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Number theory III: Diophantine geometry. (English) Zbl 0744.14012
Encyclopaedia of Mathematical Sciences. 60. Berlin etc.: Springer-Verlag. xi, 296 p. (1991).
Diophantine geometry has been the field of remarkable advances in the last decade. Faltings’ proof of the Mordell conjecture came out just when the author’s previous (and second) survey of the topic [“Fundamentals of diophantine geometry” (New York 1983; Zbl 0528.14013)] was going to press. That year also saw the solution by Gross and Zagier of the class number problem for imaginary quadratic fields via \(L\)-functions of elliptic curves, and Vojta’s thesis on an arithmetic Nevanlinna type theory. These discoveries were in turn the starting point of new directions in diophantine geometry, connecting it more tightly with several complex variables, moduli, intersection theory and schemes methods. On the other hand, the more classical approach to the field through diophantine approximations has continued to progress. The author’s present book gives a broad outlook on these different advances, with emphasis on their established or conjectural relations. We first describe the contents of its 10 chapters.
Chapter I displays the basic question of diophantine geometry: Under which geometric conditions can one expect an algebraic variety to have finitely many rational points? Its dualizing sheaf provides a first classification, and this is discussed (as in the rest of the book) both in the function field and the number field cases. [For a justification of the word ‘basic’ above, see N. Bourbaki, “Algèbre” (Paris 1952; Zbl 0049.01801); Note historique du chapitre VII, p. 71, footnote to \(\ell\)]. The chapter ends with Hilbert’s irreducibility theorem. Arithmetics proper starts in chapter II, with the notion of height and the effective conjectures (of the ‘abc’ type) it produces. Chapter III gives general pre-1983 facts on abelian varieties and the Birch- Swinnerton-Dyer conjecture. Faltings’ proof of the conjectures of Tate and Shafarevich (and, following Parshin, of Mordell) is surveyed in chapter IV, which also mentions the archimedean approach of D. Masser and G. Wüstholz (to appear in Ann. Math., II. Ser). Shimura varieties are a good testing ground in algebraic, diophantine and arithmetic geometry; their simplest version (modular curves over \(\mathbb{Q}\)) are studied in chapter V, with Mazur’s theorem on their rational points, Ribet’s reduction of Fermat’s theorem to the Taniyama-Shimura-Weil conjecture, Tunnel’s connection of the Birch-Swinnerton-Dyer conjecture to congruent numbers, and the work of Gross-Zagier and Kolyvagin on Heegner points. Back to Mordell’s conjecture in chapter VI, which gives an up-to-date account of the proofs of Grauert, Bogomolov-Miyaoka- Parshin, Manin and Samuel in the function field case. How complex analytic geometry has become a chapter of number theory is explained in chapter VII on Arakelov theory (and its multidimensional extension by Gillet-Soulé; note that its 1-dimensional appearance is Weil’s definition of heights), and in chapter VIII, which conjecture analytic obstructions for an algebraic variety to have infinitely many rational points (in parallel to the global algebraic ones of chapter 1), gives Parshin’s proof of the function field case of Mordell’s conjecture via hyperbolicity, and describes Vojta’s approach to Nevanlinna theory. Chapter IX deals with diophantine approximations: this is traditionally associated with the study of integral points, as in Siegel’s theorem (using Siegel’s lemma and Roth-Schmidt type theorems) and this context is indeed illustrated by Faltings’ recent proof of Lang’s conjecture on affine subsets of abelian varieties. But Roth’s theorem has also enabled Vojta to give an entirely new proof of both cases of Mordell’s conjecture. The chapter ends with upper bounds à la Baker-Stark-Masser- Wüstholz for minimal isogenies between 1-motives. Finally, chapter X describes recent work on Manin’s obstruction to the Hasse principle.
To say that the book is densely written is an understatement. But this is a natural price to pay for a survey of such a fastly expanding field. To the already experienced reader, this compact presentation of different approaches will be welcome. Algebraic geometers will probably be grateful for the early display in the book of the main conjectures of the subject. The newcomer, on the other hand, may find it easier to start with chapters II, III and V, then pass to Lang’s previous survey (including the remarkable appendix written by Yu. G. Zarhin and A. N. Parshin for its Russian edition (1986; Zbl 0644.14007), or to J.-P. Serre [“Lectures on the Mordell-Weil theorem” (1989; Zbl 0676.14005)] and return afterwards to the main bulk of the present book.
Some points of details, to conclude. Of course, the size of the book made it impossible to cover all aspects of the subject; similarly, proofs are as a rule omitted. But one may sometimes regret the absence of ‘lemmas’ which have a significance of their own. For instance, Mumford’s angle inequality is perhaps more important than the theorem it led to (p. 61), and it would have been interesting to state it in connection with gap principles, and as a forerunner of Vojta’s sphere packing condition on p. 231 [see P. Vojta, Ann. Math., II. Ser. 133, No. 3, 509-548 (1991)]. Similarly, Belyj’s theorem on coverings of the projective line minus 3 points, mentioned on p. 42, deserved a full statement [of course, this is easy to say in retrospect, now that we have N. D. Elkies’ deduction of Mordell from the abc conjecture via this theorem; cf. Duke Math. J. / Int. Math. Res. Not., Vol. 1991, No. 7, 95-109 (1991)]. On the other hand, some space could have been saved on certain definitions (e.g., on the decomposition group, which is defined p. 83, p. 88 and again p. 112). Finally, we mention, for those who need the state of the art on specific diophantine equations, the book of T. N. Shorey and R. Tijdeman [“Exponential diophantine equations” (Cambridge 1986; Zbl 0606.10011)].
Reviewer: D.Bertrand (Paris)

14Gxx Arithmetic problems in algebraic geometry; Diophantine geometry
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
11-02 Research exposition (monographs, survey articles) pertaining to number theory
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14G05 Rational points
11Gxx Arithmetic algebraic geometry (Diophantine geometry)
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G10 Abelian varieties of dimension \(> 1\)
11G05 Elliptic curves over global fields
11J82 Measures of irrationality and of transcendence
11Dxx Diophantine equations
14H52 Elliptic curves
14G25 Global ground fields in algebraic geometry
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14K15 Arithmetic ground fields for abelian varieties