Mathematische Annalen

, Volume 343, Issue 2, pp 285–349

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


    • Laboratoire PainlevéUniversité Lille 1, CNRS-UMR 8524

DOI: 10.1007/s00208-008-0273-9

Cite this article as:
Nicaise, J. Math. Ann. (2009) 343: 285. doi:10.1007/s00208-008-0273-9


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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© Springer-Verlag 2008