Distribution des points de petite hauteur dans les groupes multiplicatifs

Francesco Amoroso[1]; Sinnou David[2]

  • [1] U. M. R. 6139 (C. N. R. S.) Laboratoire de Mathématiques Nicolas Oresme Département de Mathématiques Université de Caen Campus II, BP 5186 14032 Caen Cédex, France
  • [2] U. M. R. 7586 (C. N. R. S.) – U. F. R. 921 Problèmes Diophantiens Département de Mathématiques Université Pierre et Marie Curie 4, Place Jussieu 75005 Paris, France

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2004)

  • Volume: 3, Issue: 2, page 325-348
  • ISSN: 0391-173X

Abstract

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We prove a new lower bound for the height of points on a subvariety  V of a multiplicative torus, which lie outside the union of torsion subvarieties of  V . Although lower bounds for the heights of these points where already known (decreasing multi-exponential function of the degree for Scmhidt and Bombieri–Zannier, [Sch], [Bo-Za], and inverse monomial in the degree by the second author of this note and P. Philippon, [Da-Phi]), our method provesup to an ε the sharpest conjectures that can be formulated.

How to cite

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Amoroso, Francesco, and David, Sinnou. "Distribution des points de petite hauteur dans les groupes multiplicatifs." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 3.2 (2004): 325-348. <http://eudml.org/doc/84532>.

@article{Amoroso2004,
abstract = {We prove a new lower bound for the height of points on a subvariety $V$ of a multiplicative torus, which lie outside the union of torsion subvarieties of $V$. Although lower bounds for the heights of these points where already known (decreasing multi-exponential function of the degree for Scmhidt and Bombieri–Zannier, [Sch], [Bo-Za], and inverse monomial in the degree by the second author of this note and P. Philippon, [Da-Phi]), our method provesup to an $\varepsilon $ the sharpest conjectures that can be formulated.},
affiliation = {U. M. R. 6139 (C. N. R. S.) Laboratoire de Mathématiques Nicolas Oresme Département de Mathématiques Université de Caen Campus II, BP 5186 14032 Caen Cédex, France; U. M. R. 7586 (C. N. R. S.) – U. F. R. 921 Problèmes Diophantiens Département de Mathématiques Université Pierre et Marie Curie 4, Place Jussieu 75005 Paris, France},
author = {Amoroso, Francesco, David, Sinnou},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {fre},
number = {2},
pages = {325-348},
publisher = {Scuola Normale Superiore, Pisa},
title = {Distribution des points de petite hauteur dans les groupes multiplicatifs},
url = {http://eudml.org/doc/84532},
volume = {3},
year = {2004},
}

TY - JOUR
AU - Amoroso, Francesco
AU - David, Sinnou
TI - Distribution des points de petite hauteur dans les groupes multiplicatifs
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2004
PB - Scuola Normale Superiore, Pisa
VL - 3
IS - 2
SP - 325
EP - 348
AB - We prove a new lower bound for the height of points on a subvariety $V$ of a multiplicative torus, which lie outside the union of torsion subvarieties of $V$. Although lower bounds for the heights of these points where already known (decreasing multi-exponential function of the degree for Scmhidt and Bombieri–Zannier, [Sch], [Bo-Za], and inverse monomial in the degree by the second author of this note and P. Philippon, [Da-Phi]), our method provesup to an $\varepsilon $ the sharpest conjectures that can be formulated.
LA - fre
UR - http://eudml.org/doc/84532
ER -

References

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  1. [Am-Da] F. Amoroso – S. David, Le problème de Lehmer en dimension supérieure, J. reine angew. Math. 513 (1999), 145-179. Zbl1011.11045MR1713323
  2. [Am-Da2] F. Amoroso – S. David, Minoration de la hauteur normalisée des hypersurfaces, Acta Arith. 92 (2000), 340-366. Zbl0948.11025MR1760242
  3. [Am-Da3] F. Amoroso – S. David, Densité des points à coordonnées multiplicativement indépendantes, Ramanujan Math. Journal. 5 (2001), 237-246. Zbl0996.11046MR1876697
  4. [Am-Da4] F. Amoroso – S. David, Minoration de la hauteur normalisée dans un tore, Journal de l’Institut de Mathématiques de Jussieu 2 (2003), 335-381. Zbl1041.11048MR1990219
  5. [Am-Dv] F. Amoroso – R. Dvornicich, A lower bound for the height in abelian extensions, J. Number Theory 80 (2000), 260-272. Zbl0973.11092MR1740514
  6. [Am-Za] F. Amoroso – U. Zannier, Minoration de la hauteur normalisée dans un tore, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), 711-727. Zbl1016.11026MR1817715
  7. [Bo-Va] E. Bombieri – J. Vaaler, Siegel’s lemma, Invent. Math. 73 (1983), 11-32. Zbl0533.10030MR707346
  8. [Bo-Za] E. Bombieri – U. Zannier, Algebraic points on subvarieties of 𝔾 m n , Int. Math. Res. Not. 7 (1995), 333-347. Zbl0848.11030MR1350686
  9. [Ch] M. Chardin, Une majoration de la fonction de Hilbert et ses conséquences pour l’interpolation algébrique, Bulletin de la Société Mathématique de France 117 (1988), 305-318. Voir aussi Contributions à l’algèbre commutative effective et à la théorie de l’élimination, Thèse de doctorat, Université de Paris VI, 1990. Zbl0709.13007MR1020108
  10. [Da–Hi] S. David – M. Hindry, Minoration de la hauteur de Néron–Tate sur les variétés abéliennes de type C.M., J. reine angew. Math. 529 (2000), 1-74. Zbl0993.11034MR1799933
  11. [Da–Ph] S. David – P. Philippon, Minorations des hauteurs normalisées des sous-variétés des tores, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), 489-543 ; Errata ibidem 29 (2000). Zbl1002.11055MR1736526
  12. [Do] E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith. 34 (1979), 391-401. Zbl0416.12001MR543210
  13. [Le] H. Lehmer, Factorisation of some cyclotomic functions, Ann. of Math. 34 (1933), 461-479. Zbl0007.19904
  14. [Ph] P. Philippon, Lemmes de zéros dans les groupes algébriques commutatifs, Bull. Soc. Math. France 114 (1986), 355-383 ; Nouveaux lemmes de zéros dans les groupes algébriques commutatifs, Rocky Mountain J. Math. (3) 26 (1996), 1069-1088. Zbl0617.14001MR1428487
  15. [Sch] W. Schmidt, Heights of points on subvarieties of 𝔾 m n , in “Number theory, Séminaire de Théorie des Nombres de Paris” 1993-1994 (S. David editeur) London Math. Soc. Ser. 235 (1996), 157-187, Cambridge University Press. Zbl0917.11023MR1628798

Citations in EuDML Documents

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  1. Enrico Bombieri, David Masser, Umberto Zannier, Intersecting a plane with algebraic subgroups of multiplicative groups
  2. Francesco Amoroso, Small points on a multiplicative group and class number problem
  3. Emmanuel Delsinne, Le problème de Lehmer relatif en dimension supérieure
  4. Gaël Rémond, Autour de la conjecture de Zilber-Pink
  5. Nicolas Ratazzi, Intersection de courbes et de sous-groupes et problèmes de minoration de hauteur dans les variétés abéliennes C.M.

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