A trace formula for rigid varieties, and motivic Weil generating series for formal schemes

We establish a trace formula for rigid varieties X over a complete discretely valued field, which relates the set of unramified points on X to the Galois action on its étale cohomology. Next, we show that the analytic Milnor fiber of a morphism f at a point x completely determines the formal germ of...

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Bibliographic Details
Published in:Mathematische annalen 2009-02, Vol.343 (2), p.285-349
Main Author: Nicaise, Johannes
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
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Summary:We establish a trace formula for rigid varieties X over a complete discretely valued field, which relates the set of unramified points on X to the Galois action on its étale cohomology. Next, we show that the analytic Milnor fiber of a morphism f at a point x completely determines the formal germ of f at x . We develop a theory of motivic integration for formal schemes of pseudo-finite type over a complete discrete valuation ring R , and we introduce the Weil generating series of a regular formal R -scheme of pseudo-finite type, via the construction of a Gelfand-Leray form on its generic fiber. When is the formal completion of a morphism f from a smooth irreducible variety to the affine line, then its Weil generating series coincides (modulo normalization) with the motivic zeta function of f . When is the formal completion of f at a closed point x of the special fiber , we obtain the local motivic zeta function of f at x .
ISSN:0025-5831
1432-1807
DOI:10.1007/s00208-008-0273-9