Generalized ambiguous logarithmic classes (Q1969615)

For a number field \(k\) and a fixed prime \(l\), the \(l\)-group of logarithmic classes \(\widetilde{Cl}_k\) has been defined by \textit{J.-F. Jaulent} in [J. Théor. Nombres Bordeux 6, 301-325 (1994; Zbl 0827.11064)]. In this note, given a cyclic \(l\)-extension \(K/k\), with \(G= \text{Gal} (K/k)\), the author considers quotients of \(\widetilde{Cl}_k\) by sub-\(G\)-modules \(\widetilde{\mathcal H}_k\) and proves a formula for the order of \((\widetilde{Cl}_k/ \widetilde{\mathcal H}_k)^G\) in the style of genus theory.