Period Spaces for Hodge Structures in Equal Characteristic

We develop the analogue in equal positive characteristic of Fontaine’s theory for crystalline Galois representations of a p-adic field. In particular we describe the analogue of Fontaine’s mysterious functor which assigns to a crystalline Galois representation a Hodge filtration. In equal characteristic the role of the Hodge filtrations is played by Hodge-Pink structures. The later were invented by Pink. Our first main result in this article is the analogue of the Colmez-Fontaine Theorem that ”weakly admissible implies admissible”. Next we construct period spaces for Hodge-Pink structures. These period spaces are analogues of the RapoportZink period spaces for Fontaine’s filtered isocrystals in mixed characteristic and likewise are rigid analytic spaces. For our period spaces we prove the analogue of a conjecture of RapoportZink stating the existence of interesting local systems on a Berkovich open subspace of the period space. As a consequence of ”weakly admissible implies admissible” this Berkovich open subspace contains every classical rigid analytic point of the period space. As the principal tool to demonstrate these results we use the analogue of Kedlaya’s Slope Filtration Theorem which we also formulate and prove here. Mathematics Subject Classification (2000): 11G09, (13A35, 14G20, 14G22)