Rational Points on Varieties

Bjorn Poonen
Publisher:
American Mathematical Society
Publication Date:
2017
Number of Pages:
337
Format:
Hardcover
Series:
Price:
83.00
ISBN:
9781470437732
Category:
Textbook
[Reviewed by
Felipe Zaldivar
, on
04/17/2018
]

Given a field $F$ and an algebraic closure $\overline{F}$ of $F$, an affine variety $X$ over $F$ is the set of common zeros in affine space ${\mathbb A}^n(\overline{F})$ of a finite family of polynomials $f_1,\ldots,f_m\in F[t_1,\ldots,t_n]$, with coefficients in the field $F$. An $F$-rational point of $X$ is a point of $X$ with coordinates in $F$. A similar definition holds for projective varieties.

For an arbitrary algebraic variety $X$ over $F$, however, the definition of rational point on $X$ requires the whole machinery of modern algebraic geometry: First, for the ring of coordinates $\overline{F}[X]$ of the affine variety $X$, one interprets the (geometric) points $x$ in $X$ as $F$-algebra morphisms $x:\overline{F}[X]\rightarrow \overline{F}$ or, equivalently, as morphisms of schemes $x:\text{Spec}\,\overline{F} \rightarrow X$ over $\text{Spec}\, F$. Then, an $F$-rational point of $X$ is a geometric point $x:\text{Spec}\,\overline{F} \rightarrow X$ of $X$ that factors through $\text{Spec}\,F$ for the morphism $\text{Spec}\,\overline{F}\rightarrow \text{Spec}\, F$ induced by the inclusion $F\rightarrow \overline{F}$. In other words, a rational point of $X$ is a section $x:\text{Spec}\, F\rightarrow X$ of the structure morphism $X\rightarrow \text{Spec}\, F$ of $X$.

With this interpretation, given an arbitrary algebraic variety $X$ over $F$, that is, a separated scheme of finite type over $F$ (which sometimes is also required to be integral), then an $F$ rational point of $X$ is a section of its structure morphism. The natural questions are: Is there an $F$-rational point on $X$? What are the obstructions for the existence of an $F$-rational point? Is it possible to estimate the number of $F$-rational points, perhaps with some additional conditions? These are classical, difficult, problems in arithmetic geometry, which remain at the forefront of research of a large portion of contemporary mathematics.

A monograph/textbook whose main goals are to introduce the interested reader to the methods and problems of arithmetic geometry and at the same time discuss open problems of interest for further research is therefore a most welcome addition to a classical subject whose literature ranges from elementary introductions to advanced monographs, even when restricted to families of specific varieties (curves, elliptic curves, abelian varieties). The choice of topics and the decisions on what to spell out and what to just barely sketch, with adequate pointers to the existing literature, make the book under review an excellent quick introduction and reference on this subject.

To accomplish these goals, the entry requirements are necessarily higher. To begin with, the proper definition of algebraic variety must assume a working knowledge of schemes, nominally up to chapter 2 of Hartshorne's textbook (Algebraic Geometry, Springer, 1977), but actually higher than that, since, for example, Hartshorne does not include the functor of points that is certainly needed for the definition of $F$-rational points or for group schemes. An acquaintance with the first few chapters of Milne's Étale Cohomology (Princeton, 1980) would certainly help to make it easier to deal with the pointers to EGA IV. On the arithmetic side, the reader must be familiar with local and global fields including their Galois theory and cohomology, for example on the level of Neukirch's book Cohomology of Number Fields (Springer, 2008). Assuming that, the first three chapters set the stage: From the fields over which one would be asking for rational points to the definition and main properties of varieties over arbitrary fields and several type of morphisms between them. The exposition is terse but precise, and flows naturally.

Next, for the cohomological methods to detect rational points, chapters four to eight give the modern interpretations of Galois descent and the local-to-global principle. For chapter four, on descent theory, the reader is advised to keep chapter six of Bosch, Lütkenbohmert and Raynaud's Néron Models (Springer, 1990) at hand. Since many algebraic varieties of interest are algebraic groups, chapter five gives a quick introduction and overview of their main properties, again in the framework of group schemes. For a more detailed and comprehensive modern treatment of algebraic groups over a field, a copy of Milne's recent book Algebraic Groups (Cambridge, 2017) should be kept close. Chapters six and eight develop the cohomological methods needed to study the obstructions to the existence of rational points. Again, for the étale cohomology, Milne's book is a good reference, and EGA IV for the fppf cohomology.

Chapter seven departs from the main narrative, since it is devoted to rational points over finite fields, but it is a nice — indeed, spectacular — application of the étale cohomological methods of chapter six.

Chapter nine gives an overview of some recent developments on the arithmetic of higher dimensional varieties, with special emphasis on del Pezzo surfaces.

Three appendices summarize some topics not treated on the main text. The one I found more useful is Appendix C, which includes two tables where some properties of morphisms and varieties are listed, with precise references to the corresponding EGAs.

The book is well structured, balancing explicit constructions, terse arguments, and precise references to the literature when needed. This approach allows to author to navigate up to some frontiers of recent research and at the same time gives a motivated graduate student a quick introduction to the methods and some conjectures on the arithmetic of algebraic varieties.

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