# 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

# The Field with One Element

##### Introduction: Analogies between Function Fields and Number Fields

As has been hinted at, and mentioned in passing, in several previous posts on this blog, there are important analogies between numbers and functions. The analogy can perhaps be made most explicit in the case of $\mathbb{Z}$ (the ring of ordinary integers) and $\mathbb{F}_{p}[t]$ (the ring of polynomials in one variable $t$ over the finite field $\mathbb{F}_{p}$). We also often say that the analogy is between $\mathbb{Q}$ (the field of rational numbers) and $\mathbb{F}_{p}(t)$ (the field of rational functions in one variable $t$ over the finite field $\mathbb{F}_{p}$), which are the respective fields of fractions of $\mathbb{Z}$ and $\mathbb{F}_{p}[t]$. Recall also from Some Basics of Class Field Theory that $\mathbb{Q}$ and $\mathbb{F}_{p}(t)$ are examples of what we call global fields, together with their respective finite extensions.

Let us go back to $\mathbb{Z}$ and $\mathbb{F}_{p}[t]$ and compare their similarities. They are both principal ideal domains, which means that all their ideals can be generated by a single element. They both have groups of units (elements which have multiplicative inverses) which are finite. They both have an infinite number of prime ideals (generated by prime numbers in the case of $\mathbb{Z}$, and by monic irreducible polynomials in the case of $\mathbb{F}_{p}[t]$), and finally, they share the property that their residue fields over these prime ideals are finite.

But of course, despite all these analogies, a rather obvious question still remains unanswered. Regarding this question we quote the words of the mathematician Christophe Soule:

“The analogy between number fields and function fields finds a basic limitation with the lack of a ground field. One says that $\text{Spec}(\mathbb{Z})$ (with a point at infinity added, as is familiar in Arakelov geometry) is like a (complete) curve, but over which field?”

This question led to the development of the idea of the “field with one element”, also written $\mathbb{F}_{1}$, or sometimes $\mathbb{F}_{\text{un}}$ (it’s a pun taken from “un”, the French word for “one”). Taken literally, there is no such thing  as a “field” with one element – the way we define a field, it must always have a “one” and a “zero”, and these two elements must be different. Instead, the idea of the “field with one element” is just a name for ideas that extend the analogy between function fields and number fields, as if this “field” really existed. The name itself has historical origins in the work of the mathematician Jacques Tits involving certain groups called Chevalley groups and Weil groups, where surprising results appear in the limit when the number of elements of the finite fields involved goes to one – but in most approaches now, the “field with one element” is not a field, and often has more than one element. The whole point is that these ideas may still work, even though the “field” itself may not even exist! As one might expect, in order to pursue these ideas one must think out of the box, and different mathematicians have approached this question in different ways.

In this post, we will look at four approaches to the field with one element, developed by the mathematicians Anton Deitmar, Christophe Soule, Bertrand Toen and Michel Vaquie, and James Borger. There are many more approaches besides these, but we will perhaps discuss them in future posts.

Note: Throughout this post it will be helpful to remind ourselves that since there exists a map from the integers $\mathbb{Z}$ to any ring, we can think of rings as $\mathbb{Z}$-algebras. One of the ways the idea of the field with one element is approached is by exploring what $\mathbb{F}_{1}$-algebras mean, if ordinary rings are $\mathbb{Z}$-algebras.

##### The Approach of Deitmar

Deitmar defines the “category of rings over $\mathbb{F}_{1}$” (this is the term Deitmar uses, but we can also think of this as the category of $\mathbb{F}_{1}$-algebras) as simply the category of monoids. A monoid $A$ is also written as $\mathbb{F}_{A}$ to emphasize its nature as a “ring over $\mathbb{F}_{1}$“. The “field with one element” $\mathbb{F}_{1}$ is simply defined to be the trivial monoid.

For an $\mathbb{F}_{1}$-ring $\mathbb{F}_{A}$ we define the base extension (see Grothendieck’s Relative Point of View) to $\mathbb{Z}$ by taking the “monoid ring” $\mathbb{Z}[A]$:

$\displaystyle \mathbb{F}_{A}\otimes_{\mathbb{F}_{1}}\mathbb{Z}=\mathbb{Z}[A]$

We may think of this monoid ring as a ring whose elements are formal sums of elements of the monoid $A$ with integer coefficients, and with a multiplication provided by the multiplication on $A$, commuting with the scalar multiplication.

Meanwhile we also have the forgetful functor $F$ which simply “forgets” the additive structure of a ring, leaving us with a monoid under its multiplication operation. The base extension functor $-\otimes_{\mathbb{F}_{1}}\mathbb{Z}$ is left adjoint to the forgetful functor $F$, i.e. for every ring $R$ and every $\mathbb{F}_{A}/\mathbb{F}_{1}$ we have $\text{Hom}_{\text{Rings}}(\mathbb{F}_{A}\otimes_{\mathbb{F}_{1}}\mathbb{Z}, R)\cong\text{Hom}_{\mathbb{F}_{1}}(\mathbb{F}_{A},F(R))$ (see also Adjoint Functors and Monads).

The important concepts of “localization” and “ideals” in the theory of rings, important to construct the structure sheaf of a variety or a scheme, have analogues in the theory of monoids. The idea is that they only make use of the multiplicative structure of rings, so we can forget the additive structure and consider monoids instead. Hence, we can define varieties or schemes over $\mathbb{F}_{1}$. Many other constructions of algebraic geometry can be replicated with only monoids instead of rings, such as sheaves of modules over the structure sheaf. Deitmar then defines the zeta function of a scheme over $\mathbb{F}_{1}$, and hopes to connect this with known ideas about zeta functions (see for example our discussion in The Riemann Hypothesis for Curves over Finite Fields).

Deitmar’s idea of using monoids is one of the earlier approaches to the idea of the field with one element, and has become somewhat of a template for other approaches. One may be able to notice the influence of Deitmar’s work in the other approaches that we will discuss in this post.

##### The Approach of Soule

Soule’s question, as phrased in his paper On the Field with One Element, is as follows:

“Which varieties over $\mathbb{Z}$ are obtained by base change from $\mathbb{F}_{1}$ to $\mathbb{Z}$?”

Soule’s approach to answering this question then makes use of three concepts. The first one is a suggestion from the early days of the development of the idea of the field with one element, apparently due to the mathematicians Andre Weil and Kenkichi Iwasawa, that the finite field extensions of the field with one element should consist of the roots of unity, together with zero.

The second concept is an important point that we only touched on briefly from Algebraic Spaces and Stacks, namely, that we may identify the functor of points of a scheme with the scheme itself. Now the functor of points of a scheme is uniquely determined by its values on affine schemes, and the category of affine schemes is the opposite category to the category of rings; therefore, we now redefine a scheme simply as a covariant functor from the category of rings to the category of sets, which is representable.

The third concept is the idea of an evaluation of a function at a point. Soule implements this concept by including a $\mathbb{C}$-algebra as part of his definition of a variety over $\mathbb{F}_{1}$, together with a natural transformation that expresses this evaluation.

We now give the details of Soule’s construction, proceeding in four steps. Taking into account the first concept mentioned earlier,  we consider the following expression, the base extension of $\mathbb{F}_{1^{n}}$ to $\mathbb{Z}$ over $\mathbb{F}_{1}$:

$\displaystyle \mathbb{F}_{1^{n}}\otimes_{\mathbb{F}_{1}}\mathbb{Z}=\mathbb{Z}[T]/(T^{n}-1)=\mathbb{Z}[\mu_{n}]$

We shall also denote this ring by $R_{n}$. We can form a category whose objects are the finite tensor products of $R_{n}$, for $n\geq 1$, and we denote this category by $\mathcal{R}$.

An affine gadget over $\mathbb{F}_{1}$ is a triple $(\underline{X},\mathcal{A}_{X},e_{X})$ where $\underline{X}$ is a covariant functor from the category $\mathcal{R}$ to the category of sets, $\mathcal{A}_{X}$ is a $\mathbb{C}$-algebra, and $e_{X}$ is a natural transformation from $\underline{X}$ to $\text{Hom}(\mathcal{A}_{X},\mathbb{C}[-])$.

A morphism of affine gadgets consists of a natural transformation $\underline{\phi}:\underline{X}\rightarrow\underline{Y}$ and a morphism of algebras $\phi^{*}:\mathcal{A}_{X}\rightarrow\mathcal{A}_{Y}$ such that $f(\underline{\phi}(P))=(\phi^{*}(f))(P)$. A morphism $(\underline{\phi}, \phi^{*})$ is also called an immersion if $\underline{\phi}$ and $\phi^{*}$ are both injective.

An affine variety over $\mathbb{F}_{1}$ is an affine gadget $X=(\underline{X},\mathcal{A}_{X},e_{X})$ over $\mathbb{F}_{1}$ such that

(i) for any object $R$ of $\mathcal{R}$, the set $\underline{X}(R)$ is finite, and

(ii) there exists an affine scheme $X_{\mathbb{Z}}=X\otimes_{\mathbb{F}_{1}}\mathbb{Z}$ of finite type over $\mathbb{Z}$ and immersion $i:X\rightarrow \mathcal{G}(X_{\mathbb{Z}})$ with the universal property that for any other affine scheme $V$ of finite type over $\mathbb{Z}$ and morphism $\varphi:X\rightarrow\mathcal{G}(V)$, there exists a unique morphism $\varphi_{\mathbb{Z}}:X_{\mathbb{Z}}\rightarrow\mathcal{G}(V)$ such that $\varphi=\mathcal{G}(\varphi_{\mathbb{Z}})\circ i$.

An object over $\mathbb{F}_{1}$ is a triple $(\underline{\underline{X}},\mathcal{A}_{X},e_{X})$ where $\underline{\underline{X}}$ is a contravariant functor from the category of affine gadgets over $\mathbb{F}_{1}$$\mathcal{A}_{X}$ is once again a $\mathbb{C}$-algebra, and $e_{X}$ is a natural transformation from $\underline{\underline{X}}$ to $\text{Hom}(\mathcal{A}_{X},\mathbb{C}[-])$.

A morphism of objects is defined in the same way as a morphism of affine gadgets.

