On non-abelian Brumer and Brumer-Stark conjecture for monomial CM-extensions

Let K/kappa be a finite Galois CM-extension of number fields whose Galois group G is monomial and S a finite set of places of kappa. Then the "Stickelberger element" theta(K/kappa,S) is defined. Concerning this element, Andreas Nickel formulated the non-abelian Brumer and Brumer-Stark conjectures and their "weak" versions. In this paper, when G is a monomial group, we prove that the weak non-abelian conjectures are reduced to the weak conjectures for abelian subextensions. We write D-4p, Q(2n+2) and A(4) for the dihedral group of order 4p for any odd prime p, the generalized quaternion group of order 2(n+2) for any natural number n and the alternating group on four letters respectively. Suppose that G is isomorphic to D-4p, Q(2n+2) or A(4)xZ/2Z. Then we prove the l-parts of the weak non-abelian conjectures, where l = 2 in the quaternion case, and l is an arbitrary prime which does not split in Q(zeta(p)) in the dihedral case and in Q(zeta(3)) in the alternating case. In particular, we do not exclude the 2-part of the conjectures and do not assume that S contains all finite places which ramify in K/kappa in contrast with Nickel's formulation.