Equivariant motivic integration on formal schemes and the motivic zeta function

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 Poincaré series, from which we deduce Denef and Loeser’s motivic zeta function including the action of the profinite group of roots of unity.

Keywords:

• Algebraic geometry
• formal schemes
• motivic integration
• motivic zeta function

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 14L30
• 14D15
• 14G22

Acknowledgements

I would like to thank Alberto Bellardini and in particular Emmanuel Bultot for discussing many technical problems with me. Moreover, I am thankful to Johannes Nicaise for his discussions, ideas and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

During the research for this article, I was supported by a research fellowship of the DFG (Aktenzeichen HA 7122/1-1).