A Trace Formula for Rigid Varieties, and Motivic Weil Generating Series for Formal Schemes
نویسنده
چکیده
We establish a trace formula for rigid varieties X over a complete discretely valued field of equicharacteristic zero, 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 analytic 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 X of pseudo-finite type, via the construction of a Gelfand-Leray form on its generic fiber. When 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 X is the formal completion of f at a closed point x of the special fiber f(0), we obtain the local motivic zeta function of f at x.
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