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For $C$ a complete algebraically closed extension of $\mathbb{Q}_p$, we showthat a one-dimensional $p$-divisible group $G/ \mathcal{O}_C$ can be definedover a complete discretely valued subfield $L \subset C$ with Hodge-Tate periodratios contained in $L$ if and only if $G$ has CM, if and only if the periodratios generate an extension of $\mathbb{Q}_p$ of degree equal to the height ofthe connected part of $G$. This is a $p$-adic analog of a classicaltranscendence result of Schneider which states that for $\tau$ in the complexupper 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 shtukaswith one paw.

Transcendence of the Hodge-Tate filtration