Non-Archimedian Period Domains

The best known example of a non-archimedean period domain is the Drinfeld upper half space $$\Omega _E^d$$ Ω E d of dimension d - 1 associated to a finite extension E of Q p (complement of all E-rational hyperplanes in the projective space Pd-1). Drinfeld [D2] interpreted this rigid-analytic space as the generic fibre of a formal scheme over O E parametrizing certain p-divisible groups. He used this to p-adically uniformize certain Shimura curves (Cherednik’s theorem) and to construct highly nontrivial étale coverings of $$\Omega _E^d$$ Ω E d . This report gives an account of joint work of Zink and myself [RZ] that generalizes the construction of Drinfeld (Sections 1–3). In the last two sections these results are put in a more general framework (Fontaine conjecture) and the problem of the computation of ℓ-adic cohomology is addressed (Kottwitz conjecture). In this report we return to the subject of Grothendieck’s talk at the Nice congress [G, esp. Section 5] where he stressed the relation between the local moduli of p-divisible groups and filtered Dieudonné modules.