Log p-divisible groups associated with log 1-motives

Würthen, Matti ; Zhao, Heer 

Canadian journal of mathematics, Tome 76 (2024) no. 3, pp. 946-983

Résumé

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.

Détail

DOI : 10.4153/S0008414X23000287

Mots-clés : Log p-divisible groups, formally log smooth, log 1-motives, log abelian varieties with constant degeneration, Serre–Tate theory