Logarithmic classes in the narrow sense
Let \(\ell\) be a prime. The concept of the logarithmic \(\ell\)-class group \(\widetilde{{\mathcal C}\ell}_K\) of a number field \(K\) has been introduced by \textit{J.-F. Jaulent} [J. Théor. Nombres Bordx. 6, 301-325 (1994; Zbl 0827.11064)]. It has its origin in studies of some \(\ell\)-extensions in connection with \(K\)-theory. A conjecture of Gross asserts that \(\widetilde{{\mathcal C}\ell}_K\) is finite. Take \(\ell=2\). The author introduces the concept of the logarithmic 2-class group in the narrow sense and investigates its properties. One of the main results is a formula for the ambiguous logarithmic classes in the case of a cyclic 2-extension \(L/K\) assuming that the finiteness conjecture is true.
scientific article; zbMATH DE number 1054311