$ is equal to the dimension of $V$, then we say that the Galois representation $V$ is de Rham. Similarly we can tensor with $B_{\mathrm{cris}}$:

$V_{\mathrm{cris}}=(V\otimes B_{\mathrm{cris}})^{\mathrm{Gal}_{\mathbb{Q}_{p}}}$

If its $V_{\mathrm{cris}}$ is equal to the dimension of $V$ , we say that $V$ is crystalline.

The idea of these “de Rham” and “crystalline” Galois representations is that if they come from the corresponding cohomologies then they will have these properties. But does the converse hold? If they are “de Rham” and “crystalline” does that mean that they come from the corresponding cohomologies (i.e. they “come from geometry”)? This is roughly the content of the Fontaine-Mazur conjecture.

Now let us say a few things about the construction of these period rings. These constructions make use of the concepts we discussed in Perfectoid Fields. We start with the ring $A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}})$, which, as we recall from Perfectoid Fields, is the ring of Witt vectors of the tilt of $\mathcal{O}_{\mathbb{C}_{p}}$. By inverting $p$ and taking the completion with respect to the canonical map $\theta: A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}}) \to\mathcal{O}_{\mathbb{C}_{p}}$, we obtain a ring which we suggestively denote by $B_{\mathrm{dR}}^{+}$.

There is a special element $t$ of $B_{\mathrm{dR}}^{+}$ which we think of as the logarithm of the element $(1, \zeta^{1/p},\zeta^{1/p},\ldots)$. Upon inverting this element $t$, we obtain the field $B_{\mathrm{dR}}$.

The field $B_{\mathrm{dR}}$ is equipped with a Galois action, carried over from the fields involved in its construction, and a filtration, given by $\mathrm{Fil}^{i}B_{\mathrm{dR}}=t^{i}B_{\mathrm{dR}}$.

To construct $B_{\mathrm{cris}}$, we once again start with $A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}})$ and invert $p$. However, to have a Frobenius, instead of completing with respect to the kernel of the map $\theta$, we take a generator of this kernel (which we shall denote by $\omega$). Then we denote by $B_{\mathrm{cris}}^{+}$ the ring formed by all the power series of the form $\sum_{n=0}^{\infty} a_{n}\omega^{n}/n!$ where the $a_{n}$‘s are elements of $A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}})[1/p]$ which converge as $n\to\infty$, under the topology of $A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}})[1/p]$ (which is not the p-adic topology!). Once again there will be an element $t$ like before; we invert $t$ to obtain $B_{\mathrm{cris}}$.

There is yet another period ring called $B_{\mathrm{st}}$, where the subscript stands for semistable; in addition to a Galois action, filtration, and Frobenius, it has a monodromy operator. Since this is less extensively discussed in introductory literature, we follow this lead and leave this topic, and the many other wonderful topics related to p-adic Hodge theory, to future posts on this blog.

References:

de Rham cohomology on Wikipedia

Hodge theory on Wikipedia

An introduction to the theory of p-adic representations by Laurent Berger

# Rigid Analytic Spaces

This blog post is inspired by and follows closely an amazing talk given by Ashwin Iyengar at the “What is a…seminar?” online seminar. I hope I can do the talk, and this wonderful topic, some justice in this blog post.

One of the most fascinating and powerful things about algebraic geometry is how closely tied it is to complex analysis, despite what the word “algebraic” might lead one to think. To state one of the more simple and common examples, we have that smooth projective curves over the complex numbers $\mathbb{C}$ are the same thing as compact Riemann surfaces. In higher dimensions we also have Chow’s theorem, which tells us that an analytic subspace of complex projective space which is topologically closed is an algebraic subvariety.

This is all encapsulated in what is known as “GAGA“, named after the foundational work “Géometrie Algébrique et Géométrie Analytique” by Jean-Pierre Serre. We refer to the references at the end of this post for the more precise statement, but for now let us think of GAGA as giving us a fully faithful functor from proper algebraic varieties over $\mathbb{C}$ to complex analytic spaces, which gives us an equivalence of categories between their coherent sheaves.

One may now ask if we can do something similar with the p-adic numbers $\mathbb{Q}_{p}$ (or more generally an extension $K$ of $\mathbb{Q}_{p}$ that is complete with respect to a valuation that extends the one on $\mathbb{Q}_{p}$) instead of $\mathbb{C}$. This leads us to the theory of rigid analytic spaces, which was originally developed by John Tate to study a p-adic version of the idea (also called “uniformization”) that elliptic curves over $\mathbb{C}$ can be described as lattices on $\mathbb{C}$.

Let us start defining these rigid analytic spaces. If we simply naively try to mimic the definition of complex analytic manifolds by having these rigid analytic spaces be locally isomorphic to $\mathbb{Q}_{p}^{m}$, with analytic transition maps, we will run into trouble because of the peculiar geometric properties of the p-adic numbers – in particular, as a topological space, the p-adic numbers are totally disconnected!

To fix this, we cannot just use the naive way because the notion of “local” would just be too “small”, in a way. We will take a cue from algebraic geometry so that we can use the notion of a Grothendieck topology to fix what would be issues if we were to just use the topology that comes from the p-adic numbers.

The Tate algebra $\mathbb{Q}_{p}\langle T_{1},\ldots,T_{n}\rangle$ is the algebra formed by power series in $n$ variables that converge on the $n$-dimensional unit polydisc $D^{n}$, which is the set of all n-tuples $(c_{1},\ldots,c_{n})$ of elements of $\mathbb{Q}_{p}$ that have p-adic absolute value less than or equal to $1$ for all $i$ from $1$ to $n$.

There is another way to define the Tate algebra, using the property of power series with coefficients in p-adic numbers that it converges on the unit polydisc $D^{n}$ if and only if its coefficients go to zero (this is not true for real numbers!). More precisely if we have a power series

$\displaystyle f(T_{1},\ldots,T_{n})=\sum_{a} c_{a}T_{1}^{a_{1}}\ldots T_{n}^{a_{n}}$

where $c_{a}\in \mathbb{Q}_{p}$ and $a=a_{1}+\ldots+a_{n}$ runs over all n-tuples of natural numbers, then $f$ converges on the unit polydisc $D^{n}$ if and only if $\lim_{a\to 0}c_{a}=0$.

The Tate algebra has many important properties, for example it is a Banach space (see also Metric, Norm, and Inner Product) with the norm of an element given by taking the biggest p-adic absolute value among its coefficients. Another property that will be very important in this post is that it satisfies a Nullstellensatz – orbits of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$ in $D^{n}$ correspond to maximal ideals of the Tate algebra. Explicitly, this correspondence is given by the evaluation map $x\in D^{n}\mapsto \lbrace f\vert f(x)=0\rbrace$.

A quotient of the Tate algebra by some ideal is referred to as an affinoid algebra. The maximal ideals of the underlying set of an affinoid algebra $A$ will be denoted $\mathrm{Max}(A)$, and this should be reminiscent of how we obtain the closed points of a scheme.

Once we have the underlying set $\mathrm{Max}(A)$ , we need two other things to be able to define “affinoid” rigid analytic spaces, which we shall later “glue” to form more general rigid analytic spaces – a topology, and a structure sheaf (again this should be reminiscent of how schemes are defined).

