Gauss sums, Jacobi sums and cyclotomic units related to torsion Galois modules

Abstract

Let $G$ be a finite group and let $N/E$ be a tamely ramified $G$-Galois extension of number fields. We show how Stickelberger's factorization of Gauss sums can be used to determine the stable isomorphism class of various arithmetic $\mathbb{Z}[G]$-modules attached to $N/E$. If $\mathcal{O}_N$ and $\mathcal{O}_E$ denote the rings of integers of $N$ and $E$ respectively, we get in particular that $\mathcal{O}_N\otimes_{\mathcal{O}_E}\mathcal{O}_N$ defines the trivial class in the class group $\mathrm{Cl}(\mathbb{Z}[G])$ and, if $N/E$ is also assumed to be locally abelian, that the square root of the inverse different (whenever it exists) defines the same class as $\mathcal{O}_N$. These results are obtained through the study of the Fr\"ohlich representatives of the classes of some torsion modules, which are independently introduced in the setting of cyclotomic number fields. Gauss and Jacobi sums, together with the Hasse-Davenport formula, are involved in this study. These techniques are also applied to recover the stable self-duality of $\mathcal{O}_N$ (as a $\mathbb{Z}[G]$-module). Finally, when $G$ is the binary tetrahedral group, we use our results in conjunction with Taylor's theorem to find a tame $G$-Galois extension whose square root of the inverse different has nontrivial class in $\mathrm{Cl}(\mathbb{Z}[G])$.