Lubin–Tate Deformation Spaces and Fields of Norms

Annie Carter; Matthias Strauch

doi:10.5802/jtnb.1166

Annie Carter ¹; Matthias Strauch ²

¹ Department of Mathematics University of California San Diego 9500 Gilman Drive # 0112 La Jolla, CA 92093-0112, U.S.A.
² Indiana University Department of Mathematics Rawles Hall Bloomington, IN 47405, U.S.A.

Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 433-458.

Abstract
Résumé

We construct a tower of fields from the rings $R_{n}$ which parametrize pairs $(X, λ)$ , where $X$ is a deformation of a fixed one-dimensional formal group $𝕏$ of finite height $h$ , together with a Drinfeld level- $n$ structure $λ$ . We choose principal prime ideals $𝔭_{n} | (p)$ in each ring $R_{n}$ in a compatible way and consider the field $K_{n}^{'}$ obtained by localizing $R_{n}$ at $𝔭_{n}$ and passing to the field of fractions of the completion. By taking the compositum $K_{n} = K_{n}^{'} K_{0}$ of $K_{n}^{'}$ with the completion $K_{0}$ of a certain unramified extension of $K_{0}^{'}$ , we obtain a tower of fields ${(K_{n})}_{n}$ which we prove to be strictly deeply ramified in the sense of Scholl. When $h = 2$ we also investigate the question of whether this is a Kummer tower.

On construit une tour de corps à partir des anneaux $R_{n}$ qui paramétrisent les couples $(X, λ)$ , où $X$ est une déformation d’un groupe formel fixé $𝕏$ de dimension un et de hauteur $h$ , muni d’une structure de Drinfeld $λ$ de niveau $n$ . On choisit des idéaux principaux premiers $𝔭_{n} | (p)$ de $R_{n}$ de manière compatible, et on considère le corps $K_{n}^{'}$ obtenu en localisant $R_{n}$ en $𝔭_{n}$ et en passant au corps des fractions de la complétion. En prenant le compositum $K_{n} = K_{n}^{'} K_{0}$ de $K_{n}^{'}$ et de la complétion $K_{0}$ d’une certaine extension non-ramifiée de $K_{0}^{'}$ , on obtient la tour de corps ${(K_{n})}_{n}$ pour laquelle on démontre qu’elle est ’strictly deeply ramified’ au sens de Scholl. Quand $h = 2$ , on étudie la question de savoir s’il s’agit d’une tour kummérienne.

Received: 2020-04-08
Revised: 2021-01-19
Accepted: 2021-04-19
Published online: 2021-09-10

DOI: 10.5802/jtnb.1166

Classification: 11F80, 11F85, 11S15, 11S31
Keywords: formal groups, Lubin–Tate deformation space, strictly deeply ramified tower, field of norms

Author's affiliations:

Annie Carter ¹; Matthias Strauch ²

¹ Department of Mathematics University of California San Diego 9500 Gilman Drive # 0112 La Jolla, CA 92093-0112, U.S.A.
² Indiana University Department of Mathematics Rawles Hall Bloomington, IN 47405, U.S.A.

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Annie Carter and Matthias Strauch},
     title = {Lubin{\textendash}Tate {Deformation} {Spaces} and {Fields} of {Norms}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {433--458},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Annie Carter; Matthias Strauch. Lubin–Tate Deformation Spaces and Fields of Norms. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 433-458. doi : 10.5802/jtnb.1166. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1166/

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