Again taking a cue from how schemes are defined, we define “rational domains”, which are analogous to the distinguished Zariski open sets of (the underlying topological space of) a scheme. Given elements $f_{1},\ldots,f_{r},g$ of the affinoid algebra $A$, the rational domain $\displaystyle A\left(\frac{f}{g}\right)$ is the set of all $x\in\mathrm{Max}(A)$ such that $f_{i}(x)\leq g(x)$ for all $1\leq i\leq r$.

The rational domains generate a topology, however this is still just the p-adic topology, and so it still does not solve the problems that we originally ran into when we tried to define rigid analytic spaces by mimicking the definition of a complex analytic manifold. The trick will be to make use of something that is more general than just a topology – a Grothendieck topology (see also More Category Theory: The Grothendieck Topos).

Let us now define the particular Grothendieck topology that we will use. Unlike other examples of a Grothendieck toplogy, the covers will involve only subsets of the space being covered (it is also referred to as a mild Grothendieck topology). Let $X=\mathrm{Max}(A)$. A subset $U$ of $X$ is called an admissible open if it can be covered by rational domains $\lbrace U_{i}\rbrace_{i\in I}$ such that for any map $Y\to X$ where $Y=\mathrm{Max}(B)$ for some affinoid algebra $B$, the covering of $Y$ given by the inverse images of the $U_{i}$‘s admit a finite subcover.

If $U$ is an admissible open covered by admissible opens $\lbrace U_{i}\rbrace_{i\in I}$, then this covering is called admissible if for any map $Y\to X$ whose image is contained in $U$, the covering of $Y$ given by the inverse images of the $U_{i}$‘s admit a finite subcover. These admissible coverings satisfy the axioms for a Grothendieck topology, which we denote $G_{X}$.

If $A$ is an affinoid algebra, and $f_{1},\ldots,f_{k},g$ are functions, we let $\displaystyle A\left\langle \frac{f}{g}\right\rangle$ denote the ring $A\langle T_{1},\ldots T_{k}\rangle/(gT_{i}-f_{i})$. By associating to a rational domain $\displaystyle A\left(\frac{f}{g}\right)$ this ring $\displaystyle A\left\langle\frac{f}{g}\right\rangle$, we can define a structure sheaf $\cal{O}_{X}$ on this Grothendieck topology.

The data consisting of the set $X=\mathrm{Max}(A)$, the Grothendieck topology $G_{X}$, and the structure sheaf $\mathcal{O}_{X}$ is what makes up an affinoid rigid analytic space. Finally, just as with schemes, we define a more general rigid analytic space as the data consisting of some set $X$, a Grothendieck topology $G_{X}$ and a sheaf $\mathcal{O}_{X}$ such that locally, with respect to the Grothendieck topology $G_{X}$, it is isomorphic to an affinoid rigid analytic space.

Under this definition, we have in fact a version of GAGA that holds for rigid analytic spaces – a fully faithful functor from proper schemes of finite type over $\mathbb{Q}_{p}$ to rigid analytic spaces over $\mathbb{Q}_{p}$ that gives an equivalence of categories between their coherent sheaves.

Finally let us now look at an example. Consider the affinoid rigid analytic space obtained from the affinoid algebra $\mathbb{Q}_{p}\langle T\rangle$. By the Nullstellensatz the underlying set is the unit disc $D$. The “boundary” of this is the rational subdomain (and therefore an admissible open) $\displaystyle D\left(\frac{1}{T}\right)$, and its complement, the “interior” is covered by rational subdomains $\displaystyle D\left(\frac{T^{n}}{p}\right)$. With this covering the interior may also be shown to be an admissible open.

While it appears that, since we found two complementary admissible opens, we can disconnect the unit disc, we cannot actually do this in the Grothendieck topology, because the set consisting of the boundary and the interior is not an admissible open! And so in this way we see that the Grothendieck topology is the difference maker that allows us to overcome the obstacles posed by the peculiar geometry of the p-adic numbers.

Since Tate’s innovation, the idea of a p-adic or nonarchimedean geometry has blossomed with many kinds of “spaces” other than the rigid analytic spaces of Tate, for example adic spaces, a special class of which generalize perfectoid fields (see also Perfectoid Fields) to spaces, or Berkovich spaces, which are honest to goodness topological spaces instead of relying on a Grothendieck topology like rigid analytic spaces do. Such spaces will be discussed on this blog in the future.

References:

Rigid analytic space on Wikipedia

Algebraic geometry and analytic geometry on Wikipedia

Several approaches to non-archimedean geometry by Brian Conrad

Reciprocity laws and Galois representations: recent breakthroughs by Jared Weinstein

p-adic families of modular forms [after Coleman, Hida, and Mazur] by Matthew Emerton

# Perfectoid Fields

Consider the field of $p$-adic numbers $\mathbb{Q}_{p}$. An element of $\mathbb{Q}_{p}$ may be written in the form

$\displaystyle \sum_{n=k}^{\infty}a_{n}p^{n}$

with each $a_{n}$ being an element of the finite field $\mathbb{F}_{p}$. Let us compare this with the field of Laurent series $\mathbb{F}_{p}((t))$ in one variable $t$ over $\mathbb{F}_{p}$. An element of $\mathbb{F}_{p}((t))$ may be written in the form

$\displaystyle \sum_{m=l}^{\infty}a_{m}t^{m}$

We see that they look very similar, even though $\mathbb{Q}_{p}$ is characteristic $0$, and $\mathbb{F}_{p}((t))$ is characteristic $p$.

How far can we push this analogy? The fact that one is in characteristic $0$, and the other is characteristic $p$ means we cannot ask for an isomorphism of fields. However, the Fontaine-Wintenberger theorem gives us another connection between $\mathbb{Q}_{p}$ and $\mathbb{F}_{p}((t))$ – if we modify them by adjoining $p$-power roots of $p$ and $t$ respectively. This theorem states that the fields $\cup_{n}\mathbb{Q}_{p}(p^{1/p^{n}})$ and $\cup_{n}\mathbb{F}_{p}((t^{1/p^{n}}))$ have the same absolute Galois group! By the fundamental theorem of Galois theory, this means the category formed by their extensions will be equivalent as well.

We now let $F$ denote the completion of $\cup_{n}\mathbb{Q}_{p}(p^{1/p^{n}})$, and we let $F^{\flat}$ suggestively denote the completion of $\cup_{n}\mathbb{F}_{p}((t^{1/p^{n}}))$. Completing these fields does not change their absolute Galois groups, so the absolute Galois groups of $F$ and $F^{\flat}$ remain isomorphic. We say that the characteristic $p$ field $F^{\flat}$ is the tilt of the characteristic $0$ field $F$, and that $F^{\flat}$ is an untilt of $F$ (note the subtle change in our choice of article – untilts are not unique).

In this post, we will explore these kinds of fields – which are called perfectoid fields – and the process of tilting and untilting that bridges the world of characteristic $0$ and characteristic $p$. After Fontaine and Wintenberger came up with their famous theorem their ideas have since been developed into even more general and even more powerful theories of perfectoid rings and perfectoid spaces – but we will leave these to future posts. For now we concentrate on the case of fields.

