Local monodromy of $p$-divisible groups

Abstract:

A $p$-divisible group over a field $K$ admits a slope decomposition; associated to each slope $\lambda$ is an integer $m$ and a representation $\mathrm {Gal}(K)\rightarrow \mathrm {GL}_m(D_\lambda )$, where $D_\lambda$ is the $\mathbb {Q}_p$-division algebra with Brauer invariant $[\lambda ]$. We call $m$ the multiplicity of $\lambda$ in the $p$-divisible group. Let $G_0$ be a $p$-divisible group over a field $k$. Suppose that $\lambda$ is not a slope of $G_0$, but that there exists a deformation of $G$ in which $\lambda$ appears with multiplicity one. Assume that $\lambda \not = (s-1)/s$ for any natural number $s>1$. We show that there exists a deformation $G/R$ of $G_0/k$ such that the representation $\mathrm {Gal}(\mathrm {Frac} R) \rightarrow \mathrm {GL}_1(D_\lambda )$ has a large image.