A variety over $\mathbb{F}_{1}$ is an object $X=(\underline{\underline{X}},\mathcal{A}_{X},e_{X})$ over $\mathbb{F}_{1}$ such that such that

(i) for any object $R$ of $\mathcal{R}$, the set $\underline{\underline{X}}(\text{Spec}(R))$ is finite, and

(ii) there exists a scheme $X_{\mathbb{Z}}=X\otimes_{\mathbb{F}_{1}}\mathbb{Z}$ of finite type over $\mathbb{Z}$ and immersion $i:X\rightarrow \mathcal{O}b(X_{\mathbb{Z}})$ with the universal property that for any other scheme $V$ of finite type over $\mathbb{Z}$ and morphism $\varphi:X\rightarrow\mathcal{O}b(V)$, there exists a unique morphism $\varphi_{\mathbb{Z}}:X_{\mathbb{Z}}\rightarrow\mathcal{O}b(V)$ such that $\varphi=\mathcal{O}b(\varphi_{\mathbb{Z}})\circ i$.

Like Deitmar, Soule constructs the zeta function of a variety over $\mathbb{F}_{1}$, and furthermore explores connections with certain kinds of varieties called “toric varieties”, which are also of interest in other approaches to the field with one element, and the theory of motives (see The Theory of Motives).

##### The Approach of Toen and Vaquie

We recall from Grothendieck’s Relative Point of View that we call a scheme $X$ a scheme “over” $S$, or an $S$-scheme, if there is a morphism of schemes from $X$ to $S$, and if $S$ is an affine scheme defined as $\text{Spec}(R)$ for some ring $R$, we also refer to it as a scheme over $R$, or an $R$-scheme. We recall also every scheme is a scheme over $\text{Spec}(\mathbb{Z})$, or a $\mathbb{Z}$-scheme. The approach of Toen and Vaquie is to construct categories of schemes “under” $\text{Spec}(\mathbb{Z})$.

From Monoidal Categories and Monoids we know that rings are the monoid objects in the monoidal category of abelian groups, and abelian groups are $\mathbb{Z}$-modules.

More generally, for a symmetric monoidal category $(\textbf{C}, \otimes, \mathbf{1})$ that is complete, cocomplete, and closed (i.e. possesses internal Homs related to the monoidal structure $\otimes$, see again Monoidal Categories and Monoids), we have in $\textbf{C}$ a notion of monoid, for such a monoid $A$ a notion of an $A$-module, and for a morphism of monoids $A\rightarrow B$ a notion of a base change functor $-\otimes_{A}B$ from $A$-modules to $B$-modules.

Therefore, if we have a category $\textbf{C}$ with a symmetric monoidal functor $\textbf{C}\rightarrow \mathbb{Z}\text{-Mod}$, we obtain a notion of a “scheme relative to $\textbf{C}$” and a base change functor to $\mathbb{Z}$-schemes. This gives us our sought-for notion of schemes under $\text{Spec}(\mathbb{Z})$.

In particular, there exists a notion of commutative monoids (associative and with unit) in $\textbf{C}$, and they form a category which we denote by $\textbf{Comm}(\textbf{C})$. We define the category of affine schemes related to $\textbf{C}$ as $\textbf{Aff}_{\textbf{C}}:= \textbf{Comm}(\textbf{C})^{\text{op}}$.

These constructions satisfy certain properties needed to define a category of schemes relative to $(\textbf{C},\otimes,\mathbf{1})$, such as a notion of Zariski topology. A relative scheme is defined as a sheaf on the site $\textbf{Aff}_{\textbf{C}}$ provided with the Zariski topology, and which has a covering by affine schemes. The category of schemes obtained is denoted $\textbf{Sch}(\textbf{C})$. It is a subcategory of the category of sheaves on $\textbf{Aff}_{\textbf{C}}$ which is closed under the formation of fiber products and disjoint unions. It contains a full subcategory of affine schemes, given by the representable sheaves, and which is equivalent to the category $\textbf{Comm}(\textbf{C})^{\text{op}}$. The purely categorical nature of the construction makes the category $\textbf{Sch}(\textbf{C})$ functorial in $\textbf{C}$.

In their paper, Toen and Vacquie give six examples of their construction, one of which is just the ordinary category of schemes, while the other five are schemes “under $\text{Spec}(\mathbb{Z})$“.

First we let $(C,\otimes,\mathbf{1})=(\mathbb{Z}\text{-Mod},\otimes,\mathbb{Z})$, the symmetric monoidal category of abelian groups (for the tensor product). The category of schemes obtained $\mathbb{Z}\text{-Sch}$ is equivalent to the category of schemes in the usual sense.

The second example will be $(C,\otimes,\mathbf{1})=(\mathbb{N}\text{-Mod},\otimes,\mathbb{N})$ the category of commutative monoids, or abelian semigroups, with the tensor product, which could also be called $\mathbb{N}$-modules. The category of schemes in this case will be denoted $\mathbb{N}\text{-Sch}$, and the subcategory of affine schemes is equivalent to the opposite category of commutative semirings.

The third example is $(C,\otimes,\mathbf{1})=(\text{Ens},\times, *)$, the symmetric monoidal category of sets with the direct product. The category of relative schemes will be denoted $\mathbb{F}_{1}\text{-Sch}$, and we can think of them as schemes or varieties defined on the field with one element. By definition, the subcategory of affine $\mathbb{F}_{1}$-schemes is equivalent to the opposite category of commutative monoids.

We have the base change functors

$-\otimes_{\mathbb{F}_{1}}\mathbb{N}:\mathbb{F}_{1}\text{-Sch}\rightarrow \mathbb{N}\text{-Sch}$

and

$-\otimes_{\mathbb{N}}\mathbb{Z}:\mathbb{N}\text{-Sch}\rightarrow \mathbb{Z}\text{-Sch}$

We can compose these base change functors and represent it with the following diagram:

$\text{Spec}(\mathbb{Z})\rightarrow\text{Spec}(\mathbb{N})\rightarrow\text{Spec}(\mathbb{F}_{1})$.

The final three examples of “schemes under $\text{Spec}(\mathbb{Z})$” given by Toen and Vaquie make use of ideas from “homotopical algebraic geometry“. Homotopical algebraic geometry is a very interesting subject that unfortunately we have not discussed much on this blog. Roughly, in homotopical algebraic geometry the role of rings in ordinary algebraic geometry is taken over by ring spectra – spectra (in the sense of Eilenberg-MacLane Spaces, Spectra, and Generalized Cohomology Theories) with a “smash product” operation. This allows us to make use of concepts from abstract homotopy theory. In this post we will only introduce some very basic concepts that we will need to discuss Toen and Vaquie’s examples, and leave the rest to the references.

We will need the concepts of $\Gamma$-spaces and simplicial sets. We define the category $\Gamma^{0}$ to be the category whose objects are “pointed” finite sets (a finite set where one element is defined to be the “basepoint”) and whose morphisms are maps of finite sets that preserve the basepoint. We also define the category $\Delta$ to be the category whose objects are finite ordered sets $[n]=\{0<1<2... and whose morphisms are monotone (non-decreasing) maps of finite ordered sets. A $\Gamma$-space is then simply a covariant functor from the category $\Gamma^{0}$ to the category of pointed sets, while a simplicial set is a covariant functor from the category $\Delta$ to the category of sets. Simplicial sets are rather abstract constructions, but they are inspired by simplices and simplicial complexes in algebraic topology (see Simplices).

Let $M$ be a $\Gamma$-space. If there is a monoid structure on $\pi_{0}M(1_{+})$ (see Homotopy Theory), then we say that $M$ is a special $\Gamma$-space. If, in addition, this structure is also an abelian group structure, then we say that $M$ is a very special $\Gamma$-space.

The category of $\Gamma$-spaces and the category of simplicial sets are both symmetric monoidal categories, which we need to define relative schemes. For the category of $\Gamma$-spaces, we have the smash product, defined by the requirement that any morphism $F_{1}\wedge F_{2}\rightarrow G$ to any functor $G$ from $\Gamma^{0}\times \Gamma^{0}$ to the category of pointed sets be a natural transformation, i.e. there are maps of pointed sets from $F_{1}\wedge F_{2}(X\wedge Y)$ to $G(X\wedge Y)$, natural in $X$ and $Y$ (here $X\wedge Y$ refers to the smash product of pointed sets obtained by taking the direct product and collapsing the wedge sum, see Eilenberg-MacLane Spaces, Spectra, and Generalized Cohomology Theories),  and the unit is the sphere spectrum $\mathbb{S}$, which is just the inclusion functor from the category of pointed finite sets to the category of pointed sets.

For the category of simplicial sets, we have the direct product, defined as the functor $X\times Y$ which sends the finite ordered set $[n]$ to the set $X([n])\times Y([n])$, for two simplicial sets $X$ and $Y$, and the unit is the functor $*$, which sends any finite ordered set to the set with a single element.

We now go back to Toen and Vaquie’s final three examples of relative schemes. The first of these examples is when one has $(C,\otimes,\mathbf{1}) = (\mathcal{GS},\wedge,\mathbb{S})$, the category of very special $\Gamma$-spaces. We thus have a category of schemes relative to $\mathcal{GS}$, which we will denote $\mathbb{S}\text{-Sch}$, where the notation $\mathbb{S}$ recalls the sphere spectrum.

The second example is $(C,\otimes,\mathbf{1})=(\mathcal{MS},\wedge,\mathbb{S}_{+})$, the category of special $\Gamma$-spaces. The category of relative schemes will be noted $\mathbb{S}_{+}\text{-Sch}$, and its affine objects are homotopical analogs of commutative semirings. The notation $\mathbb{S}_{+}$ intuitively means the semiring in spectra of positive integers, and is a homotopical version of the semiring $\mathbb{N}$.

The third example is $(C,\otimes,\mathbf{1})=(\text{SEns},\times,*)$, the category of simplicial sets with its direct product. The schemes that we obtain are homotopical versions of the $\mathbb{F}_{1}$-schemes, and will be called $\mathbb{S}_{1}$-schemes, where $\mathbb{S}_{1}$ may be thought of as the “ring spectrum with one element”, in analogy with $\mathbb{F}_{1}$, the “field with one element”.