First let us look at a much more primitive example of bridging the world of characteristic $0$ and characteristic $p$. Consider $\mathbb{Q}_{p}$ (characteristic $0$). It has a ring of integers $\mathbb{Z}_{p}$, whose residue field is $\mathbb{F}_{p}$ (characteristic $p$). To got the other way, starting from $\mathbb{F}_{p}$ we can take its ring of Witt vectors, which is $\mathbb{Z}_{p}$. Then we take its field of fractions which is $\mathbb{Q}_{p}$.

More generally, there is a correspondence between characteristic $0$ discretely valued complete fields whose uniformizer is $p$ and characteristic $p$ fields which are perfect, i.e. for which the Frobenius morphism is bijective, and the way to go from one category to the other is as in the previous paragraph.

This is a template for “bridging the world of characteristic $0$ and characteristic $p$“. However, we may want more, something like the Fontaine-Wintenberger theorem where the characteristic $0$ object and the characteristic $p$ object have isomorphic absolute Galois groups. We will be tweaking this basic bridge in order to create something like Fontaine-Winterger theorem, and these tweaks will lead us to the notion of a perfectoid field. However, we already have isolated one property that we want from such a “perfectoid” field:

The first property that we want from a perfectoid field is that it has to be nonarchimedean. This allows us to have a “ring of integers” that serves as an intermediary object between the two worlds, as we have seen above.

Now let us concentrate on the Fontaine-Wintenberger theorem. To understand this phenomenon better, we need to make use of a version of the fundamental theorem of Galois theory, which allows us to think in terms of extensions of fields instead of their Galois groups. More properly, we want an equivalence of categories between the “Galois categories” of certain extensions of these “base” fields and this will be the property of these base fields being perfectoid. Now the problem is that the extensions that we are considering may not fit into the primitive correspondence we stated above – for example the corresponding characteristic $p$ object may not be perfect, i.e. the Frobenius morphism may not be surjective.

The fix to this is a kind of “perfection”, which is the tilting functor we mentioned earlier. Let $R$ be a ring. The tilt of $R$, denoted $R^{\flat}$ is defined to be the inverse limit

$\displaystyle R^{\flat}=\varprojlim_{x\mapsto x^{p}}R/pR$

In other words, an element $x$ of $R^{\flat}$ is an infinite sequence of elements $(x_{0},x_{1},x_{2},\ldots)$ of the quotient $R/pR$ such that $x_{1}\cong x_{0}^{p}\mod p$, $x_{2}\cong x_{1}^{p}\mod p$, and so on. We want $R^{\flat}$ to be a ring, so we define it to have componentwise multiplication, i.e.

$\displaystyle (xy)_{i}=x_{i}y_{i}$

However the addition is going to be more complicated. We define it, for each component, as follows:

$\displaystyle (x+y)_{i}=\lim_{n\to\infty}(x_{i+n}+y_{i+n})^{p^n}$

At this point we take the opportunity to define another important concept in the theory of perfectoid fields (and rings). Let $W$ be the Witt vector functor (see also The Field with One Element). Then we give the Witt vectors of the tilt of $R$, $W(R^{\flat})$, a special name. We will refer to this ring as $A_{\mathrm{inf}}(R)$. It will make an appearance again later. For now we note that there is going to be a canonical map $\theta: A_{\mathrm{inf}}(R)\to R$.

As we can see, we have defined the tilt of an arbitrary ring. This is not exclusive to the ones which are “perfectoid” whatever the definition of “perfectoid” may be (we will come to this later of course). Again what makes perfectoid fields (such as our earlier examples) special though, is that if $F$ is a perfectoid field of characteristic $0$, then $F$ and its tilt $F^{\flat}$ will have isomorphic absolute Galois groups. This will actually follow from the following statement (together with some technicalities involving fiber functors and so on):

There is an equivalence of categories between the category of finite etale algebras over a perfectoid field $F$ and the category of finite etale algebras over its tilt $F^{\flat}$.

This in turn will follow from the following two statements:

1. Finite extensions of perfectoid fields are perfectoid.
2. There is an equivalence of categories between the category of perfectoid extensions of a perfectoid field $F$ and the category of perfectoid extensions over its tilt $F^{\flat}$.

This equivalence of categories is given by tilting a perfectoid extension over $F$. This will actually give us a perfectoid extension over $F^{\flat}$. However, we need a functor that goes in the other direction, a “quasi-inverse” that when composed with tilting gives us back our original perfectoid extension over $F$ (or at least something isomorphic to it, this is what the “quasi-” part means). However, we also said in an earlier paragraph that the “untilt” of a characteristic $p$ field may not be unique (two different untilts may also not be isomorphic). How do we approach this problem?

We recall again the ring $A_{\mathrm{inf}}(R)$ defined earlier as the ring of Witt vectors of the tilt of $R$, and we recall that it has a canonical map $\theta:A_{\mathrm{inf}}(R)\to R$. If we know this map, and if we know that it is surjective, then we can recover $R$ simply by quotienting out by the kernel of the map $\theta$!

The problem is that (aside from not knowing whether it is in fact surjective or not) is that we only know this map if we know that $R^{\flat}$ was obtained as the tilt of $R$. If we were simply handed some characteristic $p$ field for instance we would not be able to know this map.

However, note that we are interested in an equivalence of categories between the category of perfectoid extensions over the field $F$ and the corresponding category over its tilt $F^{\flat}$. By specifying these “bases” $F$ and $F^{\flat}$, it is in fact enough to specify unique untilts! In other words, if we have say just some perfectoid field $A$, we cannot determine a unique untilt for it, but if we say in addition that it is a perfectoid extension over $F$, and we are looking for the unique untilt of it over $F^{\flat}$, we can in fact find it, as long as the map $\theta$ is surjective.

So now how do we guarantee that $\theta$ is surjective? This brings us to our second property, which is that the Frobenius morphism from $\overline{R}$ to itself must be surjective. This is actually the origin of the word “perfectoid”; since as above a field for which the Frobenius morphism is bijective is called perfect; hence, requiring it to be surjective is a relaxation of this condition. This condition guarantees that the map $\phi:A_{\mathrm{inf}}(R)\to R$ is going to be surjective.

The final property that we want from a perfectoid field is that its valuation must be non-discretely valued. The reason for this is that we want to consider infinitely ramified extensions of $\mathbb{Q}_{p}$. The two previous conditions that we want can only be found in unramified (discretely valued) or infinitely ramified (non-discretely valued) of $\mathbb{Q}_{p}$. We have already seen above that if we only look at the ones which are unramified then our corresponding characteristic $p$ objects will be limited to perfect $\mathbb{F}_{p}$-algebras, and this is not enough to give us the Fontaine-Wintenberger theorem. Therefore we will want infinitely ramified extension of $\mathbb{Q}_{p}$, and these are non-discretely valued.

These three properties are enough to give us the Fontaine-Wintenberger theorem. To summarize – a perfectoid field is a complete, nonarchimedean field $F$ such that the Frobenius morphism from $\mathcal{O}_{F}/\mathfrak{p}$ to itself is surjective and such that its valuation is non-discretely valued.

