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

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 of pseudo-finite type, via the construction of a Gelfand-Leray form on its generic fiber. When 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 is the formal completion of f at a closed point x of the special fiber , we obtain the local motivic zeta function of f at x. The research for this article was partially supported by ANR-06-BLAN-0183.