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

By Johannes Nicaise. 2007

64 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics, Pharmacology

View Paper: https://www.semanticscholar.org/paper/e01bc64a64ddc3aefd1a6231488b1641517256e0

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In Geometry and Topology

In Mathematical Physics

In Pharmacology

Citations in the corpus

Motivic zeta functions of abelian varieties, and the monodromy conjecture (2009, methodology): “The image of S( X ; T ) under the forgetful morphism equals S( X η ,ω T ) , with ω a so-called Gelfand–Leray form on X η [25, 7.21].”
Functorial desingularization of quasi-excellent schemes in characteristic zero: the nonembedded case (2009, methodology): “ﬁnite type over a ring k[[t1,...,tn]] with k a ﬁeld. For example, this case (in its non-functorial version from [Tem1]) was used for studying log-canonical thresholds in [FEM], motivic integration in [Nic], and desingularization of meromorphic connections in [Ked]. The non-functorial method of [Tem1”
Desingularization of quasi-excellent schemes in characteristic zero (2007, methodology): “nsult loc.cit. instead of reading the rest of this section. Remark 2.1.6. Arbitrary formal blow ups (generalizing admissible formal blow ups) were deﬁned independently by J. Nicaise. In a recent work [23] on a trace formula and motivic integration, he establishes some basic properties of formal blow”