Integrality of Stickelberger elements attached to unramified extensions of imaginary quadratic fields
To each finite Galois extension \(K/k\) of number fields one can attach certain `Stickelberger elements' which are defined via values of Artin \(L\)-functions at \(s=0\). When the Galois extension is abelian, one knows that the corresponding Stickelberger elements have integral coefficients and a famous conjecture of Brumer asserts that they annihilate the class group of \(K\). For arbitrary Galois extensions several authors have meanwhile formulated conjectures on the integrality of Stickelberger elements and how one can use them to construct annihilators of the class group. These conjectures are mainly interesting when \(K/k\) is a CM-extension, i.e., \(k\) is totally real and \(K\) is a totally complex quadratic extension of a totally real field. In fact, in the work of \textit{G. Dejou} and \textit{F.-X. Roblot} [J. Number Theory 142, 51--88 (2014; Zbl 1295.11125)] and the reviewer [Ann. Inst. Fourier 61, No. 6, 2577--2608 (2011; Zbl 1246.11176)] only this case has been considered. However, there is one more case of interest; namely, when \(k\) is an imaginary quadratic field and \(K/k\) is unramified. In this paper the author considers the latter case and proves the expected integrality properties. Moreover, he shows that the Stickelberger elements generate the `noncommutative Fitting invariants' (as introduced by the reviewer [J. Algebra 323, No. 10, 2756--2778 (2010; Zbl 1222.11132)]) of certain arithmetic Galois modules and may be used to construct annihilators of corresponding ray class groups (as predicted by the obvious analogues of the conjectures of Dejou-Roblot and the reviewer). When \(K/k\) is abelian, this is not hard to show. It follows from the fact that every ideal of \(k\) becomes principal in its Hilbert class field and \(K\) is contained in the latter; this is already explained in a remark by \textit{C. Greither} and \textit{C. Popescu} [J. Algebr. Geom. 24, No. 4, 629--692 (2015; Zbl 1330.11070)]. The non-abelian case is more involved.
scientific article; zbMATH DE number 6866570