Séminaire de Géométrie Algébrique de Rennes - Exposé du 17 novembre 2005
Johannes Nicaise (Lille)
Titre : Rigid geometry and the Monodromy Conjecture
Igusa's Monodromy Conjecture predicts an intriguing relationship between the arithmetic and the complex topological properties of a polynomial f over a number field K. The motivic version of this conjecture relates the monodromy transformation on the cohomology of the Milnor fiber of f at the points of the complex hypersurface defined by f, to the structure of the jet spaces of this hypersurface. In this joint work with Julien Sebag, we describe a direct link between the jet spaces of f, and the monodromy on its Milnor fibers, using Berkovich' theory of analytic spaces. Using Loeser and Sebag's motivic Serre invariant to count rational points on non-archimedean analytic spaces, we establish Denef and Loeser's motivic zeta function as a Weil zeta function of the Milnor fiber. The corresponding trace formula generalizes a result by Denef and Loeser, relating the Lefschetz numbers of the monodromy, to the topological specialization of the zeta function.