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

top
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

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.