Equivariant motivic integration on formal schemes and the motivic zeta function

TL;DR

This paper develops an equivariant version of motivic integration for formal schemes with group actions, establishing a change of variables formula and defining an equivariant motivic zeta function that extends previous work by incorporating group symmetries.

Contribution

It introduces equivariant motivic integration for formal schemes with finite group actions and proves a change of variable formula, extending motivic zeta functions to include group symmetries.

Findings

Defined equivariant motivic integration for formal schemes.

Proved change of variable formula in the equivariant setting.

Extended motivic zeta functions to incorporate group actions.

Abstract

For a formal scheme over a complete discrete valuation ring with a good action of a finite group, we define equivariant motivic integration, and we prove a change of variable formula for that.To do so, we construct and examine an induced group action on the Greenberg scheme of such a formal scheme. Using this equivariant motivic integration, we define an equivariant volume Poincar\'e series, from which we deduce Denef and Loeser's motivic zeta function including the action of the profinite group of roots of unity.