Diophantine equations: The geometric approach. (English) Zbl 0863.14012

This is a survey on the study of diophantine equations, with emphasis on those methods that make use of the geometry of an underlying algebraic variety. In addition, the survey emphasizes the study of rational (as opposed to integral) solutions. Topics covered include lifting (of rational points to one of a finite set of covering varieties); the Hasse principle and Brauer-Manin obstruction; modular curves and the Taniyama-Shimura conjecture; the Birch-Swinnerton-Dyer conjecture; and conjectures of Manin and others on the density of rational points.
Reviewer: P.Vojta (Berkeley)


14G05 Rational points
11D72 Diophantine equations in many variables
11G35 Varieties over global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)