Asian Journal of Mathematics

Volume 27 (2023)

Number 1

On $\pi$-divisible $\mathcal{O}$-modules over fields of characteristic $p$

Pages: 1 – 56



Chuangxun Cheng (Department of Mathematics, Nanjing University, Nanjing, China)


In this paper, we construct a Dieudonné theory for $\pi$-divisible $\mathcal{O}$-modules over a perfect field of characteristic $p$. Applying this theory, we prove the existence of slope filtration of $\pi$-divisible $\mathcal{O}$-modules over an integral normal Noetherian base. We also study minimal $\pi$-divisible $\mathcal{O}$-modules over an algebraically closed field of characteristic $p$ and prove that their isomorphism classes are determined by their $\pi$-torsion parts by introducing Oort’s filtration. Moreover, after a detailed study of deformations of $\pi$-divisible $\mathcal{O}$-modules via displays, we prove the generalized Traverso’s isogeny conjecture.


$\mathcal{O}$-isocrystal, $\mathcal{O}$-crystal, Dieudonné $\mathcal{O}$-module, $\pi$-divisible $\mathcal{O}$-module, completely slope divisible $\mathcal{O}$-module, slope filtration, Oort filtration, Traverso’s isogeny conjecture

2010 Mathematics Subject Classification

14L05, 14L15

The authors were supported by NSFC grant 11701272, NSFC grant 12071221, and Grant 020314803001 of Jiangsu Province (China).

Received 2 September 2018

Accepted 1 December 2022

Published 16 June 2023