Associated orders of certain extensions arising from Lubin-Tate formal groups

Byott, Nigel P.

Byott, Nigel P.

Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 2, pp. 449-462.

Résumé
Abstract

Soit $k$ une extension finie de $ℚ_{p}, k_{1}$ et $k_{3}$ les corps de division de niveaux respectifs $1$ et $3$ associés à un groupe formel de Lubin-Tate, et soit $Γ =$ Gal( $k_{3} / k_{1}$ ). On sait que si $k \neq ℚ_{p}$ l’anneau de valuation de $k_{3}$ n’est pas libre sur son ordre associé $𝔄$ dans $K Γ$ . Nous explicitons $𝔄$ dans le cas où l’indice absolu de ramification de $k$ est assez grand.

Let $k$ be a finite extension of $ℚ_{p}$ , let $k_{1}$ , respectively $k_{3}$ , be the division fields of level $1$ , respectively $3$ , arising from a Lubin-Tate formal group over $k$ , and let $Γ =$ Gal( $k_{3} / k_{1}$ ). It is known that the valuation ring $k_{3}$ cannot be free over its associated order $𝔄$ in $K Γ$ unless $k = ℚ_{p}$ . We determine explicitly under the hypothesis that the absolute ramification index of $k$ is sufficiently large.

MR Zbl | 2 citations dans Numdam

Classification : 11S23, 11S31, 11R33
Mots-clés : associated order, Lubin-Tate formal group

@article{JTNB_1997__9_2_449_0,
     author = {Byott, Nigel P.},
     title = {Associated orders of certain extensions arising from {Lubin-Tate} formal groups},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {449--462},
     publisher = {Universit\'e Bordeaux I},
     volume = {9},
     number = {2},
     year = {1997},
     mrnumber = {1617408},
     zbl = {0902.11052},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_1997__9_2_449_0/}
}

TY  - JOUR
AU  - Byott, Nigel P.
TI  - Associated orders of certain extensions arising from Lubin-Tate formal groups
JO  - Journal de théorie des nombres de Bordeaux
PY  - 1997
SP  - 449
EP  - 462
VL  - 9
IS  - 2
PB  - Université Bordeaux I
UR  - http://www.numdam.org/item/JTNB_1997__9_2_449_0/
LA  - en
ID  - JTNB_1997__9_2_449_0
ER  -

%0 Journal Article
%A Byott, Nigel P.
%T Associated orders of certain extensions arising from Lubin-Tate formal groups
%J Journal de théorie des nombres de Bordeaux
%D 1997
%P 449-462
%V 9
%N 2
%I Université Bordeaux I
%U http://www.numdam.org/item/JTNB_1997__9_2_449_0/
%G en
%F JTNB_1997__9_2_449_0

Byott, Nigel P. Associated orders of certain extensions arising from Lubin-Tate formal groups. Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 2, pp. 449-462. http://www.numdam.org/item/JTNB_1997__9_2_449_0/

Bibliographie
Cité par

[B1] N.P. Byott, Some self-dual rings of integers not free over their associated orders, Math. Proc. Camb. Phil. Soc. 110 (1991), 5-10; Corrigendum, 116 (1994), 569. | MR

[B2] N.P. Byott, Galois structure of ideals in wildly ramified abelian p-extensions of a p-adic field, and some applications, J. de Théorie des Nombres de Bordeaux 9 (1997), 201-219. | Numdam | MR | Zbl

[C] S.-P. Chan, Galois module structure of non-Kummer extensions, Preprint, National University of Singapore (1995). | MR

[C-L] S.-P. Chan and C.-H. Lim, The associated orders of rings of integers in Lubin- Tate division fields over the p-adic number field, Ill. J. Math. 39 (1995), 30-38. | MR | Zbl

[R] P. Ribenboim, The Book of Prime Number Records, 2nd edition, Springer, 1989. | MR | Zbl

[S] J.-P. Serre, Local Class Field Theory, in Algebraic Number Theory (J.W.S. Cassels and A. Fröhlich, eds.), Academic Press, 1967. | MR

[T] M.J. Taylor, Formal groups and the Galois module structure of local rings of integers, J. reine angew. Math. 358 (1985), 97-103. | MR | Zbl