A trace formula for varieties over a discretely valued field
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Johannes Nicaise
Abstract
We study the motivic Serre invariant of a smoothly bounded algebraic or rigid variety X over a complete discretely valued field K with perfect residue field k. If K has characteristic zero, we extend the definition to arbitrary K-varieties using Bittner's presentation of the Grothendieck ring and a process of Néron smoothening of pairs of varieties.
The motivic Serre invariant can be considered as a measure for the set of unramified points on X. Under certain tameness conditions, it admits a cohomological interpretation by means of a trace formula. In the curve case, we use T. Saito's geometric criterion for cohomological tameness to obtain more detailed results. We discuss some applications to Weil–Châtelet groups, Chow motives, and the structure of the Grothendieck ring of varieties.
© Walter de Gruyter Berlin · New York 2011
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- Classifications of linear operators preserving elliptic, positive and non-negative polynomials
- Graded polynomial identities and exponential growth
- Un théorème de la masse positive pour le problème de Yamabe en dimension paire
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- A trace formula for varieties over a discretely valued field
Articles in the same Issue
- Ricci soliton solvmanifolds
- Rational normal scrolls and the defining equations of Rees algebras
- Classifications of linear operators preserving elliptic, positive and non-negative polynomials
- Graded polynomial identities and exponential growth
- Un théorème de la masse positive pour le problème de Yamabe en dimension paire
- Degenerate problems with irregular obstacles
- Twisted cyclic theory, equivariant KK-theory and KMS states
- A trace formula for varieties over a discretely valued field