We have only attempted to motivate the definition of a perfectoid field in this post, and barely gone into any sort of detail. For that one can only recommend the excellent post by Alex Youcis on his blog The Fontaine-Wintenberger Theorem: Going Full Tilt, which inspired this post, but barely does it any justice.

Aside from the Fontaine-Wintenberger theorem, the concepts we have described here – the idea behind “perfectoid”, the equivalence of categories of perfectoid extensions that gives rise to the Fontaine-Wintenberger theorem, the idea of tilting and untilting which bridges the worlds of characteristic $0$ and characteristic $p$, the ring $A_{\mathrm{inf}}(R)$, and so on, have found much application in many areas of math, from the aforementioned perfectoid rings and perfectoid spaces, to p-adic Hodge theory, and to many others.

References:

Perfectoid Space on Wikipedia

What is…a Perfectoid Space? by Bhargav Bhatt

Berkeley Lectures on p-adic Geometry by Peter Scholze and Jared Weinstein

Reciprocity Laws and Galois Representations: Recent Breakthroughs by Jared Weinstein

The Fontaine-Wintenberger Theorem: Going Full Tilt by Alex Youcis

# Galois Representations

The absolute Galois group $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ is one of the most important objects of study in mathematics. However the direct study of this group is very difficult; for instance it is an infinite group, and we know very little about it. To make it easier for us, we will often instead study representations of this group – i.e. group homomorphisms to the group $\text{GL}(V)$ of linear transformations of some vector space $V$ over some field $F$. When $V$ has finite dimension $n$, $\text{GL}(V)$ is just $\text{GL}_{n}(F)$, the group of $n\times n$ matrices with entries in $F$ and nonzero determinant. Often we will also want the field $F$ to carry a topology – this will also endow $\text{GL}_{n}(F)$ with a topology. For instance, if $F$ is the $p$-adic numbers $\mathbb{Q}_{p}$ it has a $p$-adic topology (see also Valuations and Completions). Since $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ has its own topology, we can then talk about representations which are continuous. In this post we shall consider three examples of these continuous Galois representations.

Our first example of a Galois representation is known as the $p$-adic cyclotomic character. This is a one-dimensional representation over the $p$-adic numbers $\mathbb{Q}_{p}$, i.e. a group homomorphism from $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}$ to $\text{GL}_{1}(\mathbb{Q}_{p})$, which also happens to just be the multiplicative group $\mathbb{Q}_{p}^{\times}$. Let us explain how to obtain this Galois representation.

Consider a primitive $p^{n}$-th root of unity $\zeta_{p^{n}}$. Any element $\sigma$ of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on $\zeta_{p^{n}}$ and sends it to some $p^{n}$-th root of unity, which amounts to raising it to some integer power between $1$ and $p^{n}-1$, i.e. an element of $(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}$. We now define the $p$-adic cyclotomic character $\chi$ to be the map from $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ to $\mathbb{Z}_{p}^{\times}$ which sends the element $\sigma$ to the element of $\mathbb{Z}_{p}^{\times}$ which after modding out by $p^{n}$ is precisely the integer power to which we raised $\zeta_{p^{n}}$.

Our second example of a Galois representation is known as the Tate module of an elliptic curve. We recall that we also discussed an example of a Galois representation coming from the $p$-torsion points of an elliptic curve in Elliptic Curves. The Tate module is a way to package the action of the Galois group not only the $p$-torsion points but also the $p^{n}$-torsion for any $n$, by taking an inverse limit over $n$. Now the $p^{n}$-torsion points are isomorphic to $(\mathbb{Z}/p^{n}\mathbb{Z})^{2}$, so the inverse limit is going to be isomorphic to $\mathbb{Z}_{p}^{2}$. This is not a vector space, since $\mathbb{Z}_{p}$ is not a field, so we take the tensor product with $\mathbb{Q}_{p}$ to get $\mathbb{Q}_{p}^{2}$, which is a vector space. Therefore we get a Galois representation, i.e. a homomorphism from $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ to $\text{GL}_{2}(\mathbb{Q}_{p})$. This construction also works for abelian varieties – higher dimensional analogues of elliptic curves – except that the Tate module is now $2g$-dimensional, where $g$ is the dimension of the abelian variety.

Our last example of a Galois representation is given by the $\ell$-adic cohomology (explanation of this terminology to come later) of a smooth proper algebraic variety $X$ over $\mathbb{Q}$. This is the inverse limit over $n$ of the etale cohomology (see also Cohomology in Algebraic Geometry) of $X$ with coefficients in the constant sheaf $\mathbb{Z}/p^{n}\mathbb{Z}$. These etale cohomology groups are somewhat confusingly denoted $H^{i}(X,\mathbb{Z}_{p})$ – note that they are not the etale cohomology of $X$ with $\mathbb{Z}_{p}$ coefficients! Just as in the case of the Tate module, we take the tensor product with $\mathbb{Q}_{p}$ to produce our Galois representation.

These Galois representations coming from the $\ell$-adic cohomology somewhat subsume the Tate modules discussed earlier – that is because, if $X$ is an elliptic curve or more generally an abelian variety, we have that the $\mathbb{Q}_{p}$-linear maps from the Tate module (tensored with $\mathbb{Q}_{p}$) is isomorphic to the first $\ell$-adic cohomology $H_{1}(X,\mathbb{Z}_{p})\otimes\mathbb{Q}_{p}$. We say that the first $\ell$-adic cohomology is the dual of the Tate module.

Although we discussed representations over $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ in this post, it is also often useful to make our study “local” and focus on a single prime $\ell$, and study $\text{Gal}(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})$ instead. In this case we might as well just have replaced $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ with $\text{Gal}(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})$ in the above discussion, and nothing really changes, as long as the primes $\ell$ and $p$ are different primes. In the case that they are the same prime, things become much more complicated (and the theory is far richer)!

Note: Usually, when discussing “local” Galois representations, the notation for the primes $p$ and $\ell$ are switched! In other words, our local Galois representations are group homomorphisms from $\text{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$ to $\text{GL}_{n}(\mathbb{Q}_{\ell})$. This is the reason for the terminology “$\ell$-adic cohomology”. Since we started out just discussing “global” Galois representations, I switched the notation to use $p$ instead for the only instances were we needed a prime. Hopefully this is not overly confusing. We can also study Galois representations more generally for number fields (“global”) and finite extensions of $\mathbb{Q}_{p}$ (“local”).

Finally, although we stated above that we will only discuss three examples here, let us mention a fourth example: Galois representations can also come from modular forms (see also Modular Forms). To discuss these Galois representations would require us to develop some more machinery first, so we leave this to future posts for now.

References:

Cyclotomic character on Wikipedia

Tate module on Wikipedia

Etale cohomology on Wikipedia

Reciprocity laws and Galois representations: recent breakthroughs by Jared Weinstein

# Modular Forms

We have previously mentioned modular forms in The Moduli Space of Elliptic Curves and discussed them very briefly in the context of modular curves in Shimura Varieties. In this post, we will discuss this very important and central concept in modern number theory in more detail.

