New ramification breaks and additive Galois structure

Byott, Nigel P.; Elder, G. Griffith

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 87-107.

Quels invariants d’une $p$-extension galoisienne de corps local $L/K$ (de corps résiduel de charactéristique $p$ et groupe de Galois $G$) déterminent la structure des idéaux de $L$ en tant que modules sur l’anneau de groupe ${\mathbb{Z}}_p\left[G\right]$, ${\mathbb{Z}}_p$ l’anneau des entiers $p$-adiques ? Nous considérons cette question dans le cadre des extensions abéliennes élémentaires, bien que nous considérions aussi brièvement des extensions cycliques. Pour un groupe abélien élémentaire $G$, nous proposons et étudions un nouveau groupe (dans l’anneau de groupe ${𝔽}_q\left[G\right]$ où ${𝔽}_q$ est le corps résiduel) ainsi que ses filtrations de ramification.

Which invariants of a Galois $p$-extension of local number fields $L/K$ (residue field of char $p$, and Galois group $G$) determine the structure of the ideals in $L$ as modules over the group ring ${\mathbb{Z}}_p\left[G\right]$, ${\mathbb{Z}}_p$ the $p$-adic integers? We consider this question within the context of elementary abelian extensions, though we also briefly consider cyclic extensions. For elementary abelian groups $G$, we propose and study a new group (within the group ring ${𝔽}_q\left[G\right]$ where ${𝔽}_q$ is the residue field) and its resulting ramification filtrations.

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