Small points on a multiplicative group and class number problem

Francesco Amoroso[1]

  • [1] Université de Caen Laboratoire de Mathématiques Nicolas Oresme, U.M.R. 6139 (C.N.R.S.) Campus II, BP 5186 F–14032 Caen Cedex

Journal de Théorie des Nombres de Bordeaux (2007)

  • Volume: 19, Issue: 1, page 27-39
  • ISSN: 1246-7405

Abstract

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Let V be an algebraic subvariety of a torus 𝔾 m n ↪ ℙ n and denote by V * the complement in V of the Zariski closure of the set of torsion points of V . By a theorem of Zhang, V * is discrete for the metric induced by the normalized height h ^ . We describe some quantitative versions of this result, close to the conjectural bounds, and we discuss some applications to study of the class group of some number fields.

How to cite

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Amoroso, Francesco. "Small points on a multiplicative group and class number problem." Journal de Théorie des Nombres de Bordeaux 19.1 (2007): 27-39. <http://eudml.org/doc/249982>.

@article{Amoroso2007,
abstract = {Let $V$ be an algebraic subvariety of a torus $\{\mathbb\{G\}\}_m^n\hookrightarrow \{\mathbb\{P\}\}^n$ and denote by $V^*$ the complement in $V$ of the Zariski closure of the set of torsion points of $V$. By a theorem of Zhang, $V^*$ is discrete for the metric induced by the normalized height $\hat\{h\}$. We describe some quantitative versions of this result, close to the conjectural bounds, and we discuss some applications to study of the class group of some number fields.},
affiliation = {Université de Caen Laboratoire de Mathématiques Nicolas Oresme, U.M.R. 6139 (C.N.R.S.) Campus II, BP 5186 F–14032 Caen Cedex},
author = {Amoroso, Francesco},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Lehmer's problem; heights; class numbers; exponent of class group},
language = {eng},
number = {1},
pages = {27-39},
publisher = {Université Bordeaux 1},
title = {Small points on a multiplicative group and class number problem},
url = {http://eudml.org/doc/249982},
volume = {19},
year = {2007},
}

TY - JOUR
AU - Amoroso, Francesco
TI - Small points on a multiplicative group and class number problem
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2007
PB - Université Bordeaux 1
VL - 19
IS - 1
SP - 27
EP - 39
AB - Let $V$ be an algebraic subvariety of a torus ${\mathbb{G}}_m^n\hookrightarrow {\mathbb{P}}^n$ and denote by $V^*$ the complement in $V$ of the Zariski closure of the set of torsion points of $V$. By a theorem of Zhang, $V^*$ is discrete for the metric induced by the normalized height $\hat{h}$. We describe some quantitative versions of this result, close to the conjectural bounds, and we discuss some applications to study of the class group of some number fields.
LA - eng
KW - Lehmer's problem; heights; class numbers; exponent of class group
UR - http://eudml.org/doc/249982
ER -

