Des méthodes du crible classiques en théorie analytique des nombres ont été récemment adaptées à un cadre géométrique. Dans ce nouveau cadre, les nombres premiers sont remplacés par les points fermés d’une variété algébrique sur un corps fini ou plus généralement un schéma de type fini sur . Nous présentons la méthode et certains des résultats surprenants qui en découlent. Par exemple, la probabilité qu’une courbe plane sur soit lisse est asymptotiquement quand son degré tend vers l’infini. La plus grande partie de cet article est une exposition des résultats de [Poo04] et [Ngu05].
Classical sieve methods of analytic number theory have recently been adapted to a geometric setting. In the new setting, the primes are replaced by the closed points of a variety over a finite field or more generally of a scheme of finite type over . We will present the method and some of the surprising results that have been proved using it. For instance, the probability that a plane curve over is smooth is asymptotically as its degree tends to infinity. Much of this paper is an exposition of results in [Poo04] and [Ngu05].
Mots clés : Bertini theorem, finite field, Lefschetz pencil, squarefree integer, sieve
@article{JTNB_2007__19_1_221_0, author = {Poonen, Bjorn}, title = {Sieve methods for varieties over finite fields and arithmetic schemes}, journal = {Journal de Th\'eorie des Nombres de Bordeaux}, pages = {221--229}, publisher = {Universit\'e Bordeaux 1}, volume = {19}, number = {1}, year = {2007}, doi = {10.5802/jtnb.583}, mrnumber = {2332063}, zbl = {1149.11031}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.583/} }
TY - JOUR AU - Poonen, Bjorn TI - Sieve methods for varieties over finite fields and arithmetic schemes JO - Journal de Théorie des Nombres de Bordeaux PY - 2007 DA - 2007/// SP - 221 EP - 229 VL - 19 IS - 1 PB - Université Bordeaux 1 UR - http://archive.numdam.org/articles/10.5802/jtnb.583/ UR - https://www.ams.org/mathscinet-getitem?mr=2332063 UR - https://zbmath.org/?q=an%3A1149.11031 UR - https://doi.org/10.5802/jtnb.583 DO - 10.5802/jtnb.583 LA - en ID - JTNB_2007__19_1_221_0 ER -
Poonen, Bjorn. Sieve methods for varieties over finite fields and arithmetic schemes. Journal de Théorie des Nombres de Bordeaux, Tome 19 (2007) no. 1, pp. 221-229. doi : 10.5802/jtnb.583. http://archive.numdam.org/articles/10.5802/jtnb.583/
[Del74] La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. (1974) no. 43, pp. 273-307 | Numdam | MR 340258 | Zbl 0287.14001
[Del80] La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. (1980) no. 52, pp. 137-252 | Numdam | MR 601520 | Zbl 0456.14014
[Dwo60] On the rationality of the zeta function of an algebraic variety, Amer. J. Math., Volume 82 (1960), pp. 631-648 | MR 140494 | Zbl 0173.48501
[Gab01] On space filling curves and Albanese varieties, Geom. Funct. Anal., Volume 11 (2001) no. 6, pp. 1192-1200 | MR 1878318 | Zbl 1072.14513
[Gra98] allows us to count squarefrees, Internat. Math. Res. Notices (1998) no. 19, pp. 991-1009 | MR 1654759 | Zbl 0924.11018
[Hoo67] On the power free values of polynomials, Mathematika, Volume 14 (1967), pp. 21-26 | MR 214556 | Zbl 0166.05203
[Kat73] Pinceaux de Lefschetz: théorème d’existence, Groupes de monodromie en géométrie algébrique. II (1973), pp. 212-253 Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II); Dirigé par P. Deligne et N. Katz, Lecture Notes in Mathematics, Vol. 340, Exposé XVII | Zbl 0284.14006
[Kat99] Space filling curves over finite fields, Math. Res. Lett., Volume 6 (1999) no. 5-6, pp. 613-624 | MR 1739219 | Zbl 1016.11022
[LLR05] On the Brauer group of a surface, Invent. Math., Volume 159 (2005) no. 3, pp. 673-676 | MR 2125738 | Zbl 1077.14023
[Ngu05] Whitney theorems and Lefschetz pencils over finite fields (2005-05) (Ph. D. Thesis)
[Poo03] Squarefree values of multivariable polynomials, Duke Math. J., Volume 118 (2003) no. 2, pp. 353-373 | MR 1980998 | Zbl 1047.11021
[Poo04] Bertini theorems over finite fields, Annals of Math., Volume 160 (2004) no. 3, pp. 1099-1127 | MR 2144974 | Zbl 1084.14026
[Wei49] Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc., Volume 55 (1949), pp. 497-508 | MR 29393 | Zbl 0032.39402
Cité par Sources :