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

Abstract

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" $\theta_{K/k,S}$ 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 $D_{4p},\ Q_{2^{n+2}}$ 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 4 letters respectively.\ Suppose that $G$ is isomorphic to $D_{4p}$,\ $Q_{2^{n+2}}$ or $A_4 \times \mathbb{Z}/2\mathbb{Z}$.\ 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 $\mathbb{Q}(\zeta_p)$ in the dihedral case and in $\mathbb{Q}(\zeta_3)$ 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 $K/k$ in contrast with Nickel's formulation.\