The Brumer-Stark conjecture in some families of extensions of specified degree

Abstract

Abstract. As a starting point, an important link is established between Brumer's conjecture and the Brumer-Stark conjecture which allows one to translate recent progress on the former into new results on the latter. For example, if K/F is an abelian extension of relative degree 2p, p an odd prime, we prove the l-part of the Brumer-Stark conjecture for all odd primes l = p with F belonging to a wide class of base fields. In the same setting, we study the 2-part and p-part of Brumer-Stark with no special restriction on F and are left with only two well-defined specific classes of extensions that elude proof. Extensive computations were carried out within these two classes and a complete numerical proof of the Brumer-Stark conjecture was obtained in all cases. Overview and resultsAn important conjecture due to Brumer predicts that specific group ring elements constructed from the values of partial zeta-functions at s = 0 annihilate the ideal class groups of certain number fields. Recent progress has been made on this conjecture ([Gr1], [Wi]) and this will be used to obtain new results on the related conjecture of Brumer-Stark where progress thus far has been more restricted (see Section 1 of [RT] for the present status of the Brumer-Stark conjecture). The setting for these conjectures is the following: K/F is a relative Galois extension, with G = Gal(K/F ) abelian, K totally complex, and F totally real. Both conjectures predict that certain elements of Z[G] annihilate the ideal class group Cl K of K.Our results fall naturally into two parts. The first part (Sections 1-3) presents our theoretical results, which we now briefly state:(1) (Probably well known) There is the following localization principle: The Brumer-Stark conjecture (BS) holds for K/F if and only if, for all prime numbers l, an "l-primary analog" (BS) l holds for K/F .(2) (Semi-simple case) If F is an abelian extension of Q with the l-part of Gal(F/Q) cyclic, K is a CM field, and l = 2 is prime to the order of G, then (BS) l holds. For example, given this setup, if G is abelian of order 2p, p an odd prime, then it suffices to prove (BS) 2 and (BS) p in order to establish (BS) for K/F . (The condition that F be abelian over Q can be relaxed somewhat; see Proposition 1.3 for details.) (3) If G is abelian of order 2p, p an odd prime, then (BS) p holds unless K/F is of type or . Here means that K contains a primitive p-th root of unity ζ p and

On non-abelian Stark-type conjectures

“…By a similar argument we can prove a partial converse of Lemma 2.12 which is a non-abelian analogue of [17], Prop. 1.2.…”

“…[17], remark preceding Prop. 1.1): Let a be a fractional ideal whose class in cl L has ppower order.…”

A Brumer–Stark conjecture for non-abelian Galois extensions

“…We refer the interested reader to [12] for more precise statements. In [7], a local version of the conjecture is stated and is proved in some cases and numerically studied in some other. The recent results of Greither and Popescu [6] implies that the local abelian BrumerStark conjecture at p holds provided that S contains all the primes above p and some appropriate Iwasawa µ-invariant vanishes.…”

Computing Fitting ideals of Iwasawa modules

“…In a final section ( 8) we show that under condition NTZ and some other less serious conditions, we can deduce the p-part of the Brumer-Stark conjecture from the preceding results. Some cases that were handled by computer in [10] are recaptured here.…”

“…(1) For a precise definition of the p-part of the Brumer-Stark conjecture, and an explanation of the fact that all these p-parts together are equivalent to the Brumer-Stark conjecture, see [10]. (2) The examples of type ( ) studied in [10] often satisfy the assumptions of 8.2. In loc.cit., p is chosen to be 3, F is real quadratic of discriminant d, and K is obtained by composing F (ζ 3 ) with a cubic extension K + /F which is unramified outside 3.…”