On the p-adic local invariant cycle theorem

Abstract:  This is a joint work with Matthias Flach. The aim is to construct a p-adic analogue of the local invariant cycle theorem for regular arithmetic schemes. We construct the specialization map from the rigid cohomology of the geometric special fiber to $D_{crys}$ of the p-adic etale cohomology of the geometric generic fiber of a proper flat and generically smooth scheme over the ring of integers of a local field. The construction is via descent from the semistable reduction case, and we can prove the morphism is an isomorphism in slope [0,1) part. We outline a possible approach to the proof in slope [0,1) part in the general regular case using the trace maps recently constructed by Berthelot, Esnault and Ruelling.