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 . An example of this is the scheme , which is two-dimensional, and hence also referred to as an arithmetic surface.
We recall that the points of an affine scheme , for some ring , are given by the prime ideals of . Therefore the scheme has one point for every prime ideal – one “closed point” for every prime number , and a “generic point” given by the prime ideal .
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 , 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 instead of , where is the ring of integers of a number field . In this case we may have several infinite primes, corresponding to the embediings of into the real and complex numbers. In this post, however, we will consider only and one infinite prime.
How do we describe an arithmetic scheme when the scheme has been “compactified” with the infinite prime? Let us look at the fibers of the arithmetic scheme. The fiber of an arithmetic scheme at a finite prime is given by the scheme defined by the same homogeneous polynomials as , but with the coefficients taken modulo , so that they are elements of the finite field . The fiber over the generic point is given by taking the tensor product of the coordinate ring of 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 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 be a smooth projective complex equidimensional variety with complex dimension . The space of differential forms (see Differential Forms) of degree on has the following decomposition:
We say that is the vector space of complex-valued differential forms of type . We have differential operators
We let be the dual to the vector space , and we write to denote . We refer to an element of as a current of type . We have an inclusion map
mapping a differential form of type to a current of type , given by
for all .
The differential operators , , , and induce maps , , and on . We define the maps , , and on by
We also define
For every irreducible analytic subvariety of codimension , we define the current by
for all , where is the nonsingular locus of .
A Green current for a codimension analytic subvariety is defined to be an element of such that
for some .
Let be the resolution of singularities of . This means that there exists a proper map such that is smooth, is a divisor with normal crossings (this means that each irreducible component of is nonsingular, and whenever they meet at a point their local equations are linearly independent) whenever contains the singular locus of , and is an isomorphism.
A smooth form on is said to be of logarithmic type along if there exists a projective map such that is a divisor with normal crossings, is smooth, and is the direct image by of a form on satisfying the following equation
where is a local equation of for every in , are and closed smooth forms, and is a smooth form.
For every irreducible subvariety there exists a smooth form on of logarithmic type along such that is a Green current for :
where w is smooth on X. We say that is a Green current of logarithmic type.
We now proceed to discuss this intersection theory on the arithmetic scheme. We consider a vector bundle on the arithmetic scheme , a holomorphic vector bundle (a complex vector bundle such that the projection map is holomorphic) on the fibers at the infinite prime, and a smooth hermitian metric (a sesquilinear form with the property that ) on which is invariant under the complex conjugation on . We refer to this collection of data as a hermitian vector bundle on .
Given an arithmetic scheme and a hermitian vector bundle on , 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 is a pair where is an algebraic cycle on , i.e. a linear combination of closed irreducible subschemes of , of some fixed codimension , with integer coefficients , and is a Green current for , i.e. satisfies the equation
for differential forms and of appropriate degree.
We define the arithmetic Chow group as the group of arithmetic cycles modulo the subgroup generated by the pairs and , where and are currents of appropriate degree and is some rational function on some irreducible closed subscheme of codimension in .
Next we want to have an intersection product on Chow groups, i.e. a bilinear pairing
We now define this intersection product. Let and . Assume that and are irreducible. Let , and . If and intersect properly, i.e. , then we have
where is just the usual intersection product of algebraic cycles, and is the -product of Green currents, defined for a Green current of logarithmic type and a Green current , where and are closed irreducible subsets of with not contained in , as
for is the resolution of singularities of composed with the inclusion of into .
In the case that and do not intersect properly, there is a rational function on such that and intersect properly, and if is another rational function such that and intersect properly, the cycle
is in the subgroup . Here the notation refers to the pair .
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 .
In order to define the arithmetic analogue of the Grothendieck-Riemann-Roch theorem, one must then define the direct image map for a proper flat map of arithmetic varieties. This involves constructing a canonical line bundle on , whose fiber at is the determinant of cohomology of , i.e.
as well as a metric , called the Quillen metric, on . 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 :
where denotes the arithmetic Chern character, denotes the arithmetic Todd class, is the relative tangent bundle of , is the map from
to sending the element in to the class of in , and
Later on Gillet and Soule formulated the arithmetic Grothendieck-Riemann-Roch theorem in higher degree as
Aside from the work of Gillet and Soule, there is also the work of the mathematician Amaury Thuillier making use of ideas from -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.
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