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

TL;DR

This paper establishes an equivalence between categories of log $p$-divisible groups and semistable Galois representations, extending the understanding of $p$-adic Hodge theory in a log geometric setting.

Contribution

It proves that the generic fiber functor induces an equivalence between log $p$-divisible groups and semistable $p$-divisible groups, also relating them to Galois representations with Hodge-Tate weights.

Findings

The generic fiber functor is an equivalence between categories of log $p$-divisible groups and semistable $p$-divisible groups.

Log $p$-divisible groups correspond to semistable Galois representations with weights in {0,1}.

The equivalences preserve monodromy actions.

Abstract

Let $\mathscr{O}_K$ be a henselian DVR with field of fractions $K$ and residue field of characteristic $p>0$. Let $S$ denote $\mathop{\mathrm{Spec}} \mathscr{O}_K$ endowed with the canonical log structure. We show that the generic fiber functor $\mathbf{BT}_{S, {\mathrm{d}}}^{\log}\to \mathbf{BT}^{\mathrm{st}}_K$ 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 $\mathscr{O}_K$ 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.