The Igusa local zeta functions of superelliptic curves

Abstract

Let $K$ be a local field and $f(x)\in K[x]$ be a non-constant polynomial. The local zeta function $Z_f(s,\chi )$ was first introduced by Weil, then studied in detail by Igusa. When $\mathrm{char}(K)=0$, Igusa proved that $Z_f(s,\chi )$ is a rational function of $q^{-s}$ by using the resolution of singularities. Later on, Denef gave another proof of this remarkable result. However, if $\mathrm{char}(K)>0$, the question of rationality of $Z_f(s,\chi )$ is still kept open. Actually, there are only a few known results so far. In this paper, we investigate the local zeta functions of two-variable polynomial $g(x,y)$, where $g(x,y)=0$ is the superelliptic curve with coefficients in a non-archimedean local field of positive characteristic. By using the notable Igusa's stationary phase formula and with the help of some results due to Denef and Z$\acute{\mathrm{u}}$$\tilde{\mathrm{n}}$iga-Galindo, and developing a detailed analysis, we prove the rationality of these local zeta functions and also describe explicitly all their candidate poles.

18 pages