Poles of Zeta Functions on Normal Surfaces

Abstract

Let $(S, 0)$ be a normal surface germ and Let $f$ a non-constant regular function on Let $(S, 0)$ with Let $f(0) = 0$. Using any additive invariant on complex algebraic varieties one can associate a zeta function to these data, where the topological and motivic zeta functions are the roughest and the finest zeta functions, respectively. In this paper we are interested in a geometric determination of the poles of these functions. The second author has already provided such a determination for the topological zeta function in the case of non-singular surfaces. Here we give a complete answer for all normal surfaces, at least on the motivic level. The topological zeta function however seems to be too rough for this purpose, although for negative poles, which are the only ones in the non-singular case, we are able to prove exactly the same result as for non-singular surfaces.

We also give and verify a (natural) definition for when a rational number is a pole of the motivic zeta function.