Several of the recent advances in number theory in the last decades are tied to its interactions with algebraic geometry, the aptly named *arithmetic algebraic geometry. *The proofs of the Weil, Mordell, and Fermat conjectures are precisely at this junction. The book under review is devoted to one important facet of this many-sided gem, *Arakelov geometry*, which was originally developed and used to prove the Mordell conjecture in the 1980s.

The starting point is with the most arithmetic of fields and rings, the finite extensions of the field of rational numbers and their rings of integers. The geometric viewpoint immediately notes that these rings are Dedekind domains and thus have Krull dimension one and are normal. That is, rings of integers are arithmetic analogues of normal (smooth) algebraic curves. Some elementary textbooks just work with these rings of integers, that is, with smooth arithmetic curves, neglecting somehow the important case of singular algebraic curves, whose arithmetic analogues are the *orders* in a number field. An *order* in a number field is a one-dimensional noetherian subdomain, and rings of integers are just maximal orders.

Since several cohomology theories are relevant for the study of number fields, and since some of these theories need singular schemes for their very construction, the importance of orders is something that cannot be neglected. A nice treatment of orders in this context can be found in pp. 72–93 of Neukirch’s *Algebraic Number Theory* (Springer, 1998).

All geometric notions, such as divisors, have a classical interpretation in the arithmetic context. Of course, these analogies between number fields and their rings of integers on one side, and complete algebraic curves or function fields on the other, are classical, as emphasized by Weil in his 1939a paper in Oeuvres Scientifiques-Collected Papers Vol. I (1926-1951), pp. 236–240. One example of these analogies is given by Cauchy’s residue formula for a non-constant meromorphic function on a given curve, which corresponds to the product formula for the valuations of a non-zero element of the number field.

Now, completeness of the curve requires to add points at infinity to the corresponding affine curve, the Spec of the corresponding order, and these points at infinity correspond naturally to the Archimedean places of the number field. Classically, a *divisor* for a number field is a formal finite linear combination, with integer coefficients, of non-Archimedean primes. The generalization in this context defines an *Arakelov divisor as *a finite linear combination of non-Archimedean primes (with integer coefficients) plus a finite linear combination of Archimedean primes (allowing now real coefficients). The set of Arakelov divisors is a commutative group that decomposes as a direct product of the classical divisor group and a direct sum of copies of the real field, with as many copies of this field as the number of Archimedean primes in the given number field. It is natural to give the discrete topology to the first factor and the classical topology to the second one. Hence, the Arakelov divisor group becomes a locally compact topological group.

Minkowski theory affords a natural map from the group of units K* to the Arakelov divisor group, whose kernel is the group of roots of unity in K and whose image, a discrete subgroup, is the group of principal Arakelov divisors. The *Arakelov class group* of the given field is the quotient of the Arakelov divisor group by the subgroup of principal Arakelov divisors. There is natural generalization of the degree map to the Arakelov class group and the main result in this context is that the kernel of the degree map is a compact group. An immediate consequence of this theorem is a conceptual and unified proof of two of the main theorems of algebraic number theory, the finiteness of the class number and Dirichlet’s unit theorem.

The unifying thread running through these isomorphisms is an instance of a fundamental result of arithmetic geometry that provides natural morphisms from the Chow groups to the algebraic K-theory groups of a general scheme X. Now, for the classical (local) congruence arguments in Diophantine equations to work in this new context a notion of height must be introduced for the points of the corresponding curve. Arakelov’s remarkable intuition was to consider Hermitian metrics to take into account the Archimedean places.

In the first three chapters of the book under review we find a quick overview of Arakelov theory in the case of curves, with preliminary notions treated in the first two chapters. It goes without saying that, since this is a monograph, motivations are absent and everything is about orders. The classical case of arithmetic surfaces is treated in chapter four, up to and including Falting’s Riemann-Roch formula.

The remaining chapters deal with the higher dimensional case, with preliminaries on intersection theory on arithmetic varieties treated in chapter five, up to the formulation of Gillet-Soulé’s arithmetic Riemann-Roch formula, whose proof may be found, for example, in Soulé’s *Lectures on Arakelov Geometry* (Cambridge, 1992). The main contributions of this monograph are in chapters six to nine, which give an exposition of recent results on birational Arakelov geometry, with special emphases on the existence of small sections and estimations for the number of such sections. Many important results are presented for the first time in a book, such as the arithmetic Nakai-Moishezon criterion or the arithmetic Bogomolov inequality. This is a timely monograph that should appeal to researchers in this important area of mathematics.

Buy Now

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.