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

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Abstract

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 \({\mathfrak{X}}\) of pseudo-finite type, via the construction of a Gelfand-Leray form on its generic fiber. When \({\mathfrak{X}}\) 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 \({\mathfrak{X}}\) 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.

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  1. Laboratoire Painlevé, Université Lille 1, CNRS-UMR 8524, Cité Scientifique, 59655, Villeneuve d’Ascq Cédex, France

    Johannes Nicaise

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The research for this article was partially supported by ANR-06-BLAN-0183.

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Nicaise, J. A trace formula for rigid varieties, and motivic Weil generating series for formal schemes. Math. Ann. 343, 285–349 (2009). https://doi.org/10.1007/s00208-008-0273-9

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