On non-abelian Stark-type conjectures
type: Item
instance of: scholarly article
title: On non-abelian Stark-type conjectures
zbMATH Open document ID: 1246.11176
author: Andreas Nickel
publication date: 2012-06-06T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/219691
reviewed by: Cornelius Greither
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 6043088
zbMATH Keywords: Brumer conjecture
zbMATH Keywords: Brumer-Stark
zbMATH Keywords: annihilators
zbMATH Keywords: Fitting ideals
zbMATH Keywords: noncommutative rings
zbMATH Keywords: maximal orders
zbMATH Keywords: reduced norms
MaRDI profile type: Publication
OpenAlex ID: W2324217161
Wikidata QID: Q123146291
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DOI: 10.5802/AIF.2683
published in: Annales de l’institut Fourier
review text: One starts by considering \(L/K\), a finite Galois CM extension of number fields with Galois group \(G\). In the abelian case, there are well-established conjectures (B) and (BS), named after Brumer, and Brumer-Stark respectively. Both involve \(\Theta_S(L/K) \in \mathbb Q[G]\), a certain equivariant L-value at zero (which is for good reasons also called a generalized Stickelberger element), and predict annihilation properties of the class group \(cl_L\) as a \(\mathbb Z[G]\)-module. (B) claims that \(Ann(\mu_L)\Theta_S(L/K)\) annihilates \(cl_L\), and (BS) claims that the smaller ideal \(|\mu_L|\Theta_S(L/K)\) annihilates even in a stricter sense. More precisely, for any nonzero ideal \(\mathfrak a\) of \(O_L\), the ideal \(\mathfrak a^{|\mu_L|\Theta_S(L/K)}\) is supposed not only to be principal (so as to represent zero in the class group), but to admit a generator \(x\) such that \(L(x^{1/|\mu_L|})\) is abelian over \(K\). It is known that (BS) implies (B), and that both conjectures split up into their \(p\)-parts, \(p\) running over all primes.NEWLINENEWLINEIn the present paper, the author formulates non-abelian generalizations of both conjectures, and he brings in a third conjecture called the Strong BS property (SBSP). It claims that \(\Theta_S(L/K)\) lies in the Fitting ideal of the class group. Of course one needs a non-commutative analog of the generalized Stickelberger element, and of the Fitting ideal. Both are available, the latter thanks to previous work of Nickel. The main technical problem is the following. For non-commutative group rings it is no longer true that the Fitting ideal of a module \(M\) is contained in its annihilator, simply because the usual proof using adjoint matrices fails and cannot be salvaged. The price to pay is that the putative annihilators in (B) and (BS) have to be multiplied by an extra factor \(\mathcal H\), which is a certain ideal in \(\mathbb Z[G]\) that measures the failure of the following property (MA): if \(H\) is a …
description: scientific article; zbMATH DE number 6043088