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

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 6= 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. 0. Overview and results An 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 ClK 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 6= 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 ζp and Received by the editor December 20, 2001. 2000 Mathematics Subject Classification. Primary 11R42; Secondary 11R29, 11R80, 11Y40.