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

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Chern Classes and Generalized Riemann-Roch Theorems

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

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

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

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

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

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

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

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

We can also define the total Chern class

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

and the Chern polynomial

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

Chern classes satisfy the following important properties:

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

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

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

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

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

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

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

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

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

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

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

and the Todd class as

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

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

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

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

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

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

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

The Hirzebruch-Riemann-Roch theorem says that

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

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

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

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

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

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

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

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

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

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

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

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

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

References:

Chern Class on Wikipedia

Projective Bundle on Wikipedia

Hirzebruch-Riemann-Roch Theorem on Wikipedia

Grothendieck-Riemann-Roch Theorem on Wikipedia

Chern Classes: Part 1 on Rigorous Trivialities

Chern Classes: Part 2 on Rigorous Trivialities

The Chow Ring and Chern Classes on Rigorous Trivialities

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

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

Algebraic Geometry by Andreas Gathmann

Algebraic Geometry by Robin Hartshorne

 

Functions of Complex Numbers

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\displaystyle f'(0)=1

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

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

\displaystyle f'(0)=-1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

References:

Complex Analysis on Wikipedia

Analytic Function on Wikipedia

Holomorphic Function on Wikipedia

Cauchy-Riemann Equations on Wikipedia

Cauchy’s Integral Theorem on Wikipedia

Cauchy’s Integral Formula on Wikipedia

Residue Theorem on Wikipedia

Complex Analysis by Lars Ahlfors

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

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...<n\} 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}

form an adjoint triple.

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 Basics of Class Field Theory

Class field theory is one of the crown jewels of modern algebraic number theory. It can be viewed as a generalization of the “reciprocity laws” discovered by Carl Friedrich Gauss and other number theorists of the 19th century. However, as we have not yet discussed reciprocity laws in this blog, we will leave that point of view to the references for now. Instead, in order to describe class field theory, we rely on the following quote from the mathematician Claude Chevalley:

“The object of class field theory is to show how the abelian extensions of an algebraic number field K can be determined by objects drawn from our knowledge of K itself; or, if one prefers to present things in dialectic terms, how a field contains within itself the elements of its own transcending.”

Our approach in this post, as with much of the modern literature, will start from the “local” point of view, and then we will put together all these local pieces in order to have a “global” theory. Recall that there is an analogy between geometry and number theory, and that primes play the role of points in this analogy. Therefore, as in geometry “local” means “zooming in” on a point, in number theory “local” means “zooming in” on a prime. Putting these pieces together is akin to what we have done in Adeles and Ideles. This will become more clear when we define local fields and global fields.

Let K be a (nonarchimedean) local field. This means that K is complete with respect to a discrete valuation (see Valuations and Completions and Adeles and Ideles) and that its residue field is finite. These are either the fields \mathbb{Q}_{p} (the p-adic numbers), \mathbb{F}_{p}((t)) (the field of formal power series over a finite field \mathbb{F}_{p}), or their finite extensions. Let L be a finite extension of K.

We define the norm homomorphism as

\displaystyle N_{L|K}(x)=\prod_{\sigma}\sigma x

for x\in L and \sigma\in \text{Gal}(L|K) (note that there are many notations for the action of \sigma on x; in the book Algebraic Number Theory by Jurgen Neukirch, the notation x^{\sigma} is used instead). We let N_{L|K}L^{\times} stand for the image of the norm homomorphism in K. Then local class field theory tells us that we have the following isomorphism:

\displaystyle K^{\times}/N_{L|K}L^{\times}\xrightarrow{\sim}\text{Gal}(L|K)^{\text{ab}}.

We see that everything in the left-hand side belongs to the field K. This is related to Chevalley’s quote earlier. However, there is even more to this, as we shall see later.

Understanding more about this isomorphism requires the theory of Galois cohomology (see Etale Cohomology of Fields and Galois Cohomology). Namely, the Galois cohomology group H^{2}(\text{Gal}(L|K),L^{\times}) is isomorphic as a group to the group homomorphisms from \text{Gal}(L|K)^{\text{ab}} to K^{\times}/N_{L|K}L^{\times}. It is cyclic of degree equal to the degree of L over K.

