The book under review is divided into five parts. The first part is a long introduction which consists of an overview of some basic arithmetic and geometric problems in a unified context entitled, by the author, unlikely intersections. More specifically, let \(X\) be a fixed algebraic variety and let \(Y\) run through a denumerable set \(\mathcal{Y}\) of algebraic varieties, chosen in advance independently of \(X\), with a certain structure one may concern, and such that \(\text{dim}(X)+\text{dim}(Y)\) is less than an integer \(n\) which is the dimension of a whole space containing \(X\) and \(Y\). Then the author expects that only for a small subset of \(Y\in \mathcal{Y}\) one will have \(X\cap Y\neq\emptyset\), unless there is a special structure relating \(X\) with \(\mathcal{Y}\) which forces the contrary to happen. If such a special structure really exists, the variety \(X\) is generally called a special variety. When \(X\) is nonspecial, the expected smallness may be measured in terms of the union of the intersections \(\bigcup_{Y\in \mathcal{Y}}(X\cap Y)\) and the interesting point is that how is this set distributed in \(X\), for instance, one may ask is this set finite? A typical example is Lang’s problem on roots of unity. This problem proposes the following question: Suppose that \(X: f(x,y)=0\) is a complex plane irreducible curve containing infinitely many points \((\zeta,\theta)\) whose coordinates are roots of unity, then what can be said of the polynomial \(f\)? To fit the unlikely intersection problem the author concerns, the set \(\mathcal{Y}\) consists of the points with roots of unity coordinates which are precisely the torsion points in the algebraic group \(\mathbb{G}_m^2\). The special structure which forces \(X\) to contain infinitely many torsion points can be formulating as \(X\) being a translate of an algebraic subgroup by a torsion point. Then the result foreseen by Lang can be rephrased by stating that an irreducible curve contains infinitely many torsion points if and only if it is a special curve.
The second part of the book under review discusses unlikely intersections in commutative multiplicative groups over a field of characteristic zero and the Zilber’s conjecture. The problem of unlikely intersections in multiplicative groups is a natural generalization of Lang’s problem to higher dimensional case. The final result the author mentioned is that the Zariski closure of a set of torsion points \(\Sigma\) in \(\mathbb{G}_m^n(\overline{\mathbb{Q}})\) is a finite union of torsion cosets. Here, the torsion cosets are by definition the translates of algebraic subgroups by a torsion point. The author also introduces Bombieri and Masser’s theorems which generalize this problem to higher multiplicative rank, by intersecting a given variety \(X\subset \mathbb{G}_m^n\) not merely with the set of torsion points but with the family of algebraic subgroup of \(\mathbb{G}_m^n\) up to any given dimension. And the Zilber’s conjecture, arising from model theory, implies uniformity in quantitative versions of the said results. In the third part of this book, the author presents arithmetical analogues of some contents of the second part.
The fourth and the last parts of the book under review are devoted to discuss unlikely intersections in elliptic surfaces, the problems of Masser, and the André-Oort conjecture: the Zariski closure of a set of special points in a Shimura variety is a special subvariety. These three problems were studied in a unified method introduced by Pila and the author in [J. Pila and U. Zannier, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 19, No. 2, 149–162 (2008; Zbl 1164.11029)].