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Borne générique pour le problème de Mordell-Lang

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Abstract

Given an abelian variety A over a number field and an integer D, we prove that there is only a finite number up to translation of curves on A with degree D with more than D7 dimA rational points. We describe a more general result for higher dimensional varieties on semi-abelian varieties. This extends work of J.-H. Evertse on linear equations.

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References

  1. Bombieri, E. et Zannier, U.: Heights of algebraic points on subvarieties of abelian varieties. Ann. Scuola Norm. Sup. Pisa Série IV. 23, 779–792 (1996)

    Google Scholar 

  2. Caporaso, L., Harris, J. et Mazur, B.: Uniformity of rational points. J. Am. Math. Soc. 10, 1–35 (1997)

    Article  Google Scholar 

  3. David, S. et Philippon, P.: Sous-variétés de torsion des variétés semi-abéliennes. C. R. Acad. Sci. Paris. 331, 587–592 (2000)

    Google Scholar 

  4. Evertse, J.-H.: Points on subvarieties of tori. A Panorama in Number Theory or the View from Baker’s Garden, Proc. Conf. Number Theory in honour of the 60th birthday of Prof. Alan Baker, Zurich 1999 (édité par G. Wüstholz). Cambridge Univ. Press. 2002, 214–230

  5. Evertse, J.-H.: Linear equations with unknowns from a multiplicative group whose solutions lie in a small number of subspaces. Indig. math. (NS) 15, 347–355 (2004)

    Google Scholar 

  6. Evertse, J.-H., Győry, K., Stewart, C. et Tijdeman, R.: S-unit equations in two unknowns. Invent. Math. 92, 461–477 (1988)

    Article  Google Scholar 

  7. Hrushovski, E.: The Manin-Mumford conjecture and the model theory of difference fields. Ann. Pure Appl. Logic 112, 43–115 (2001)

    Article  Google Scholar 

  8. Kollár, J.: Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 32. Springer-Verlag. Berlin. 1996

  9. McQuillan, M.: Division points on semi-abelian varieties. Invent. Math. 120, 143–159 (1995)

    Article  Google Scholar 

  10. Poonen, B.: Mordell-Lang plus Bogomolov. Invent. Math. 137, 413–425 (1999)

    Article  Google Scholar 

  11. Rémond, G.: Décompte dans une conjecture de Lang. Invent. Math. 142, 513–545 (2000)

    Article  Google Scholar 

  12. Rémond, G.: Sur les sous-variétés des tores. Comp. Math. 134, 337–366 (2002)

    Article  Google Scholar 

  13. Rémond, G.: Approximation diophantienne sur les variétés semi-abéliennes. Ann. Sci. École Norm. Sup. 36, 191–212 (2003)

    Google Scholar 

  14. Scanlon, T.: Automatic uniformity. Internat. Math. Res. Notices 62, 3317–3326 (2004)

    Article  MathSciNet  Google Scholar 

  15. Vojta, P.: Integral points on subvarieties of semi-abelian varieties, I. Invent. Math. 126, 133–181 (1996)

    Article  Google Scholar 

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  1. Institut Fourier, UMR 5582, BP 74, 38402, Saint-Martin-d’Hères Cedex, France

    Gaël Rémond

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  1. Gaël Rémond
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Correspondence to Gaël Rémond.

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Rémond, G. Borne générique pour le problème de Mordell-Lang. manuscripta math. 118, 85–97 (2005). https://doi.org/10.1007/s00229-005-0581-2

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