Rational points. Seminar Bonn/Wuppertal 1983/84. 3. enlarged ed. (English) Zbl 0753.14019

Aspects of Mathematics. E 6. Braunschweig: Friedr. Vieweg & Sohn. 311 p. (1992).

This is the third edition of this book the first edition of which appeared eight years ago [1984; Zbl 0588.14027); see also the second edition 1986; Zbl 0636.14019)]. A lot of new ideas and techniques have been developed since 1984, for instance several different approaches to the Mordell conjecture, the foundation of the arithmetic intersection theory and the proof of the arithmetic Riemann-Roch theorem.
This third edition has a new appendix written by G. Wüstholz [“New developments in diophantine and arithmetic algebraic geometry”]. The appendix contains an overview on the new developments, especially E. Bombieri’s beautiful proof of Faltings theorem [see Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 17, No. 4, 615-640 (1990; Zbl 0722.14010)].

Reviewer: G.Pfister (Berlin)

MSC:

14G40	Arithmetic varieties and schemes; Arakelov theory; heights
14G05	Rational points
14-02	Research exposition (monographs, survey articles) pertaining to algebraic geometry
14H52	Elliptic curves
14K15	Arithmetic ground fields for abelian varieties

Keywords:

proof of Tate conjecture; proof of Shafarevich conjecture; proof of Mordell conjecture; arithmetic intersection theory; arithmetic Riemann- Roch theorem

Citations:

Zbl 0588.14027; Zbl 0636.14019; Zbl 0722.14010