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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