Local zeta functions and Meuser's invariant functions
type: Item
instance of: scholarly article
title: Local zeta functions and Meuser's invariant functions
zbMATH Open document ID: 0734.11071
DOI: 10.1016/0022-314X(91)90020-C
author: Johan Pas
published in: Journal of Number Theory
publication date: 1991-01-01T00:00:00Z
review text: Let \(K_1\) be a fixed finite field extension of \(\mathbb{Q}_p\), the field of \(p\)-adic numbers. For a positive integer \(d\), let \(K_d\) denote the unique unramified extension of \(K_1\) of degree \(d\) with ring of integer \(\mathcal O_{K_d}\). Let \(h(x)\) be a polynomial in \(m\) variables \(x=(x_1,x_2,\ldots,x_m)\) with coefficients in \(\mathcal O_{K_1}\). Igusa's local zeta function of \(h(x)\) on the field \(K_d\) is defined as \[ Z(s,K_d)=\int_{\mathcal O^m_{K_d}} | h(x)|^s_d\, | dx|_d\text{ for } s\in\mathbb{R},\quad s>0, \] where \(| dx|_d\) is the Haar measure on \(K_d^m\) such that \(\mathcal O^m_{K_d}\) has measure \(1\). The author shows that a generalization of Igusa's local zeta function to integrals over sets defined in a first order language of valued fields is an invariant function (in the sense of Meuser) of the class of unramified extensions of a \(p\)-adic field. As a corollary the author proves the invariance of certain Poincaré series which was conjectured by Fried and Jarden.
Mathematics Subject Classification ID: 11S40
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 11U09
zbMATH DE Number: 4214159
zbMATH Keywords: Igusa local zeta function
zbMATH Keywords: first order language
zbMATH Keywords: valued fields
zbMATH Keywords: invariant function
zbMATH Keywords: unramified extensions
zbMATH Keywords: p-adic field
zbMATH Keywords: Poincaré series
reviewed by: Masakazu Muro
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4214159