Local monodromy of p-divisible groups

Abstract

A $p$-divisible group over a field $K$ admits a slope decomposition; associated to each slope $λ$ is an integer $m$ and a representation $\gal(K) \ra \gl_m(D_λ)$, where $D_λ$ is the $\rat_p$-division algebra with Brauer invariant $[λ]$. We call $m$ the multiplicity of $λ$ in the $p$-divisible group. Let $G_0$ be a $p$-divisible group over a field $k$. Suppose that $λ$ is not a slope of $G_0$, but that there exists a deformation of $G$ in which $λ$ appears with multiplicity one. Assume that $λ\not= (s-1)/s$ for any natural number $s>1$. We show that there exists a deformation $G/R$ of $G_0/k$ such that the representation $\gal(\Frac R) \ra \gl_1(D_λ)$ has large image.