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.

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

References in corpus (4)

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