Contents:\ Gerd Faltings, Some historical notes (pp. 1--8); Gerd Faltings,
Finiteness theorems for abelian varieties over number fields (pp. 9--27);
Stephen S. Shatz, Group schemes, formal groups, and $p$-divisible groups
(pp.\ 29--78); Michael Rosen, Abelian varieties over $ C$ (pp. 79--101);
J. S. Milne, Abelian varieties (pp. 103--150); Joseph H. Silverman, The
theory of height functions (pp. 151--166); J. S. Milne, Jacobian varieties
(pp. 167--212); M. Artin, Néron models (pp. 213--230); Ching-Li
Chai, Siegel moduli schemes and their compactifications over $ C$ (pp.
231--251); Joseph H. Silverman, Heights and elliptic curves (pp. 253--265);
M. Artin, Lipman's proof of resolution of singularities for surfaces (pp.\
267--287); T. Chinburg, An introduction to Arakelov intersection theory
(pp. 289--307); T. Chinburg, Minimal models for curves over Dedekind rings
(pp. 309--326); Benedict H. Gross, Local heights on curves (pp. 327--339);
Paul Vojta, A higher-dimensional Mordell conjecture (pp. 341--353). The book starts with a brief historical sketch by Faltings of the path to the proof. The second article is an English translation of Faltings' original paper. There is then the article "Group schemes, formal groups, and $p$-divisible groups" by Shatz. This concentrates on finite group schemes and includes a proof of the group scheme analogue of the Sylow theorems. There are three articles on abelian varieties; the first by Rosen is on the analytic theory, while the other two by Milne are on the geometric theory in arbitrary characteristics and on Jacobian varieties, respectively. These are the three main approaches to the theory of abelian varieties, and to have all three represented in one place is very pleasant. For example, at the elementary level, the reader can compare the proof that a connected compact complex Lie group is commutative, in Rosen's article, with the proof that a complete group variety is commutative, in the article "Abelian varieties" by Milne. (Note that Section 16 describes Zarkhin's trick, which is used in Faltings' paper.) The article "Jacobian varieties " by Milne is a modern treatment of the subject, and as such helps fill an important gap in the expository literature, since \n D. Mumford\en never wrote the second volume of his book Abelian varieties \ref[Oxford Univ. Press, London, 1970; MR 44 #219], and the book Abelian varieties by \n S. Lang\en \ref[Interscience, New York, 1959; MR 21 #4959] was written in the language of Weil. The bibliographical notes to Milne's article are well worth reading; it is a pity that this kind of thing is not done more often. The moduli and degenerations of abelian varieties and algebraic curves are discussed in the article on Néron models by Artin, the article on minimal models by Chinburg and the article on Siegel moduli schemes by Chai. This last is a brief survey of the results contained in his book Compactifications of Siegel moduli schemes \ref[Cambridge Univ. Press, Cambridge, 1985; MR 88b:32074]. This together with Faltings' more recent work on moduli of abelian varieties over the integers \ref[in Workshop Bonn 1984, 321--383, Lecture Notes in Math., 1111, Springer, Berlin, 1985; MR 87c:14050] should allow one to avoid the appeal to the moduli of stable curves in Faltings' paper on the Mordell conjecture. At various times people have tried to use Arakelov theory directly to give proofs of Mordell's and related conjectures. So far this approach has not met with success; however, it played a key role, via the theory of canonical heights, in Faltings' proof. The approach to heights via Arakelov theory is explained in the first article of Silverman. His second article gives an explicit formula, in terms of classical invariants, of the canonical height of an elliptic curve. Here we should note that the canonical height may be constructed as an "Arakelov intersection number" on the moduli space of elliptic curves. Chinburg gives an introduction to Arakelov intersection theory, which includes a proof of the adjunction formula in the Arakelov context; this last is of interest because the statement and proof of the formula given by Arakelov himself \ref[S. Yu. Arakelov, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 1179--1192; MR 57 #12505] were valid only in a special case of the theorem. Another corrected proof was given by \n P. Hriljac\en \ref["The Arakelov adjunction formula", to appear]. The article of Gross on local heights describes the Néron height on a curve, and interprets Arakelov's theory as a generalization of Néron's pairing. It also contains a description of the local pairing over a non-Archimedean prime via $p$-adic analysis on the Mumford curve. The last article is not background to Faltings' theorem, but rather an outline by Vojta of his generalization (motivated by Nevanlinna theory) of the Mordell conjecture to higher dimensions. A detailed exposition has recently appeared \ref[Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Math., 1239, Springer, Berlin, 1987]. Finally, it would have been nice to have an index. Reviewed by H. Gillet |