First we recall some facts about the group $\text{SL}_{2}(\mathbb{Z})$, which is so important that it is given the special name of the modular group. It is defined as the group of $2\times 2$ matrices with integer coefficients and determinant equal to $1$, and it acts on the upper half-plane (the set of complex numbers with positive imaginary part) in the following manner. Suppose an element $\gamma$ of $\text{SL}_{2}(\mathbb{Z})$ is written in the form $\left(\begin{array}{cc}a&b\\ c&d\end{array}\right)$. Then for $\tau$ an element of the upper half-plane we write

$\displaystyle \gamma(\tau)=\frac{a\tau+b}{c\tau+d}$

A modular form (with respect to $\text{SL}_{2}(\mathbb{Z}))$ is a holomorphic function on the upper half-plane such that

$\displaystyle f(\gamma(\tau))=(c\tau+d)^{k}f(\tau)$

for some $k$ and such that $f(\tau)$ is bounded as the imaginary part of $\tau$ goes to infinity. The number $k$ is called the weight of the modular form. If the function is not required to be bounded as the imaginary part of $\tau$ goes to infinity it is a weakly modular form, and if furthermore it is merely required to be meromorphic, , it is a meromorphic modular form. A meromorphic modular form of weight $0$ is just a meromorphic function on the upper half-plane which is invariant under the action of $\text{SL}_{2}(\mathbb{Z})$ (and bounded as the imaginary part of its argument goes to infinity) – we also call it a modular function.

We denote the set of modular forms of weight $k$ with respect to $\text{SL}_{2}(\mathbb{Z})$ by $\mathcal{M}_{k}(\text{SL}_{2}(\mathbb{Z}))$. Adding together two modular forms of the same weight gives another modular form of the same weight, and modular forms can be scaled by a complex number, so $\mathcal{M}_{k}(\text{SL}_{2}(\mathbb{Z}))$ actually forms a vector space. We can also multiple a modular form of weight $k$ with a modular form of weight $l$ to get a modular form of weight $k+l$, so modular forms of a certain weight form a graded piece of a graded ring $\mathcal{M}(\text{SL}_{2}(\mathbb{Z})$:

$\displaystyle \mathcal{M}(\text{SL}_{2}(\mathbb{Z}))=\bigoplus_{k}\mathcal{M}_{k}(\text{SL}_{2}(\mathbb{Z}))$

Modular functions are actually functions on the moduli space of elliptic curves – but what about modular forms of higher weight? It turns out that he modular forms of weight $2$ correspond to coefficients of differential forms on this space. To see this, consider $d\tau$ and how the group $\text{SL}(\mathbb{Z})$ acts on it:

$\displaystyle d\gamma(\tau)=\gamma'(\tau)d\tau=(c\tau+d)^{-2}d\tau$

where $\gamma'(\tau)$ is just the usual derivative of he action of $\gamma$ as describe earlier. For a general differential form given by $f(\tau)d\tau$ to be invariant under the action of $\text{SL}(\mathbb{Z})$ we must therefore have

$\displaystyle f(\gamma(\tau))=(c\tau+d)^{2}f(\tau)$.

The modular forms of weight greater than $2$ arise when we consider products of these differential forms. More technically, modular forms are sections of line bundles on modular curves, which come about when we compactify moduli spaces of elliptic curves (possibly with extra structure).

Let us now look at some examples of modular forms. Since modular forms “live on” moduli spaces of elliptic curves, we will keep in mind elliptic curves as we look at these examples. Our first family of examples are Eisenstein series of weight $k$, denoted by $G_{k}(\tau)$ which is of the form

$\displaystyle G_{k}(\tau)=\sum_{(m,n)\in\mathbb{Z}^{2}\setminus (0,0)}\frac{1}{(m+n\tau)^{k}}$

Any modular form can in fact be written in terms of Eisenstein series $G_{4}(\tau)$ and $G_{6}(\tau)$.

Now, let us relate this to elliptic curves. An elliptic curve over the complex numbers may be written as a Weierstrass equation

$\displaystyle y^{2}=4x^{3}-g_{2}x-g_{3}$

The coefficients on the right-hand side $g_{2}$ and $g_{3}$ are in fact modular forms, of weight $4$ and weight $6$ respectively, given in terms of the Eisenstein series by $g_{2}(\tau)=60G_{4}(\tau)$ and $g_{3}(\tau)=140G_{6}(\tau)$.

Another example of a modular form is the modular discriminant of an elliptic curve, as a modular form denoted $\Delta(\tau)$. It is a modular form of weight $12$, and can be expressed via the elliptic curve coefficients that we defined earlier:

$\Delta(\tau)=(g_{2}(\tau))^{3}-27(g_{3}(\tau))^{2}$.

Our final example in this post is not of a modular form, but a meromorphic modular form of weight $0$, i.e. a modular function. It is holomorphic on the upper half-plane, but goes to infinity as the imaginary part of $\tau$ goes to infinity. It is the j-invariant associated to an elliptic curve. Once again we may express it in terms of the elliptic curve coefficients $g_{2}$ and $g_{3}$:

$\displaystyle j(\tau)=1728\frac{(g_{2}(\tau))^{3}}{(g_{2}(\tau))^{3}-27(g_{3}(\tau))^{2}}$

Note that the denominator is also the modular discriminant.  The points of the moduli space of elliptic curves correspond to isomorphism classes of elliptic curves, and since the j-invariant is an honest-to-goodness holomorphic function on the moduli space of elliptic curves over $\mathbb{C}$, we can see that isomorphic elliptic curves will have the same j-invariant. This is not the case for the other modular forms we described above, which are not modular functions, i.e. they have nonzero weight! Why is this so? Let us recall that an elliptic curve over $\mathbb{C}$ corresponds to a lattice. Acting on a basis of this lattice by an element of $\text{SL}_{2}(\mathbb{Z})$ changes the basis, but preserves the lattice. This will be reflected as “admissible changes of coordinates” in the Weierstrass equations, and also changes these modular forms associated to the elliptic curves even though the elliptic curves are still isomorphic. But they change in a predictable way, according to the definition of modular forms.

A modular form $f(\tau)$ is also called a cusp form if the limit of $f(\tau)$ is zero as the imaginary part of $\tau$ approaches infinity. We denote the set of cusp forms of weight $k$ by $\mathcal{S}_{k}(\text{SL}_{2}(\mathbb{Z})$. They are a vector subspace of $\mathcal{M}_{k}(\text{SL}_{2}(\mathbb{Z})$ and the graded ring formed by their direct sum for all $k$, denoted $\mathcal{S}_{k}(\text{SL}_{2}(\mathbb{Z})$, is an ideal of the graded ring $\mathcal{M}(\text{SL}_{2}(\mathbb{Z})$. Cusp forms form a very important part of modern research, but we will not discuss them much in this introductory post and leave them for the future.

Let us now discuss congruence subgroups of $\text{SL}_{2}(\mathbb{Z})$ (we have also discussed this briefly in Shimura Varieties), so that we can define more general modular forms with respect to such a congruence subgroup instead of just $\text{SL}_{2}(\mathbb{Z})$. Given an integer $N$, the principal congruence subgroup $\Gamma(N)$ of $\text{SL}_{2}(\mathbb{Z})$ is the subgroup consisting of the elements which reduce to the identity when we reduce the entries modulo $N$. A congruence subgroup is any subgroup $\Gamma$ that contains the principal congruence subgroup $\Gamma(N)$. We refer to $N$ as the level of the congruence subgroup.

