Kiran Kedlaya

Title: Slope filtrations for relative Frobenius: prelude to a (Phi, Gamma)-module theory in families

Abstract: I'll describe a new version of the slope filtration theorem for Phi-modules over the Robba ring (a certain power series ring over a p-adic field). This version incorporates a number of simplifications and clarifications, including a direct characterization of the Frobenius slopes (e.g., one can define isoclinic phi-modules without having to refer to the Dieudonné-Manin classification), and systematic use of faithfully flat descent to extract the theorem from an analogue of the DM classification. But the main new feature is that the theorem now covers "relative Frobenius", in the sense that it no longer matters how the Frobenius acts on coefficients of the power series. I'll try to say (based on limited advance information) how this figures into some exciting new work of Berger and Colmez, in which they develop a theory of (Phi, Gamma)-modules for p-adic families of local Galois representations.