There is an injective map from H^{2}(\text{Gal}(L|K),L^{\times}) to the quotient \mathbb{Q}/\mathbb{Z}, and the element of H^{2}(\text{Gal}(L|K),L^{\times}) that gets mapped to 1/n, where n is the degree of L over K, is precisely the element that corresponds to the inverse of the isomorphism K^{\times}/N_{L|K}L^{\times}\xrightarrow{\sim}\text{Gal}(L|K)^{\text{ab}}.

Now let K be a global field, which means that it is a finite extension either of \mathbb{Q} (the rational numbers) or of \mathbb{F}_{p}(t) (the function field over a finite field \mathbb{F}_{p}). Let L be a finite extension of K. Let C_{K} and C_{L} denote the idele class groups (see Adeles and Ideles) of K and L respectively. As in the local case, we will need a norm homomorphism, but this time it will be for idele class groups.

We will define this norm homomorphism “componentwise”. Writing an idele as (z_{w}), we take the norm N_{L_{w}|K_{v}}(z_{w}), and take the product for all primes w above v. We do this for every prime v of K, and thus we obtain an element of the group of ideles of K, and then we take the quotient to obtain an element of the idele class group of K. We denote by N_{L|K}C_{L} the image of this norm homomorphism in C_{K}.

Then global class field theory tells us that we have the following isomorphism:

\displaystyle C_{K}/N_{L|K}C_{L}\sim\text{Gal}(L|K)^{\text{ab}}

Again, as in the local case, everything in the left-hand side belongs to C_{K}.

As we have said earlier, we can obtain this isomorphism by putting together the local pieces from local class field theory, i.e. homomorphisms from K_{v}^{\times} to \text{Gal}(L_{w}|K_{v})^{\text{ab}} which come from the isomorphisms from K_{v}^{\times}/N_{L|K}L_{w}^{\times}, as ideles have components which are local fields, and then taking the quotient to obtain the isomorphism for idele class groups, similar to what we have done for the norm homomorphism.

However, in order to obtain the desired isomorphism, the map (called the Artin map)

\psi:I_{K}^{\times}\rightarrow\text{Gal}(L|K)^{\text{ab}}

from the group of ideles I_{K} of K to the group \text{Gal}(L|K)^{\text{ab}}, which is obtained from putting together the local pieces (before taking the quotients) must satisfy three properties:

(i) It has to be continuous with respect to the topologies on I_{K} and \text{Gal}(L|K) (the topology on the group of ideles is discussed in Adeles and Ideles, while the topology on \text{Gal}(L|K) is the so-called Krull topology – the latter is part of the theory of profinite groups).

(ii) The image of K^{\times} (as embedded in its group of ideles I_{K}) is equal to the identity.

(iii) It is equal to the Frobenius morphism for elements in I_{K}^{S} (see again Adeles and Ideles for the explanation of this notation), where S consists of the archimedean primes and those primes which are ramified in L (see Splitting of Primes in Extensions).

It is a challenging task in itself to prove that the Artin map does indeed satisfy these properties, and for now we leave it to the references. Instead, we mention a few more properties of class field theory. In the local case, class field theory also classifies the subgroups of K^{\times} which are of the form N_{L|K}L^{\times}, which correspond to the open subgroups of finite index in K^{\times}. Since the finite abelian extension L of K also obviously corresponds to the subgroup N_{L|K}L^{\times}, we then obtain a classification of the finite abelian extensions of K. Similarly, in the global case, class field theory classifies the subgroups of C_{K} which are of the form N_{L|K}C_{L}, which correspond to the open subgroups of finite index in C_{K}. The field which corresponds to the such a subgroup is called its class field. In the case that L is the maximal unramified abelian extension of K, L is called the Hilbert class field of K, and there we have the result that the ideal class group (see Algebraic Numbers) of K is isomorphic to the Galois group \text{Gal}(L|K). With the ideas discussed in this last paragraph, the goal of class field theory as expressed in the quote of Chevalley is fulfilled; we are able to describe the abelian extensions of K from knowledge only of K itself.

References:

Class Field Theory on Wikipedia

Artin Reciprocity Law on Wikipedia

Profinite Group on Wikipedia

Class Field Theory by J.S. Milne

Algebraic Number Theory by Jurgen Neukirch

Algebraic Number Theory by J. W. S. Cassels and A. Frohlich

A Panorama of Pure Mathematics by Jean Dieudonne

Primes of the Form x^{2}+ny^{2} by David A. Cox

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“:

250px-TrefoilKnot_01.svg

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.

