Motivic Serre invariants, ramification, and the analytic Milnor fiber
type: Item
instance of: scholarly article
title: Motivic Serre invariants, ramification, and the analytic Milnor fiber
zbMATH Open document ID: 1136.14010
author: Johannes Nicaise
author: Julien Sebag
published in: Inventiones Mathematicae
publication date: 2007-10-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0703217
review text: The Igusa local zeta function is defined as a \(p\)-adic integral \[ Z_p(f,s)=\int_{\mathbb{Z}_p^n} \left| f\left( x\right) \right| _{p}^{s}\left| dx\right| , \] where \(s\) is a complex variable, \(f\) is a polynomial over \(\mathbb{Z}p\) in \(n\) variables, \(\left| \cdot\right| \) is the \(p\)-adic norm, and \(\left| dx\right| \) denotes the Haar measure on the compact group \(\mathbb{Z}_{p}^{n}\). \textit{J.-i. Igusa} proved, using resolution of singularities, that \(Z_{p}(f,s)\) has a meromorphic continuation to the complex plane as a rational function of \(p^{-s}\) [An introduction to the theory of local zeta functions. AMS/IP Studies in Advanced Mathematics. 14. Providence, RI: American Mathematical Society (2000; Zbl 0959.11047)]. A complete list of possible poles for the meromorphic continuation of \(Z_{p}(f,s)\) can be given in terms of the numerical data of an embedded resolution for \(f\), however, since this list depends on the chosen resolution, a lot of these candidate poles will not be actual poles of \(Z_{p}(f,s)\). In the case in which \(f\) is defined on \(\mathbb{Q}\), there are several conjectures relating the poles of \(Z_{p}(f,s)\) with the topology of the complex singularity \(f^{-1}\left( 0\right) \). One of these conjectures is Igusa's Monodromy Conjecture, which predicts an intriguing connection between the eigenvalues of the monodromy at complex points of \(f^{-1}\left( 0\right) \subset\) \(\mathbb{C}^{n}\), and the real parts of the poles of \(Z_{p}(f,s)\), for almost all primes \(p\). This conjecture establishes a remarkable connection between the arithmetic -- \(Z_{p}(f,s)\) contains information about the number of solutions of the congruence \(f(x)\equiv0\) mod \(p^{m}\) -- and complex topology. The exploration of this analogy is a key feature of the paper: ``the monodromy conjecture establishes a fascinating bridge between arithmetic properties and complex topology, much like the Weil conjectures. In fact, we will show in this paper …
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14G20
zbMATH DE Number: 5204080
zbMATH Keywords: motivic integration
zbMATH Keywords: Igusa's local zeta function
zbMATH Keywords: Milnor fiber
zbMATH Keywords: monodromy
reviewed by: W. A. Zúñiga-Galindo
MaRDI profile type: Publication
OpenAlex ID: W3105411107
arXiv ID: math/0703217
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DOI: 10.1007/S00222-006-0029-7
description: scientific article