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

Naumann, Niko1
1 Institut für Mathematik, Einsteinstrasse 62, 48149 Münster, Germany
Comptes Rendus. Mathématique, Tome 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
Naumann, Niko 1

1 Institut für Mathematik, Einsteinstrasse 62, 48149 Münster, Germany 
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Naumann, Niko. On the irreducibility of the two variable zeta-function for curves over finite fields. Comptes Rendus. Mathématique, Tome 336 (2003) no. 4, pp. 289-292. doi : 10.1016/S1631-073X(03)00039-6. https://www.numdam.org/articles/10.1016/S1631-073X(03)00039-6/

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

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