Similar to the earlier cases, we also have the base change functors

$-\otimes_{\mathbb{S}_{1}}\mathbb{S}_{+}:\mathbb{S}_{1}\text{-Sch}\rightarrow \mathbb{S}_{+}\text{-Sch}$

and

$-\otimes_{\mathbb{S}_{+}}\mathbb{S}:\mathbb{S}_{+}\text{-Sch}\rightarrow \mathbb{S}\text{-Sch}$

which we can also compose and represent it with the following diagram:

$\text{Spec}(\mathbb{S})\rightarrow\text{Spec}(\mathbb{S}_{+})\rightarrow\text{Spec}(\mathbb{S}_{1})$.

Moreover, we also have the following functors:

$-\otimes_{\mathbb{S}_{1}}\mathbb{F}_{1}:\mathbb{S}_{1}\text{-Sch}\rightarrow \mathbb{F}_{1}\text{-Sch}$

$-\otimes_{\mathbb{S}_{+}}\mathbb{N}:\mathbb{S}_{+}\text{-Sch}\rightarrow \mathbb{N}\text{-Sch}$

and

$-\otimes_{\mathbb{S}}\mathbb{Z}:\mathbb{S}\text{-Sch}\rightarrow \mathbb{Z}\text{-Sch}$

which relate the “homotopical” relative schemes to the ordinary relative schemes. And so, all these schemes, both the new schemes “under $\text{Spec}(\mathbb{Z})$” as well as the ordinary schemes over $\text{Spec}(\mathbb{Z})$, are related to each other.

##### The Approach of Borger

Borger’s approach makes use of the idea of adjoint triples (see Adjoint Functors and Monads). Before we discuss the field with one element in this approach, let us first discuss something more elementary. Consider a field $K$ and and a field extension $L$ of $K$, and let $G=\text{Gal}(L/K)$. We have the following adjoint triple:

$\displaystyle \text{Weil restrict}:L\textbf{-Alg}\rightarrow K\textbf{-Alg}$

$\displaystyle -\otimes_{K}L: K\textbf{-Alg}\rightarrow L\textbf{-Alg}$

$\displaystyle \text{forget base}:L\textbf{-Alg}\rightarrow K\textbf{-Alg}$

Grothendieck’s abstract reformulation of Galois theory says that there is an equivalence of categories between the category of $K$-algebras and the category of $L$-algebras with an action of $G$. This means that we can also consider the above adjoint triple in the following sense:

$\displaystyle A\rightarrow\otimes_{G}A:L\textbf{-Alg}\rightarrow L\textbf{-Alg}\text{ (with }G\text{-action)}$

$\displaystyle \text{fgt}: L\textbf{-Alg}\text{ (with }G\text{-action)}\rightarrow L\textbf{-Alg}$

$\displaystyle A\rightarrow\prod_{G}A:L\textbf{-Alg}\rightarrow L\textbf{-Alg}\text{ (with }G\text{-action)}$

Let us now go back to the field with one element. We want to construct the following adjoint triple:

$\displaystyle \text{Weil restrict}:\mathbb{Z}\textbf{-Alg}\rightarrow\mathbb{F}_{1}\textbf{-Alg}$

$\displaystyle -\otimes_{\mathbb{F}_{1}}\mathbb{Z}:\mathbb{F}_{1}\textbf{-Alg}\rightarrow\mathbb{Z}\textbf{-Alg}$

$\displaystyle \text{forget base}:\mathbb{Z}\textbf{-Alg}\rightarrow\mathbb{F}_{1}\textbf{-Alg}$

Following the above example of the field $K$ and the field extension $L$ of $K$, we will approach the construction of this adjoint triple by considering instead the following adjoint triple:

$\displaystyle \Lambda\odot-:\mathbf{Rings}\rightarrow\Lambda\textbf{-}\mathbf{Rings}$

$\displaystyle \text{fgt}:\Lambda\textbf{-}\mathbf{Rings}\rightarrow\mathbf{Rings}$

$\displaystyle W(-):\mathbf{Rings}\rightarrow\Lambda\textbf{-}\mathbf{Rings}$

We must now discuss the meaning of the concepts involved in the last adjoint triple. In particular, the must define the category of $\Lambda$-rings, as well as the adjoint functors $\Lambda\odot-$, $\text{fgt}$, and $W(-)$ that form the adjoint triple.

Let $R$ be a ring and let $p$ be a prime number. A Frobenius lift is a ring homomorphism $\psi_{p}:R\rightarrow R$ such that $F\circ q=q\circ\psi_{p}$ where $q:R\rightarrow R/pR$ is the quotient map and $F:R/pR\rightarrow R/pR$ is the Frobenius map which sends an element $x$ to the element $x^{p}$.

Closely related to the idea of Frobenius lifts is the idea of $p$-derivations. If the terminology is reminiscent of differential calculus, this is because Borger’s approach is closely related to the mathematician Alexandru Buium’s theory of “arithmetic differential equations“. If numbers are like functions, then what Buium wants to figure out is what the analogue of a derivative of a function should be for numbers.

Let

$\displaystyle \psi_{p}(x)=x^{p}+p\delta_{p}(x)$.

Being a ring homomorphism means that $\psi$ satisfies the following properties:

(1) $\psi_{p}(x+y)=\psi_{p}(x)+\psi_{p}(y)$

(2) $\psi_{p}(xy)=\psi_{p}(x)\psi_{p}(y)$

(3) $\psi_{p}(1)=1$

(4) $\psi_{p}(0)=0$

Recalling that $\psi_{p}(x)=x^{p}+p\delta_{p}(x)$, this means that $\delta_{p}(x)$ must satisfy the following properties corresponding to the above properties for $\psi_{p}(x)$:

(1) $\delta_{p}(x+y)=\delta_{p}(x)+\delta_{p}(y)-\frac{1}{p}\sum_{i=1}^{p-1}\binom{p}{i}x^{p-i}y^{i}$

(2) $\delta_{p}(xy)=x^{p}\delta_{p}(y)+y^{p}\delta_{p}(x)+p\delta_{p}(x)\delta_{p}(y)$

(3) $\delta_{p}(1)=0$

(4) $\delta_{p}(0)=0$.

Let

$\displaystyle \Lambda_{p}\odot A=\mathbb{Z}[\delta_{p}^{\circ n}(a)|n\geqslant 0,a\in A]/\sim$

where $\sim$ is the equivalence relation given by the “Liebniz rule”, i.e.

$\displaystyle \delta_{p}^{\circ 0}(x+y)=\delta_{p}^{\circ 0}(x)+\delta_{p}^{\circ 0}(y)$

$\displaystyle \delta_{p}^{\circ 0}(xy)=\delta_{p}^{\circ 0}(x)\delta_{p}^{\circ 0}(y)$

$\displaystyle \delta_{p}^{\circ 1}(x+y)=\delta_{p}^{\circ 1}(x)+\delta_{p}^{\circ 1}(y)-\frac{1}{p}\sum_{i=1}^{p-1}\binom{p}{i}x^{p-i}y^{i}$

$\displaystyle \delta_{p}^{\circ 1}(xy)=\delta_{p}^{\circ 1}(x)\delta_{p}^{\circ 1}(y)+p\delta_{p}^{\circ 1}(x)\delta_{p}^{\circ 1}(y)$

and so on.

We now discuss the closely related (and an also important part of modern mathematical research)  notion of Witt vectors. We define the ring of Witt vectors of the ring $A$ by

$\displaystyle W_{p}(A)=A\times A\times...$

with ring operations given by

$\displaystyle (a_{0},a_{1},...)+(b_{0},b_{1},...)=(a_{0}+b_{0},a_{1}+b_{1}-\sum_{i=1}^{p-1}\frac{1}{p}\binom{p}{i}a_{0}^{i}b_{0}^{p-i},...)$

$\displaystyle (a_{0},a_{1},...)(b_{0},b_{1},...)=(a_{0}b_{0},a_{0}^{p}b_{1}+a_{1}b_{0}^{p}+pa_{1}b_{1},...)$

$\displaystyle 0=(0,0,...)$

$\displaystyle 1=(1,0,...)$

The functors

$\displaystyle \Lambda_{p}\odot-:\mathbf{Rings}\rightarrow\mathbf{\delta_{p}}\textbf{-}\mathbf{Rings}$

$\displaystyle \text{fgt}:\mathbf{\delta_{p}}\textbf{-}\mathbf{Rings}\rightarrow\mathbf{Rings}$

$\displaystyle W_{p}(-):\mathbf{Rings}\rightarrow\mathbf{\delta_{p}}\textbf{-}\mathbf{Rings}$

A $\Lambda_{p}$-ring is defined to be the smallest $\Lambda_{p}^{'}$-ring that contains $e$, where a $\Lambda_{p}^{'}$-ring is in turn defined to be a $p$-torsion free ring together with a Frobenius lift. But it so happens that a $\Lambda_{p}$-ring is also the same thing as a $\delta_{p}$-ring, so we also have the following adjoint triple:

$\displaystyle \Lambda_{p}\odot-:\mathbf{Rings}\rightarrow\Lambda_{p}\textbf{-}\mathbf{Rings}$

$\displaystyle \text{fgt}:\Lambda_{p}\textbf{-}\mathbf{Rings}\rightarrow\mathbf{Rings}$

$\displaystyle W_{p}(-):\mathbf{Rings}\rightarrow\Lambda_{p}\textbf{-}\mathbf{Rings}$

Now that we know the basics of a “$p$-typical” $\Lambda$-ring, which is a ring together with a Frobenius morphism $\psi_{p}$ for one fixed $p$, we can also consider a ring together with a Frobenius morphism $\psi_{p}$ for every prime $p$, to form a “big” $\Lambda$-ring. We will then obtain the following adjoint triple:

$\displaystyle \Lambda\odot-:\mathbf{Rings}\rightarrow\Lambda\textbf{-}\mathbf{Rings}$

$\displaystyle \text{fgt}:\Lambda\textbf{-}\mathbf{Rings}\rightarrow\mathbf{Rings}$

$\displaystyle W(-):\mathbf{Rings}\rightarrow\Lambda\textbf{-}\mathbf{Rings}$

This is just the adjoint triple that we were looking for in the beginning of this section. In other words, we now have what we need to construct rings over $\mathbb{F}_{1}$, or $\mathbb{F}_{1}$-algebras and moreover, we have an adjoint triple that relates them to ordinary rings (or $\mathbb{Z}$-algebras).

