2018/02/01 by Qiuyu Yin, Yin, Qiuyu, Shaofang Hong +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1802.00318
openalex publication_date 2018/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let K be a local field and f(x)∈ K[x] be a non-constant polynomial. The local zeta function Zf(s, χ) was first introduced by Weil, then studied in detail by Igusa. When \rm char(K)=0, Igusa proved that Zf(s, χ) 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 Zf(s, χ) 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 \acuteu\rmniga-Galindo, and developing a detailed analysis, we prove the rationality of these local zeta functions and also describe explicitly all their candidate poles.