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The Subspace Theorem of W. M. Schmidt. (English) Zbl 0812.11039

Edixhoven, Bas (ed.) et al., Diophantine approximation and abelian varieties. Introductory lectures. Papers of the conference, held in Soesterberg, Netherlands, April 12-16, 1992. Berlin: Springer-Verlag. Lect. Notes Math. 1566, 31-50 (1993).
[For a review of the entire collection see Zbl 0811.14019.]
The celebrated finiteness theorem of K. F. Roth on the approximation of algebraic (real) integers by rationals, which was published in 1955, has been significantly generalized by W. M. Schmidt (1972), and, later on, by Schlickewei (1977) and Vojta (1987). The present article provides a detailed discussion of this development, in the first two sections, and then a fairly complete proof of W. M. Schmidt’s far-reaching generalization of Roth’s theorem, which is well-known as “the subspace theorem”.
The author’s presentation is a skillfully outlined version of W. M. Schmidt’s original approach [Diophantine approximation, Lect. Notes Math. 785 (1980; Zbl 0421.10019)], and reflects the full depth and beauty of this central result in diophantine approximation theory. In the course of the discussion of the subspace theorem, the author also explains the recent progress made in confirming S. Lang’s conjecture [Fundamentals of diophantine geometry (Springer (1983; Zbl 0528.14013))], which is essentially due to M. Laurent [Invent. Math. 78, 299-327 (1984; Zbl 0554.10009)] and G. Faltings [The general case of S. Lang’s conjecture, Perspect. Math. 15, 175-182 (1994)]. This already gives, to some extent, the connection with the second main topic of these lecture notes, namely the diophantine approximation on abelian varieties.
For the entire collection see [Zbl 0811.14019].

MSC:

11J25 Diophantine inequalities
11D75 Diophantine inequalities
11D72 Diophantine equations in many variables
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