A trace formula for varieties over a discretely valued field

Abstract

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.

Geometric criteria for tame ramification

“…We expect that the trace formula is valid if X is geometrically connected and cohomologically tame, and X(K t ) non-empty. We've proven this if k has characteristic zero [Ni11,6.5], if X is a curve [Ni11,§7], and if X is an abelian variety [Ni09b, 2.9]. Our formula for the error term shows that (assuming the existence of an sncd-model), our conjecture is equivalent to a partial generalization of Saito's criterion to arbitrary dimension (Question 4.2.4).…”

“…In particular, s(X) = 0 if X(K) = ∅, since in this case, X is a weak Néron model of itself. In [Ni11], we've shown that under a certain tameness condition on X, the value s(X) admits a cohomological interpretation in terms of a trace formula. To study this formula, we introduce the following definition.…”

“…The condition that X(K t ) is non-empty can not be omitted, by [Ni11,7.7]. If X is not cohomologically tame, it would be quite interesting to relate ε(X) to other measures of wild ramification.…”

“…If X is not cohomologically tame, it would be quite interesting to relate ε(X) to other measures of wild ramification. However, by the example in [Ni11,7.7], the value ε(X) can not always be computed from the Chow motive of X, if we don't impose the condition X(K t ) = ∅.…”

“…As a second application, in Section 4, we compute the error term in the trace formula for X on an sncd-model X of X over R. The trace formula was introduced in [NS07b] and further studied in [Ni09a,Ni09b,Ni11]. It expresses a certain measure for the set of rational points on X in terms of the Galois action on the tame ℓ-adic cohomology of X.…”