Self-Dual Integral Normal Bases and Galois Module Structure

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 \(\Z[G]\)-module, however this question remains open. In this paper we prove that \(\mathcal{A}\) is free as a \(\Z[G]\)-module assuming that \(N/F\) is weakly ramified and under the hypothesis that for every prime \(\wp\) of \({\mathfrak O}\) 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 \(\Q\).