Log $p$-divisible groups and semi-stable representations

Abstract

Let ${\mathcal{O}}_K$ be a henselian DVR with field of fractions $K$ and residue field of characteristic $p>0$. Let $S$ denote $\mathrm{Spec}{\mathcal{O}}_K$ endowed with the canonical log structure. We show that the generic fiber functor ${\mathbf{BT}}_{S,\mathrm{d}}^{\log }\to {\mathbf{BT}}_K^{\mathrm{st}}$ between the category of dual representable log $p$-divisible groups over $S$ and the category of $p$-divisible groups with semistable reduction over $K$ is an equivalence. If ${\mathcal{O}}_K$ is further complete with perfect residue field and of mixed characteristic, we show that ${\mathbf{BT}}_{S,\mathrm{d}}^{\log }$ is also equivalent to the category of semistable Galois ${\mathbb{Z}}_p$-representations with Hodge-Tate weights in $\{0,1\}$. Finally, we show that the above equivalences respect monodromies.

Minor changes in the introduction