  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.