We can then generalize these constructions from rings to schemes. The definition of a $\Lambda$-structure on general schemes is complicated and left to the references, but when the scheme $X$ is flat over $\mathbb{Z}$ (see The Hom and Tensor Functors), a $\Lambda$-structure on $X$ is simply defined to be a commuting family of endomorphisms $\psi_{p}$, one for each prime $p$, such that they agree with the $p$-th power Frobenius map on the fibers $X\times_{\text{Spec}(\mathbb{Z})}\mathbb{F}_{p}$.

One may notice that in Borger’s approach an $\mathbb{F}_{1}$-scheme has more structure than a $\mathbb{Z}$-scheme, whereas in Deitmar’s approach $\mathbb{F}_{1}$-schemes, being commutative monoids, have less structure than $\mathbb{Z}$-schemes. One may actually think of the $\Lambda$-structure as “descent data” to $\mathbb{F}_{1}$. In other words, the $\Lambda$-structure tells us how a scheme defined over $\mathbb{Z}$ is defined over $\mathbb{F}_{1}$. There is actually a way to use a monoid $M$ to construct a $\Lambda$-ring $\mathbb{Z}[M]$, where $\mathbb{Z}[M]$ is just the monoid ring as described earlier in the approach of Deitmar, and the Frobenius lifts are defined by $\psi_{p}=m^{p}$ for $m\in M$. We therefore have some sort of connection between Deitmar’s approach (which is also easily seen to be closely related to Soule’s and Toen and Vaquie’s approach) with Borger’s approach.

##### Conclusion

We have mentioned only four approaches to the idea of the field with one element in this rather lengthy post. There are many others, and these approaches are often related to each other. In addition, there are other approaches to uncovering even more analogies between function fields and number fields that are not commonly classified as being part of this circle of ideas. To end this post, we just mention that many open problems in mathematics, such as the abc conjecture and the Riemann hypothesis, have function field analogues that have already been solved (we have already discussed the function field analogue of the Riemann hypothesis in The Riemann Hypothesis for Curves over Finite Fields) – perhaps an investigation of these analogies would lead to the solution of their number field analogues – or, in the other direction, perhaps work on these problems would help uncover more aspects of these mysterious and beautiful analogies.

References:

Field with One Element on Wikipedia

Field with One Element on the nLab

Function Field Analogy on the nLab

Schemes over F1 by Anton Deitmar

Lectures on Algebraic Varieties over F1 by Christophe Soule

Les Varieties sur le Corps a un Element by Christophe Soule

On the Field with One Element by Christophe Soule

Under Spec Z by Bertrand Toen and Michel Vaquie

Lambda-Rings and the Field with One Element by James Borger

Witt Vectors, Lambda-Rings, and Arithmetic Jet Spaces by James Borger

Mapping F1-Land: An Overview of Geometries over the Field with One Element by Javier Lopez-Pena and Oliver Lorscheid

Geometry and the Absolute Point by Lieven Le Bruyn

This Week’s Finds in Mathematical Physics (Week 259) by John Baez

Algebraic Number Theory by Jurgen Neukirch

The Local Structure of Algebraic K-Theory by Bjorn Ian Dundas, Thomas G. Goodwillie, and Randy McCarthy

# Some Useful Links: Knots in Physics and Number Theory

In modern times, “knots” have been important objects of study in mathematics. These “knots” are akin to the ones we encounter in ordinary life, except that they don’t have loose ends. For a better idea of what I mean, consider the following picture of what is known as a “trefoil knot“:

More technically, a knot is defined as the embedding of a circle in 3-dimensional space. For more details on the theory of knots, the reader is referred to the following Wikipedia pages:

Knot on Wikipedia

Knot Theory on Wikipedia

One of the reasons why knots have become such a major part of modern mathematical research is because of the work of mathematical physicists such as Edward Witten, who has related them to the Feynman path integral in quantum mechanics (see Lagrangians and Hamiltonians).

Witten, who is very famous for his work on string theory (see An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry) and for being the first, and so far only, physicist to win the prestigious Fields medal, himself explains the relationship between knot theory and quantum mechanics in the following article:

Knots and Quantum Theory by Edward Witten

But knots have also appeared in other branches of mathematics. For example, in number theory, the result in etale cohomology known as Artin-Verdier duality states that the integers are similar to a 3-dimensional object in some sense. In particular, because it has a trivial etale fundamental group (which is kind of an algebraic analogue of the fundamental group discussed in Homotopy Theory and Covering Spaces), it is similar to a 3-sphere (recall the common but somewhat confusing notation that the ordinary sphere we encounter in everyday life is called the 2-sphere, while a circle is also called the 1-sphere).

Note: The fact that a closed 3-dimensional space with a trivial fundamental group is a 3-sphere is the content of a very famous conjecture known as the Poincare conjecture, proved by Grigori Perelman in the early 2000’s.  Perelman refused the million-dollar prize that was supposed to be his reward, as well as the Fields medal.

The prime numbers, because their associated finite fields have one cover for every integer, are like circles, and recalling the definition of knots mentioned above, are therefore like knots on this 3-sphere. This analogy, originally developed by David Mumford and Barry Mazur, is better explained in the following post by Lieven le Bruyn on his blog neverendingbooks:

What is the Knot Associated to a Prime on neverendingbooks

Finally, given what we have discussed, could it be that knot theory can “tie together” (pun intended) physics and number theory? This is the motivation behind the new subject called “arithmetic Chern-Simons theory” which is introduced in the following paper by Minhyong Kim:

Arithmetic Chern-Simons Theory I by Minhyong Kim

Of course, it must also be clarified that this is not the only way by which physics and number theory are related. It is merely another way, a new and not yet thoroughly explored one, by which the unity of mathematics manifests itself via its many different branches helping one another.

# Algebraic Spaces and Stacks

We introduced the concept of a moduli space in The Moduli Space of Elliptic Curves, and constructed explicitly the moduli space of elliptic curves, using the methods of complex analysis. In this post, we introduce the concepts of algebraic spaces and stacks, far-reaching generalizations of the concepts of varieties and schemes (see Varieties and Schemes Revisited), that are very useful, among other things, for constructing “moduli stacks“, which are an improvement over the naive notion of moduli space, namely in that one can obtain from it all “families of objects” by pulling back a “universal object”.

We need first the concept of a fibered category (also spelled fibred category). Given a category $\mathcal{C}$, we say that some other category $\mathcal{S}$ is a category over $\mathcal{C}$ if there is a functor $p$ from $\mathcal{S}$ to $\mathcal{C}$ (this should be reminiscent of our discussion in Grothendieck’s Relative Point of View).

If $\mathcal{S}$ is a category over some other category $\mathcal{C}$, we say that it is a fibered category (over $\mathcal{C}$) if for every object $U=p(x)$ and morphism $f: V\rightarrow U$ in $\mathcal{C}$, there is a strongly cartesian morphism $\phi: f^{*}x\rightarrow x$ in $\mathcal{S}$ with $f=p(\phi)$.

This means that any other morphism $\psi: z\rightarrow x$ whose image $p(\psi)$ under the functor $p$ factors as $p(\psi)=p(\phi)\circ h$ must also factor as $\psi=\phi\circ \theta$ under some unique morphism $\theta: z\rightarrow f^{*}x$ whose image under the functor $p$ is $h$. We refer to $f^{*}x$ as the pullback of $x$ along $f$.

Under the functor $p$, the objects of $\mathcal{S}$ which get sent to $U$ in $\mathcal{C}$ and the morphisms of $\mathcal{S}$ which get sent to the identity morphism $i_{U}$ in $\mathcal{C}$ form a subcategory of $\mathcal{S}$ called the fiber over $U$. We will also write it as $\mathcal{S}_{U}$.

An important example of a fibered category is given by an ordinary presheaf on a category $\mathcal{C}$, i.e. a functor $F:\mathcal{C}^{\text{op}}\rightarrow (\text{Set})$; we can consider it as a category fibered in sets $\mathcal{S}_{F}\rightarrow\mathcal{C}$.

A special kind of fibered category that we will need later on is a category fibered in groupoids. A groupoid is simply a category where all morphisms have inverses, and a category fibered in groupoids is a fibered category where all the fibers are groupoids. A set is a special kind of groupoid, since it may be thought of as a category whose only morphisms are the identity morphisms (which are trivially their own inverses). Hence, the example given in the previous paragraph, that of a presheaf, is also an example of a category fibered in groupoids, since it is fibered in sets.

Now that we have the concept of fibered categories, we next want to define prestacks and stacks. Central to the definition of prestacks and stacks is the concept known as descent, so we have to discuss it first. The theory of descent can be thought of as a formalization of the idea of “gluing”.

Let $\mathcal{U}=\{f_{i}:U_{i}\rightarrow U\}$ be a covering (see Sheaves and More Category Theory: The Grothendieck Topos) of the object $U$ of $\mathcal{C}$. An object with descent data is a collection of objects $X_{i}$ in $\mathcal{S}_{U}$ together with transition isomorphisms $\varphi_{ij}:\text{pr}_{0}^{*}X_{i}\simeq\text{pr}_{1}^{*}X_{j}$ in $\mathcal{S}_{U_{i}\times_{U}U_{j}}$, satisfying the cocycle condition

$\displaystyle \text{pr}_{02}^{*}\varphi_{ik}=\text{pr}_{01}^{*}\varphi_{ij}\circ \text{pr}_{12}^{*}\varphi_{jk}:\text{pr}_{0}^{*}X_{i}\rightarrow \text{pr}_{2}^{*}X_{k}$

The morphisms $\text{pr}_{0}:U_{i}\times_{U}U_{j}\rightarrow U_{i}$ and the $\text{pr}_{1}:U_{i}\times_{U}U_{j}\rightarrow U_{j}$ are the projection morphisms. The notations $\text{pr}_{0}^{*}X_{i}$ and $\text{pr}_{1}^{*}X_{j}$ means that we are “pulling back” $X_{i}$ and $X_{j}$ from $\mathcal{S}_{U_{i}}$ and $\mathcal{S}_{U_{j}}$, respectively, to $\mathcal{S}_{U_{i}\times_{U}U_{j}}$.

