Log p-divisible groups associated to log 1-motives

TL;DR

This paper advances the theory of log p-divisible groups by providing detailed classification, defining standard extensions, proving formal smoothness, and establishing local standardness and Serre-Tate theorem for log abelian varieties.

Contribution

It offers a detailed proof of Kato's classification theorem, introduces the notion of standard extensions in log schemes, and proves key properties like formal log smoothness and local standardness, along with a Serre-Tate theorem for log abelian varieties.

Findings

Kato's classification theorem of log p-divisible groups is rigorously proved.

Log p-divisible groups are shown to be formally log smooth.

Finite Kummer flat group log schemes of log 1-motives are locally standard extensions.

Abstract

We first provide a detailed proof of Kato's classification theorem of log $p$-divisible groups over a noetherian henselian local ring. Exploring Kato's idea further, we then define the notion of a standard extension of a classical finite \'etale group scheme (resp. classical \'etale $p$-divisible group) by a classical finite flat group scheme (resp. classical $p$-divisible group) in the category of finite Kummer flat group log schemes (resp. log $p$-divisible groups), with respect to a given chart on the base. These results are then used to prove that log $p$-divisible groups are formally log smooth. We then study the finite Kummer flat group log schemes $T_n(\mathbf{M}):=H^{-1}(\mathbf{M}\otimes_{\mathbb{Z}}^L\mathbb{Z}/n\mathbb{Z})$ (resp. the log $p$-divisible group $\mathbf{M}[p^{\infty}]$) of a log 1-motive $\mathbf{M}$ over an fs log scheme and show that they are \'etale locally standard extensions. Lastly, we give a proof of the Serre-Tate theorem for log abelian varieties with constant degeneration.