2007/03/31 by Johannes Nicaise
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Alkaloids: synthesis and pharmacology #math.AG #msc:14B05 #msc:14G22 #msc:32S55
paper · pdf · doi:10.1007/s00208-008-0273-9
To appear in Math. Ann. The original publication is available at http://www.springerlink.com
openalex publication_date 2008/08/18 · arxiv created 2008/09/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
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. 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 \mathfrakX of pseudo-finite type, via the construction of a Gelfand-Leray form on its generic fiber. Our trace formula yields a cohomological interpretation of this Weil generating series. When \mathfrakX 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 \mathfrakX is the formal completion of f at a closed point x of the special fiber f-1(0), we obtain the local motivic zeta function of f at x. In the latter case, the generic fiber of \mathfrakX is the so-called analytic Milnor fiber of f at x; we show that it completely determines the formal germ of f at x.