On F-crystalline representations

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 $F\mathrm{/}{\mathbb{Q}}_p$, and an arbitrary finite extension $K\mathrm{/}F$, we construct a general class of infinite and totally wildly ramified extensions $K_{\mathrm{\infty }}\mathrm{/}K$ so that the functor $V\mapsto V|{G{K_\infty}}$isfully-faithfullonthecategoryof$F$-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.