On the irreducibility of the two variable zeta-function for curves over finite fields. (Q1408070)

In [Arithmetic, Geometry and Coding Theory 4, Luminy 1993, 175--184 (1996; Zbl 1019.11016)], \textit{R. Pellikaan} introduced a two variable zeta-function for curves over finite fields. Inspired by this, \textit{G. van der Geer} and \textit{R. Schoof} [Sel. Math., New Ser. 6, 377--398 (2000; Zbl 1030.11063)] defined a two variable zeta-function for number fields. This led to the study of the zero locus of this function. \textit{J. C. Lagarias} and \textit{E. Rains} [Ann. Inst. Fourier 53, No. 1, 1--68 (2003; Zbl 1106.11036)] asked whether this locus is the closure of a single multplicity one irreducible complex analytic variety. In the present paper an anologous question for the two variable zeta-function for curves over finite fields is answered affirmatively: Pellikaan's two variable zeta-function, viewed as a polynomial in two variables over~\({\mathbb C}\), is irreducible.