Log p-divisible groups associated to log 1-motives

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 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 $T_n(\mathbf{M}):=H^{-1}(\mathbf{M}{\otimes }_{\mathbb{Z}}^L\mathbb{Z}/n\mathbb{Z})$ (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 étale locally standard extensions. Lastly, we give a proof of the Serre-Tate theorem for log abelian varieties with constant degeneration.

Published online at Canadian Journal of Mathematics. Slightly different from the published version in format! 38 pages

References in corpus (3)

• Extending tamely ramified strict 1-motives into ket log 1-motives
• Logarithmic abelian varieties, Part VII: Moduli
• Extending finite subgroup schemes of semi-stable abelian varieties via log abelian varieties