Integration over spaces of non-parametrized arcs and motivic versions of the monodromy zeta function

We study the scheme of formal arcs on a singular algebraic variety and its images under truncations. We prove a rationality result for the Poincare series of these images which is an analogue of the rationality of the Poincare series associated to p-adic points on a p-adic variety. The main tools which are used are semi-algebraic geometry in spaces of power series and motivic integration (a notion introduced by M. Kontsevich). In particular we develop the theory of motivic in...

A notion of Milnor fibration for meromorphic functions and the corresponding concepts of monodromy and monodromy zeta function have been introduced in [GZLM1]. In this article we define the topological zeta function for meromorphic germs and we study its poles in the plane case. We show that the poles do not behave as in the holomorphic case but still do satisfy a generalization of the monodromy conjecture.

Let $(X,x)$ be a germ of real or complex analytic space and $\mathcal{A}_{(X,x)}$ the space of germs of arcs on $(X,x)$. Let us consider $F_{x}: (X,x) \to (Y,y)$ a germ of a morphism and denote by $\mathcal{F}_{x}: \mathcal{A}_{(X,x)} \to \mathcal{A}_{(Y,y)}$ the induced morphism at the level of arcs. In this paper, we try to emphasize the analogies between the metric or local topological properties of $F_{x}$ and those of $\mathcal{F}_{x}$. We then define the notions of Nash...

For a formal scheme over a complete discrete valuation ring with a good action of a finite group, we define equivariant motivic integration, and we prove a change of variable formula for that.To do so, we construct and examine an induced group action on the Greenberg scheme of such a formal scheme. Using this equivariant motivic integration, we define an equivariant volume Poincar\'e series, from which we deduce Denef and Loeser's motivic zeta function including the action of...

We investigate connections between Lipschitz geometry of real algebraic varieties and properties of their arc spaces. For this purpose we develop motivic integration in the real algebraic set-up. We construct a motivic measure on the space of real analytic arcs. We use this measure to define a real motivic integral which admits a change of variables formula not only for the birational but also for generically one-to-one Nash maps. As a consequence we obtain an inverse mappi...

The connection between Lefschetz formulae and zeta function is explained. As a particular example the theory of the generalized Selberg zeta function is presented. Applications are given to the theory of Anosov flows and prime geodesic theorems.

Combining the idea of motivic zeta function, due to Kapranov, and Pellikaan's definition of a two- variable zeta function for curves over finite fields in the present note we introduce a motivic two- variable zeta function for curves over arbitrary fields and prove the generalizations of Pellikaan's results in this context.

We show how formal and rigid geometry can be used in the theory of complex singularities, and in particular in the study of the Milnor fibration and the motivic zeta function. We introduce the so-called analytic Milnor fiber associated to the germ of a morphism f from a smooth complex algebraic variety X to the affine line. This analytic Milnor fiber is a smooth rigid variety over the field of Laurent series C((t)). Its etale cohomology coincides with the singular cohomology ...

In this article, we compute the motivic Igusa zeta function of a space monomial curve that appears as the special fiber of an equisingular family whose generic fiber is a complex plane branch. To this end, we determine the irreducible components of the jet schemes of such a space monomial curve. This approach does not only yield a closed formula for the motivic zeta function, but also allows to determine its poles. We show that, while the family of the jet schemes of the fibe...

We lift the splicing formula of N\'emethi and Veys, which deals with polynomials in two variables, to the motivic level. After defining the motivic zeta function and the monodromic motivic zeta function with respect to a differential form, we prove a splicing formula for them, which specializes to this formula of N\'emethi and Veys. We also show that we cannot introduce a monodromic motivic zeta functions in terms of a (splice) diagram since it does not contain all the necess...