The Igusa local zeta functions of superelliptic curves

TL;DR

This paper proves the rationality of Igusa local zeta functions for superelliptic curves over non-archimedean local fields of positive characteristic, providing explicit descriptions of their candidate poles.

Contribution

It extends the rationality results of Igusa's local zeta functions to superelliptic curves in positive characteristic fields, using stationary phase formula and detailed analysis.

Findings

Proved the rationality of local zeta functions for superelliptic curves in positive characteristic.

Explicitly described all candidate poles of these zeta functions.

Applied Igusa's stationary phase formula and built on results by Denef and Zúñiga-Galindo.

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 ${\rm char}(K)=0$, 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 ${\rm char}(K)>0$, 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${\rm \acute{u}}$${\rm\tilde{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.