Formal deformations of algebraic spaces and generalizations of the motivic Igusa zeta function

TL;DR

This paper extends the concept of the motivic Igusa zeta function to formal deformations of algebraic spaces, introducing a new canonical series that accounts for all algebraic coordinate transformations.

Contribution

It generalizes the auto-Igusa zeta function to a broader setting involving formal deformations and introduces the canonical auto-Igusa zeta function with quotient stack coefficients.

Findings

Generalization of the auto-Igusa zeta function to formal deformations.

Introduction of the canonical auto-Igusa zeta function.

Discussion of current research and future directions.

Abstract

We generalize the notion of the auto-Igusa zeta function to formal deformations of algebraic spaces. By incorporating data from all algebraic transformations of local coordinates, this function can be viewed as a generalization of the traditional motivic Igusa zeta function. Furthermore, we introduce a new series, which we term the canonical auto-Igusa zeta function, whose coefficients are given by the quotient stacks formed from the coefficients of the auto-Igusa zeta function modulo change of coordinates. We indicate the current state of the literature on these generalized Igusa-zeta functions and offer directions for future research.