Comptes Rendus
no. 4
Number Theory/Algebraic Geometry
On the irreducibility of the two variable zeta-function for curves over finite fields
[Sur l'irréductibilité de la fonction zêta d'une courbe définie sur un corps fini]

Présenté par : Christophe Soulé

Niko Naumann1
1 Institut für Mathematik, Einsteinstrasse 62, 48149 Münster, Germany
Comptes Rendus. Mathématique, Volume 336 (2003) no. 4, pp. 289-292.

R. Pellikaan (Arithmetic, Geometry and Coding Theory, Vol. 4, Walter de Gruyter, Berlin, 1996, pp. 175–184) a introduit une fonction zêta Z(t,u) en deux variables pour une courbe définie sur un corps fini 𝔽q. Pour u=q on obtient la fonction zêta habituelle et Pellikaan démontre que Z(t,u) est une fonction rationelle : Z(t,u)=(1−t)−1(1−ut)−1P(t,u) où P(t,u)∈ℤ[t,u]. Nous démontrons que P(t,u) est absolument irréductible. Nous avons été motivés par une question de J. Lagarias et E. Rains concernant une fonction zêta en deux variables analogue pour des corps de nombres.

R. Pellikaan (Arithmetic, Geometry and Coding Theory, Vol.  4, Walter de Gruyter, Berlin, 1996, pp. 175–184) introduced a two variable zeta-function Z(t,u) for a curve over a finite field 𝔽q which, for u=q, specializes to the usual zeta-function and he proved rationality: Z(t,u)=(1−t)−1(1−ut)−1P(t,u) with P(t,u)∈ℤ[t,u]. We prove that P(t,u) is absolutely irreducible. This is motivated by a question of J. Lagarias and E. Rains about an analogous two variable zeta-function for number fields.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00039-6

Niko Naumann 1

1 Institut für Mathematik, Einsteinstrasse 62, 48149 Münster, Germany 
@article{CRMATH_2003__336_4_289_0,
     author = {Niko Naumann},
     title = {On the irreducibility of the two variable zeta-function for curves over finite fields},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {289--292},
     publisher = {Elsevier},
     volume = {336},
     number = {4},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00039-6},
     language = {en},
}
TY  - JOUR
AU  - Niko Naumann
TI  - On the irreducibility of the two variable zeta-function for curves over finite fields
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 289
EP  - 292
VL  - 336
IS  - 4
PB  - Elsevier
DO  - 10.1016/S1631-073X(03)00039-6
LA  - en
ID  - CRMATH_2003__336_4_289_0
ER  - 
%0 Journal Article
%A Niko Naumann
%T On the irreducibility of the two variable zeta-function for curves over finite fields
%J Comptes Rendus. Mathématique
%D 2003
%P 289-292
%V 336
%N 4
%I Elsevier
%R 10.1016/S1631-073X(03)00039-6
%G en
%F CRMATH_2003__336_4_289_0
Niko Naumann. On the irreducibility of the two variable zeta-function for curves over finite fields. Comptes Rendus. Mathématique, Volume 336 (2003) no. 4, pp. 289-292. doi : 10.1016/S1631-073X(03)00039-6. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00039-6/

[1] C. Deninger, Two-variable zeta functions and regularized products, Preprint, 2002

[2] D. Eisenbud Commutative Algebra with a View Towards Algebraic Geometry, Graduate Texts in Math., 150, Springer, New York, 1995

[3] G. van der Geer; R. Schoof Effectivity of Arakelov divisors and the theta divisor of a number field, Selecta Math. (N.S.), Volume 6 (2000) no. 4, pp. 377-398

[4] R. Hartshorne Algebraic Geometry, Graduate Texts in Math., 52, Springer, New York, 1977

[5] J. Lagarias, E. Rains, On a two-variable zeta function for number fields, , v5, 7 July 2002 | arXiv

[6] R. Pellikaan On special divisors and the two variable zeta function of algebraic curves over finite fields, Arithmetic, Geometry and Coding Theory, Vol. 4, Luminy, 1993, W. de Gruyter, Berlin, 1996, pp. 175-184

[7] J. Tate Endomorphisms of Abelian varieties over finite fields, Invent. Math., Volume 2 (1966), pp. 143-144

  • Dino Lorenzini Two-Variable Zeta-Functions on Graphs and Riemann–Roch Theorems, International Mathematics Research Notices, Volume 2012 (2012) no. 22, p. 5100 | DOI:10.1093/imrn/rnr227
  • Machiel van Frankenhuijsen The two-variable zeta function and the Riemann hypothesis for function fields, Expositiones Mathematicae, Volume 26 (2008) no. 3, pp. 249-260 | DOI:10.1016/j.exmath.2007.11.001 | Zbl:1213.11163

Cité par 2 documents. Sources : Crossref, zbMATH

Commentaires - Politique