Self-dual integral normal bases and Galois module structure
Résumé
Let $N/F$ be an odd degree Galois extension of number fields with Galois group $G$ and rings of integers $O_N$ and $O_F=O$ respectively. Let $\A$ be the unique fractional $O_N$-ideal with square equal to the inverse different of $N/F$. Erez has shown that $\A$ is a locally free $O[G]$-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 $\A$ as a $\Z[G]$-module, however this question remains open. In this paper we prove that $\A$ is free as a $\Z[G]$-module assuming that $N/F$ is weakly ramified and under the hypothesis that for every prime $\wp$ of $O$ 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 $\Q$.