Log p-divisible groups associated with log 1-motives

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 étale group scheme (resp. classical étale 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 (resp. the log p-divisible group) of a log 1-motive over an fs log scheme and show that they are étale locally standard extensions. Lastly, we give a proof of the Serre-Tate theorem for log abelian varieties with constant degeneration.