Algebraic varieties are geometric objects defined by polynomials. When the polynomials have coefficients in a ring or field of arithmetic interest (e.g., the ring of rational integers Z or a finite extension of the field of rational numbers Q) these geometric objects carry a lot of arithmetic baggage. The study of these properties is often called “arithmetic geometry.”
A well-developed area of research is essentially devoted to counting the number of points on varieties defined over finite fields. Arithmetic geometry in the second half of the twentieth century was, in great part, driven by a remarkable set of conjectures of André Weil on the zeta functions of those varieties that carry all the information on the number of points on all finite extensions of the base field of definition. Étale, ℓ-adic and crystalline cohomology theories were specifically created to deal with these and related topics.
Switching now to algebraic varieties defined over a number field or its ring of integers, some conditions have to be imposed on the integral or rational points that are to be counted, since a priori these sets could be infinite. A natural condition is to consider only points with bounded height, which essentially means that we want to bound the number of digits that appear in the numerator and denominator or their coordinates. The book under review considers the distribution of integral or rational points of bounded height on (projective) algebraic varieties. More specifically, the author considers the problem of describing the behavior of the counting function N(f;B) of points of height bounded by B on the locus V(f) of non-zero integer solutions of given (homogeneous) polynomials f(x1,…,xn) in n variables with integer coefficients. He also wants to see what happens when we let the bound B go to infinity. More generally, for a projective variety V defined over the ring of integers Z and a given bound B we consider the function NV(B) that counts the number of rational points in V with height bounded by B. The main goal is to decide how large NV(B) is.
With this goal in mind, the author focuses on varieties of dimension at least two, since for curves the situation can be summarized as follows: For curves of genus g = 0, if the curve has a rational point, then it has infinitely many; for curves of genus g = 1 with a rational point, i.e., for elliptic curves, the number of rational points of bounded height on the curve depends on the bound B and the rank of the curve (Néron); for curves of genus greater than 1, Faltings has proved that the set of rational points is finite and so the asymptotic behavior of NV(B) is not that interesting.
The monograph under review is devoted to the application of the analytic number theory tools that are suited to the task at hand, in particular the Hardy-Littlewood circle method. In some instances, using the circle method it can be shown that N(f;B) goes to infinity as B does, and therefore the locus V(f) is infinite.
Although this monograph describes recent and deep work by the author and some of his collaborators, the book is well-written and well-organized. He includes, when appropriate, the heuristics of results before these are stated or proved. There is a chapter devoted to a discussion of the circle method with some applications to illustrate its use, such as treating the case of quartic hypersurfaces or diagonal cubic surfaces. Most of the monograph describes recent work on a conjecture of Manin relating the asymptotic behavior of counting functions of points of bounded height to the geometry of the given variety, focusing on a specific family of algebraic projective varieties known as del Pezzo surfaces (of small degree).
Introductory material is discussed when appropriate, motivation and context are provided when necessary, and there are even small sets of exercises at the end of every chapter, making the book suitable for self or guided study, giving a well-organized introduction to this relatively new area of Diophantine geometry.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org