The Galois Brumer-Stark conjecture for SL2(F_3)-extensions.

Gaëlle Dejou (1) , Xavier-François Roblot (2)

1 ICJ - Institut Camille Jordan
2 CTN - Combinatoire, théorie des nombres

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 SL2(F3). 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 2. We also prove the conjecture for 59 SL2(F3)-extensions of Q 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.

Mots clés: Brumer–Stark conjecture; Galois extensions; Artin L-functions; ideal class group annihilators

Domaines: Théorie des nombres [math.NT]

Gaëlle Dejou, Xavier-François Roblot. The Galois Brumer-Stark conjecture for SL2(F_3)-extensions.. International Journal of Number Theory, 2016, 12 (1), pp.165-188. ⟨10.1142/S179304211650010X⟩. ⟨hal-01657677⟩