Monodromy Eigenvalues and Zeta Functions with Differential Forms

For a complex polynomial or analytic function f , there is a strong correspondence between poles of the so-called local zeta functions or complex powers ∫ |f |2sω, where the ω are C∞ differential forms with compact support, and eigenvalues of the local monodromy of f . In particular Barlet showed that each monodromy eigenvalue of f is of the form exp(2π −1s0), where s0 is such a pole. We prove an analogous result for similar p-adic complex powers, called Igusa (local) zeta functions, but mainly for the related algebro-geometric topological and motivic zeta functions.