Splitting of Primes in Extensions

In Algebraic Numbers we discussed how ideals factorize in an algebraic number field (recall that we had to look at factorization of ideals since the elements in the ring of integers of more general algebraic number fields may no longer factorize uniquely). In this post, we develop some more terminology related to this theory, and we also discuss how in the case of a so-called “Galois extension” the Galois group (see Galois Groups) may express information related to the factorization of ideals in an algebraic number field.

Let \mathfrak{p} be a prime ideal of the ring of integers \mathcal{O}_{K} of an algebraic number field K (we will sometimes also refer to \mathfrak{p} as a prime ideal of K – this is common practice and hopefully will not cause any confusion). In an algebraic number field L which contains K (we also say that L is an extension of K, and write L|K), this prime ideal \mathfrak{p} decomposes into a product of prime ideals \mathfrak{P}_{1},\mathfrak{P}_{2}...\mathfrak{P}_{r} in \mathcal{O}_L, with respective exponents e_{1},e_{2}...e_{r}, i.e.

\displaystyle \mathfrak{p}=\mathfrak{P}_{1}^{e_{1}}\mathfrak{P}_{2}^{e_{2}}...\mathfrak{P}_{n}^{e_{r}}.

The exponents e_{1},e_{2}...e_{r} are called the ramification indices of the prime ideals \mathfrak{P}_{1},\mathfrak{P}_{2},...\mathfrak{P}_{r}. If e_{i}=1, and the residue field extension \kappa(\mathfrak{P}_{i})|\kappa(\mathfrak{p}) (see below) is separable, we say that \mathfrak{P}_{i} is unramified over K. If e_{1}=e_{2}=...e_{r}=1, we say that the prime \mathfrak{p} is unramified. If all primes of K are unramified in L, we say that the extension L|K is unramified.

In the rest of this post we will continue to assume the factorization of \mathfrak{p} as shown above. The residue fields \kappa(\mathfrak{P}_{i}) and \kappa(\mathfrak{p}) of \mathcal{O}_{L} and \mathcal{O}_{K} at the primes \mathfrak{P}_{i} and \mathfrak{p} are defined as the quotients \mathcal{O}_{L}/\mathfrak{P}_{i} and \mathcal{O}_{K}/\mathfrak{p}, and the inertia degrees f_{i} are defined as the degrees of the fields \kappa(\mathfrak{P}_{i}) with respect to the field \kappa(\mathfrak{p}) (i.e. the dimensions of the vector spaces \kappa(\mathfrak{P}_{i}) over the field of scalars \kappa(\mathfrak{p})), i.e.

\displaystyle f_{i}=[\kappa(\mathfrak{P}_{i}):\kappa(\mathfrak{p})].

The ramification indices e_{i}, the inertia degrees f_{i}, and the degree n=[L:K] of the field extension L with respect to K are related by the following “fundamental identity“:

\displaystyle \sum_{i=1}^{r}e_{i}f_{i}=n

In order to understand these concepts better, we can look at the following “extreme” cases:

If e_{i}=1 and f_{i}=1 for all i, then r=n, and we say that the prime \mathfrak{p} splits completely in L.

If r=1 and f_{1}=1, then e_{1}=n, and we say that the prime \mathfrak{p} ramifies completely in L.

If r=1 and e_{1}=1, then f_{1}=n, and we say that the prime \mathfrak{p}  is inert in L.

Consider for example, the field \mathbb{Q}(i) as a field extension of the field \mathbb{Q}. The ring of integers of \mathbb{Q}(i) is the ring of Gaussian integers \mathbb{Z}[i] (see The Fundamental Theorem of Arithmetic and Unique Factorization), while the ring of integers of \mathbb{Q} is the ring of ordinary integers \mathbb{Z}. The degree [\mathbb{Q}(i):\mathbb{Q}] is equal to 2. The prime ideal (5) of \mathbb{Z} splits completely as the product (2+i)(2-i) in \mathbb{Z}[i], the prime ideal (2) of \mathbb{Q} ramifies completely as (1+i)^{2} in \mathbb{Z}[i], while the prime ideal (3) of \mathbb{Z} is inert in \mathbb{Z}[i].

