Duke Mathematical Journal

Sur la non-densité des points entiers

Pascal Autissier

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Résumé

On donne des résultats de non-densité pour les points entiers sur des variétés affines, dans l'esprit de la conjecture de Lang-Vojta. En particulier, soit X une variété projective de dimension d2 sur un corps de nombres K (resp., sur C). Soit H la somme de 2d diviseurs amples sur X qui se coupent proprement. On montre que tout ensemble de points quasi-entiers (resp., toute courbe entière) sur XH est non Zariski-dense.

Abstract

We give nondensity results for integral points on affine varieties, in the spirit of the Lang-Vojta conjecture. In particular, let X be a projective variety of dimension d2 over a number field K (resp., over C). Let H be the sum of 2d properly intersecting ample divisors on X. We show that any set of quasi-integral points (resp., any integral curve) on XH is not Zariski-dense.

Article information

Source
Duke Math. J., Volume 158, Number 1 (2011), 13-27.

Dates
First available in Project Euclid: 3 May 2011

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1304429492

Digital Object Identifier
doi:10.1215/00127094-1276292

Mathematical Reviews number (MathSciNet)
MR2794367

Zentralblatt MATH identifier
1217.14020

Subjects
Primary: 14G25: Global ground fields
Secondary: 11J97: Analogues of methods in Nevanlinna theory (work of Vojta et al.) 11G35: Varieties over global fields [See also 14G25]

Citation

Autissier, Pascal. Sur la non-densité des points entiers. Duke Math. J. 158 (2011), no. 1, 13--27. doi:10.1215/00127094-1276292. https://projecteuclid.org/euclid.dmj/1304429492


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References

  • P. Autissier, Géométrie, points entiers et courbes entières, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 221–239.
  • P. Corvaja, A. Levin et U. Zannier, Integral points on threefolds and other varieties, Tohoku Math. J. 61 (2009), 589–601.
  • P. Corvaja et U. Zannier, A subspace theorem approach to integral points on curves, C. R. Math. Acad. Sci. Paris 334 (2002), 267–271.
  • —, On integral points on surfaces, Ann. of. Math. (2) 160 (2004), 705–726.
  • W. Fulton, Intersection Theory, 2nd ed., Ergeb. Math. Grenzgeb. 3, Springer, Berlin, 1998.
  • S. Lang, Number Theory, III: Diophantine Geometry, Encyclopaedia Math. Sci. 60, Springer, Berlin, 1991.
  • R. Lazarsfeld, Positivity in Algebraic Geometry, I: Classical Setting–-Line Bundles and Linear Series, Ergeb. Math. Grenzgeb. 48, Springer, Berlin, 2004.
  • A. Levin, Generalizations of Siegel's and Picard's theorems, Ann. of Math. (2) 170 (2009), 609–655.
  • H. Matsumura, Commutative Algebra, 2nd ed., Math. Lecture Note Ser. 56, Benjamin/Cummings, Reading, Mass., 1980.
  • H. P. Schlickewei, The $p$-adic Thue-Siegel-Roth-Schmidt theorem, Arch. Math. (Basel) 29 (1977), 267–270.
  • W. M. Schmidt, Diophantine approximation, Lecture Notes in Math. 785, Springer Berlin, 1980.
  • J.-P. Serre, Lectures on the Mordell-Weil Theorem, 3rd ed., Aspects Math., Friedr. Vieweg and Sons, Braunschweig, 1997.
  • P. Vojta, Diophantine Approximations and Value Distribution Theory, Lecture Notes in Math. 1239, Springer, Berlin, 1987.
  • —, A refinement of Schmidt's subspace theorem, Amer. J. Math. 111 (1989), 489–518.
  • —, Integral points on subvarieties of semiabelian varieties, I, Invent. Math. 126 (1996), 133–181.
  • —, On Cartan's theorem and Cartan's conjecture, Amer. J. Math. 119 (1997), 1–17.
  • S. Zhang, Small points and adelic metrics, J. Algebraic Geom. 4 (1995), 281–300.