A trace formula for varieties over a discretely valued field

Details A trace formula for varieties over a discretely valued field A trace formula for varieties over a discretely valued field

2008-05-09

arxiv.org/abs/0805.1323v2

Mathematics

Algebraic Geometry

Presentation reorganized; minor errors corrected

Scientific paper

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 N\'eron 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-Ch\^atelet groups, Chow motives, and the structure of the Grothendieck ring.

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