There are two important kinds of congruence subgroups of $\text{SL}_{2}(\mathbb{Z})$, denoted by $\Gamma_{0}(N)$ and $\Gamma_{1}(N)$. The subgroup $\Gamma_{0}(N)$ consists of the elements that become upper triangular after reduction modulo $N$, while the subgroup $\Gamma_{1}(N)$ consists of the elements that become upper triangular with ones on the diagonal after reduction modulo $N$. As we discussed in Shimura Varieties, these are related to moduli spaces of “elliptic curves with level structure”.

Now we can define the modular forms of weight $k$ with respect to such a congruence subgroup $\Gamma$. We shall once again require them to be holomorphic functions on the upper half-plane, and we require that for $\gamma\in \Gamma$ written as $\left(\begin{array}{cc}a&b\\ c&d\end{array}\right)$ we must have

$\displaystyle f(\gamma(\tau))=(c\tau+d)^{k}f(\tau)$.

However, the condition that the function be bounded as the imaginary part of $\tau$ goes to infinity must be modified. The reason is that the “point at infinity” is a cusp, a point of the modular curve that does not correspond to an elliptic curve over $\mathbb{C}$ but rather to a “degeneration” of it (this point is therefore not a part of the usual moduli space of elliptic curves –  we can think of it as a “puncture” in this space).

We recall that the construction of the moduli space of elliptic curves over $\mathbb{C}$ starts with the upper half-plane, then we quotient out by the action of $\text{SL}_{2}(\mathbb{Z})$. The cusps come from taking the union of the rational numbers with the upper half-plane, as well as the point at infinity. When we take the quotient by $\text{SL}_{2}(\mathbb{Z})$ this all gets sent to the same point, therefore the usual moduli space has only one cusp. But if we take the quotient by a congruence subgroup, we may have several cusps. Therefore, what we really require is for the modular form to be “holomorphic at the cusps“. We can still express this condition in familiar terms by requiring that not $f(\tau)$, but rather $(c\tau+d)^{-k}f(\gamma(\tau))$ for $\gamma\in \text{SL}_{2}(\mathbb{Z})$ be bounded as the imaginary part of $\tau$ goes to infinity. We can then define cusp forms with respect to $\Gamma$ by requiring vanishing at the cusps instead. The set of modular forms (resp. cusp forms) of weight $k$ with respect to $\Gamma$ are denoted $\mathcal{M}_{k}(\Gamma)$ (resp. $\mathcal{S}_{k}(\Gamma)$), and they also have the same structures of being vector spaces and being graded pieces of graded rings as the ones for $\text{SL}_{2}(\mathbb{Z})$.

Having only discussed the very basics of modular forms we end the post here, with the hope  that in the near future we will be able to discuss things such as Hecke operators, modular curves and their Jacobians, and their associated Galois representations. We redirect the interested reader to the references for now.

References:

Modular Form on Wikipedia

Eisenstein Series in Wikipedia

j-invariant on Wikipedia

Modular Form on Wikipedia

Congruence Subgroups on Wikipedia

A First Course in Modular Forms by Fred Diamond and Jerry Shurman

Advanced Topics in the Arithmetic of Elliptic Curves by Joseph H. Silverman

# Shimura Varieties

In The Moduli Space of Elliptic Curves we discussed how to construct a space whose points correspond to isomorphism classes of elliptic curves over $\mathbb{C}$. This space is given by the quotient of the upper half-plane by the special linear group $\text{SL}_{2}(\mathbb{Z})$. Shimura varieties kind of generalize this idea. In some cases their points may correspond to isomorphism classes of abelian varieties over $\mathbb{C}$, which are higher-dimensional generalizations of elliptic curves in that they are projective varieties whose points form a group, possibly with some additional information.

Using the orbit-stabilizer theorem of group theory, the upper half-plane can also be expressed as the quotient $\text{SL}_{2}(\mathbb{R})/\text{SO}(2)$. Therefore, the moduli space of elliptic curves over $\mathbb{C}$ can be expressed as

$\displaystyle \text{SL}_{2}(\mathbb{Z})\backslash\text{SL}_{2}(\mathbb{R})/\text{SO}(2)$.

If we wanted to parametrize “level structures” as well, we could replace $\text{SL}_{2}(\mathbb{Z})$ with a congruence subgroup $\Gamma(N)$, a subgroup which contains the matrices in $\text{SL}_{2}(\mathbb{Z})$ which reduce to an identity matrix when we mod out b some natural number $N$ which is greater than $1$. Now we obtain a moduli space of elliptic curves over $\mathbb{C}$ together with a basis of their $N$-torsion:

$Y(N)=\Gamma(N)\backslash\text{SL}_{2}(\mathbb{R})/\text{SO}(2)$

We could similarly consider the subgroup $\Gamma_{0}(N)$, the subgroup of $\text{SL}_{2}(\mathbb{Z})$ containing elements that reduce to an upper-triangular matrix mod $N$, to parametrize elliptic curves over $\mathbb{C}$ together with a cyclic $N$-subgroup, or $\Gamma_{1}(N)$, the subgroup of $\text{SL}_{2}(\mathbb{Z})$ which contains elements that reduce to an upper-triangular matrix with $1$ on every diagonal entry mod $N$, to parametrize elliptic curves over $\mathbb{C}$ together with a point of order $N$. These give us

$Y_{0}(N)=\Gamma_{0}(N)\backslash\text{SL}_{2}(\mathbb{R})/\text{SO}(2)$

and

$Y_{1}(N)=\Gamma_{1}(N)\backslash\text{SL}_{2}(\mathbb{R})/\text{SO}(2)$

Let us discuss some important properties of these moduli spaces, which will help us generalize them. The space $\text{SL}_{2}(\mathbb{R})/\text{SO}(2)$, i.e. the upper-half plane, is an example of a Riemannian symmetric space. This means it is a Riemannian manifold whose group of automorphisms act transitively – in layperson’s terms, every point looks like every other point – and every point has an associated involution fixing only that point in its neighborhood.

These moduli spaces almost form smooth projective curves, but they have missing points called “cusps” that do not correspond to an isomorphism class of elliptic curves but rather to a “degeneration” of such. We can fill in these cusps to “compactify” these moduli spaces, and we get modular curves $X(N)$, $X_{0}(N)$, and $X_{1}(N)$. On these modular curves live cusp forms, which are modular forms satisfying certain conditions at the cusps. Traditionally these modular forms are defined as functions on the upper-half plane satisfying certain conditions under the action of $\text{SL}_{2}(\mathbb{Z})$, but when they are cusp forms we may also think of them as sections of line bundles on these modular curves. In particular the cusp forms of “weight $2$” are the differential forms on a modular curve.

These modular curves are equipped with Hecke operators, $T_{p}$ and $\langle p\rangle$ for every $p$ not equal to $N$. These are operators on modular forms, but may also be thought of in terms of Hecke correspondences. We recall that elliptic curves over $\mathbb{C}$ are lattices in $\mathbb{C}$. Take such a lattice $\Lambda$. The $p$-th Hecke correspondence is a sum over all the index $p$ sublattices of $\Lambda$. It is a multivalued function from the modular curve to itself, but the better way to think of such a multivalued function is as a correspondence, a curve inside the product of the modular curve with itself.