A morphism between two objects with descent data is a a collection of morphisms $\psi_{i}:X_{i}\rightarrow X'_{i}$ in $\mathcal{S}_{U_{i}}$ such that $\varphi'_{ij}\circ\text{pr}_{0}^{*}\psi_{i}=\text{pr}_{1}^{*}\psi_{j}\circ\varphi_{ij}$. Therefore we obtain a category, the category of objects with descent data, denoted $\mathcal{DD}(\mathcal{U})$.

We can define a functor $\mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U})$ by assigning to each object $X$ of $\mathcal{S}_{U}$ the object with descent data given by the pullback $f_{i}^{*}X$ and the canonical isomorphism $\text{pr}_{0}^{*}f_{i}^{*}X\rightarrow\text{pr}_{1}^{*}f_{j}^{*}X$. An object with descent data that is in the essential image of this functor is called effective.

Before we give the definitions of prestacks and stacks, we recall some definitions from category theory:

A functor $F:\mathcal{A}\rightarrow\mathcal{B}$ is faithful if the induced map $\text{Hom}_{\mathcal{A}}(x,y)\rightarrow \text{Hom}_{\mathcal{B}}(F(x),F(y))$ is injective for any two objects $x$ and $y$ of $\mathcal{A}$.

A functor $F:\mathcal{A}\rightarrow\mathcal{B}$ is full if the induced map $\text{Hom}_{\mathcal{A}}(x,y)\rightarrow \text{Hom}_{\mathcal{B}}(F(x),F(y))$ is surjective for any two objects $x$ and $y$ of $\mathcal{A}$.

A functor $F:\mathcal{A}\rightarrow\mathcal{B}$ is essentially surjective if any object $y$ of $\mathcal{B}$ is isomorphic to the image $F(x)$ of some object $x$ in $\mathcal{A}$ under $F$.

A functor which is both faithful and full is called fully faithful. If, in addition, it is also essentially surjective, then it is called an equivalence of categories.

Now we give the definitions of prestacks and stacks using the functor $\mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U})$ we have defined earlier.

If the functor $\mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U})$ is fully faithful, then the fibered category $\mathcal{S}\rightarrow\mathcal{C}$ is a prestack.

If the functor $\mathcal{S}_{U}\rightarrow\mathcal{DD}(\mathcal{U})$ is an equivalence of categories, then the fibered category $\mathcal{S}\rightarrow\mathcal{C}$ is a stack.

Going back to the example of a presheaf as a fibered category, we now look at what it means when it satisfies the conditions for being a prestack, or a stack:

(i) $F$ is a prestack if and only if it is a separated functor,

(ii) $F$ is stack if and only if it is a sheaf.

We now have the abstract idea of a stack in terms of category theory. Next we want to have more specific examples of interest in algebraic geometry, namely, algebraic spaces and algebraic stacks. For this we need first the idea of a representable functor (and the closely related idea of a representable presheaf). The importance of representability is that this will allow us to “transfer” interesting properties of morphisms between schemes such as being surjective, etale, or smooth, to functors between categories or natural transformations between functors. Therefore we will be able to say that a functor or natural transformation is surjective, or etale, or smooth, which is important, because we will define algebraic spaces and stacks as functors and categories, respectively, but we want them to still be closely related, or similar enough, to schemes.

A representable functor is a functor from $\mathcal{C}$ to $\textbf{Sets}$ which is naturally isomorphic to the functor which assigns to any object $X$ the set of morphisms $\text{Hom}(X,U)$, for some fixed object $U$ of $\mathcal{C}$.

A representable presheaf is a contravariant functor from $\mathcal{C}$ to $\textbf{Sets}$ which is naturally isomorphic to the functor which assigns to any object $X$ the set of morphisms $\text{Hom}(U,X)$, for some fixed object $U$ of $\mathcal{C}$. If $\mathcal{C}$ is the category of schemes, the latter functor is also called the functor of points of the object $U$.

We take this opportunity to emphasize a very important concept in modern algebraic geometry. The functor of points $h_{U}$ of a scheme $U$ may be identified with $U$ itself. There are many advantages to this point of view (which is also known as functorial algebraic geometry); in particular we will need it later when we give the definition of algebraic spaces and stacks.

We now have the idea of a representable functor. Next we want to have an idea of a representable natural transformation (or representable morphism) of functors. We will need another prerequisite, that of a fiber product of functors.

Let $F,G,H:\mathcal{C}^{\text{op}}\rightarrow \textbf{Sets}$ be functors, and let $a:F\rightarrow G$ and $b:H\rightarrow G$ be natural transformations between these functors. Then the fiber product $F\times_{a,G,b}H$ is a functor from $\mathcal{C}^{\text{op}}$ to $\textbf{Sets}$, and is given by the formula

$\displaystyle (F\times_{a,G,b}H)(X)=F(X)\times_{a_{X},G(X),b_{X}}H(X)$

for any object $X$ of $\mathcal{C}$.

Let $F,G:\mathcal{C}^{\text{op}}\rightarrow \textbf{Sets}$ be functors. We say that a natural transformation $a:F\rightarrow G$ is representable, or that $F$ is relatively representable over $G$ if for every $U\in\text{Ob}(\mathcal{C})$ and any $\xi\in G(U)$ the functor $h_{U}\times_{G}F$ is representable.

We now let $(\text{Sch}/S)_{\text{fppf}}$ be the site (a category with a Grothendieck topology –  see also More Category Theory: The Grothendieck Topos) whose underlying category is the category of $S$-schemes, and whose coverings are given by families of flat, locally finitely presented morphisms. Any etale covering or Zariski covering is an example of this “fppf covering” (“fppf” stands for fidelement plate de presentation finie, which is French for faithfully flat and finitely presented).

An algebraic space over a scheme $S$ is a presheaf

$\displaystyle F:((\text{Sch}/S)_{\text{fppf}})^{\text{op}}\rightarrow \textbf{Sets}$

with the following properties

(1) The presheaf $F$ is a sheaf.

(2) The diagonal morphism $F\rightarrow F\times F$ is representable.

(3) There exists a scheme $U\in\text{Ob}((\text{Sch}/S)_{\text{fppf}})$ and a map $h_{U}\rightarrow F$ which is surjective, and etale (This is often written simply as $U\rightarrow F$). The scheme $U$ is also called an atlas.

The diagonal morphism being representable implies that the natural transformation $h_{U}\rightarrow F$ is also representable, and this is what allows us to describe it as surjective and etale, as has been explained earlier.

An algebraic space is a generalization of the notion of a scheme. In fact, a scheme is simply the case where, for the third condition, we have $U$ is the disjoint union of affine schemes $U_{i}$ and where the map $h_{U}\rightarrow F$ is an open immersion. We recall that a scheme may be thought of as being made up of affine schemes “glued together”. This “gluing” is obtained using the Zariski topology. The notion of an algebraic space generalizes this to the etale topology.

Next we want to define algebraic stacks. Unlike algebraic spaces, which we defined as presheaves (functors), we will define algebraic stacks as categories, so we need to once again revisit the notion of representability in terms of categories.

Let $\mathcal{C}$ be a category. A category fibered in groupoids $p:\mathcal{S}\rightarrow\mathcal{C}$ is called representable if there exists an object $X$ of $\mathcal{C}$ and an equivalence $j:\mathcal{S}\rightarrow \mathcal{C}/X$ (The notation $\mathcal{C}/X$ signifies a slice category, whose objects are morphisms $f:U\rightarrow X$ in $\mathcal{C}$, and whose morphisms are morphisms $h:U\rightarrow V$ in $\mathcal{C}$ such that $f=g\circ h$, where $g:U\rightarrow X$).

We give two specific special cases of interest to us (although in this post we will only need the latter):

Let $\mathcal{X}$ be a category fibered in groupoids over $(\text{Sch}/S)_{\text{fppf}}$. Then $\mathcal{X}$ is representable by a scheme if there exists a scheme $U\in\text{Ob}((\text{Sch}/S)_{\text{fppf}})$ and an equivalence $j:\mathcal{X}\rightarrow (\text{Sch}/U)_{\text{fppf}}$ of categories over $(\text{Sch}/S)_{\text{fppf}}$.

A category fibered in groupoids $p : \mathcal{X}\rightarrow (\text{Sch}/S)_{\text{fppf}}$ is representable by an algebraic space over $S$ if there exists an algebraic space $F$ over $S$ and an equivalence $j:\mathcal{X}\rightarrow \mathcal{S}_{F}$ of categories over $(\text{Sch}/S)_{\text{fppf}}$.

Next, following what we did earlier for the case of algebraic spaces, we want to define the notion of representability (by algebraic spaces) for morphisms of categories fibered in groupoids (these are simply functors satisfying some compatibility conditions with the extra structure of the category). We will need, once again, the notion of a fiber product, this time of categories over some other fixed category.

Let $F:\mathcal{X}\rightarrow\mathcal{S}$ and $G:\mathcal{Y}\rightarrow\mathcal{S}$ be morphisms of categories over $\mathcal{C}$. The fiber product $\mathcal{X}\times_{\mathcal{S}}\mathcal{Y}$ is given by the following description:

(1) an object of $\mathcal{X}\times_{\mathcal{S}}\mathcal{Y}$ is a quadruple $(U,x,y,f)$, where $U\in\text{Ob}(\mathcal{C})$, $x\in\text{Ob}(\mathcal{X}_{U})$, $y\in\text{Ob}(\mathcal{Y}_{U})$, and $f : F(x)\rightarrow G(y)$ is an isomorphism in $\mathcal{S}_{U}$,

(2) a morphism $(U,x,y,f) \rightarrow (U',x',y',f')$ is given by a pair $(a,b)$, where $a:x\rightarrow x'$ is a morphism in $X$, and $b:y\rightarrow y'$ is a morphism in $Y$ such that $a$ and $b$ induce the same morphism $U\rightarrow U'$, and $f'\circ F(a)=G(b)\circ f$.

