Galois module structure of the square root of the inverse different in even degree tame extensions of number fields

Let G be a finite group and let N/E be a tamely ramified G-Galois extension of number fields whose inverse different CN/E is a square. Let AN/E denote the square root of CN/E. Then AN/E is a locally free Z[G]-module, which is in fact free provided N/E has odd order, as shown by Erez. Using M. Taylor's theorem, we can rephrase this result by saying that, when N/E has odd degree, the classes of AN/E and ON (the ring of integers of N) in Cl(Z[G]) are equal (and in fact both trivial). We show that the above equality of classes still holds when N/E has even order, assuming that N/E is locally abelian. This result is obtained through the study of the Fröhlich representatives of the classes of some torsion modules, which are independently introduced in the setting of cyclotomic number fields. Jacobi sums, together with the Hasse–Davenport formula, are involved in this study. Finally, when G is the binary tetrahedral group, we use our result in conjunction with Taylor's theorem to exhibit a tame G-Galois extension whose square root of the inverse different has nontrivial class in Cl(Z[G]).