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

TL;DR

This paper proves that for certain monomial Galois CM-extensions, the non-abelian Brumer and Brumer-Stark conjectures can be reduced to abelian cases, and verifies parts of these conjectures for specific groups and primes.

Contribution

It reduces the weak non-abelian conjectures to abelian subextensions for monomial groups and verifies the conjectures for specific non-abelian groups and primes.

Findings

Weak non-abelian conjectures reduce to abelian subextensions for monomial groups.

Verification of the conjectures for dihedral, quaternion, and alternating groups under certain prime conditions.

No need to assume all ramified places are in set S or exclude the 2-part of the conjectures.

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.\