With these properties as our guide, let us now proceed to generalize these concepts. One generalization is through the concept of an arithmetic manifold. This is a double coset space

$\Gamma\backslash G(\mathbb{R})/K$

where $G$ is a semisimple algebraic group over $\mathbb{Q}$, $K$ is a maximal compact subgroup of $G(\mathbb{R})$, and $\Gamma$ is an arithmetic subgroup, which means that it is intersection with $G(\mathbb{Z})$ has finite index in both $\Gamma$ and $G(\mathbb{Z})$. A theorem of Margulis says that, with a handful of exceptions, $G(\mathbb{R})/K$ is a Riemannian symmetric space. Arithmetic manifolds are equipped with Hecke correspondences as well.

Arithmetic manifolds can be difficult to study. However, in certain cases, they form algebraic varieties, in which case we can use the methods of algebraic geometry to study them. For this to happen, the Riemannian symmetric space $G(\mathbb{R})/K$ must have a complex structure compatible with its Riemannian structure, which makes it into a Hermitian symmetric space. The Baily-Borel theorem guarantees that the quotient of a Hermitian symmetric space by an arithmetic subgroup of $G(\mathbb{Q})$ is an algebraic variety. This is what Shimura varieties accomplish.

To motivate this better, we discuss the idea of Hodge structures. Let $V$ be an $n$-dimensional real vector space. A (real) Hodge structure on $V$ is a decomposition of its complexification $V\otimes\mathbb{C}$ as follows:

$\displaystyle V\otimes\mathbb{C}=\bigoplus_{p,q} V^{p,q}$

such that $V^{q,p}$ is the complex conjugate of $V^{p,q}$. The set of pairs $(p,q)$ for which $V^{p,q}$ is nonzero is called the type of the Hodge structure. Letting $V_{n}=\bigoplus_{p+q=n} V^{p,q}$, the decomposition $V=\bigoplus_{n} V_{n}$ is called the weight decomposition. An integral Hodge structure is a $\mathbb{Z}$-module $V$ together with a Hodge structure on $V_{\mathbb{R}}$ such that the weight decomposition is defined over $\mathbb{Q}$. A rational Hodge structure is defined similarly but with $V$ a finite-dimensional vector space over $\mathbb{Q}$.

An example of a Hodge structure is given by the singular cohomology of a smooth projective variety over $\mathbb{C}$:

$\displaystyle H^{n}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{C}=\bigoplus_{i+j=n}H^{j}(X,\Omega_{X/\mathbb{C}}^{i})$

In particular for an abelian variety $A$, the integral Hodge structure of type $(1,0),(0,1)$ given by the first singular cohomology $H^{1}(A(\mathbb{C}),\mathbb{Z})$ gives an integral Hodge structure of type $(-1,0),(0,-1)$ on its dual, the first singular homology $H_{1}(A(\mathbb{C}),\mathbb{Z})$. Specifying such an integral Hodge structure of type $(-1,0),(0,-1)$ on $H_{1}(A(\mathbb{C}),\mathbb{Z})$ is also the same as specifying a complex structure on $H_{1}(A(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}} \mathbb{R}$. In fact, the category of integral Hodge structures of type $(-1,0),(0,-1)$ is equivalent to the category of complex tori.

Let $\mathbb{S}$ be the group $\text{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{\text{m}}$. It is the Tannakian group for Hodge structures on finite-dimensional real vector spaces, which basically means that the category of Hodge structures on finite-dimensional real vector spaces are equivalent to the category of representations of $\mathbb{S}$ on finite-dimensional real vector spaces. This lets us redefine Hodge structures as a pair $(V,h)$ where $V$ is a finite-dimensional real vector space and $h$ is a map from $\mathbb{S}$ to $\text{GL}(V)$.

We have earlier stated that the category of integral Hodge structures of type $(-1,0),(0,-1)$ is equivalent to the category of complex tori. However, not all complex tori are abelian varieties. To obtain an equivalence between some category of Hodge structures and abelian varieties, we therefore need a notion of polarizable Hodge structures. We let $\mathbb{R}(n)$ denote the Hodge structure on $\mathbb{R}$ of type $(-n,-n)$ and define $\mathbb{Q}(n)$ and $\mathbb{Z}(n)$ analogously. A polarization on a real Hodge structure $V$ of weight $n$ is a morphism $\Psi$ of Hodge structures from $V\times V$ to $\mathbb{R}(-n)$ such that the bilinear form defined by $(u,v)\mapsto \Psi(u,h(i)v)$ is symmetric and positive semidefinite.

A polarizable Hodge structure is a Hodge structure that can be equipped with a polarization, and it turns out that the functor that assigns to an abelian variety $A$ its first singular homology $H_{1}(X,\mathbb{Z})$ defines an equivalence of categories between the category of abelian varieties over $\mathbb{C}$ and the category of polarizable integral Hodge structures of type $(-1,0),(0,-1)$.

A Shimura datum is a pair $(G,X)$ where $G$ is a connected reductive group over $\mathbb{Q}$, and $X$ is a $G(\mathbb{R})$ conjugacy class of homomorphisms from $\mathbb{S}$ to $G$, satisfying the following conditions:

• The composition of any $h\in X$ with the adjoint action of $G(\mathbb{R})$ on its Lie algebra $\mathfrak{g}$ induces a Hodge structure of type $(-1,1)(0,0)(1,-1)$ on $\mathfrak{g}$.
• For any $h\in X$, $h(i)$ is a Cartan involution on $G(\mathbb{R})^{\text{ad}}$.
• $G^{\text{ad}}$ has no factor defined over $\mathbb{Q}$ whose real points form a compact group.

Let $(G,X)$ be a Shimura datum. For $K$ a compact open subgroup of $G(\mathbb{A}_{f})$ where $\mathbb{A}_{f}$ is the finite adeles (the restricted product of completions of $\mathbb{Q}$ over all finite places, see also Adeles and Ideles), the Shimura variety $\text{Sh}_{K}(G,X)$ is the double quotient

$\displaystyle G(\mathbb{Q})\backslash (X\times G(\mathbb{A}_{f})/K)$

The introduction of adeles serves the purpose of keeping track of the level structures all at once. The space $\text{Sh}_{K}(G,X)$ is a disjoint union of locally symmetric spaces of the form $\Gamma\backslash X^{+}$, where $X^{+}$ is a connected component of $X$ and $\Gamma$ is an arithmetic subgroup of $G(\mathbb{Q})^{+}$. By the Baily-Borel theorem, it is an algebraic variety. Taking the inverse limit of over compact open subgroups $K$ gives us the Shimura variety at infinite level $\text{Sh}(G,X)$.

Let us now look at some examples. Let $G=\text{GL}_{2}$, and let $X$ be the conjugacy class of the map

$\displaystyle h:a+bi\to\left(\begin{array}{cc}a&b\\ -b&a\end{array}\right)$

There is a $G(\mathbb{R})$-equivariant bijective map from $X$ to $\mathbb{C}\setminus \mathbb{R}$ that sends $h$ to $i$. Then the Shimura varieties $\text{Sh}_{K}(G,X)$ are disjoint copies of modular curves and the Shimura variety at infinite level $\text{Sh}(G,X)$ classifies isogeny classes of elliptic curves with full level structure.

