p-adic cohomology over local fields of characteristic p

If K is a local field of residue characteristic p, then every l-adic representation (l different from p) of G_K is potentially semistable - this is Grothendieck’s l-adic local monodromy theorem. For p-adic representations, and K of characteristic 0, potential semistablility is a condition that one needs to impose, and one must then prove that representations coming from geometry are potentially semistable. When K is of characteristic p, then the natural replacement for p-adic Galois representations that one encounters when looking at the cohomology of varieties over K are (phi,nabla) modules over the Amice ring, and the theory that produces them is rigid cohomology. We propose a condition on these modules analogous to potential semistability, and outline work in progress to show that all (phi,nabla) modules arising from geometry satisfy this condition. The idea is to replace rigid cohomology by a relative version by looking at compactifications over the ring of integers O_K of K. For varieties over K for which one can show finiteness of this theory (currently, for smooth curves), one can then attach Weil-Delinge representations to their cohomology, exactly as in the l-adic and mixed characteristic p-adic cases. This is joint work with Ambrus Pal.