Small points on a multiplicative group and class number problem
Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 27-39.

Let V be an algebraic subvariety of a torus 𝔾 m n 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.

Soit V une sous-variété algébrique du tore 𝔾 m n n et notons V * le complémentaire dans V de l’adhérence de Zariski de l’ensemble des points de torsion de V. Par un théorème de Zhang, V * est discrète pour la métrique induite par la hauteur normalisée h ^. Nous décrirons certaines versions quantitatives de ce résultat, proche des conjectures les plus précises que l’on puisse formuler, et ses applications à l’étude du groupe de classes d’idéaux de certains corps de nombres.

Received: 2005-12-31
Published online: 2008-12-03
DOI: https://doi.org/10.5802/jtnb.571
@article{JTNB_2007__19_1_27_0,
     author = {Francesco Amoroso},
     title = {Small points on a multiplicative group and class number problem},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {1},
     year = {2007},
     pages = {27-39},
     doi = {10.5802/jtnb.571},
     zbl = {1131.11044},
     mrnumber = {2332051},
     language = {en},
     url={jtnb.centre-mersenne.org/item/JTNB_2007__19_1_27_0/}
}
Amoroso, Francesco. Small points on a multiplicative group and class number problem. Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 27-39. doi : 10.5802/jtnb.571. https://jtnb.centre-mersenne.org/item/JTNB_2007__19_1_27_0/

[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. | MR 1713323 | Zbl 1011.11045

[3] F. Amoroso, S. David, Densité des points à cordonnées multiplicativement indépendantes. Ramanujan J. 5 (2001), 237–246. | MR 1876697 | Zbl 0996.11046

[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. | Zbl 1041.11048

[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. | Numdam | MR 2075986

[6] F. Amoroso, S. David, Points de petite hauteur sur une sous-variété d’un tore (2005). Compos. Math. (to appear). | Zbl 05042485

[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. | MR 1963737 | Zbl 1040.11077

[8] F. Amoroso, R. Dvornicich, A Lower Bound for the Height in Abelian Extensions. J. Number Theory 80, no. 2 (2000), 260–272. | MR 1740514 | Zbl 0973.11092

[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). | Zbl 05124517

[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. | Numdam | Zbl 1016.11026

[11] E. Bombieri, U. Zannier, Algebraic points on subvarieties of 𝔾 m n . Internat. Math. Res. Notices, 7 (1995), 333–347. | MR 1350686 | Zbl 0848.11030

[12] P. Borwein, E. Dobrowolski, M. Mossinghoff, Lehmer’s problem for polynomials with odd coefficients. Bull. London Math. Soc., 36 (2004), 332–338.

[13] T. Chinburg, On the arithmetic of two constructions of Salem numbers. J. Reine Angew. Math. 348 (1984), 166-179. | MR 733929 | Zbl 0517.12001

[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. | Numdam | MR 1736526 | Zbl 1002.11055

[15] E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arith. 34 (1979), 391–401. | MR 543210 | Zbl 0416.12001

[16] J. C. Lagarias, A. M. Odlyzko, Effective versions of the C ˇebotarev density theorem. Algebraic Number Fields, Durham Symposium, Academic Press, 1977. | MR 447191 | Zbl 0362.12011

[17] M. Laurent, Equations diophantiennes exponentielles. Invent. Math. 78 (1984), 299–327. | MR 767195 | Zbl 0554.10009

[18] D. H. Lehmer, Factorization of certain cyclotomic functions. Ann. of Math. 34 (1933), 461–479. | MR 1503118 | Zbl 0007.19904

[19] S. Louboutin, R.  Okazaki, Exponents of the ideal class groups of CM number fields. Math. Z. 243, no.1 (2003), 155–159. | MR 1953054 | Zbl 1049.11122

[20] M. Mignotte, Estimations élémentaires effectives sur les nombres algébriques. Publications I. R. M. A., Strasbourg, 1979.

[21] A. M. Odlyzko, Some analytic estimates of class numbers and discriminants. Invent. Math. 29 (1975), 275–286. | MR 376613 | Zbl 0306.12005

[22] P. Philippon, Critères pour l’indépendance algébrique. Inst. Hautes Etudes Sci. Publ. Math. 64 (1986), 5–52. | Numdam | Zbl 0615.10044

[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. | Zbl 05001260

[24] C. Pontreau, Points de petite hauteur d’une surface. Canadian Journal of Mathematics. À paraître.

[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. | MR 360515 | Zbl 0275.12004

[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. | MR 1628798 | Zbl 0917.11023

[27] J. H. Silverman, Lower bounds for height functions. Duke Math. J. 51 (1984), 395–403. | MR 747871 | Zbl 0579.14035

[28] D. Simon, The index of nonmonic polynomials. Indag. Math. New Ser. 12, no.4 (2001), 505–517. | MR 1908878 | Zbl 1020.11065

[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. | MR 289451 | Zbl 0235.12003

[30] S. Zhang, Positive line bundles on arithmetic varieties. J. Amer. Math. Soc. 8, no. 1 (1995), 187–221. | MR 1254133 | Zbl 0861.14018