Let $S$ be a scheme. A morphism $f:\mathcal{X}\rightarrow \mathcal{Y}$ of categories fibered in groupoids over $(\text{Sch}/S)_{\text{fppf}}$ is called representable by algebraic spaces if for any $U\in\text{Ob}((\text{Sch}/S)_{\text{fppf}})$ and any $y:(\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{Y}$ the category fibered in groupoids

$\displaystyle (\text{Sch}/U)_{\text{fppf}}\times_{y,\mathcal{Y}}\mathcal{X}$

over $(\text{Sch}/U)_{\text{fppf}}$ is representable by an algebraic space over $U$.

An algebraic stack (or Artin stack) over a scheme $S$ is a category

$\displaystyle p:\mathcal{X}\rightarrow (\text{Sch}/S)_{\text{fppf}}$

with the following properties:

(1) The category $\mathcal{X}$ is a stack in groupoids over $(\text{Sch}/S)_{\text{fppf}}$ .

(2) The diagonal $\Delta:\mathcal{X}\rightarrow \mathcal{X}\times\mathcal{X}$ is representable by algebraic spaces.

(3) There exists a scheme $U\in\text{Ob}((\text{Sch/S})_{\text{fppf}})$ and a morphism $(\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{X}$ which is surjective and smooth (This is often written simply as $U\rightarrow\mathcal{X}$). Again, the scheme $U$ is called an atlas.

If the morphism $(\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{X}$ is surjective and etale, we have a Deligne-Mumford stack.

Just as an algebraic space is a generalization of the notion of a scheme, an algebraic stack is also a generalization of the notion of an algebraic space (recall that that a presheaf can be thought of as category fibered in sets, which themselves are special cases of groupoids). Therefore, the definition of an algebraic stack closely resembles the definition of an algebraic space given earlier, including the requirement that the diagonal morphism (which in this case is a functor between categories) be representable, so that the functor $(\text{Sch}/U)_{\text{fppf}}\rightarrow\mathcal{X}$ is also representable, and we can describe it as being surjective and smooth (or surjective and etale).

As an example of an application of the ideas just discussed, we mention the moduli stack of elliptic curves (which we denote by $\mathcal{M}_{1,1}$ – the reason for this notation will become clear later). A family of elliptic curves over some “base space” $B$ is a fibration $\pi:X\rightarrow B$ with a section $O:B\rightarrow X$ such that the fiber $\pi^{-1}(b)$ over any point $b$ of $B$ is an elliptic curve with origin $O(b)$.

Ideally what we want is to be able to obtain every family $X\rightarrow B$ by pulling back a “universal object” $E\rightarrow\mathcal{M}_{1,1}$ via the map $B\rightarrow\mathcal{M}_{1,1}$. This is something that even the notion of moduli space that we discussed in The Moduli Space of Elliptic Curves cannot do (we suggestively denote that moduli space by $M_{1,1}$). So we need the concept of stacks to construct this “moduli stack” that has this property. A more thorough discussion would need the notion of quotient stacks and orbifolds, but we only mention that the moduli stack of elliptic curves is in fact a Deligne-Mumford stack.

More generally, we can construct the moduli stack of curves of genus $g$ with $\nu$ marked points, denoted $\mathcal{M}_{g,\nu}$. The moduli stack of elliptic curves is simply the special case $\mathcal{M}_{1,1}$. Aside from just curves of course, we can construct moduli stacks for many more mathematical objects, such subschemes of some fixed scheme, or vector bundles, also on some fixed scheme.

The subject of algebraic stacks is a vast one, as may perhaps be inferred from the size of one of the main references for this post, the open-source reference The Stacks Project, which consists of almost 6,000 pages at the time of this writing. All that has been attempted in this post is but an extremely “bare bones” introduction to some of its more basic concepts. Hopefully more on stacks will be featured in future posts on the blog.

References:

Stack on Wikipedia

Algebraic Space on Wikipedia

Fibred Category on Wikipedia

Descent Theory on Wikipedia

Stack on nLab

Grothendieck Fibration on nLab

Algebraic Space on nLab

Algebraic Stack on nLab

Moduli Stack of Elliptic Curves on nLab

Stacks for Everybody by Barbara Fantechi

What is…a Stack? by Dan Edidin

Notes on the Construction of the Moduli Space of Curves by Dan Edidin

Notes on Grothendieck Topologies, Fibered Categories and Descent Theory by Angelo Vistoli

Lectures on Moduli Spaces of Elliptic Curves by Richard Hain

The Stacks Project

Algebraic Spaces and Stacks by Martin Olsson

Fundamental Algebraic Geometry: Grothendieck’s FGA Explained by Barbara Fantechi, Lothar Gottsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and Angelo Vistoli

# The Theory of Motives

The theory of motives originated from the observation, sometime in the 1960’s, that in algebraic geometry there were several different cohomology theories (see Homology and Cohomology and Cohomology in Algebraic Geometry), such as Betti cohomology, de Rham cohomology, $l$-adic cohomology, and crystalline cohomology. The search for a “universal cohomology theory”, such that all these other cohomology theories could be obtained from such a universal cohomology theory is what led to the theory of motives.

The four cohomology theories enumerated above are examples of what is called a Weil cohomology theory. A Weil cohomology theory, denoted $H^{*}$, is a functor (see Category Theory) from the category $\mathcal{V}(k)$ of smooth projective varieties over some field $k$ to the category $\textbf{GrAlg}(K)$ of graded $K$-algebras, for some other field $K$ which must be of characteristic zero, satisfying the following axioms:

(1) (Finite-dimensionality) The homogeneous components $H^{i}(X)$ of $H^{*}(X)$ are finite dimensional for all $i$, and $H^{i}(X)=0$ whenever $i<0$ or $i>2n$, where $n$ is the dimension of the smooth projective variety $X$.

(2) (Poincare duality) There is an orientation isomorphism $H^{2n}\cong K$, and a nondegenerate bilinear pairing $H^{i}(X)\times H^{2n-i}(X)\rightarrow H^{2n}\cong K$.

(3) (Kunneth formula) There is an isomorphism

$\displaystyle H^{*}(X\times Y)\cong H^{*}(X)\otimes H^{*}(Y)$.

(4) (Cycle map) There is a mapping $\gamma_{X}^{i}$ from $C^{i}(X)$, the abelian group of algebraic cycles of codimension $i$ on $X$ (see Algebraic Cycles and Intersection Theory), to $H^{i}(X)$, which is functorial with respect to pullbacks and pushforwards, has the multiplicative property $\gamma_{X\times Y}^{i+j}(Z\times W)=\gamma_{X}^{i}(Z)\otimes \gamma_{Y}^{j}(W)$, and such that $\gamma_{\text{pt}}^{i}$ is the inclusion $\mathbb{Z}\hookrightarrow K$.

(5) (Weak Lefschetz axiom) If $W$ is a smooth hyperplane section of $X$, and $j:W\rightarrow X$ is the inclusion, the induced map $j^{*}:H^{i}(X)\rightarrow H^{i}(W)$ is an isomorphism for $i\leq n-2$, and a monomorphism for $i\leq n-1$.

(6) (Hard Lefschetz axiom) The Lefschetz operator

$\displaystyle \mathcal{L}:H^{i}(X)\rightarrow H^{i+2}(X)$

given by

$\displaystyle \mathcal{L}(x)=x\cdot\gamma_{X}^{1}(W)$

for some smooth hyperplane section $W$ of $X$, with the product $\cdot$ provided by the graded $K$-algebra structure of $H^{*}(X)$, induces an isomorphism

$\displaystyle \mathcal{L}^{i}:H^{n-i}(X)\rightarrow H^{n+i}(X)$.

The idea behind the theory of motives is that all Weil cohomology theories should factor through a “category of motives”, i.e. any Weil cohomology theory

$\displaystyle H^{*}: \mathcal{V}(k)\rightarrow \textbf{GrAlg}(K)$

can be expressed as the following composition of functors:

$\displaystyle H^{*}: \mathcal{V}(k)\xrightarrow{h} \mathcal{M}(k)\rightarrow\textbf{GrAlg}(K)$

where $\mathcal{M}(k)$ is the category of motives. We can get different Weil cohomology theories, such as Betti cohomology, de Rham cohomology, $l$-adic cohomology, and crystalline cohomology, via different functors (called realization functors) from the category of motives to a category of graded algebras over some field $K$. This explains the term “motive”, which actually comes from the French word “motif”, which itself is already used in music and visual arts, among other things, as some kind of common underlying “theme” with different possible manifestations.

Let us now try to construct this category of motives. This category is often referred to in the literature as a “linearization” of the category of smooth projective varieties. This means that we obtain it from some sense starting with the category of smooth projective varieties, but we also want to modify it so that it we can do linear algebra, or more properly homological algebra, in some sense. In other words, we want it to behave like the category of modules over some ring. With this in mind, we want the category to be an abelian category, so that we can make sense of notions such as kernels, cokernels, and exact sequences.

An abelian category is a category that satisfies the following properties:

(1) The morphisms form an abelian group.

(2) There is a zero object.

(3) There are finite products and coproducts.

(4) Every morphism $f:X\rightarrow Y$ has a kernel and cokernel, and satisfies a decomposition

$\displaystyle K\xrightarrow{k} X\xrightarrow{i} I\xrightarrow{j} Y\xrightarrow{c} K'$

where $K$ is the kernel of $f$, $K'$ is the cokernel of $f$, and $I$ is the kernel of $c$ and the cokernel of $k$ (not to be confused with our notation for fields).

In order to proceed with our construction of the category of motives, which we now know we want to be an abelian category, we discuss the notion of correspondences.

The group of correspondences of degree $r$ from a smooth projective variety $X$ to another smooth projective variety $Y$, written $\text{Corr}^{r}(X,Y)$, is defined to be the group of algebraic cycles of $X\times Y$ of codimension $n+r$, where $n$ is the dimension of $X$, i.e.

$\text{Corr}^{r}(X,Y)=C^{n+r}(X\times Y)$

A morphism (of varieties, in the usual sense) $f:Y\rightarrow X$ determines a correspondence from $X$ to $Y$ of degree $0$ given by the transpose of the graph of $f$ in $X\times Y$. Therefore we may think of correspondences as generalizations of the usual concept of morphisms of varieties.

