Motivic Serre invariants, ramification, and the analytic Milnor fiber
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 that this analogy is more than just philosophical.'' \textit{J. Denef} and \textit{F. Loeser} [J. Am. Math. Soc. 5, No. 4, 705--720 (1992; Zbl 0777.32017)] introduced the topological zeta function \(Z_{\text{top}}(f,s)\), which is a geometric version of \(Z_{p}(f,s)\). The topological zeta function is a rational function of \(s\) defined in terms of an embedded resolution of singularities of \(f\). Denef and Loeser established \(Z_{\text{top}}(f,s)\) as a limit of \(p\)-adic zeta functions, and thus they showed that it does not depend on the choice of a resolution of \(f\). Based on the theory of motivic integration of Kontsevich, Denef and Loeser introduced a naïve motivic zeta function, \(Z_{\text{mot}}(f,s)\), which specializes to \(Z_{\text{top}}\), and to \(Z_{p}\) for almost all primes \(p\). \textit{J. Denef} and \textit{F. Loeser} [in: ECM 2000, Vol. I. Prog. Math. 201, 327--348 (2001; Zbl 1079.14003)] defined the motivic nearby cycles of \(f\) as a limit of the motivic zeta function. It is an object in an appropriate Grothendieck ring of varieties over the complex hypersurface \(X_{s}=f^{-1}(0)\), and the fiber over each point \(x\in\) \(X_{s}\) has the same Hodge polynomial as the Milnor fiber of \(f\) at \(x\). The motivic monodromy conjecture states that each pole of the naïve motivic zeta function induces an eigenvalue of the monodromy at some point of \(X_{s}\). A central goal of the paper is to develop a theoretical framework to attack ``directly'' the Monodromy Conjecture: ``Most research on this conjecture made a `detour' via a resolution of singularities, studying the geometry of the exceptional locus in order to eliminate fake candidate poles, and applying A'Campo's formula for the monodromy zeta function [Comment. Math. Helv. 50, 233--248 (1975; Zbl 0333.14008)]. This approach has yielded proofs in particular cases, and inspired nice results concerning the geometry of embedded resolutions, but it seems difficult to apply this technique to the general case. In this article, we try to establish a more direct link between the motivic zeta function of \(f\), and the Milnor fibration, using \textit{V. G. Berkovich}'s étale cohomology for analytic spaces [Publ. Math., Inst. Hautes Étud. Sci. 78, 5--161 (1993; Zbl 0804.32019)], and the theory of motivic integration on rigid spaces [\textit{F. Loeser} and \textit{J. Sebag}, Duke Math. J. 119, No. 2, 315--344 (2003; Zbl 1078.14029)]. More precisely, we introduce the analytic Milnor fiber, a rigid variety over \(\mathbb{C}((t))\) with the `same' cohomology as the topological Milnor fiber, and whose points are closely related to the arc spaces used to define the motivic zeta function.''
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