Self-Dual Integral Normal Bases and Galois Module Structure

Let $N\mathrm{/}F$ be an odd degree Galois extension of number fields with Galois group $G$ and rings of integers ${\mathfrak{O}}_N$ and ${\mathfrak{O}}_F=\text{bo}$ respectively. Let $\mathcal{A}$ be the unique fractional ${\mathfrak O}_N$-idealwithsquareequaltotheinversedifferentof$N/F$. Erez has shown that $\mathcal{A}$ is a locally free $\mathfrak{O}[G]$-module if and only if $N\mathrm{/}F$ 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 $N\mathrm{/}F$ is weakly ramified and under the hypothesis that for every prime $\mathrm{\wp }$ of $\mathfrak{O}$ which ramifies wildly in $N\mathrm{/}F$, the decomposition group is abelian, the ramification group is cyclic and $\mathrm{\wp }$ is unramified in $F\mathrm{/}\text{Q}$. 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 $\text{Q}$.