We now bring in Galois groups. We assume that L is a Galois extension of K. This means that the order of G(L|K), the Galois group of L over K, is equal to the degree of L over K. In this case, it turns out that we will have

\displaystyle e_{1}=e_{2}=...=e_{r}

and

\displaystyle f_{1}=f_{2}=...=f_{r}.

The fundamental identity then becomes

efr=n.

This is but the first of many simplifications we obtain whenever we are dealing with Galois extensions.

Given a prime ideal \mathfrak{P} of \mathcal{O}_{K}, we define the decomposition group G_{\mathfrak{P}} as the subgroup of the Galois group G that fixes \mathfrak{P}, i.e.

\displaystyle G_{\mathfrak{P}}=\{\sigma\in G|\sigma\mathfrak{P=\mathfrak{P}}\}.

The elements of L that are fixed by the decomposition group G_{\mathfrak{P}} form what is called the decomposition field of K over \mathfrak{P}, denoted Z_{\mathfrak{P}}:

 \displaystyle Z_{\mathfrak{P}}=\{x\in L|\sigma x=x,\forall\sigma\in G_{\mathfrak{P}}\}

Every element \sigma of G_{\mathfrak{P}} automorphism \bar{\sigma} of \kappa(\mathfrak{P}) sending the element given by a\text{ mod }\mathfrak{P} to the element given by \sigma a\text{ mod }\mathfrak{P}. The residue field of the decomposition field Z_{\mathfrak{P}} with respect to \mathfrak{p} is the same as the residue field of the field K with respect to \mathfrak{p}, which is \kappa(\mathfrak{p}). Therefore we have a surjective homomorphism

\displaystyle G_{\mathfrak{P}}\rightarrow G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p}))

which sends the element \sigma of G_{\mathfrak{P}} to the element \bar{\sigma} of G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p})). The kernel of this homorphism is called the inertia group of \mathfrak{P} over K. Once again, the elements of L fixed by the inertia group I_{\mathfrak{P}} form what we call the inertia field of K over \mathfrak{P}, denoted T_{\mathfrak{P}}:

 \displaystyle T_{\mathfrak{P}}=\{x\in K|\sigma x=x,\forall\sigma\in I_{\mathfrak{P}}\}

The groups G_{\mathfrak{P}}, I_{\mathfrak{P}}, G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p})) are related by the following exact sequence:

\displaystyle 0\rightarrow I_{\mathfrak{P}}\rightarrow G_{\mathfrak{P}}\rightarrow G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p}))\rightarrow 0

Meanwhile, the relationship between the fields K, Z_{\mathfrak{P}}, T_{\mathfrak{P}}, and L can be summarized as follows:

\displaystyle K\subseteq Z_{\mathfrak{P}}\subseteq T_{\mathfrak{P}}\subseteq L

The ramification index, inertia degree, and the number of primes in K into which a prime \mathfrak{p} in L splits are given in terms of the degrees of the aforementioned fields as follows:

\displaystyle e=[L:T_{\mathfrak{P}}]

\displaystyle f=[T_{\mathfrak{P}}:Z_{\mathfrak{P}}]

\displaystyle r=[Z_{\mathfrak{P}}:K]

Let \mathfrak{P}_{Z}=\mathfrak{P}\cap Z_{\mathfrak{P}}, and \mathfrak{P}_{T}=\mathfrak{P}\cap T_{\mathfrak{P}}. We also refer to \mathfrak{P}_{Z} (resp. \mathfrak{P}_{T}) as the prime ideal of Z_{\mathfrak{P}} (resp. T_{\mathfrak{P}}) below \mathfrak{P}.

The ramification index of \mathfrak{P} over \mathfrak{P}_{T} is equal to e, and its inertia degree is equal to 1. Meanwhile, the ramification index of \mathfrak{P}_{T} over \mathfrak{P}_{Z} is equal to 1, and its inertia degree is equal to e. Finally, the ramification index and inertia degree of \mathfrak{P}_{Z} over \mathfrak{p} are both equal to 1.

