Local Monodromy of 1-Dimensional p-Divisible Groups

TL;DR

This paper investigates the ramification and Lie filtrations of Galois representations associated with 1-dimensional p-divisible groups over complete discrete valuation rings, extending classical theorems and proving irreducibility of certain representations.

Contribution

It establishes a relation between ramification and Lie filtrations for 1-dimensional p-divisible groups, generalizes Sen's theorem, and proves irreducibility of the associated Galois representation.

Findings

Relation between ramification and Lie filtrations in the 1-dimensional case

Generalization of Sen's theorem to equicharacteristic setting

Irreducibility of the Galois representation of the étale part

Abstract

Let $G$ be a $p$-divisible group over a complete discrete valuation ring $R$ of characteristic $p$. The generic fiber of $G$ determines a Galois representation $\rho$. The image of $\rho$ admits a ramification filtration and a Lie filtration. We relate these filtrations in the case $G$ is one dimensional, giving an equicharacteristic version of Sen's theorem in this setting. This result generalizes a result of Gross. Additionally, we prove that the representation associated to the \'etale part of $G$ is irreducible, generalizing a result of Chai.