References

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  1. F. Amoroso, Une minoration pour l’exposant du groupe des classes d’un corps engendré par un nombre de Salem. International Journal of Number Theory 3, no. 2 (2007), 1–13. 
  2. F. Amoroso, S. David, Le problème de Lehmer en dimension supérieure. J. Reine Angew. Math. 513 (1999), 145–179. Zbl1011.11045MR1713323
  3. F. Amoroso, S. David, Densité des points à cordonnées multiplicativement indépendantes. Ramanujan J. 5 (2001), 237–246. Zbl0996.11046MR1876697
  4. F. Amoroso, S. David, Minoration de la hauteur normalisée dans un tore. Journal de l’Institut de Mathématiques de Jussieu 2, no. 3 (2003), 335–381. Zbl1041.11048
  5. F. Amoroso, S. David, Distribution des points de petite hauteur dans les groupes multiplicatifs. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 3, no. 2 (2004), 325–348. MR2075986
  6. F. Amoroso, S. David, Points de petite hauteur sur une sous-variété d’un tore (2005). Compos. Math. (to appear). Zbl1116.11045
  7. F. Amoroso, R. Dvornicich, Lower bounds for the height and size of the ideal class group in CM fields. Monatsh. Math. 138, no.2 (2003), 85–94. Zbl1040.11077MR1963737
  8. F. Amoroso, R. Dvornicich, A Lower Bound for the Height in Abelian Extensions. J. Number Theory 80, no. 2 (2000), 260–272. Zbl0973.11092MR1740514
  9. F. Amoroso, F. Nuccio, Algebraic Numbers of Small Weil’s height in CM-fields: on a Theorem of Schinzel (2005). J. Number Theory (to appear). Zbl1111.11032
  10. F. Amoroso, U. Zannier, A relative Dobrowolski’s lower bound over abelian extensions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29, no 3 (2000), 711–727. Zbl1016.11026
  11. E. Bombieri, U. Zannier, Algebraic points on subvarieties of 𝔾 m n . Internat. Math. Res. Notices, 7 (1995), 333–347. Zbl0848.11030MR1350686
  12. P. Borwein, E. Dobrowolski, M. Mossinghoff, Lehmer’s problem for polynomials with odd coefficients. Bull. London Math. Soc., 36 (2004), 332–338. Zbl1172.11034
  13. T. Chinburg, On the arithmetic of two constructions of Salem numbers. J. Reine Angew. Math. 348 (1984), 166-179. Zbl0517.12001MR733929
  14. S. David, P. Philippon, Minorations des hauteurs normalisées des sous-variétés des tores. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28, no. 3 (1999), 489-543; Errata, ibidem 29, no 3 (2000), 729-731. Zbl1002.11055MR1736526
  15. E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arith. 34 (1979), 391–401. Zbl0416.12001MR543210
  16. J. C. Lagarias, A. M. Odlyzko, Effective versions of the C ˇ ebotarev density theorem. Algebraic Number Fields, Durham Symposium, Academic Press, 1977. Zbl0362.12011MR447191
  17. M. Laurent, Equations diophantiennes exponentielles. Invent. Math. 78 (1984), 299–327. Zbl0554.10009MR767195
  18. D. H. Lehmer, Factorization of certain cyclotomic functions. Ann. of Math. 34 (1933), 461–479. Zbl0007.19904MR1503118
  19. S. Louboutin, R.  Okazaki, Exponents of the ideal class groups of CM number fields. Math. Z. 243, no.1 (2003), 155–159. Zbl1049.11122MR1953054
  20. M. Mignotte, Estimations élémentaires effectives sur les nombres algébriques. Publications I. R. M. A., Strasbourg, 1979. Zbl0502.10016
  21. A. M. Odlyzko, Some analytic estimates of class numbers and discriminants. Invent. Math. 29 (1975), 275–286. Zbl0306.12005MR376613
  22. P. Philippon, Critères pour l’indépendance algébrique. Inst. Hautes Etudes Sci. Publ. Math. 64 (1986), 5–52. Zbl0615.10044
  23. C. Pontreau, Minoration effective de la hauteur des points d’une courbe de 𝔾 m 2 . Acta Arith. 120, no. 1 (2005), 1–26. Zbl1155.11331
  24. C. Pontreau, Points de petite hauteur d’une surface. Canadian Journal of Mathematics. À paraître. Zbl1239.11072
  25. A. Schinzel, On the product of the conjugates outside the unit circle of an algebraic number. Acta Arith. 24 (1973), 385–399. Addendum; ibid. 26 (1973), 329–361. Zbl0275.12004MR360515
  26. W. M. Schmidt, Heights of points on subvarieties of 𝔾 m n . In Number Theory 93–94. S. David editor, London Math. Soc. Ser., volume 235, Cambridge University Press, 1996. Zbl0917.11023MR1628798
  27. J. H. Silverman, Lower bounds for height functions. Duke Math. J. 51 (1984), 395–403. Zbl0579.14035MR747871
  28. D. Simon, The index of nonmonic polynomials. Indag. Math. New Ser. 12, no.4 (2001), 505–517. Zbl1020.11065MR1908878
  29. C. J. Smyth, On the product of the conjugates outside the unit circle of an algebraic number. Bull. London Math. Soc. 3 (1971), 169–175. Zbl0235.12003MR289451
  30. S. Zhang, Positive line bundles on arithmetic varieties. J. Amer. Math. Soc. 8, no. 1 (1995), 187–221. Zbl0861.14018MR1254133

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