On non-abelian Stark-type conjectures
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 square matrix over the group ring, then its adjoint matrix \(H^*\) has again coefficients in the group ring. One only knows that the coefficients lie in a maximal order. In the abelian setting (MA) is of course true. There is a fairly canonical definition for the adjoint in the noncommutative setting, via reduced norms, but even the adjoints of scalar matrices are not trivial to find, in contrast to the abelian case where the adjoint of a \(1\times 1\) matrix is simply \((1)\). At least one can estimate \(\mathcal H\) from below by the so-called central conductor \(\mathcal F\), which measures the ``distance'' between the center of the group ring and the center of a maximal order. This term \(\mathcal F\) has one advantage and one disadvantage: it is know explicitly (while \(\mathcal H\) is not), but it is a fairly small ideal in general, resulting in weak annihilation statements. (We should note here, however, that Johnston and Nickel managed to determine \(\mathcal H\) in some interesting special cases, see their recent preprint ``Noncommutative Fitting ideals and annihilation properties'', in particular Example 6.22.) All the conjectures (B), (BS), (SBSP) have ``weakened'' counterparts; in the first two, one replaces \(\mathcal H\) by \(\mathcal F\), and in the third one replaces \(\mathbb Z[G]\) b a maximal order.NEWLINENEWLINEThe principal results of the paper describe relations among these conjectures, and three more conjectures in the literature: (ETNC), a Leading Term Conjecture due to Burns, and the so-called Strong Stark conjecture. (Terminology is getting a little complicated at this point; perhaps one could have called (SBSP) a Refined BS conjecture, as done by Popescu and the reviewer, so as to avoid the double use of the word ``strong''.)NEWLINENEWLINEWe will just list the results here, giving an occasional comment; we also suppress a lot of technical detail. -- (i) Various lemmas show that every conjecture implies its ``weakened'' counterpart, and that (BS) implies (B). -- (ii) (Theorem 3.5) The validity of Burns' leading term conjecture (which we do not explain here) implies the weakened versions of both (B) and (BS). -- (iii) (Theorem 4.1) The strong Stark conjecture at \(p\) implies the weakened (SBSP). -- (iv) Theorem 5.1: The minus part of the \(p\)-part of ETNC (again we will not explain this) implies (BS).NEWLINENEWLINEThis nice result ``lifts'' a previous result [J. Algebra 323, No. 10, 2756--2778 (2010; Zbl 1222.11132)] of the author, which says the same with (BS) replaced by (B). We say ``lift'' since (BS) implies (B); so the new results fills a triangle of implications. The author assumes here that the \(p\)-part of the roots of unity in \(L\) is \(G\)-cohomologically trivial. It is interesting to note that under this hypothesis in the commutative case, one may establish ``(B) implies (BS)'' in a rather direct way (see Prop.~1.2 in [\textit{C. Greither, X.-F. Roblot} and \textit{B. A. Tangedal}, Math. Comput. 73, No. 245, 297--315 (2004; Zbl 1094.11043)]). The author proves a non-commutative analogue of this (his Prop.~3.10), but it involves an extra undesired factor, so this alternative approach towards Theorem 5.1 would not work in general. -- (v) The last main result is Theorem 5.3, which gives an implication from ETNC to the Strong Brumer-Stark property, this time with some hypotheses on ramification in \(L/K\).NEWLINENEWLINEBecause of the multitude of interrelated conjectures, papers in this subject are not always easy to read. The author of the paper under review does a good job of making things as clear as possible.
scientific article; zbMATH DE number 6043088