Sieve methods for varieties over finite fields and arithmetic schemes
Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 221-229.

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 𝔽 2 is smooth is asymptotically 21/64 as its degree tends to infinity. Much of this paper is an exposition of results in [Poo04] and [Ngu05].

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 𝔽 2 soit lisse est asymptotiquement 21/64 quand son degré tend vers l’infini. La plus grande partie de cet article est une exposition des résultats de [Poo04] et [Ngu05].

DOI: 10.5802/jtnb.583
Keywords: Bertini theorem, finite field, Lefschetz pencil, squarefree integer, sieve
Poonen, Bjorn 1

1 Department of Mathematics University of California Berkeley, CA 94720-3840, USA
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Poonen, Bjorn. Sieve methods for varieties over finite fields and arithmetic schemes. Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 221-229. doi : 10.5802/jtnb.583. http://www.numdam.org/articles/10.5802/jtnb.583/

[Del74] Deligne, Pierre La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. (1974) no. 43, pp. 273-307 | Numdam | MR | Zbl

[Del80] Deligne, Pierre La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. (1980) no. 52, pp. 137-252 | Numdam | MR | Zbl

[Dwo60] Dwork, Bernard On the rationality of the zeta function of an algebraic variety, Amer. J. Math., Volume 82 (1960), pp. 631-648 | MR | Zbl

[Gab01] Gabber, O. On space filling curves and Albanese varieties, Geom. Funct. Anal., Volume 11 (2001) no. 6, pp. 1192-1200 | MR | Zbl

[Gra98] Granville, Andrew ABC allows us to count squarefrees, Internat. Math. Res. Notices (1998) no. 19, pp. 991-1009 | MR | Zbl

[Hoo67] Hooley, C. On the power free values of polynomials, Mathematika, Volume 14 (1967), pp. 21-26 | MR | Zbl

[Kat73] Katz, Nicholas M. Pinceaux de Lefschetz: théorème d’existence, Groupes de monodromie en géométrie algébrique. II, Springer-Verlag (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

[Kat99] Katz, Nicholas M. Space filling curves over finite fields, Math. Res. Lett., Volume 6 (1999) no. 5-6, pp. 613-624 | MR | Zbl

[LLR05] Liu, Qing; Lorenzini, Dino; Raynaud, Michel On the Brauer group of a surface, Invent. Math., Volume 159 (2005) no. 3, pp. 673-676 | MR | Zbl

[Ngu05] Nguyen, Nghi Huu Whitney theorems and Lefschetz pencils over finite fields, University of California at Berkeley (2005-05) (Ph. D. Thesis)

[Poo03] Poonen, Bjorn Squarefree values of multivariable polynomials, Duke Math. J., Volume 118 (2003) no. 2, pp. 353-373 | MR | Zbl

[Poo04] Poonen, Bjorn Bertini theorems over finite fields, Annals of Math., Volume 160 (2004) no. 3, pp. 1099-1127 | MR | Zbl

[Wei49] Weil, André Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc., Volume 55 (1949), pp. 497-508 | MR | Zbl

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