Equivariant motivic integration on special formal schemes

Abstract

In this article, we construct an equivariant version of motivic integration on special formal schemes that generalizes our previous work for algebraic varieties. Pointing out the existence of an equivariant Néron smoothening for a flat generically smooth special formal scheme, we prove a change of variables formula in this integration. Finally, the article introduces the motivic Milnor fiber of a formal power series. It predicts that this quantity is the right one to define the motivic Milnor fiber of a germ of complex analytic functions.

Minor change: $R$ is always assumed to be a complete discrete valuation ring whose fraction field $K$ and residue field $k$ have the same characteristic. The final version is accepted for publication in Mathematische Zeitschrift