Transcendence of the Hodge–Tate filtration

Howe, Sean

Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 2, pp. 671-680.

Soit $C$ une extension algébriquement close et complète de ${\mathbb{Q}}_p$. Nous démontrons qu’un groupe $p$-divisible $G/{𝒪}_C$ de dimension $1$ est défini sur un sous-corps $L\subset C$ complet pour une valuation discrète et contenant les ratios entre les périodes de Hodge–Tate si et seulement si $G$ est de type CM et si et seulement si les ratios entre les périodes engendrent une extension de ${\mathbb{Q}}_p$ de degré égal à la hauteur de la composante connexe neutre de $G$. C’est un analogue $p$-adique du résultat classique de transcendance de Schneider qui dit que, pour $\tau$ dans le demi-plan complexe supérieur, $\tau$ et $j\left(\tau \right)$ sont tous les deux algèbriques sur $\mathbb{Q}$ si et seulement si $\tau$ appartient à une extension quadratique de $\mathbb{Q}$. Nous discutons aussi brièvement d’une conjecture qui généralise ce résultat aux shtukas à une patte.

For $C$ a complete algebraically closed extension of ${\mathbb{Q}}_p$, we show that a one-dimensional $p$-divisible group $G/{𝒪}_C$ 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 $j\left(\tau \right)$ 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.

Classification : 14L05, 14G22
Mots-clés : Hodge–Tate filtration, p-divisible groups, formal groups, $j$-invariant, transcendence, transcendentals, Lubin–Tate tower, Hodge–Tate period map, Gross–Hopkins period map

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