The meromorphic continuation of a zeta function of Weil and Igusa type
type: Item
instance of: scholarly article
title: The meromorphic continuation of a zeta function of Weil and Igusa type
zbMATH Open document ID: 0571.12009
DOI: 10.1007/BF01390327
author: Diane Meuser
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143379
review text: Let \(K_1\) denote a finite algebraic extension of \({\mathbb{Q}}_p\). Let \(K_d\) denote the unique unramified extension of \(K_1\) of degree \(d\). For \(d\ge 1\) let \(R_d\) denote the ring of integers of \(K_d\) and \(P_d\) its unique maximal ideal. Let \(q= \operatorname{card} R_1/P_1\). Then \(R_d/P_d\simeq {\mathbb{F}}_{q^d}\), the finite field with \(q^d\) elements. Let \(f(x)=f(x_1,\ldots,x_n)\) denote a polynomial in \(n\) variables with coefficients in \(R_1\). Let \[ N_{e,d} = \operatorname{card}\{x \pmod{P^e_d)^{(n)}}\mid f(x)\equiv 0 \pmod {P^e_d} \] and define \(P(w,z)\) by \(P(w,z)=\sum_{d\ge 1}\sum_{e\ge 0} N_{e,d} q^{-nde} w^dz^e\). The above series is a holomorphic function on the product of the two open unit discs in \({\mathbb{C}}^ 2\). The coefficient of \(z\) is \(Q_1(w)=\sum_{d\ge 1} N_{1,d} q^{-nd} w^d\), which is rational as a consequence of the rationality of the Weil zeta function. The coefficient of \(w^d\) is \(P_d(z)=\sum_{e\ge 0} N_{e,d} q^{-nde} z^e\) which is the Igusa zeta function for the field \(K_d\) and hence is rational. In this paper we show that \(P(w,z)\) has a meromorphic continuation to \({\mathbb{C}}^ 2\). This is a best possible result since \(P(w,z)\) is not in general rational. We completely characterize those polynomials for which it is rational. In the process of proving the above result we also prove two other results. The first is that a zeta function which is analogous to the Weil zeta function, namely \(Z_e(T)=\exp \sum_{d\ge 1} N_{e,d} T^d/d\) for any \(e>1\), is rational. The second is that the Igusa zeta function is an invariant of \(f(x)\) which only depends upon the degree \(d\) of the extension in a simple way.
Mathematics Subject Classification ID: 11S40
Mathematics Subject Classification ID: 11S05
Mathematics Subject Classification ID: 32A20
Mathematics Subject Classification ID: 14G10
zbMATH DE Number: 3910511
zbMATH Keywords: polynomial in several variables
zbMATH Keywords: Weil zeta function
zbMATH Keywords: Igusa zeta function
zbMATH Keywords: meromorphic continuation
zbMATH Keywords: rationality of zeta functions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3910511