  For $C$ a complete algebraically closed extension of $\mathbb{Q}_p$, we show
that a one-dimensional $p$-divisible group $G/ \mathcal{O}_C$ can be defined
over a complete discretely valued subfield $L \subset C$ with Hodge-Tate period
ratios contained in $L$ if and only if $G$ has CM, if and only if the period
ratios generate an extension of $\mathbb{Q}_p$ of degree equal to the height of
the connected part of $G$. This is a $p$-adic analog of a classical
transcendence result of Schneider which states that for $\tau$ in the complex
upper half plane, $\tau$ and $j(\tau)$ are simultaneously algebraic over
$\mathbb{Q}$ if and only if $\tau$ is contained in a quadratic extension of
$\mathbb{Q}$. We also briefly discuss a conjectural generalization to shtukas
with one paw.
