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Vojta’s inequality in higher dimension. (Inégalité de Vojta en dimension supérieure.) (French) Zbl 0972.11052

Vojta’s proof of Mordell’s conjecture (now a theorem of Faltings) has been revisited by E. Bombieri [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 17, No. 4, 615-640 (1990; Zbl 0722.14010); ibid. 18, No. 3, 473 (1991; Zbl 0763.14007)] who derived the finiteness of rational points on a curve \(C\) of genus \(\geq 2\) from two inequalities: for distinct algebraic points \(x\) and \(w\) on \(C\) satisfying \(|w|\geq |z|\geq \gamma\) and \(\langle z,w\rangle\geq (3/4)|z|\cdot |w|\), Mumford’s inequality asserts \(|w|\geq 2|z|\) and Vojta’s inequality is \(|w|\leq \gamma |z|\). Here \(\gamma\) depends only on \(C\) and \(|\cdot|\), \(\langle \cdot,\cdot\rangle\) refer to the Neron-Tate height on the Jacobian of \(C\).
G. Faltings [The general case of S. Lang’s conjecture. Barsotti symposium in algebraic geometry, 1991, Perspect. Math. 15, 175-182 (1994; Zbl 0823.14009)] also proved a generalization of Mordell’s conjecture which was conjectured by Lang: for a subvariety \(X\) of an abelian variety \(A\) over a number field \(K\), there are subvarieties \(B_{1},\ldots,B_{n}\) and points \(x_{1},\ldots,x_{n}\) in \(X(K)\) such that \(X(K)=\bigcup_{i=1}^{n} (x_{i}+B_{i}(K))\).
The main goal of this paper is to generalize Vojta’s inequality to this more general context. The author succeeds in producing a simplified proof of Faltings’ Theorem solving Lang’s Conjecture where Néron models are no more needed. Also the approach is more effective: it is the first time the constants are so explicitely computed. The final estimates display some uniformity which yields new results for family of curves. The result of this paper is one of the main tools in the subsequent work by the author [Invent. Math. 142, No. 3, 513-545 (2000; Zbl 0972.11053)] where he also extends Mumford’s inequality to higher dimension.
Among the main new ingredients of this important paper are bounds for the height of multiplication formulae by an integer on an abelian variety; such a estimates are required for using Siegel’s lemma and producing a section with a small height.

MSC:

11G10 Abelian varieties of dimension \(> 1\)
14G05 Rational points
11D45 Counting solutions of Diophantine equations
14K15 Arithmetic ground fields for abelian varieties
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References:

[1] E. Bombieri , The Mordell conjecture revisited , Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 ( 1990 ) 615 - 640 . - Erratum. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18 ( 1991 ) 473 . Numdam | MR 1093712 | Zbl 0763.14007 · Zbl 0763.14007
[2] E. Bombieri , On G-functions, In: ”Recent progress in analytic number theory” , vol. II ( Durham 1979 ), H. Halberstam et C. Hooley (eds), Academic Press , 1981 , pp. 1 - 67 . MR 637359 | Zbl 0461.10031 · Zbl 0461.10031
[3] J.-B. Bost - H. Gillet - C. Soulé , Heights ofprojective varieties and positive Green forms , J. Amer. Math. Soc. 7 ( 1994 ) 903 - 1027 . MR 1260106 | Zbl 0973.14013 · Zbl 0973.14013
[4] M. Chardin , ”Contributions à l’algèbre commutative effective et à la théorie de l’élimination” , Thèse. Univ. Paris VI , 1990 .
[5] T. De Diego , Points rationnels sur les familles de courbes de genre au moins 2 , J. of Number Theory 67 ( 1997 ) 85 - 114 . MR 1485428 | Zbl 0896.11025 · Zbl 0896.11025
[6] B. Edixhoven - J.-H. Evertse , ”Diophantine approximation and abelian varieties” , Lecture Notes in Mathematics 1566 , Springer-Verlag , Berlin , 1994 . MR 1288998 | Zbl 0811.14019 · Zbl 0811.14019
[7] C. Faber , Geometric part of Faltings’s proof , In: ”[EE]”, Chapitre IX, pp. 83 - 91 . MR 1289007 | Zbl 0811.14023 · Zbl 0811.14023
[8] G. Faltings , Diophantine approximation on abelian varieties , Ann. of Math. 133 ( 1991 ) 549 - 576 . MR 1109353 | Zbl 0734.14007 · Zbl 0734.14007
[9] G. Faltings , The general case of S. Lang’s conjecture, In: ”Barsotti Symposium in Algebraic Geometry” ( Abano Terme , 1991 ). Perspect. Math. 15. Academic Press , San Diego , 1994 , pp. 175 - 182 . MR 1307396 | Zbl 0823.14009 · Zbl 0823.14009
[10] R. Hartshorne , ” Algebraic Geometry ”, Springer-Verlag , 1977 . MR 463157 | Zbl 0367.14001 · Zbl 0367.14001
[11] ] M. Hindry , Autour d’une conjecture de Serge Lang , Invent. Math. 94 ( 1988 ) 575 - 603 . MR 969244 | Zbl 0638.14026 · Zbl 0638.14026
[12] M. Hindry , Sur les conjectures de Mordell et de Lang [d’après Vojta, Faltings et Bombieri] , Astérisque 209 ( 1992 ) 39 - 56 . MR 1211002 | Zbl 0792.14009 · Zbl 0792.14009
[13] D. Mumford , A remark on Mordell’s conjecture , Amer. J. Math. 87 ( 1965 ) 1007 - 1016 . MR 186624 | Zbl 0151.27301 · Zbl 0151.27301
[14] F. Oort , ”The” general case of S. Lang’s conjecture (after Faltings ), In : ”[EE]” Chapitre XIII, pp. 117 - 122 . Zbl 0811.14025 · Zbl 0811.14025
[15] P. Philippon , Sur des hauteurs alternatives III , J. Math. Pures Appl. 74 ( 1995 ) 345 - 365 . MR 1341770 | Zbl 0878.11025 · Zbl 0878.11025
[16] G. Rémond , Géométrie diophantienne multiprojective, Chapitre 7 de : ”Introduction to Algebraic Independence Theory” , Y. Nesterenko - P. Philippon (eds), à paraître dans Lecture Notes in Mathematic s, Springer-Verlag . MR 1837822
[17] G. Rémond , Sur le théorème du produit, XXIèmes Journées Arithmétiques , Rome , 1999 .
[18] P. Vojta , ” Applications of arithmetic algebraic geometry to diophantine approximations ”, Lecture Notes in Mathematics , 1553 , Springer-Verlag . MR 1338861 | Zbl 0846.14009 · Zbl 0846.14009
[19] O. Zariski - P. Samuel , ” Commutative Algebra I ”, Van Nostrand , New York , 1958 . MR 90581 | Zbl 0081.26501 · Zbl 0081.26501
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