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.

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 $\Gamma =$ Gal(${k}_{3}/{k}_{1}$). On sait que si $k\ne {ℚ}_{p}$ l’anneau de valuation de ${k}_{3}$ n’est pas libre sur son ordre associé $𝔄$ dans $K\Gamma$. 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 $\Gamma =$ Gal(${k}_{3}/{k}_{1}$). It is known that the valuation ring ${k}_{3}$ cannot be free over its associated order $𝔄$ in $K\Gamma$ unless $k={ℚ}_{p}$. We determine explicitly under the hypothesis that the absolute ramification index of $k$ is sufficiently large.

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},
zbl = {0902.11052},
mrnumber = {1617408},
language = {en},
url = {http://archive.numdam.org/item/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://archive.numdam.org/item/JTNB_1997__9_2_449_0/

[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 1104596

[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 1469668 | Zbl 0889.11040

[C] S.-P. Chan, Galois module structure of non-Kummer extensions, Preprint, National University of Singapore (1995). | MR 1611192

[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 1299647 | Zbl 0816.11061

[R] P. Ribenboim, The Book of Prime Number Records, 2nd edition, Springer, 1989. | MR 1016815 | Zbl 0642.10001

[S] J.-P. Serre, Local Class Field Theory, in Algebraic Number Theory (J.W.S. Cassels and A. Fröhlich, eds.), Academic Press, 1967. | MR 220701

[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 797677 | Zbl 0582.12008