The Galois Brumer–Stark conjecture for SL2(????3)-extensions

In a previous work, we stated a conjecture, called the Galois Brumer–Stark conjecture, that generalizes the (abelian) Brumer–Stark conjecture to Galois extensions. We also proved that, in several cases, the Galois Brumer–Stark conjecture holds or reduces to the abelian Brumer–Stark conjecture. The first open case is the case of extensions with Galois group isomorphic to [Formula: see text]. This is the case studied in this paper. These extensions split naturally into two different types. For the first type, we prove the conjecture outside of [Formula: see text]. We also prove the conjecture for [Formula: see text] [Formula: see text]-extensions of [Formula: see text] using computations. The version of the conjecture that we study is a stronger version, called the Refined Galois Brumer–Stark conjecture, that we introduce in the first part of the paper.