Period spaces for Hodge structures in equal characteristic

Abstract

We develop the analog in equal positive characteristic of Fontaine’s theory for crystalline Galois representations of a $p$-adic field. In particular we describe the analog of Fontaine’s functor which assigns to a crystalline Galois representation an isocrystal with a Hodge filtration. In equal characteristic the role of isocrystals and Hodge filtrations is played by $z$-isocrystals and Hodge-Pink structures. The latter were invented by Pink. Our first main result in this article is the analog of the Colmez-Fontaine Theorem that “weakly admissible implies admissible”. Next we construct period spaces for Hodge-Pink structures on a fixed $z$-isocrystal. These period spaces are analogs of the Rapoport-Zink period spaces for Fontaine’s filtered isocrystals in mixed characteristic and likewise are rigid analytic spaces. For our period spaces we prove the analog of a conjecture of Rapoport and Zink stating the existence of a “universal local system” 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 analog of Kedlaya’s Slope Filtration Theorem which we also formulate and prove here.