We can therefore see that in the case of a Galois extension, the theory of the splitting of primes becomes simple and elegant. Before we end this post, there is one more concept that we will define. Let \mathfrak{P} be a prime that is unramified over K. Then G_{\mathfrak{P}} is isomorphic to G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p})), it is cyclic, and it is generated by the unique automorphism

\displaystyle \varphi_{\mathfrak{P}}\equiv a^{q}\text{ mod }\mathfrak{P}    for all    \displaystyle a\in \mathcal{O}_{K}

where q=[\kappa(\mathfrak{P}):\kappa(\mathfrak{p})]. The automorphism \varphi_{\mathfrak{P}} is called the Frobenius automorphism, and it is a very important concept that shows up in many aspects of algebraic number theory.

References:

Splitting of Prime Ideals in Galois Extensions on Wikipedia

A Classical Introduction to Modern Number Theory by Kenneth Ireland and Michael Rosen

Number Fields by Daniel Marcus

Algebraic Theory of Numbers by Pierre Samuel

Algebraic Number Theory by Jurgen Neukirch

SEAMS School Manila 2017: Topics on Elliptic Curves

A few days ago, from July 17 to 25, I attended the SEAMS (Southeast Asian Mathematical Society) School held at the Institute of Mathematics, University of the Philippines Diliman, discussing topics on elliptic curves. The school was also partially supported by CIMPA (Centre International de Mathematiques Pures et Appliquees, or International Center for Pure and Applied Mathematics), and I believe also by the Roman Number Theory Association and the Number Theory Foundation. Here’s the official website for the event:

Southeast Asian Mathematical Society (SEAMS) School Manila 2017: Topics on Elliptic Curves

There were many participants from countries all over Southeast Asia, including Indonesia, Malaysia, Philippines, and Vietnam, as well as one participant from Austria and another from India. The lecturers came from Canada, France, Italy, and Philippines.

Jerome Dimabayao and Michel Waldschmidt started off the school, introducing the algebraic and analytic aspects of elliptic curves, respectively. We have tackled these subjects in this blog, in Elliptic Curves and The Moduli Space of Elliptic Curves, but the school discussed them in much more detail; for instance, we got a glimpse of how Karl Weierstrass might have come up with the function named after him, which relates the equation defining an elliptic curve to a lattice in the complex plane. This requires some complex analysis, which unfortunately we have not discussed that much in this blog yet.

Francesco Pappalardi then discussed some important theorems regarding rational points on elliptic curves, such as the Nagell-Lutz theorem and the famous Mordell-Weil theorem. Then, Julius Basilla discussed the counting of points of elliptic curves over finite fields, often making use of the Hasse-Weil inequality which we have discussed inThe Riemann Hypothesis for Curves over Finite Fields, and the applications of this theory to cryptography. Claude Levesque then introduced to us the fascinating theory of quadratic forms, which can be used to calculate the class number of a quadratic number field (see Algebraic Numbers), and the relation of this theory to elliptic curves.

Richell Celeste discussed the reduction of elliptic curves modulo primes, a subject which we have also discussed here in the post Reduction of Elliptic Curves Modulo Primes, and two famous problems related to elliptic curves, Fermat’s Last Theorem, which was solved by Andrew Wiles in 1995, and the still unsolved Birch and Swinnerton-Dyer conjecture regarding the rank of the group of rational points of elliptic curves. Fidel Nemenzo then discussed the classical problem of finding “congruent numbers“, rational numbers forming the sides of a right triangle whose area is given by an integer, and the rather surprising connection of this problem to elliptic curves.

On the last day of the school, Jerome Dimabayao discussed the fascinating connection between elliptic curves and Galois representations, which we have given a passing mention to at the end of the post Elliptic Curves. Finally, Jared Guissmo Asuncion gave a tutorial on the software PARI which we can use to make calculations related to elliptic curves.

Participants were also given the opportunity to present their research work or topics they were interested in. I gave a short presentation discussing certain aspects of algebraic geometry related to number theory, focusing on the spectrum of the integers, and a mention of related modern mathematical research, such as Arakelov theory, and the view of the integers as a curve (under the Zariski topology) and as a three-dimensional manifold (under the etale topology).

