Log prismatic Dieudonn\'e theory for log pp-divisible groups over OK\mathcal{O}_{K}

Let $\mathcal{O}_{K}$ be a complete discrete valuation ring of mixed characteristic with perfect residue field, endowed with its canonical log-structure. We prove that log $p$-divisible groups over $\mathcal{O}_{K}$ correspond to Dieudonn\'e crystals on the absolute log-prismatic site of $\mathcal{O}_{K}$ endowed with the Kummer log-flat topology. The proof uses log-descent to reduce the problem to the classical prismatic correspondence, recently established by Ansch\"utz-Le Bras.Comment: 19 page

Log p-divisible groups associated to log 1-motives

We first provide a detailed proof of Kato's classification theorem of log $p$-divisible groups over a 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. We show that the finite Kummer flat group log scheme $T_n(M):=H^{-1}(M\otimes_{\mathbb{Z}}^L\mathbb{Z}/n\mathbb{Z})$ (resp. the log $p$-divisible group $M[p^{\infty}]$) of a log 1-motive $M$ over an fs log scheme is \'etale locally a standard extension. We further show that $M[p^{\infty}]$ and its infinitesimal deformations are formally log smooth. At last, as an application of these formal log smoothness results, we give a proof of the Serre-Tate theorem for log abelian varieties with constant degeneration.Comment: 36 page

Divided prismatic Frobenius crystals of small height and the category MF\mathcal{M}\mathcal{F}

Let $\mathcal{X}$ be a smooth $p$-adic formal scheme over a mixed characteristic complete discrete valuation ring $\mathcal{O}_{K}$ with perfect residue field. We introduce a general category $\mathcal{M}\mathcal{F}_{[0, p-2]}^{tor-free}(\mathcal{X})$ of $p$-torsion free crystalline coefficient objects and show that this category is equivalent to the category of completed prismatic Frobenius crystals of height $p-2$, recently introduced by Du-Liu-Moon-Shimizu. In particular this shows that the category $\mathcal{M}\mathcal{F}^{tor-free}_{[0, p-2]}(\mathcal{X})$ is equivalent to the category of crystalline $\mathbb{Z}_p$-local systems on $\mathcal{X}$ with Hodge-Tate weights in $\{0,\ldots , p-2\}$, which generalizes the crystalline part of a theorem of Breuil-Liu to higher dimensions.Comment: 41 pages, comments welcome