Transcendence of the Hodge–Tate filtration
Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 671-680.

For C a complete algebraically closed extension of p , we show that a one-dimensional p-divisible group G/𝒪 C can be defined over a complete discretely valued subfield LC 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 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 τ in the complex upper half plane, τ and j(τ) are simultaneously algebraic over if and only if τ is contained in a quadratic extension of . We also briefly discuss a conjectural generalization to shtukas with one paw.

Soit C une extension algébriquement close et complète de p . Nous démontrons qu’un groupe p-divisible G/𝒪 C de dimension 1 est défini sur un sous-corps LC 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 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 τ dans le demi-plan complexe supérieur, τ et j(τ) sont tous les deux algèbriques sur si et seulement si τ appartient à une extension quadratique de . Nous discutons aussi brièvement d’une conjecture qui généralise ce résultat aux shtukas à une patte.

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DOI: 10.5802/jtnb.1044
Classification: 14L05, 14G22
Keywords: Hodge–Tate filtration, p-divisible groups, formal groups, $j$-invariant, transcendence, transcendentals, Lubin–Tate tower, Hodge–Tate period map, Gross–Hopkins period map
Howe, Sean 1

1 Stanford University Department of Mathematics Building 380 Stanford, California 94305, USA
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Howe, Sean. Transcendence of the Hodge–Tate filtration. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 671-680. doi : 10.5802/jtnb.1044. http://www.numdam.org/articles/10.5802/jtnb.1044/

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