Aside from the lectures, we also had an excursion to the mountainous province of Rizal, which is a short distance away from Manila, but provides a nice getaway from the environment of the big city. We visited a couple of art museums (one of which was also a restaurant serving traditional Filipino cuisine), an underground cave system, and a waterfall. We used this time to relax and talk with each other, for instance about our cultures, and many other things. Of course we still talked about mathematics, and during this trip I learned about many interesting things from my fellow participants, such as the class field theory problem and the subject of real algebraic geometry .

I believe lecture notes will be put up on the school website at some point by some of the participants of the school. For now, some of the lecturers have put up useful references for their lectures.

SEAMS School Manila 2017 was actually the first summer school or conference of its kind that I attended in mathematics, and I enjoyed very much the time I spent there, not only in learning about elliptic curves but also making new friends among the mathematicians in attendance. At some point I also hope to make some posts on this blog regarding the interesting things I have learned at that school.

Some Useful Links: Quantum Gravity Seminar by John Baez

I have not been able to make posts tackling physics in a while, since I have lately been focusing my efforts on some purely mathematical stuff which I’m trying very hard to understand. Hence my last few posts have been quite focused mostly on algebraic geometry and category theory. Such might perhaps be the trend in the coming days, although of course I still want to make more posts on physics at some point.

Of course, the “purely mathematical” stuff I’ve been posting about is still very much related to physics. For instance, in this post I’m going to link to a webpage collecting notes from seminars by mathematical physicist John Baez on the subject of quantum gravity – and much of it involves concepts from subjects like category theory and algebraic topology (for more on the basics of these subjects from this blog, see Category TheoryHomotopy Theory, and Homology and Cohomology).

Here’s the link:

Seminar by John Baez

As Baez himself says on the page, however, quantum gravity is not the only subject tackled on his seminars. Other subjects include topological quantum field theory, quantization, and gauge theory, among many others.

John Baez also has lots of other useful stuff on his website. One of the earliest mathematics and mathematical physics blogs on the internet is This Week’s Finds in Mathematical Physics, which apparently goes back all the way to 1995, and is one of the inspirations for this blog:

This Week’s Finds in Mathematical Physics by John Baez

Many of the posts on This Week’s Finds in Mathematical Physics show the countless fruitful, productive, and beautiful interactions between mathematics and physics. This is also one of the main goals of this blog – reflected even by the posts which have been focused on mostly “purely mathematical” stuff.

Monoidal Categories and Monoids

A monoid is a concept in mathematics similar to that of a group (see Groups), except that every element need not have an inverse. Therefore, a monoid is a set, equipped with a law of composition which is associative, and an identity element. An example of a monoid is the natural numbers (including zero) with the law of composition given by addition.

In this post, we will introduce certain concepts in category theory (see Category Theory) that are abstractions of the classical idea of a monoid.

A monoidal category is given by a category \mathbf{C}, a bifunctor \Box: \mathbf{C}\times\mathbf{C}\rightarrow\mathbf{C}, an object I of \mathbf{C}, and three natural isomorphisms \alpha (also known as the associator), \lambda (also known as the left unitor), and \rho (also known as the right unitor), with components

\displaystyle \alpha_{A,B,C}:A\Box (B\Box C)\cong (A\Box B)\Box C

\displaystyle \lambda_{A}:I\Box A\cong A

\displaystyle \rho_{A}:A\Box I\cong A

satisfying the conditions

\displaystyle 1_{A}\Box\alpha_{A,B,C}\circ\alpha_{A,B\Box C,D}\circ\alpha_{A,B,C}\Box 1_{D}=\alpha_{A,B,C\Box D}\circ\alpha_{A\Box B,C,D}

for any four objects A, B, C, and D of \mathbf{C}, and

\displaystyle \alpha_{A,I,B}\circ 1_{A}\Box \lambda_{B}=\rho_{A}\Box 1_{B}

for any two objects A and B in \mathbf{C}.

The following “commutative diagrams” courtesy of user IkamusumeFan of Wikipedia may help express these conditions better (the symbol \otimes is used here instead of \Box to denote the bifunctor; this is very common notation, but we use \Box following the book Categories for the Working Mathematician by Saunders Mac Lane in order to differentiate it from the tensor product, which is just one specific example of the bifunctor in question; I hope this will not cause any confusion):

Monoidal_category_pentagon.svg

Monoidal_category_triangle.svg

If the natural isomorphisms \alpha, \lambda, and \rho are identities, then we have a strict monoidal category.

