A Note on heights in certain infinite extensions of Q

Enrico Bombieri; Umberto Zannier

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2001)

  • Volume: 12, Issue: 1, page 5-14
  • ISSN: 1120-6330

Abstract

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We study the behaviour of the absolute Weil height of algebraic numbers in certain infinite extensions of Q . In particular, we obtain a Northcott type property for infinite abelian extensions of finite exponent and also a Bogomolov type property for certain fields which are a p -adic analog of totally real fields. Moreover, we obtain a non-archimedean analog of a uniform distribution theorem of Bilu in the archimedean case.

How to cite

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Bombieri, Enrico, and Zannier, Umberto. "A Note on heights in certain infinite extensions of $\mathbb{Q}$." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 12.1 (2001): 5-14. <http://eudml.org/doc/252378>.

@article{Bombieri2001,
abstract = {We study the behaviour of the absolute Weil height of algebraic numbers in certain infinite extensions of $\mathbb\{Q\}$. In particular, we obtain a Northcott type property for infinite abelian extensions of finite exponent and also a Bogomolov type property for certain fields which are a $p$-adic analog of totally real fields. Moreover, we obtain a non-archimedean analog of a uniform distribution theorem of Bilu in the archimedean case.},
author = {Bombieri, Enrico, Zannier, Umberto},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Algebraic number theory; Heights; Uniform distribution; algebraic number theory; heights; uniform distribution},
language = {eng},
month = {3},
number = {1},
pages = {5-14},
publisher = {Accademia Nazionale dei Lincei},
title = {A Note on heights in certain infinite extensions of $\mathbb\{Q\}$},
url = {http://eudml.org/doc/252378},
volume = {12},
year = {2001},
}

TY - JOUR
AU - Bombieri, Enrico
AU - Zannier, Umberto
TI - A Note on heights in certain infinite extensions of $\mathbb{Q}$
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2001/3//
PB - Accademia Nazionale dei Lincei
VL - 12
IS - 1
SP - 5
EP - 14
AB - We study the behaviour of the absolute Weil height of algebraic numbers in certain infinite extensions of $\mathbb{Q}$. In particular, we obtain a Northcott type property for infinite abelian extensions of finite exponent and also a Bogomolov type property for certain fields which are a $p$-adic analog of totally real fields. Moreover, we obtain a non-archimedean analog of a uniform distribution theorem of Bilu in the archimedean case.
LA - eng
KW - Algebraic number theory; Heights; Uniform distribution; algebraic number theory; heights; uniform distribution
UR - http://eudml.org/doc/252378
ER -

References

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  1. Amoroso, F. - Dvornicich, R., A lower bound for the height in abelian extensions. J. Number Th., 80, 2000, 260-272. Zbl0973.11092MR1740514DOI10.1006/jnth.1999.2451
  2. Amoroso, F. - Zannier, U., A relative Dobrowolski’s lower bound over abelian extensions. Preprint 1999; Annali Sc. Norm. Sup. Pisa, to appear. Zbl1016.11026
  3. Bilu, Y., Limit distribution of small points on algebraic tori. Duke Math. J., 89, 1997, 465-476. Zbl0918.11035MR1470340DOI10.1215/S0012-7094-97-08921-3
  4. Lang, S., Algebra. 3rd ed., Addison-Wesley, 1994, xv + 912 pp. Zbl0712.00001MR197234
  5. Mahler, K., An inequality for the discriminant of a polynomial. Michigan Math. J., 11, 1964, 257-262. Zbl0135.01702MR166188
  6. Narkiewicz, W., Elementary and Analytic Theory of Algebraic Numbers. PWN - Polish Scientific Publishers & Springer-Verlag, Warszawa1990, xiv + 746 pp. Zbl0717.11045MR1055830
  7. Northcott, D.G., An inequality on the theory of arithmetic on algebraic varieties. Proc. Cambridge Philos. Soc., 45, 1949, 502-509. Zbl0035.30701MR33094
  8. Rumely, R., On Bilu’s equidistribution theorem. In: Spectral problems in geometry and arithmetic (Iowa City IA 1997). Contemp. Math., 237, AMS, Providence, RI1999, 159-166. Zbl1029.11030MR1710794DOI10.1090/conm/237/1710794
  9. Schinzel, A., On the product of the conjugates outside the unit circle of an algebraic number. Acta Arith., 24, 1973, 385-399. Addendum ibidem, 26, 1973, 329-361. Zbl0275.12004MR360515
  10. Smyth, C.J., On the measure of totally real algebraic numbers. (I). J. Austral. Math. Soc., Ser. A, 30, 1980-81, 137-149. Zbl0457.12001MR607924
  11. Smyth, C.J., On the measure of totally real algebraic numbers. (II). Math. Comp., 37, 1981, 205-208. Zbl0475.12001MR616373DOI10.2307/2007513

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