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Self-dual integral normal bases and Galois module structure

Part of: Algebraic number theory: global fields Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  10 May 2013

Erik Jarl Pickett  and
Stéphane Vinatier
Erik Jarl Pickett
Affiliation:
53 Parkfield Crescent, Kimpton, Herts, SG4 8EQ, UK email erik_pickett@hotmail.com
Stéphane Vinatier
Affiliation:
XLIM-DMI, 123 avenue Albert Thomas, 87060 Limoges, France email stephane.vinatier@unilim.fr
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Abstract

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Let $N/ F$ be an odd-degree Galois extension of number fields with Galois group $G$ and rings of integers ${\mathfrak{O}}_{N} $ and ${\mathfrak{O}}_{F} = \mathfrak{O}$. Let $ \mathcal{A} $ be the unique fractional ${\mathfrak{O}}_{N} $-ideal with square equal to the inverse different of $N/ F$. B. Erez showed that $ \mathcal{A} $ is a locally free $\mathfrak{O}[G] $-module if and only if $N/ F$ is a so-called weakly ramified extension. Although a number of results have been proved regarding the freeness of $ \mathcal{A} $ as a $ \mathbb{Z} [G] $-module, the question remains open. In this paper we prove that $ \mathcal{A} $ is free as a $ \mathbb{Z} [G] $-module provided that $N/ F$ is weakly ramified and under the hypothesis that for every prime $\wp $ of $\mathfrak{O}$ which ramifies wildly in $N/ F$, the decomposition group is abelian, the ramification group is cyclic and $\wp $ is unramified in $F/ \mathbb{Q} $. We make crucial use of a construction due to the first author which uses Dwork’s exponential power series to describe self-dual integral normal bases in Lubin–Tate extensions of local fields. This yields a new and striking relationship between the local norm-resolvent and the Galois Gauss sum involved. Our results generalise work of the second author concerning the case of base field $ \mathbb{Q} $.

Keywords

Galois modulesquare root of the inverse differentweakly ramified extensionsDwork’s power seriesself-dual integral normal basisGalois Gauss sum

MSC classification

Secondary: 11R04: Algebraic numbers; rings of algebraic integers 11R33: Integral representations related to algebraic numbers; Galois module structure of rings of integers 11S31: Class field theory; $p$-adic formal groups 11R20: Other abelian and metabelian extensions 11R29: Class numbers, class groups, discriminants 11R32: Galois theory 11S15: Ramification and extension theory 11S20: Galois theory 11S80: Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 7 , July 2013 , pp. 1175 - 1202
Copyright
© The Author(s) 2013 

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