Lubin–Tate Deformation Spaces and Fields of Norms
Journal de Théorie des Nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 433-458.

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.

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Accepted:
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DOI: 10.5802/jtnb.1166
Classification: 11F80,  11F85,  11S15,  11S31
Keywords: formal groups, Lubin–Tate deformation space, strictly deeply ramified tower, field of norms
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.
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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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