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On the application of ramified coverings in the theory of diophantine equations. (English. Russian original) Zbl 0702.14017

Math. USSR, Sb. 66, No. 1, 249-264 (1990); translation from Mat. Sb. 180, No. 2, 244-259 (1989).
In this article is discussed the geometric case of the Van de Van- Bogomolov-Miyaoka-Yau inequality and it is given its arithmetic analogue. The author shows that the validity of the mentioned inequality yields a positive solution of the following problems of diophantine geometry of algebraic curves having
(I) genus \(=1:\)
(a) the boundedness of the torsion over any fixed base field,
(b) the lower bound of the Tate height of the infinite order points,
(c) an estimation of the number of integral points in terms of the rank of the curve,
(d) a new effective proof of Siegel theorem on integral points;
(II) genus \(>1:\)
(a) an effective proof of Mordell conjecture about rational points.
(b) the nonexistence of nontrivial points on Fermat curves having sufficiently large prime degree.
Reviewer: S.Kotov

MSC:

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14E20 Coverings in algebraic geometry
14H25 Arithmetic ground fields for curves
14G05 Rational points
11D41 Higher degree equations; Fermat’s equation
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