Let K/k be a finite Galois CM-extension of number fields whose Galois group G is monomial and S a finite set of places of k. Then the "Stickelberger element" θK/k,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 D4p, Q2n+2 and A4 for the dihedral group of order 4p for any odd prime p, the generalized quaternion group of order 2n+2 for any natural number n and the alternating group on four letters respectively. Suppose that G is isomorphic to D4p, Q2n+2 or A4 × ℤ/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(ζp) in the dihedral case and in ℚ(ζ3) 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/k in contrast with Nickel's formulation.
On the elliptic Stark conjecture at primes of multiplicative reduction
