A trace formula for varieties over a discretely valued field

2008/05/09 by Johannes Nicaise, Nicaise, Johannes · 2 citations
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.0805.1323

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.

Cited by

• Trace formula for component groups of Néron models
• Motivic zeta functions of abelian varieties, and the monodromy conjecture

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