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Seminar on arithmetic bundles: the Mordell conjecture (held at the École Normale Supérieure, Paris 1983–1984). (Séminaire sur les pinceaux arithmétiques: La conjecture de Mordell.) (French) Zbl 0588.14028
Astérisque, 127. Paris: Société Mathématique de France. x, 287 pp. (1985).
The book under review contains the éxposés of a seminar held at École Normale Supérieur, Paris in 1983/84 whose main aim was to get a self-contained exposition of Faltings’ proof of the Mordell conjecture [Invent. Math. 73, 349–366 (1983; Zbl 0588.14026)], and also to present some new results or improvements (especially concerning the ”effectiveness”). Thus, there are available now two books [this one and the book ”Rational points”, edited by G. Faltings and G. Wüstholz (1984; Zbl 0588.14027)], which together give a complete picture of difficult and fundamental results of arithmetic algebraic geometry.
Faltings’ proof of the Mordell conjecture consists mainly of four main themes, namely: modular heights, isogenies and the Tate conjecture, the Shafarevich conjecture and the construction of Kodaira-Parshin.
The concept of modular height is introduced and studied in exposés I, IV and V. The first exposé (by L. Szpiro) presents the idea of Arakelov to use hermitian metrics in order to define the height of a rational point. Then one introduces the concept of modular height of an abelian variety over a number field and one shows that for a semi-stable family of abelian varieties, the modular height has ”logarithmic singularities”. The aim of the exposés IV (by L. Moret-Bailly) and V (by P. Deligne) is to prove that the set of polarized abelian varieties over a fixed number field of a given dimension and of bounded modular height, is finite. Roughly speaking, this is done by interpreting the modular height as a height over a certain compactification of the moduli space of polarized abelian varieties, associated to an ample sheaf.
The second theme is taken up in the exposés VI, VII and VIII. In exposé VI (by L. Illusie) one proves (à la Grothendieck) an existence theorem of infinitesimal extensions of Barsotti-Tate truncated groups (which is to be used in the next exposé). In exposé VII (by M. Raynaud) one proves that the variation of the modular height in an isogeny class is effectively bounded. In exposé VIII (by M. Flexor) one proves the Tate conjecture, using some arguments of Zarkhin and Tate to reduce this conjecture to the fact that there are only finitely many abelian varieties over a number field in an isogeny class.
The third and the fourth theme are the Shafarevich conjecture and Parshin’s proof that it implies the Mordell conjecture, to which the exposés IX (by P. Deligne) and X (by M. Martin-Deschamps) are devoted. In exposé IX one proves (following the author) that there are only finitely many isogeny-classes of abelian varieties over a number field with fixed bad reduction. This - together with the results of exposé VIII - prove the Shafarevich conjecture: if $$K$$ is a number field, $$S$$ a finite set of places of $$K$$, $$g$$ a positive integer, then the set of all classes of $$K$$-isomorphic abelian varieties over $$K$$ of dimension $$g$$ and having good reduction off $$S$$ is finite (actually, the original Shafarevich conjecture is obtained from this fact via Torelli theorem).
In exposé X one presents Parshin’s proof that the Mordell conjecture is a consequence of the Shafarevich conjecture via the construction of Kodaira-Parshin. This construction uses the intersection theory of Arakelov. This intersection theory (together with the subsequent developments due to the author) are presented in the exposés I (L. Szpiro), II (by L. Moret-Bailly), III (by R. Elkik), and XI (by L. Szpiro).
Since the points of view of the two books (reviewed here and above) are rather different, both of them are useful for people who want to understand such kind of difficult arithmetic questions.
Contents:
Exposé I: Lucien Szpiro: ”Degrés, intersections, hauteurs” (p. 11–28). - Exposé II: Laurent Moret-Bailly: ”Métriques permises” (p. 29–87). - Exposé III: Renée Elkik: ”Fonctions de Green, volumes de Faltings, application aux surfaces arithmétiques” (p. 89– 112). - Exposé IV: Laurent Moret-Bailly: ”Compactifications, hauteurs et finitude” (p. 113–129). - Exposé V: Pierre Deligne: ”Le lemme de Gabber” (p. 131–150). - Exposé VI: Luc Illusie: ”Déformation des groupes de Barsotti-Tate” (p. 151–198). - Exposé VII: Michel Raynaud: ”Hauteurs et isogénies” (p. 199–234). - Exposé VIII: Marguerite Flexor: ”Endomorphismes des variétés abéliennes” (p. 235–248). - Exposé IX: Pierre Deligne: ”Représentations $$\ell$$-adiques”; Appendice: Mireille Martin-Deschamps: ”Conjecture de Shafarevich pour les corps de fonctions sur $${\mathbb Q}''$$ (p. 249–255; 256–259). - Exposé X: Mireille Martin-Deschamps: ”La construction de Kodaira-Parshin” (p. 261–273). - Expose XI: Lucien Szpiro: ”Un peu d’effectivité” (p. 275–287).
Reviewer: L. Bădescu

MSC:
 14-06 Proceedings, conferences, collections, etc. pertaining to algebraic geometry 11-06 Proceedings, conferences, collections, etc. pertaining to number theory 00B25 Proceedings of conferences of miscellaneous specific interest 11G30 Curves of arbitrary genus or genus $$\ne 1$$ over global fields 11G05 Elliptic curves over global fields 14K15 Arithmetic ground fields for abelian varieties 14G05 Rational points 14H45 Special algebraic curves and curves of low genus 14H52 Elliptic curves
Citations:
Zbl 0588.14027; Zbl 0588.14026