On $F$-crystalline representations

Abstract

We extend the theory of Kisin modules and crystalline representations to allow more general coefficient fields and lifts of Frobenius. In particular, for a finite and totally ramified extension $F/{\mathbb{Q}}_p$, and an arbitrary finite extension $K/F$, we construct a general class of infinite and totally wildly ramified extensions $K_{\infty }/K$ so that the functor $V\mapsto V{|}_{G_{K_{\infty }}}$ is fully-faithfull on the category of $F$-crystalline representations $V$. We also establish a new classification of $F$-Barsotti–Tate groups via Kisin modules of height 1 which allows more general lifts of Frobenius.