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Applications of arithmetic algebraic geometry to diophantine approximations. (English) Zbl 0846.14009

Colliot-Thélène, Jean-Louis et al., Arithmetic algebraic geometry. Lectures given at the 2nd session of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Trento, Italy, June 24 - July 2, 1991. Berlin: Springer-Verlag. Lect. Notes Math. 1553, 164-208 (1993).
In the first part of this note a detailed survey of two proofs of Faltings’ theorem, formerly known as Mordell’s conjecture, using diophantine approximations (instead of moduli spaces of abelian varieties) is given: The first one is due to the author [Ann. Math., II. Ser. 133, No. 3, 509-548 (1991; Zbl 0774.14019)]; it makes use of arithmetic intersection theory introduced by H. Gillet and C. Soulé. The second one is due to E. Bombieri [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 17, No. 4, 615-640 (1990; Zbl 0722.14010)]; it is a simplification of Vojta’s approach.
In the second part a detailed description of Faltings’ proof of the following conjecture of S. Lang is presented: Let \(X\) be a subvariety of an abelian variety \(A\), both being defined over a number field \(k\). Then, the set \(X(k)\) of \(k\)-rational points on \(X\) is contained in a finite union \(\bigcup^n_{j = 1} B_j (k)\), where each \(B_j\) \((j = 1, \dots, n)\) is a translate of an abelian subvariety of \(A\) contained in \(X\). G. Faltings’ original proof of this theorem appeared as “The general case of S. Lang’s conjecture” [in: Barsotti symposium in algebraic geometry, Perspect. Math. 15, 175-182 (1994; Zbl 0823.14009)]. The main ideas of the proof are already contained in G. Faltings’ article “Diophantine approximation on abelian varieties”, Ann. Math., II. Ser. 133, No. 3, 549-576 (1991; Zbl 0734.14007).
For the entire collection see [Zbl 0780.00022].
Reviewer: J.Kramer (Berlin)

MSC:

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14G05 Rational points
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