The book contains an exposition of the author’s results on an application of differential algebraic geometry to diophantine problems. It has seven chapters. The first chapter gives a short survey of classical diophantine problems over function fields: Mordell-Weil and De Francis finiteness theorems, Lang conjecture. The core of the book are Chapters 2-5 where the main notions and results of differential algebraic geometry are presented. Chapter 2 contains a detailed survey of Ritt’s differential algebra. It serves as a necessary “commutative algebra” background for the theory of algebraic varieties defined over differential fields (\(\delta\)-fields, i.e. fields with a fixed derivation \(\delta\)). Their theory is developed in the next chapter where the author introduces the \(\delta\)-analogues of the main notions and constructions from the “usual” algebraic geometry (\(\delta\)-regular functions and maps, \(\delta\)-open sets and \(\delta\)-closure, \(\delta\)-dimension, \(\delta\)-tangent space and so on).
Chapters 4 and 5 contain some results on rational and abelian varieties from this point of view. In particular, for unirational varieties the rational points are \(\delta\)-dense in the set of all points. The author also discusses a \(\delta\)-analogue of the Lüroth problem. The results on abelian varieties are concentrated around a descent theory. The author considers the case of elliptic curves referring to his papers for arbitrary abelian varieties. Here the important notion of \(\delta\)-Hodge structure is introduced.
The last two chapters really deal with diophantine problems. In Chapter 6 a reformulation of the classical finiteness problems from Chapter 1 is given for the differential situation. It is proven that for any curve \(X\) from an algebraic group \(G\) over a \(\delta\)-field the intersection \(X \cap \Sigma\) is finite. Here \(\Sigma\) is a \(\delta\)-closed subgroup of \(\delta\)-dimension zero. Again the case of subvarieties \(X\) of higher dimension is considered in another place A. Buium [Ann. Math., II. Ser. 136, 583-593 (1992; Zbl 0817.14021)].
The second important result in this chapter is a \(\delta\)-version of the Isogeny theorem for elliptic curves. It says that \(\delta\)-closures of isogeny classes of two elliptic curves over a \(\delta\)-field coincide iff their \(\delta\)-Hodge structures are isomorphic. In the last chapter the author returns to the classical diophantine problems over function fields (Lang’s conjecture, Mordell and Siegel conjectures for curves) and shows how to get their proof by “differential” methods.
The book is written very clearly with many motivations and examples and serves as an excellent introduction to the whole subject.