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

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.