As we have learned in Algebraic Cycles and Intersection Theory, whenever we are dealing with algebraic cycles, it is often useful to consider them only up to some equivalence relation. In the aforementioned post we introduced the notion of rational equivalence. This time we consider also homological equivalence and numerical equivalence between algebraic cycles.

We say that two algebraic cycles $Z_{1}$ and $Z_{2}$ are homologically equivalent if they have the same image under the cycle map, and we say that they are numerically equivalent if the intersection numbers $Z_{1}\cdot Z$ and $Z_{2}\cdot Z$ are equal for all $Z$ of complementary dimension. There are other such equivalence relations on algebraic cycles, but in this post we will only mostly be using rational equivalence, homological equivalence, and numerical equivalence.

Since correspondences are algebraic cycles, we often consider them only up to these equivalence relations, and denote the quotient group we obtain by $\text{Corr}_{\sim}^{r}(X,Y)$, where $\sim$ is the equivalence relation imposed, for example, for numerical equivalence we write $\text{Corr}_{\text{num}}^{r}(X,Y)$.

Taking the tensor product of the abelian group $\text{Corr}_{\sim}^{r}(X,Y)$ with the rational numbers $\mathbb{Q}$, we obtain the vector space

$\displaystyle \text{Corr}_{\sim}^{r}(X,Y)_{\mathbb{Q}}=\text{Corr}_{\sim}^{r}(X,Y)\otimes_{\mathbb{Z}}\mathbb{Q}$

To obtain something closer to an abelian category (more precisely, we will obtain what is known as a pseudo-abelian category, but in the case where the equivalence relation is numerical equivalence, we will actually obtain an abelian category), we need to consider “projectors”, correspondences $p$ of degree $0$ from a variety $X$ to itself such that $p^{2}=p$. So now we form a category, whose objects are $h(X,p)$ for a variety $X$ and projector $p$, and whose morphisms are given by

$\displaystyle \text{Hom}(h(X,p),h(Y,q))=q\circ\text{Corr}_{\sim}^{0}(X,Y)_{\mathbb{Q}}\circ p$.

We call this category the category of pure effective motives, and denote it by $\mathcal{M}_{\sim}^{\text{eff}}(k)$. The process described above is also known as passing to the pseudo-abelian (or Karoubian) envelope.

We write $h^{i}(X,p)$ for the objects of $\mathcal{M}_{\sim}^{\text{eff}}(k)$ that map to $H^{i}(X)$. In the case that $X$ is the projective line $\mathbb{P}^{1}$, and $p$ is the diagonal $\Delta_{\mathbb{P}^{1}}$, we find that

$h(\mathbb{P}^{1},\Delta_{\mathbb{P}^{1}})=h^{0}\mathbb{P}^{1}\oplus h^{2}\mathbb{P}^{1}$

which can be rewritten also as

$\displaystyle h(\mathbb{P}^{1},\Delta_{\mathbb{P}^{1}})=\mathbb{I}\oplus\mathbb{L}$

where $\mathbb{I}$ is the image of a point in the category of pure effective motives, and $\mathbb{L}$ is known as the Lefschetz motive. It is also denoted by $\mathbb{Q}(-1)$. The above decomposition corresponds to the projective line $\mathbb{P}^{1}$ being a union of the affine line $\mathbb{A}^{1}$ and a “point at infinity”, which we may denote by $\mathbb{A}^{0}$:

$\displaystyle \mathbb{P}^{1}=\mathbb{A}^{0}\cup\mathbb{A}^{1}$

More generally, we have

$\displaystyle h(\mathbb{P}^{n},\Delta_{\mathbb{P}^{n}})=\mathbb{I}\oplus\mathbb{L}\oplus...\oplus\mathbb{L}^{n}$

corresponding to

$\displaystyle \mathbb{P}^{n}=\mathbb{A}^{0}\cup\mathbb{A}^{1}\cup...\cup\mathbb{A}^{n}$.

The category of effective pure motives is an example of a tensor category. This means it has a bifunctor $\otimes: \mathcal{M}_{\sim}^{\text{eff}}\times\mathcal{M}_{\sim}^{\text{eff}}\rightarrow\mathcal{M}_{\sim}^{\text{eff}}$ which generalizes the usual notion of a tensor product, and in this particular case it is given by taking the product of two varieties. We can ask for more, however, and construct a category of motives which is not just a tensor category but a rigid tensor category, which provides us with a notion of duals.

By formally inverting the Lefschetz motive (the formal inverse of the Lefschetz motive is then known as the Tate motive, and is denoted by $\mathbb{Q}(1)$), we can obtain this rigid tensor category, whose objects are triples $h(X,p,m)$, where $X$ is a variety, $e$ is a projector, and $m$ is an integer. The morphisms of this category are given by

$\displaystyle \text{Hom}(h(X,p,n),h(Y,q,m))=q\circ\text{Corr}_{\sim}^{n-m}(X,Y)_{\mathbb{Q}}\circ p$.

This category is called the category of pure motives, and is denoted by $\mathcal{M}_{\sim}(k)$. The category $\mathcal{M}_{\text{rat}}(k)$ is called the category of Chow motives, while the category $\mathcal{M}_{\text{num}}(k)$ is called the category of Grothendieck (or numerical) motives.

The category of Chow motives has the advantage that it is known to be “universal”, in the sense that every Weil cohomology theory factors through it, as discussed earlier; however, in general it is not even abelian, which is a desirable property we would like our category of motives to have. Meanwhile, the category of Grothendieck motives is known to be abelian, but it is not yet known if it is universal. If the so-called “standard conjectures on algebraic cycles“, which we will enumerate below, are proved, then the category of Grothendieck motives will be known to be universal.

We have seen that the category of pure motives forms a rigid tensor category. Closely related to this concept, and of interest to us, is the notion of a Tannakian category. More precisely, a Tannakian category is a $k$-linear rigid tensor category with an exact faithful functor (called a fiber functor) to the category of finite-dimensional vector spaces over some field extension $K$ of $k$.

One of the things that makes Tannakian categories interesting is that there is an equivalence of categories between a Tannakian category $\mathcal{C}$ and the category $\text{Rep}_{G}$ of finite-dimensional linear representations of the group of automorphisms of its fiber functor, which is also known as the Tannakian Galois group, or, if the Tannakian category is a “category of motives” of some sort, the motivic Galois group. This aspect of Tannakian categories may be thought of as a higher-dimensional analogue of the classical theory of Galois groups, which can be stated as an equivalence of categories between the category of finite separable field extensions of a field $k$ and the category of finite sets equipped with an action of the Galois group $\text{Gal}(\bar{k}/k)$, where $\bar{k}$ is the algebraic closure of $k$.

So we see that being a Tannakian category is yet another desirable property that we would like our category of motives to have. For this not only do we have to tweak the tensor product structure of our category, we also need certain conjectural properties to hold. These are the same conjectures we have hinted at earlier, called the standard conjectures on algebraic cycles, formulated by Alexander Grothendieck at around the same time he initially developed the theory of motives.

These conjectures have some very important consequences in algebraic geometry, and while they remain unproved to this day, the search for their proof (or disproof) is an important part of modern mathematical research on the theory of motives. They are the following:

(A) (Standard conjecture of Lefschetz type) For $i\leq n$, the operator $\Lambda$ defined by

$\displaystyle \Lambda=(\mathcal{L}^{n-i+2})^{-1}\circ\mathcal{L}\circ (\mathcal{L}^{n-i}):H^{i}\rightarrow H^{i-2}$

$\displaystyle \Lambda=(\mathcal{L}^{n-i})\circ\mathcal{L}\circ (\mathcal{L}^{n-i+2})^{-1}:H^{2n-i+2}\rightarrow H^{2n-i}$

is induced by algebraic cycles.

(B) (Standard conjecture of Hodge type) For all $i\leq n/2$, the pairing

$\displaystyle x,y\mapsto (-1)^{i}(\mathcal{L}x\cdot y)$

is positive definite.

(C) (Standard conjecture of Kunneth type) The projectors $H^{*}(X)\rightarrow H^{i}(X)$ are induced by algebraic cycles in $X\times X$ with rational coefficients. This implies the following decomposition of the diagonal:

$\displaystyle \Delta_{X}=\pi_{0}+...+\pi_{2n}$

which in turn implies the decomposition

$\displaystyle h(X,\Delta_{X},0)=h(X,\pi_{0},0)\oplus...\oplus h(X,\pi_{2n},0)$

which, writing $h(X,\Delta_{X},0)$ as $hX$ and $h(X,\pi_{i},0)$ as $h^{i}(X)$, we can also compactly and suggestively write as

$\displaystyle hX=h^{0}X\oplus...\oplus h^{2n}X$.

In other words, every object $hX=h(X,\Delta_{X},0)$ of our “category of motives” decomposes into graded “pieces” $h^{i}(X)=h(X,\pi_{i},0)$ of pure “weight$i$. We have already seen earlier that this is indeed the case when $X=\mathbb{P}^{n}$. We will need this conjecture to hold if we want our category to be a Tannakian category.

(D) (Standard conjecture on numerical equivalence and homological equivalence) If an algebraic cycle is numerically equivalent to zero, then its cohomology class is zero. If the category of Grothendieck motives is to be “universal”, so that every Weil cohomology theory factors through it, this conjecture must be satisfied.

In Algebraic Cycles and Intersection Theory and Some Useful Links on the Hodge Conjecture, Kahler Manifolds, and Complex Algebraic Geometry, we have made mention of the two famous conjectures in algebraic geometry known as the Hodge conjecture and the Tate conjecture. In fact, these two closely related conjectures can be phrased in the language of motives as the conjectures stating that the realization functors from the category of motives to the category of pure Hodge structures and continuous $l$-adic representations of $\text{Gal}(\bar{k}/k)$, respectively, be fully faithful. These conjectures are closely related to the standard conjectures on algebraic cycles as well.

