On the irreducibility of the two variable zeta-function for curves over finite fields

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 $\text{F}{}_{\mathrm{q}}$ 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 $\text{P(t,u)∈}\text{Z}\text{[t,u]}$. 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. To cite this article: N. Naumann, C. R. Acad. Sci. Paris, Ser. I 336 (2003).