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

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