Relative p-adic Hodge theory and Rapoport-Zink period domains

As an example of relative p-adic Hodge theory, we sketch the construction of the universal admissible filtration of an isocrystal (\phi$-module) over the completion of the maximal unramified extension of Q_p, together with the associated universal crystalline local system.Comment: 20 page

On commutative nonarchimedean Banach fields

We study the problem of whether a commutative nonarchimedean Banach ring which is algebraically a field can be topologized by a multiplicative norm. This can fail in general, but it holds for uniform Banach rings under some mild extra conditions. Notably, any perfectoid ring whose underlying ring is a field is a perfectoid field.Comment: 14 pages; v7: includes corrections to published versio