Transcendence of the Hodge-Tate filtration

TL;DR

This paper characterizes when one-dimensional p-divisible groups over algebraically closed fields have Hodge-Tate periods in a subfield, linking this to complex multiplication and a p-adic analog of Schneider's transcendence theorem.

Contribution

It establishes a criterion connecting CM, period ratios, and definability over subfields for p-divisible groups, extending classical transcendence results to the p-adic setting.

Findings

Characterization of p-divisible groups with periods in subfields

Connection between CM and period ratios generating specific extensions

Analogy with classical transcendence results of Schneider

Abstract

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.