Problems and results on the distribution of algebraic points on algebraic varieties
Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 1, pp. 41-57.

Cet article est un exposé de plusieurs résultats sur la distribution des points algébriques sur les variétés algébriques.

This is a survey paper on the distribution of algebraic points on algebraic varieties.

DOI : 10.5802/jtnb.656
Bombieri, Enrico 1

1 Institute for Advanced Study 1 Einstein Drive Princeton, NJ 08540, USA
@article{JTNB_2009__21_1_41_0,
     author = {Bombieri, Enrico},
     title = {Problems and results on the distribution of algebraic points on algebraic varieties},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {41--57},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {1},
     year = {2009},
     doi = {10.5802/jtnb.656},
     mrnumber = {2537702},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.656/}
}
TY  - JOUR
AU  - Bombieri, Enrico
TI  - Problems and results on the distribution of algebraic points on algebraic varieties
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2009
SP  - 41
EP  - 57
VL  - 21
IS  - 1
PB  - Université Bordeaux 1
UR  - http://www.numdam.org/articles/10.5802/jtnb.656/
DO  - 10.5802/jtnb.656
LA  - en
ID  - JTNB_2009__21_1_41_0
ER  - 
%0 Journal Article
%A Bombieri, Enrico
%T Problems and results on the distribution of algebraic points on algebraic varieties
%J Journal de théorie des nombres de Bordeaux
%D 2009
%P 41-57
%V 21
%N 1
%I Université Bordeaux 1
%U http://www.numdam.org/articles/10.5802/jtnb.656/
%R 10.5802/jtnb.656
%G en
%F JTNB_2009__21_1_41_0
Bombieri, Enrico. Problems and results on the distribution of algebraic points on algebraic varieties. Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 1, pp. 41-57. doi : 10.5802/jtnb.656. http://www.numdam.org/articles/10.5802/jtnb.656/

[1] F. Amoroso, S. David, Le problème de Lehmer en dimension supérieure. J. reine angew. Math. 513 (1999), 145–179. | MR | Zbl

[2] F. Amoroso, R. Dvornicich, A lower bound for the height in an abelian extension. J. Number Th. 80 (2000), 260–272. | MR | Zbl

[3] V.V. Batyrev, Yu.I. Manin, Sur le nombre des points rationnels de hauteur bornée des variétés algébriques. Math. Annalen 286 (1980), 27–43. | MR | Zbl

[4] Yu.F. Bilu, Limit distribution of small points on algebraic tori. Duke Math. J. 89 (1997), 465–476. | MR | Zbl

[5] E. Bombieri, W. Gubler, Heights in Diophantine Geometry, Cambridge Univ. Press 2006, xvi+652pp. | MR | Zbl

[6] E. Bombieri, H.P.F. Swinnerton-Dyer, On the local zeta function of a cubic threefold. Ann. Scuola Norm. Super. Pisa Sci. Fis. Mat. (3) 21 (1967), 1–29. | Numdam | MR | Zbl

[7] E. Bombieri, U. Zannier, A note on heights in infinite extensions of . Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 12 (2001), 5–14. | MR | Zbl

[8] C. Christensen, W. Gubler, Der relative Satz von Schanuel. Manuscripta Math. 126 (2008), 505–525. | MR | Zbl

[9] H. Clemens, P.A. Griffiths, The intermediate Jacobian of the cubic threefold. Ann. of Math. (2) 95 (1972), 281–356. | MR | Zbl

[10] 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: ibid. (4) 29 (2000), 729–731. | Numdam | MR | Zbl

[11] R. de la Bretèche, Points rationnels sur la cubique de Segre. Proc. London Math. Soc. (3) 95 (2007), 69–155. | MR | Zbl

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

[13] G. Faltings, Diophantine approximation on abelian varieties. Ann. of Math. (2) 133 (1991), 549–576. | MR | Zbl

[14] K.B. Ford, New estimates for mean values of Weyl sums. Internat. Math. Res. Notices (1995), 155–171. | MR | Zbl

[15] J. Franke, Yu.I. Manin, Y. Tschinkel, Rational points of bounded height on Fano varieties. Inventiones Math. 95 (1989), 421–435. Erratum:“Rational points of bounded height on Fano varieties”. ibid. 105 (1990), 463. | MR | Zbl

[16] X. Gao, On Northcott’s theorem. Ph. D. Thesis, University of Colorado (1995).

[17] C. Hooley, On some topics connected with Waring’s problem. J. Reine Angew. Math. 369 (1986), 110–153. | MR | Zbl

[18] R. Louboutin, Sur la mesure de Mahler d’un nombre algébrique. C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), 707–708. | MR | Zbl

[19] D. Masser, J.D. Vaaler, Counting algebraic numbers with large height. Trans. Amer. Math. Soc. 359 (2007), 427–445. | MR

[20] E.Peyre, Points de hauteur bornée et géométrie des variétés (d’après Y. Manin et al.). Séminaire Bourbaki, Vol. 2000/2001, 323–344. | Numdam | MR | Zbl

[21] E. Peyre, Hauteurs et measure de Tamagawa sur les variétés de Fano. Duke Math. J. 79 (1995), 101–218. | MR | Zbl

[22] P. Salberger, Tamagawa measures on universal torsors and points of bounded height on Fano varieties. Astérisque 251 (1998), 91–258. | Numdam | MR | Zbl

[23] S.H. Schanuel, Heights in number fields. Bull. Soc. Math. France 107 (1979), 433–449. | Numdam | MR | Zbl

[24] A. Schinzel, On the product of conjugates outside the unit circle of an algebraic integer. Acta Arith. XXIV (1973), 385–399. Addendum ibid. XXVI (1974/75), 329–331. | MR | Zbl

[25] W.M. Schmidt, Asymptotic formulae for point lattices of bounded determinant and subspaces of bounded height. Duke Math. J. 35 (1968), 327–339. | MR | Zbl

[26] W.M. Schmidt, Northcott’s theorem on heights II. The quadratic case. Acta Arith. LXX (1995), 343–375. | MR | Zbl

[27] C.J. Smyth, On the product of conjugates outside the unit circle of an algebraic integer. Bull. London Math. Soc. 3 (1971), 169–175. | MR | Zbl

[28] C. Smyth, The Mahler measure of algebraic numbers: A survey. In Number Theory & Polynomials conference proceedings, London Math. Soc., Lecture Note Ser. 352, Cambridge Univ. Press, Cambridge 2008, 322–349. | MR

[29] C.J. Smyth, On the measure of totally real algebraic numbers, I. J. Austral. Math. Soc. Ser. A 30 (1980/81), 137–149; II, Math. of Comp. 37 (1981), 205–208. | Zbl

[30] L. Szpiro, E. Ullmo, S. Zhang, Equirépartition des petits points. Inventiones Math. 127 (1997), 337–347. | MR | Zbl

[31] J.L.Thunder, Asymptotic estimates for the number of rational points of bounded height on flag varieties. Compos. Math. 88 (1993), 155–186. | Numdam | MR | Zbl

[32] R.C. Vaughan, T.D. Wooley, On a certain nonary cubic form and related equations. Duke Math. J. 80 (1995), 669–735. | MR | Zbl

[33] S. Zhang, Positive line bundles on arithmetic surfaces. Ann. of Math. (2) 136 (1995), 569–587. | MR | Zbl

Cité par Sources :