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

Nigel P. Byott

Journal de théorie des nombres de Bordeaux (1997)

  • Volume: 9, Issue: 2, page 449-462
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Byott, Nigel P.. "Associated orders of certain extensions arising from Lubin-Tate formal groups." Journal de théorie des nombres de Bordeaux 9.2 (1997): 449-462. <http://eudml.org/doc/247989>.

@article{Byott1997,
abstract = {Let $k$ be a finite extension of $\mathbb \{Q\}_p$, 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 $\mathfrak \{A\}$ in $K \Gamma $ unless $k = \mathbb \{Q\}_p$. We determine explicitly under the hypothesis that the absolute ramification index of $k$ is sufficiently large.},
author = {Byott, Nigel P.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {associated order; Lubin-Tate formal group; associated orders; Galois modules; ramification; Lubin-Tate extensions; divisibility properties of binomial coefficients},
language = {eng},
number = {2},
pages = {449-462},
publisher = {Université Bordeaux I},
title = {Associated orders of certain extensions arising from Lubin-Tate formal groups},
url = {http://eudml.org/doc/247989},
volume = {9},
year = {1997},
}

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
PB - Université Bordeaux I
VL - 9
IS - 2
SP - 449
EP - 462
AB - Let $k$ be a finite extension of $\mathbb {Q}_p$, 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 $\mathfrak {A}$ in $K \Gamma $ unless $k = \mathbb {Q}_p$. We determine explicitly under the hypothesis that the absolute ramification index of $k$ is sufficiently large.
LA - eng
KW - associated order; Lubin-Tate formal group; associated orders; Galois modules; ramification; Lubin-Tate extensions; divisibility properties of binomial coefficients
UR - http://eudml.org/doc/247989
ER -

References

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  1. [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. Zbl0737.11037MR1104596
  2. [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 Bordeaux9 (1997), 201-219. Zbl0889.11040MR1469668
  3. [C] S.-P. Chan, Galois module structure of non-Kummer extensions, Preprint, National University of Singapore (1995). MR1611192
  4. [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. Zbl0816.11061MR1299647
  5. [R] P. Ribenboim, The Book of Prime Number Records, 2nd edition, Springer, 1989. Zbl0642.10001MR1016815
  6. [S] J.-P. Serre, Local Class Field Theory, in Algebraic Number Theory (J.W.S. Cassels and A. Fröhlich, eds.), Academic Press, 1967. MR220701
  7. [T] M.J. Taylor, Formal groups and the Galois module structure of local rings of integers, J. reine angew. Math.358 (1985), 97-103. Zbl0582.12008MR797677

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