Galois structure of ideals in wildly ramified abelian $p$-extensions of a $p$-adic field, and some applications

Nigel P. Byott

Galois structure of ideals in wildly ramified abelian

p

-extensions of a

p

-adic field, and some applications

Nigel P. Byott

Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 1, pp. 201-219.

Résumé
Abstract

Soit $K$ une extension finie de $ℚ_{p}$ d’indice de ramification $e$ , et soit $L / K$ une $p$ -extension abélienne finie de groupe de Galois $Γ$ et d’indice de ramification $p^{n}$ . Nous donnons un critère en termes des nombres de ramification $t_{i}$ permettant de décider lorsqu’un idéal fractionnaire $𝔓^{h}$ de l’anneau de valuation $S$ de $L$ peut être libre sur son ordre associé $𝔄 (K Γ; 𝔓^{h})$ . En particulier, si $t_{n} - [t_{n} / p] < p^{n - 1} e$ , la codifférente ne peut être libre sur son ordre associé que si $t_{i} \equiv - 1$ (mod $p^{n}$ ) pour tout $i$ . Nous déduisons de cela trois conséquences. Premièrement, si $𝔄 (K Γ; S)$ est un ordre de Hopf et si $S / R$ est une $𝔄 (K Γ; S)$ -extension galoisienne, où $R$ est l’anneau de valuation de $K$ , alors $t_{i} \equiv - 1$ (mod $p^{n}$ ) pour tout $i$ . Deuxièmement, si $K = k_{r}$ et $L = k_{m + r}$ sont des corps de points de division d’un groupe de Lubin-Tate, avec $m > r$ et $k \neq ℚ_{p}$ , alors $S$ n’est pas libre sur $𝔄 (K Γ; S)$ . Troisièmement, ces extensions $k_{m + r} / k_{r}$ possèdent deux structures galoisiennes de Hopf différentes, mettant en évidence des comportements différents au niveau des entiers.

Let $K$ be a finite extension of $ℚ_{p}$ with ramification index $e$ , and let $L / K$ be a finite abelian $p$ -extension with Galois group $Γ$ and ramification index $p^{n}$ . We give a criterion in terms of the ramification numbers $t_{i}$ for a fractional ideal $𝔓^{h}$ of the valuation ring $S$ of $L$ not to be free over its associated order $𝔄 (K Γ; 𝔓^{h})$ . In particular, if $t_{n} - [t_{n} / p] < p^{n - 1} e$ then the inverse different can be free over its associated order only when $t_{i} \equiv - 1$ (mod $p^{n}$ ) for all $i$ . We give three consequences of this. Firstly, if $𝔄 (K Γ; S)$ is a Hopf order and $S$ is $𝔄 (K Γ; S)$ -Galois then $t_{i} \equiv - 1$ (mod $p^{n}$ ) for all $i$ . Secondly, if $K = k_{r} L = k_{m + r}$ are Lubin-Tate division fields, with $m > r$ and $k \neq ℚ_{p}$ , then $S$ is not free over ( $𝔄 (K Γ; S)$ . Thirdly, these extensions $k_{m + r} / k_{r}$ admit two Hopf Galois structures exhibiting different behaviour at integral level.

MR Zbl

Classification : 11S23, 11R33, 11S31, 16W30
Mots clés : Galois module structure, associated order, Hopf order, Lubin-Tate formal group

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     author = {Nigel P. Byott},
     title = {Galois structure of ideals in wildly ramified abelian $p$-extensions of a $p$-adic field, and some applications},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {201--219},
     publisher = {Universit\'e Bordeaux I},
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}

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Nigel P. Byott. Galois structure of ideals in wildly ramified abelian $p$-extensions of a $p$-adic field, and some applications. Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 1, pp. 201-219. https://jtnb.centre-mersenne.org/item/JTNB_1997__9_1_201_0/

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