We have now constructed the category of pure motives, for smooth projective varieties. For more general varieties and schemes, there is an analogous idea of “mixed motives“, which at the moment remain conjectural, although there exist several related constructions which are the closest thing we currently have to such a theory of mixed motives.

If we want to construct a theory of mixed motives, instead of Weil cohomology theories we must instead consider what are known as “mixed Weil cohomology theories“, which are expected to have the following properties:

(1) (Homotopy invariance) The projection $\pi:X\rightarrow\mathbb{A}^{1}$ induces an isomorphism

$\displaystyle \pi^{*}:H^{*}(X)\xrightarrow{\cong}H^{*}(X\times\mathbb{A}^{1})$

(2) (Mayer-Vietoris sequence) If $U$ and $V$ are open coverings of $X$, then there is a long exact sequence

$\displaystyle ...\rightarrow H^{i}(U\cap V)\rightarrow H^{i}(X)\rightarrow H^{i}(U)\oplus H^{i}(V)\rightarrow H^{i}(U\cap V)\rightarrow...$

(3) (Duality) There is a duality between cohomology $H^{*}$ and cohomology with compact support $H_{c}^{*}$.

(4) (Kunneth formula) This is the same axiom as the one in the case of pure motives.

We would like a category of mixed motives, which serves as an analogue to the category of pure motives in that all mixed Weil cohomology theories factor through it, but as mentioned earlier, no such category exists at the moment. However, the mathematicians Annette Huber-Klawitter, Masaki Hanamura, Marc Levine, and Vladimir Voevodsky have constructed different versions of a triangulated category of mixed motives, denoted $\mathcal{DM}(k)$.

A triangulated category $\mathcal{T}$ is an additive category with an automorphism $T: \mathcal{T}\rightarrow\mathcal{T}$ called the “shift functor” (we will also denote $T(X)$ by $X[1]$, and $T^{n}(X)$ by $X[n]$, for $n\in\mathbb{Z}$) and a family of “distinguished triangles

$\displaystyle X\rightarrow Y\rightarrow Z\rightarrow X[1]$

which satisfies the following axioms:

(1) For any object $X$ of $\mathcal{T}$, the triangle $X\xrightarrow{\text{id}}X\rightarrow 0\rightarrow X[1]$ is a distinguished triangle.

(2) For any morphism $u:X\rightarrow Y$ of $\mathcal{T}$, there is an object $Z$ of $\mathcal{T}$ such that $X\xrightarrow{u}Y\rightarrow Z\rightarrow X[1]$ is a distinguished triangle.

(3) Any triangle isomorphic to a distinguished triangle is a distinguished triangle.

(4) If $X\rightarrow Y\rightarrow Z\rightarrow X[1]$ is a distinguished triangle, then the two “rotations” $Y\rightarrow Z\rightarrow Z[1]\rightarrow Y[1]$ and $Z[-1]\rightarrow X\rightarrow Y\rightarrow Z$ are also distinguished triangles.

(5) Given two distinguished triangles $X\xrightarrow{u}Y\xrightarrow{v}Z\xrightarrow{w}X[1]$ and $X'\xrightarrow{u'}Y'\xrightarrow{v'}Z'\xrightarrow{w'}X'[1]$ and morphisms $f:X\rightarrow X'$ an $g:Y\rightarrow Y'$ such that the square “commutes”, i.e. $u'\circ f=g\circ u$, there exists a morphisms $h:Z\rightarrow Z$ such that all other squares commute.

(6) Given three distinguished triangles $X\xrightarrow{u}Y\xrightarrow{j}Z'\xrightarrow{k}X[1]$$Y\xrightarrow{v}Z\xrightarrow{l}X'\xrightarrow{i}Y[1]$, and $X\xrightarrow{v\circ u}Z\xrightarrow{m}Y'\xrightarrow{n}X[1]$, there exists a distinguished triangle $Z'\xrightarrow{f}Y'\xrightarrow{g}X'\xrightarrow{h}Z'[1]$ such that “everything commutes”.

A $t$-structure on a triangulated category $\mathcal{T}$ is made up of two full subcategories $\mathcal{T}^{\geq 0}$ and $\mathcal{T}^{\leq 0}$ satisfying the following properties (writing $\mathcal{T}^{\leq n}$ and $\mathcal{T}^{\leq n}$ to denote $\mathcal{T}^{\leq 0}[-n]$ and $\mathcal{T}^{\geq 0}[-n]$ respectively):

(1) $\mathcal{T}^{\leq -1}\subset \mathcal{T}^{\leq 0}$ and $\mathcal{T}^{\geq 1}\subset \mathcal{T}^{\geq 0}$

(2) $\displaystyle \text{Hom}(X,Y)=0$ for any object $X$ of $\mathcal{T}^{\leq 0}$ and any object $Y$ of $\mathcal{T}^{\geq 1}$

(3) for any object $Y$ of $\mathcal{T}$ we have a distinguished triangle

$\displaystyle X\rightarrow Y\rightarrow Z\rightarrow X[1]$

where $X$ is an object of $\mathcal{T}^{\leq 0}$ and $Z$ is an object of $\mathcal{T}^{\geq 1}$.

The full subcategory $\mathcal{T}^{0}=\mathcal{T}^{\leq 0}\cap\mathcal{T}^{\geq 0}$ is called the heart of the $t$-structure, and it is an abelian category.

It is conjectured that the category of mixed motives $\mathcal{MM}(k)$ is the heart of the $t$-structure of the triangulated category of mixed motives $\mathcal{DM}(k)$.

Voevodsky’s construction proceeds in a manner somewhat analogous to the construction of the category of pure motives as above, starting with schemes (say, over a field $k$, although a more general scheme may be used) as objects and correspondences as morphisms, but then makes use of concepts from abstract homotopy theory, such as taking the bounded homotopy category of bounded complexes, and localization with respect to a certain subcategory, before passing to the pseudo-abelian envelope and then formally inverting the Tate object $\mathbb{Z}(1)$. The triangulated category obtained is called the category of geometric motives, and is denoted by $\mathcal{DM}_{\text{gm}}(k)$. The schemes and correspondences involved in the construction of $\mathcal{DM}_{\text{gm}}(k)$ are required to satisfy certain properties which eliminates the need to consider the equivalence relations which form a large part of the study of the category of pure motives.

Closely related to the triangulated category of mixed motives is motivic cohomology, which is defined in terms of the former as

$\displaystyle H^{i}(X,\mathbb{Z}(m))=\text{Hom}_{\mathcal{DM}(k)}(X,\mathbb{Z}(m)[i])$

where $\mathbb{Z}(m)$ is the tensor product of $m$ copies of the Tate object $\mathbb{Z}(1)$, and the notation $\mathbb{Z}(m)[i]$ tells us that the shift functor of the triangulated category is applied to the object $\mathbb{Z}(m)$ $i$ times.

Motivic cohomology is related to the Chow group, which we have introduced in Algebraic Cycles and Intersection Theory, and also to algebraic K-theory, which is another way by which the ideas of homotopy theory are applied to more general areas of abstract algebra and linear algebra. These ideas were used by Voevodsky to prove several related theorems, from the Milnor conjecture to its generalization, the Bloch-Kato conjecture (also known as the norm residue isomorphism theorem).

Historically, one of the motivations for Grothendieck’s attempt to obtain a universal cohomology theory was to prove the Weil conjectures, which is a higher-dimensional analogue of the Riemann hypothesis for curves over finite fields first proved by Andre Weil himself (see The Riemann Hypothesis for Curves over Finite Fields). In fact, if the standard conjectures on algebraic cycles are proved, then a proof of the Weil conjectures would follow via an approach that closely mirrors Weil’s original proof (since cohomology provides a Lefschetz fixed-point formula –  we have mentioned in The Riemann Hypothesis for Curves over Finite Fields that the study of fixed points is an important part of Weil’s proof). The last of the Weil conjectures were eventually proved by Grothendieck’s student Pierre Deligne, but via a different approach that bypassed the standard conjectures. A proof of the standard conjectures, which would lead to a perhaps more elegant proof of the Weil conjectures, is still being pursued to this day.

The theory of motives is not only related to analogues of the Riemann hypothesis, which concerns the location of zeroes of L-functions, but to L-functions in general. For instance, it is also related to the Langlands program, which concerns another aspect of L-functions, namely their analytic continuation and functional equation, and to the Birch and Swinnerton-Dyer conjecture, which concerns their values at special points.

We recall in The Riemann Hypothesis for Curves over Finite Fields that the Frobenius morphism played an important part in counting the points of a curve over a finite field, which in turn we needed to define the zeta function (of which the L-function can be thought of as a generalization) of the curve. The Frobenius morphism is an element of the Galois group, and we recall that a category of motives which is a Tannakian category is equivalent to the category of representations of its motivic Galois group. Therefore we can see how we can define “motivic L-functions” using the theory of motives.

As the L-functions occupy a central place in many areas of modern mathematics, the theory of motives promises much to be gained from its study, if only we could make progress in deciphering the many mysteries that surround it, of which we have only scratched the surface in this post. The applications of motives are not limited to L-functions either – the study of periods, which relate Betti cohomology and de Rham cohomology, and lead to transcendental numbers which can be defined using only algebraic concepts, is also strongly connected to the theory of motives. Recent work by the mathematicians Alain Connes and Matilde Marcolli has also suggested applications to physics, particularly in relation to Feynman diagrams in quantum field theory. There is also another generalization of the theory of motives, developed by Maxim Kontsevich, in the context of noncommutative geometry.

References:

Weil Cohomology Theory on Wikipedia

Motive on Wikipedia

Standard Conjectures on Algebraic Cycles on Wikipedia

Motive on nLab

Pure Motive on nLab

Mixed Motive on nLab

The Tate Conjecture over Finite Fields on Hard Arithmetic

What is…a Motive? by Barry Mazur

Motives – Grothendieck’s Dream by James S. Milne

Noncommutative Geometry, Quantum Fields, and Motives by Alain Connes and Matilde Marcolli

Algebraic Cycles and the Weil Conjectures by Steven L. Kleiman

The Standard Conjectures by Steven L. Kleiman

Feynman Motives by Matilde Marcolli

Une Introduction aux Motifs (Motifs Purs, Motifs Mixtes, Periodes) by Yves Andre