Sieve methods for varieties over finite fields and arithmetic schemes

Bjorn Poonen[1]

  • [1] Department of Mathematics University of California Berkeley, CA 94720-3840, USA

Journal de Théorie des Nombres de Bordeaux (2007)

  • Volume: 19, Issue: 1, page 221-229
  • ISSN: 1246-7405

Abstract

top
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 ℤ . 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].

How to cite

top

Poonen, Bjorn. "Sieve methods for varieties over finite fields and arithmetic schemes." Journal de Théorie des Nombres de Bordeaux 19.1 (2007): 221-229. <http://eudml.org/doc/249972>.

@article{Poonen2007,
abstract = {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 $\mathbb\{Z\}$. 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 $\{\mathbb\{F\}\}_2$ 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].},
affiliation = {Department of Mathematics University of California Berkeley, CA 94720-3840, USA},
author = {Poonen, Bjorn},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Bertini theorem; finite field; Lefschetz pencil; squarefree integer; sieve; Whitney embedding theorem},
language = {eng},
number = {1},
pages = {221-229},
publisher = {Université Bordeaux 1},
title = {Sieve methods for varieties over finite fields and arithmetic schemes},
url = {http://eudml.org/doc/249972},
volume = {19},
year = {2007},
}

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
PB - Université Bordeaux 1
VL - 19
IS - 1
SP - 221
EP - 229
AB - 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 $\mathbb{Z}$. 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 ${\mathbb{F}}_2$ 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].
LA - eng
KW - Bertini theorem; finite field; Lefschetz pencil; squarefree integer; sieve; Whitney embedding theorem
UR - http://eudml.org/doc/249972
ER -

References

top
  1. Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. (1974), 273-307 Zbl0287.14001MR340258
  2. Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. (1980), 137-252 Zbl0456.14014MR601520
  3. Bernard Dwork, On the rationality of the zeta function of an algebraic variety, Amer. J. Math. 82 (1960), 631-648 Zbl0173.48501MR140494
  4. O. Gabber, On space filling curves and Albanese varieties, Geom. Funct. Anal. 11 (2001), 1192-1200 Zbl1072.14513MR1878318
  5. Andrew Granville, A B C allows us to count squarefrees, Internat. Math. Res. Notices (1998), 991-1009 Zbl0924.11018MR1654759
  6. C. Hooley, On the power free values of polynomials, Mathematika 14 (1967), 21-26 Zbl0166.05203MR214556
  7. Nicholas M. Katz, Pinceaux de Lefschetz: théorème d’existence, Groupes de monodromie en géométrie algébrique. II (1973), 212-253, Springer-Verlag Zbl0284.14006
  8. Nicholas M. Katz, Space filling curves over finite fields, Math. Res. Lett. 6 (1999), 613-624 Zbl1016.11022MR1739219
  9. Qing Liu, Dino Lorenzini, Michel Raynaud, On the Brauer group of a surface, Invent. Math. 159 (2005), 673-676 Zbl1077.14023MR2125738
  10. Nghi Huu Nguyen, Whitney theorems and Lefschetz pencils over finite fields, (2005-05) 
  11. Bjorn Poonen, Squarefree values of multivariable polynomials, Duke Math. J. 118 (2003), 353-373 Zbl1047.11021MR1980998
  12. Bjorn Poonen, Bertini theorems over finite fields, Annals of Math. 160 (2004), 1099-1127 Zbl1084.14026MR2144974
  13. André Weil, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55 (1949), 497-508 Zbl0032.39402MR29393

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.