Integrality of Stickelberger elements attached to unramified extensions of imaginary quadratic fields
type: Item
instance of: scholarly article
title: Integrality of Stickelberger elements attached to unramified extensions of imaginary quadratic fields
zbMATH Open document ID: 1404.11131
author: Jiro Nomura
published in: Journal of Number Theory
publication date: 2018-05-04T00:00:00Z
review text: 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.
reviewed by: Andreas Nickel
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 6866570
zbMATH Keywords: class groups
zbMATH Keywords: \(L\)-values
zbMATH Keywords: Stickelberger elements
zbMATH Keywords: imaginary quadratic fields
zbMATH Keywords: Brumer's conjecture
zbMATH Keywords: unramified extensions
MaRDI profile type: Publication
full work available at URL: j.jnt.2017.11.003
OpenAlex ID: W2774817916
cites work: On derivatives of Artin \(L\)-series
cites work: Tamagawa numbers for motives with (non-commutative) coefficients
cites work: A non-abelian Stickelberger theorem
cites work: On zeta elements for \(\mathbb G_m\)
cites work: Q3922829
cites work: A Brumer-Stark conjecture for non-abelian Galois extensions
cites work: Stickelberger elements, Fitting ideals of class groups of CM-fields, and dualisation
cites work: An Equivariant Main Conjecture in Iwasawa theory and applications
cites work: Q3686794
cites work: Noncommutative Fitting invariants and improved annihilation results.
cites work: Ideal class groups of CM-fields with non-cyclic Galois action
cites work: Non-commutative Fitting invariants and annihilation of class groups
cites work: On the equivariant Tamagawa number conjecture in tame CM-extensions
cites work: On the equivariant Tamagawa number conjecture in tame CM-extensions, II
cites work: On non-abelian Stark-type conjectures
cites work: On non-abelian Brumer and Brumer–Stark conjecture for monomial CM-extensions
cites work: A note on the Galois Brumer-Stark conjecture
cites work: The Stark conjectures on Artin \(L\)-functions at \(s=0\). Lecture notes of a course in Orsay edited by Dominique Bernardi and Norbert Schappacher.
DOI: 10.1016/J.JNT.2017.11.003
description: scientific article; zbMATH DE number 6866570