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.

Published online:
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.
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     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},
     volume = {33},
     number = {2},
     year = {2021},
     doi = {10.5802/jtnb.1166},
     language = {en},
     url = {}
AU  - Annie Carter
AU  - Matthias Strauch
TI  - Lubin–Tate Deformation Spaces and Fields of Norms
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2021
SP  - 433
EP  - 458
VL  - 33
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  -
DO  - 10.5802/jtnb.1166
LA  - en
ID  - JTNB_2021__33_2_433_0
ER  - 
%0 Journal Article
%A Annie Carter
%A Matthias Strauch
%T Lubin–Tate Deformation Spaces and Fields of Norms
%J Journal de théorie des nombres de Bordeaux
%D 2021
%P 433-458
%V 33
%N 2
%I Société Arithmétique de Bordeaux
%R 10.5802/jtnb.1166
%G en
%F JTNB_2021__33_2_433_0
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.

[1] Michael F. Atiyah; Ian G. Macdonald Introduction to commutative algebra, Addison-Wesley Publishing Group, 1969, ix+128 pages | MR | Zbl

[2] Nicolas Bourbaki Elements of mathematics. Commutative algebra, Actualités Scientifiques et Industrielles, Hermann; Addison-Wesley Publishing Group, 1972, xxiv+625 pages (Translated from the French) | MR | Zbl

[3] Vladimir G. Drinfelʼd Elliptic modules, Math. USSR, Sb., Volume 23 (1976) no. 4, pp. 561-592 | DOI | MR

[4] Michael D. Fried; Moshe Jarden Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 11, Springer, 2008, xxiv+792 pages (Revised by Jarden) | MR | Zbl

[5] Michael Harris; Richard Taylor The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, 151, Princeton University Press, 2001, viii+276 pages (With an appendix by Vladimir G. Berkovich) | MR | Zbl

[6] Jan Kohlhaase Admissible ϕ-modules and p-adic unitary representations, Math. Z., Volume 270 (2012) no. 3-4, pp. 839-869 | DOI | MR | Zbl

[7] Jonathan Lubin; John Tate Formal moduli for one-parameter formal Lie groups, Bull. Soc. Math. Fr., Volume 94 (1966), pp. 49-59 | DOI | Numdam | MR | Zbl

[8] Anthony J. Scholl Higher fields of norms and (ϕ,Γ)-modules, Doc. Math., Volume Extra Vol. (2006), pp. 685-709 | MR

[9] Peter Scholze On the p-adic cohomology of the Lubin–Tate tower, Ann. Sci. Éc. Norm. Supér., Volume 51 (2018) no. 4, pp. 811-863 (With an appendix by Michael Rapoport) | DOI | MR | Zbl

[10] Jean-Pierre Serre Local class field theory, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, 1967, pp. 128-161 | MR

[11] Jean-Pierre Serre Local fields, Graduate Texts in Mathematics, 67, Springer, 1979, viii+241 pages (Translated from the French by Marvin Jay Greenberg) | MR | Zbl

[12] Matthias Strauch Deformation spaces of one-dimensional formal modules and their cohomology, Adv. Math., Volume 217 (2008) no. 3, pp. 889-951 | DOI | MR | Zbl

[13] Matthias Strauch Geometrically connected components of Lubin–Tate deformation spaces with level structures, Pure Appl. Math. Q., Volume 4 (2008) no. 4, Special Issue: In honor of Jean-Pierre Serre. Part 1, pp. 1215-1232 | DOI | MR | Zbl

[14] Matthias Strauch Galois actions on torsion points of one-dimensional formal modules, J. Number Theory, Volume 130 (2010) no. 3, pp. 528-533 | DOI | MR | Zbl

[15] Charles A. Weibel An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, 1994, xiv+450 pages | DOI | MR | Zbl

Cited by Sources: