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Fundamentals of Diophantine geometry. (Osnovy diofantovoj geometrii). Transl. from the English by D. G. Markushevich. Transl. ed. and with a preface by Yu. I. Manin. With an appendix by Yu. G. Zarkhin and A. N. Parshin. (Russian) Zbl 0644.14007

Moskva: Mir. 448 p. R. 2.90 (1986).
The English original of this book appeared 1983 (see Zbl 0528.14013) and an earlier version 1962 (see Zbl 0115.38701). This translation contains a 70-page appendix by Zarkhin and Parshin entitled “Finiteness problems in Diophantine geometry”. This appendix is a presentation of G. Faltings’ proof of the Shafarevich, Tate and Mordell conjectures. Zarkhin and Parshin follow G. Faltings’ proof [Invent. Math. 73, 349-366 (1983; Zbl 0588.14026), erratum: 75, 381 (1984)] and incorporate some of the improvements to be found in L. Szpiro’s Paris seminar [“Séminaire sur les pinceaux arithmétiques: La conjecture de Mordell”, Astérisque 127 (1985; Zbl 0588.14028)] and some of their own. Section 1 contains an introduction, with some historical comments; p. 376 contains a useful schematic of the proof. The remaining five sections contain details and basic information needed to understand the proof: canonical heights, \(\ell\)-divisible groups, Tate modules, isogenies of abelian varieties over local fields, Galois modules behavior of the height function under isogeny. Some unsolved questions are mentioned at the end.

MSC:

14Gxx Arithmetic problems in algebraic geometry; Diophantine geometry
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14K15 Arithmetic ground fields for abelian varieties
11D41 Higher degree equations; Fermat’s equation
14G05 Rational points