×

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
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] G. Faltings , Diophantine approximation on abelian varieties . To appear in Annals of Math. ( 2 ) 133 ( 1991 ). Zbl 0734.14007 · Zbl 0734.14007 · doi:10.2307/2944319
[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 · doi:10.2307/2373258
[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 · doi:10.1112/S0025579300000644
[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 · doi:10.2307/2944318
[8] A. Weil , Arithmetic on algebraic varieties . Annals of Math. ( 2 ) 53 ( 1951 ), pp. 412 - 444 . Zbl 0043.27002 · Zbl 0043.27002 · doi:10.2307/1969564
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.