A trace formula for varieties over a discretely valued field

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 a...

Full description

Saved in:
Bibliographic Details
Published in:Journal für die reine und angewandte Mathematik 2011-01, Vol.2011 (650), p.193-238
Main Author: Nicaise, Johannes
Format: Article
Language:English
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:0075-4102
1435-5345
DOI:10.1515/crelle.2011.008