Let’s look at another example. Let $V$ be a $2n$-dimensional symplectic space over $\mathbb{Q}$ with symplectic form $\psi$. Let $G$ be the group of symplectic similitudes $\text{GSp}_{2n}$, i.e. for $k$ a $\mathbb{Q}$-algebra

$\displaystyle G(k)=\lbrace g\in \text{GL}(V\otimes k)\vert \psi(gu,gv)=\nu(g)\psi(u,v)\rbrace$

where $\nu:G\to k^{\times}$ is called the similitude character. Let $J$ be a complex structure on $V_{\mathbb{R}}$ compatible with the symplectic form $\psi$ and let $X$ be the conjugacy class of the map $h$ that sends $a+bi$ to the linear transformation $v\mapsto av+bJv$. Then the conjugacy class $X$ is the set of complex structures polarized by $\pm\psi$. The Shimura varieties $Sh_{K}(G,X)$ are called Siegel modular varieties and they parametrize isogeny classes of $n$-dimensional principally polarized abelian varieties with level structure.

There are many other kinds of Shimura varieties, which parametrize abelian varieties with other kinds of extra structure. Just like modular curves, Shimura varieties also have many interesting aspects, from Galois representations (related to their having Hecke correspondences), to certain special points related to the theory of complex multiplication, to special cycles with height pairings generalizing results such as the Gross-Zagier formula in the study of special values of L-functions and their derivatives. There is also an analogous local theory; in this case, ideas from $p$-adic Hodge theory come into play, where we can further relate the $p$-adic analogue of Hodge structures and Galois representations. The study of Shimura varieties is a very fascinating aspect of modern arithmetic geometry.

References:

Shimura variety on Wikipedia

Reciprocity Laws and Galois Representations: Recent Breakthroughs by Jared Weinstein

Perfectoid Shimura Varieties by Ana Caraiani

Introduction to Shimura Varieties by J.S. Milne

Lecture Notes for Advanced Number Theory by Jared Weinstein

# The Lubin-Tate Formal Group Law

A (one-dimensional, commutative) formal group law $f(X,Y)$ over some ring $A$ is a formal power series in two variables with coefficients in $A$ satisfying the following axioms that among other things makes it behave like an abelian group law:

• $f(X,Y)=X+Y+\text{higher order terms}$
• $f(X,Y)=f(Y,X)$
• $f(f(X,Y),Z)=f(X,f(Y,Z))$

A homomorphism of formal group laws $g:f_{1}(X,Y)\to f_{2}(X,Y)$ is another formal power series in two variable such $f_{1}(g(X,Y))=g(f_{2}(X,Y))$. An endomorphism of a formal group law is a homomorphism of a formal group law to itself.

As basic examples of formal group laws, we have the additive formal group law $\mathbb{G}_{a}(X,Y)=X+Y$, and the multiplicative group law $\mathbb{G}_{m}(X,Y)=X+Y+XY$. In this post we will focus on another formal group law called the Lubin-Tate formal group law.

Let $F$ be a nonarchimedean local field and let $\mathcal{O}_{F}$ be its ring of integers. Let $A$ be an $\mathcal{O}_{F}$-algebra with $i:\mathcal{O}_{F}\to A$ its structure map. A formal $\mathcal{O}_{F}$-module law over $A$ over $A$ is a formal group law $f(X,Y)$ such that for every element $a$ of $\mathcal{O}_{F}$ we have an associated endomorphism $[a]$ of $f(X,Y)$, and such that the linear term of this endomorphism as a power series is $i(a)X$.

Let $\pi$ be a uniformizer (generator of the unique maximal ideal) of $\mathcal{O}_{F}$. Let $q=p^{f}$ be the cardinality of the residue field of $\mathcal{O}_{F}$. There is a unique (up to isomorphism) formal $\mathcal{O}_{F}$-module law over $\mathcal{O}_{F}$ such that as a power series its linear term is $\pi X$ and such that it is congruent to $X^{q}$ mod $\pi$. It is called the Lubin-Tate formal group law and we denote it by $\mathcal{G}(X,Y)$.

The Lubin-Tate formal group law was originally studied by Jonathan Lubin and John Tate for the purpose of studying local class field theory (see Some Basics of Class Field Theory). The results of local class field theory state that the Galois group of the maximal abelian extension of $F$ is isomorphic to the profinite completion $\widehat{F}^{\times}$. This profinite completion in turn decomposes into the product $\mathcal{O}_{F}^{\times}\times \pi^{\widehat{\mathbb{Z}}}$.

The factor isomorphic to $\mathcal{O}_{F}^{\times}$ fixes the maximal unramified extension $F^{\text{nr}}$ of $F$, the factor isomorphic to $\pi^{\widehat{\mathbb{Z}}}$ fixes an infinite, totally ramified extension $F_{\pi}$ of $F$, and we have that $F=F^{\text{nr}}F_{\pi}$. The theory of the Lubin-Tate formal group law was developed to study $F_{\pi}$, taking inspiration from the case where $F=\mathbb{Q}_{p}$. In this case $\pi=p$ and the infinite totally ramified extension $F_{p}$ is obtained by adjoining to $\mathbb{Q}_{p}$ all $p$-th power roots of unity, which is also the $p$-th power torsion of the multiplicative group $\mathbb{G}_{m}$. We want to generalize $\mathbb{G}_{m}$, and this is what the Lubin-Tate formal group law accomplishes.

Let $\mathcal{G}[\pi^{n}]$ be the set of all elements in the maximal ideal of some separable extension $\mathcal{O}_{F}$ such that its image under the endomorphism $[\pi^{n}]$ is zero. This takes the place of the $p$-th power roots of unity, and adjoining to $F$ all the $\mathcal{G}[\pi^{n}]$ for all $n$ gives us the field $F_{\pi}$.

Furthermore, Lubin and Tate used the theory they developed to make local class field theory explicit in this case. We define the $\pi$-adic Tate module $T_{\pi}(\mathcal{G})$ as the inverse limit of $\mathcal{G}[\pi^{n}]$ over all $n$. This is a free $\mathcal{O}_{F}$-module of rank $1$ and its automorphisms are in fact isomorphic to $\mathcal{O}_{F}^{\times}$. Lubin and Tate proved that this is isomorphic to the Galois group of $F_{\pi}$ over $F$ and explicitly described the reciprocity map of local class field theory in this case as the map from $F^{\times }$ to $\text{Gal}(F_{\pi}/F)$ sending $\pi$ to the identity and an element of $\mathcal{O}_{F}^{\times}$ to the image of its inverse under the above isomorphism.

To study nonabelian extensions, one must consider deformations of the Lubin-Tate formal group. This will lead us to the study of the space of these deformations, called the Lubin-Tate space. This is intended to be the subject of a future blog post.

References:

Lubin-Tate Formal Group Law on Wikipedia

Formal Group Law on Wikipedia

The Geometry of Lubin-Tate Spaces by Jared Weinstein

A Rough Introduction to Lubin-Tate Spaces by Zhiyu Zhang

Formal Groups and Applications by Michiel Hazewinkel

# 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