A monoid object, or monoid in a monoidal category (\mathbf{C},\Box,I) is an object M of \mathbf{C} together with two morphisms \mu:M\Box M\rightarrow M and \eta:I\rightarrow M satisfying the conditions

\displaystyle \mu\circ 1\Box\mu\circ\alpha=\mu\circ\mu\Box 1

\displaystyle \mu\circ \eta\Box 1=\lambda

\displaystyle \mu\circ 1\Box\eta=\rho

Again we can use the following commutative diagrams made by User IkamusumeFan of Wikipedia to help express these conditions:

Monoid_multiplication.svg

Monoid_unit_svg.svg

As examples of monoidal categories, we have the following:

\displaystyle (\mathbf{Set},\times,1)

\displaystyle (\mathbf{Ab},\otimes,\mathbb{Z})

\displaystyle (K\mathbf{-Mod},\otimes_{K},K)

(\mathbf{Cat},\times,\mathbf{1})

(\mathbf{C}^{\mathbf{C}},\circ,\text{Id})    (\mathbf{C}^{\mathbf{C}} denotes the category of functors from \mathbf{C} to itself)

The monoids in these monoidal categories are given respectively by the following:

Ordinary monoids

Rings

K-algebras

Strict monoidal categories

Monads (see Adjoint Functors and Monads)

Among the important kinds of monoidal categories with extra structure are braided monoidal categories and symmetric monoidal categories. A braided monoidal category \mathbf{C} is a monoidal category equipped with a natural isomorphism \gamma (also known as a commutativity constraint) with components \gamma_{A,B}:A\Box B\cong B\Box A satisfying the following coherence conditions

\displaystyle \alpha_{B,C,A}\circ\gamma_{A,B\Box C}\circ\alpha_{A,B,C}=1_{B}\Box\gamma_{A,C}\circ\alpha_{B,A,C}\circ \gamma_{A,B}\Box 1_{C}

\displaystyle \alpha_{C,A,B}^{-1}\circ\gamma_{A\Box B,C}\circ\alpha_{A,B,C}^{-1}=\gamma_{A,C}\Box 1_{B}\circ\alpha_{A,C,B}^{-1}\circ 1\Box\gamma_{A}\gamma_{A,B}

which can be expressed in the following commutative diagrams (once again credit goes to User IkamusumeFan of Wikipedia):

Braid_category_hexagon.svg

Braid_category_inverse_hexagon.svg

The category \mathbf{C} is a symmetric monoidal category if the isomorphisms \gamma_{A,B} satisfy the condition \gamma_{B,A}\circ\gamma_{A,B}=1_{A\Box B}. We have already encountered an example of this category in The Theory of Motives in the form of tensor categories, defined as a symmetric monoidal categories whose Hom-sets (the sets of morphisms from a fixed object A to another object B) form a vector space (the term “tensor category” is sometimes used to refer to other concepts in mathematics though, including symmetric monoidal categories themselves).

Another important kind of monoidal category is a closed monoidal category. A closed monoidal category is a monoidal category where the functor -\Box B has a right adjoint (see Adjoint Functors and Monads) also known as the “internal Hom functor”, which is like a Hom functor that takes values in the category itself instead of in sets, and is denoted by (\ )^{B}. We have already seen an example of a closed monoidal category in Adjoint Functors and Monads, given by the category of R-modules for a fixed commutative ring R. There A^{B} was given by \text{Hom}(A,B) (this is the set of R-linear transformations from A to B, which itself is an R-module).

We see therefore that the concepts of monoidal categories and monoids can be found everywhere in mathematics. Studying these structures are not only interesting for their own sake, but can also help us find or construct other useful new concepts in mathematics.

References:

Monoidal Category on Wikipedia

Monoid on Wikipedia

Braided Monoidal Category on Wikipedia

Symmetric Monoidal Category on Wikipedia

Closed Monoidal Category on Wikipedia

Image by User IkamusumeFan of Wikipedia

Image by User IkamusumeFan of Wikipedia

Image by User IkamusumeFan of Wikipedia

Image by User IkamusumeFan of Wikipedia

Image by User IkamusumeFan of Wikipedia

Image by User IkamusumeFan of Wikipedia

Categories for the Working Mathematician by Saunders Mac Lane