Stickelberger elements and cotangent numbers (Q1191724)

Let \(K\) be an abelian number field and put \(G=\text{Gal}(K/\mathbb{Q})\). As usual, let \(\sigma_{-1}\in G\) denote the complex conjugation. The author finds a connection between the Stickelberger elements of the group ring \(\mathbb{Q}[G]\) and the cotangent numbers \(\eta_{m,K}\in K\) introduced by \textit{K. Girstmair} [J. Number Theory 32, 100-110 (1989; Zbl 0675.12002)]. The connection is given by Leopoldt's ``Basiszahl'' which induces (by multiplication) an isomorphism from \(\mathbb{Q}[G]\) onto \(K\). In fact, a suitable sum of modified Stickelberger elements is mapped to a modified cotangent number. Here ``modified'' means that the original definitions are to be changed slightly. As an application the author proves a formula for the module index \((R^{\pm}: \Theta_ mR^{\pm})\), where \(\Theta_ m\) is a product of Stickelberger elements of degree \(m\) and \(R^ \pm\) is any sublattice, with maximal rank, of either \((1\pm\sigma_{-1})\mathbb{Q}[G]\) or \((1\pm\sigma_{-1})K\). This implies a formula in terms of \((R^ -:\;\Theta_ 1R^ -)\) for the relative class number of \(K\).