Breuil's classification of $p$-divisible groups over regular local rings of arbitrary dimension

TL;DR

This paper extends Breuil's classification of p-divisible groups to a broader class of regular local rings with arbitrary dimension, generalizing previous results for the case when r=0.

Contribution

The paper provides a classification of p-divisible groups over certain regular local rings of arbitrary dimension, generalizing Breuil's conjecture and Kisin's proof for the case r=0.

Findings

Classification applies to rings with complex structure involving power series and polynomial relations.

Generalizes Breuil's classification from zero-dimensional to higher-dimensional rings.

Builds on and extends prior work by Breuil and Kisin.

Abstract

Let $k$ be a perfect field of characteristic $p \geq 3$. We classify $p$-divisible groups over regular local rings of the form $W(k)[[t_1,...,t_r,u]]/(u^e+pb_{e-1}u^{e-1}+...+pb_1u+pb_0)$, where $b_0,...,b_{e-1}\in W(k)[[t_1,...,t_r]]$ and $b_0$ is an invertible element. This classification was in the case $r = 0$ conjectured by Breuil and proved by Kisin.