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- Title
A trace formula for varieties over a discretely valued field.
- Authors
Nicaise, Johannes
- 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 Nééron 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--Chââtelet groups, Chow motives, and the structure of the Grothendieck ring of varieties.
- Subjects
VALUED fields; REPRESENTATIONS of groups (Algebra); AUTOMORPHIC forms; TRACE formulas; TOPOLOGICAL fields; DISCONTINUOUS groups
- Publication
Journal für die Reine und Angewandte Mathematik, 2011, Vol 2011, Issue 650, p193
- ISSN
0075-4102
- Publication type
Article
- DOI
10.1515/CRELLE.2011.008