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

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.

Keywords

$\mathcal{O}$-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

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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