**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.