This monograph is a systematic treatise on period domains over finite and over p-adic fields. The theory we present here has developed over the past fifteen years. Part of it has already appeared in various research articles or announcements, sometimes without detailed proofs. Our goal here is to present the theory as a whole and to provide complete proofs of the basics of the theory, so that these research articles can be accessed more easily. As it turned out, when working out the details, we had to change the very foundations of the theory quite a bit in some places, especially to accomodate isocrystals over non-algebraically closed fields, and also isocrystals with G-structure. Our hope is that our book can serve as the basis of future research in this exciting area.

Period domains over p-adic fields arose historically at the confluence of two theories: on the one hand, of Fontaine's theory of the “mysterious functor” conjectured by Grothendieck, which relates p-adic Galois representations of p-adic local fields and filtered isocrystals; on the other hand, of the theory of formal moduli spaces of p-divisible groups and their associated period maps. Via the latter theory, they are naturally related to local Langlands correspondences between ℓ-adic representations of the Galois groups of p-adic fields and smooth representations of p-adic Lie groups.