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

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. 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. Our trace formula yields a cohomological interpretation of this Weil generating series. 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 $x$. In the latter case, the generic fiber of $\mathfrak{X}$ is the so-called analytic Milnor fiber of $f$ at $x$; we show that it completely determines the formal germ of $f$ at $x$.

To appear in Math. Ann. The original publication is available at http://www.springerlink.com

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• A note on motivic integration in mixed characteristic
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• Trace formula for component groups of Néron models
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• The Thom-Sebastiani theorem for the Euler characteristic of cyclic L-infinity algebras
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• Equivariant motivic integration and proof of the integral identity conjecture for regular functions
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• Triangulations of non-archimedean curves, semi-stable reduction, and ramification
• Uniformly rigid spaces and Néron models of formally finite type
• Motivic zeta functions of abelian varieties, and the monodromy conjecture
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