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.
Categories: math.AG math.NT