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Distribution of points of small height in multiplicative groups. (Distribution des points de petite hauteur dans les groupes multiplicatifs.) (French) Zbl 1150.11021
The authors enhance a main result of their earlier paper [J. Reine Angew. Math. 513, 145–179 (1999; Zbl 1011.11045)]. Suppose \(n\) is a positive integer and \(V\subset{\mathbb G}_m^n\) is an algebraic subvariety of \({\mathbb G}_m^n\) defined over a cyclotomic extension \(K\) of \({\mathbb Q}\). Denote by \(V^*\) the complement of the torsion subvarieties, i.e. of translates of proper subtori of \({\mathbb G}_m^n\) by torsion points. Then there is a universal effectively computable positive real constant \(c(n)\) such that if \(V\) is the intersection of hypersurfaces of degree at most \(\delta\), then for all \(\alpha\in V^*\) \[ h(\alpha)\geq c(n)^{-1}\delta^{-1} \left[\log\big(3[K:{\mathbb Q}]\delta\big)\right]^{-\kappa(n)}, \] where \(\kappa(n)=2n(n+1)!^n-1\) (this value corrects a minor mistake in the paper cited above) and \(h\) is the height, given by some fixed projective embedding of \({\mathbb G}_m^n\). For \(K={\mathbb Q}\) this is close to being best possible.
The proofs closely follow arguments from (loc. cit.), making the changes needed to replace \({\mathbb Q}\) by \(K\) and some other refinements.

MSC:
11G10 Abelian varieties of dimension \(> 1\)
11J81 Transcendence (general theory)
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
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References:
[1] F. Amoroso - S. David, Le problème de Lehmer en dimension supérieure, J. reine angew. Math. 513 (1999), 145-179. Zbl1011.11045 MR1713323 · Zbl 1011.11045 · doi:10.1515/crll.1999.058
[2] F. Amoroso - S. David, Minoration de la hauteur normalisée des hypersurfaces, Acta Arith. 92 (2000), 340-366. Zbl0948.11025 MR1760242 · Zbl 0948.11025 · eudml:207392
[3] F. Amoroso - S. David, Densité des points à coordonnées multiplicativement indépendantes, Ramanujan Math. Journal. 5 (2001), 237-246. Zbl0996.11046 MR1876697 · Zbl 0996.11046 · doi:10.1023/A:1012966409353
[4] F. Amoroso - S. David, Minoration de la hauteur normalisée dans un tore, Journal de l’Institut de Mathématiques de Jussieu 2 (2003), 335-381. Zbl1041.11048 MR1990219 · Zbl 1041.11048 · doi:10.1017/S1474748003000094
[5] F. Amoroso - R. Dvornicich, A lower bound for the height in abelian extensions, J. Number Theory 80 (2000), 260-272. Zbl0973.11092 MR1740514 · Zbl 0973.11092 · doi:10.1006/jnth.1999.2451
[6] F. Amoroso - U. Zannier, Minoration de la hauteur normalisée dans un tore, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), 711-727. Zbl1016.11026 MR1817715 · Zbl 1016.11026 · numdam:ASNSP_2000_4_29_3_711_0 · eudml:84424
[7] E. Bombieri - J. Vaaler, Siegel’s lemma, Invent. Math. 73 (1983), 11-32. Zbl0533.10030 MR707346 · Zbl 0533.10030 · doi:10.1007/BF01393823 · eudml:143033
[8] E. Bombieri - U. Zannier, Algebraic points on subvarieties of \({\mathbb{G}}_m^n\), Int. Math. Res. Not. 7 (1995), 333-347. Zbl0848.11030 MR1350686 · Zbl 0848.11030 · doi:10.1155/S1073792895000250
[9] 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. Zbl0709.13007 MR1020108 · Zbl 0709.13007 · numdam:BSMF_1989__117_3_305_0 · eudml:87582
[10] 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. Zbl0993.11034 MR1799933 · Zbl 0993.11034 · doi:10.1515/crll.2000.096
[11] 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). Zbl1002.11055 MR1736526 · Zbl 1002.11055 · numdam:ASNSP_1999_4_28_3_489_0 · eudml:84386
[12] E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith. 34 (1979), 391-401. Zbl0416.12001 MR543210 · Zbl 0416.12001 · eudml:205618
[13] H. Lehmer, Factorisation of some cyclotomic functions, Ann. of Math. 34 (1933), 461-479. Zbl0007.19904 · Zbl 0007.19904 · doi:10.2307/1968172
[14] 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. Zbl0617.14001 MR1428487 · Zbl 0617.14001 · numdam:BSMF_1986__114__355_0 · eudml:87515
[15] W. Schmidt, Heights of points on subvarieties of \({\mathbb{G}}_m^n\), 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. Zbl0917.11023 MR1628798 · Zbl 0917.11023
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