Self-Dual Integral Normal Bases and Galois Module Structure

Abstract

Let $N/F$ be an odd degree Galois extension of number fields with Galois group $G$ and rings of integers ${\mathfrak{O}}_N$ and ${\mathfrak O}_F=\bo$ respectively. Let $\mathcal{A}$ be the unique fractional ${\mathfrak{O}}_N$-ideal with square equal to the inverse different of $N/F$. Erez has shown that $\mathcal{A}$ is a locally free $\mathfrak{O}[G]$-module if and only if $N/F$ is a so called weakly ramified extension. There have been a number of results regarding the freeness of $\mathcal{A}$ as a $\mathbb{Z}[G]$-module, however this question remains open. In this paper we prove that $\mathcal{A}$ is free as a $\mathbb{Z}[G]$-module assuming 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 $F/\Q$. We make crucial use of a construction due to the first named 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 Galois Gauss sum involved. Our results generalise work of the second named author concerning the case of base field $\Q$.

References in corpus (2)

• Explicit Construction of Self-Dual Integral Normal Bases for the Square-Root of the Inverse Different
• Construction of Self-Dual Integral Normal Bases in Abelian Extensions of Finite and Local Fields

Cited by in corpus (6)

• Equivariant epsilon constant conjectures for weakly ramified extensions
• On the Galois module structure of the square root of the inverse different in abelian extensions
• On refined metric and hermitian structures in arithmetic, I: Galois-Gauss sums and weak ramification
• Exponential power series, Galois module structure and differential modules
• The epsilon constant conjecture for higher dimensional unramified twists of ${\mathbb{Z}}_p^r(1)$
• Gauss sums, Jacobi sums and cyclotomic units related to torsion Galois modules