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The Mordell conjecture revisited. (English) Zbl 0722.14010

This paper presents a simplification of a recent proof by the reviewer [Ann. Math. (2) 133, No. 3, 509–548 (1991; Zbl 0774.14019)] of Mordell’s conjecture (first proved by Faltings in 1983), also using ideas of G. Faltings [Invent. Math. 73, 349–366 (1983; Zbl 0588.14026)]. In particular, the use of arithmetic algebraic geometry is completely removed, replaced by more classical methods using Weil’s theory of heights and the Riemann- Roch theorem for complex algebraic surfaces. In addition to its simplicity, this variant also has the advantage that explicit computations are much easier.

MSC:

14G05 Rational points
14H25 Arithmetic ground fields for curves
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11D41 Higher degree equations; Fermat’s equation
14K15 Arithmetic ground fields for abelian varieties
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References:

[1] G. Faltings , Diophantine approximation on abelian varieties . To appear in Annals of Math. ( 2 ) 133 ( 1991 ). Zbl 0734.14007 · Zbl 0734.14007
[2] A.O. Gelfond , Transcendental and Algebraic Numbers . (English translation by L. F. Boron.) Dover Publications , New York 1960 . Zbl 0090.26103 · Zbl 0090.26103
[3] S. Lang , Fundamentals of Diophantine Geometry . Springer-Verlag , New York , Berlin , Heidelberg , Tokyo 1983 . Zbl 0528.14013 · Zbl 0528.14013
[4] D. Mumford , A remark on Mordell’s conjecture . Amer. J. Math. 87 ( 1965 ), pp. 1007 - 1016 . Zbl 0151.27301 · Zbl 0151.27301
[5] D. Mumford , Algebraic Geometry I. Complex Projective Varieties . Springer-Verlag , Berlin , Heidelberg , New York 1976 . Zbl 0356.14002 · Zbl 0356.14002
[6] K.F. Roth , Rational approximations to algebraic numbers. Mathematika 2 ( 1955 ), pp. 1 - 20 . Zbl 0064.28501 · Zbl 0064.28501
[7] R. Vojta , Siegel’s theorem in the compact case . To appear in Annals of Math. ( 2 ) 133 ( 1991 ). Zbl 0774.14019 · Zbl 0774.14019
[8] A. Weil , Arithmetic on algebraic varieties . Annals of Math. ( 2 ) 53 ( 1951 ), pp. 412 - 444 . Zbl 0043.27002 · Zbl 0043.27002
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