Motivic Igusa zeta functions

TL;DR

This paper introduces motivic versions of Igusa's local zeta functions valued in a Grothendieck group of Chow motives, exploring their properties, specializations, and relation to motivic nearby cycles and Hodge spectra.

Contribution

It defines motivic Igusa zeta functions, studies their fundamental properties, and connects them to existing concepts like p-adic and topological zeta functions and motivic nearby cycles.

Findings

Establishment of functional equations for motivic Igusa zeta functions

Demonstration of specialization to p-adic and topological zeta functions

Recovery of Hodge spectrum from motivic zeta functions

Abstract

We define motivic analogues of Igusa's local zeta functions. These functions take their values in a Grothendieck group of Chow motives. They specialize to p-adic Igusa local zeta functions and to the topological zeta functions we introduced several years ago. We study their basic properties, such as functional equations, and their relation with motivic nearby cycles. In particular the Hodge spectrum of a singular point of a function may be recovered from the Hodge realization of these zeta functions.