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

Byott, Nigel P.

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

Soit $K$ une extension finie de ${\mathbb{Q}}_p$ d’indice de ramification $e$, et soit $L/K$ une $p$-extension abélienne finie de groupe de Galois $\Gamma$ 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 $𝔄(K\Gamma ;{𝔓}^h)$. 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 $𝔄(K\Gamma ;S)$ est un ordre de Hopf et si $S/R$ est une $𝔄(K\Gamma ;S)$-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 $k\ne {\mathbb{Q}}_p$, alors $S$ n’est pas libre sur $𝔄(K\Gamma ;S)$. 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 ${\mathbb{Q}}_p$ with ramification index $e$, and let $L/K$ be a finite abelian $p$-extension with Galois group $\Gamma$ 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 $𝔄(K\Gamma ;{𝔓}^h)$. 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 $𝔄(K\Gamma ;S)$ is a Hopf order and $S$ is $𝔄(K\Gamma ;S)$-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 $k\ne {\mathbb{Q}}_p$, then $S$ is not free over ($𝔄(K\Gamma ;S)$. Thirdly, these extensions $k_{m+r}/k_r$ admit two Hopf Galois structures exhibiting different behaviour at integral level.

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