A Brumer–Stark conjecture for non-abelian Galois extensions

Abstract

Abstract. The Brumer-Stark conjecture deals with abelian extensions of number fields and predicts that a group ring element, called the Brumer-Stickelberger element constructed from special values of L-functions associated to the extension, annihilates the ideal class group of the extension under consideration. Moreover it specifies that the generators obtained have special properties. The aim of this article is to propose a generalization of this conjecture to non-abelian Galois extensions that is, in spirit, very similar to the original conjecture.

The Galois Brumer–Stark conjecture for SL2(𝔽3)-extensions

“…Proof. First, note that in all cases for which Proposition 6.5 of [2] applies and reduces BS Gal (K/k, S) to BS(K ab /k, S), we also have that RBS Gal (K/k, S) reduces to BS(K ab /k, S) since then the non-linear Brumer-Stickelberger element is zero. Cases (1),…”

“…It follows from the principal rank zero Stark conjecture, proved by Tate [15], that the Brumer-Stickelberger element lies in Q[G]. The first conjecture introduced in [2] gives a denominator for this element. Recall that the commutator subgroup [Γ, Γ] is the subgroup of Γ generated by the commutators [γ 1 , γ 2 ] := γ 1 γ 2 γ −1 1 γ −1 2 with γ 1 , γ 2 ∈ Γ.…”

“…is integral using the expression given by Theorem 7.1 of [2] and the results of Proposition 7.3, ibid. A close look at the proof of Theorem 7.4, ibid., shows that d G θ (>1) K/k,S annihilates the class group of K and that one can find a generator in K • .…”

“…The Brumer-Stark conjecture [14] predicts that a group ring element called the Brumer-Stickelberger element and constructed from special values of L-functions associated to K/k, annihilates (after multiplication by a suitable factor) the ideal class group of K and specifies special properties for the generators obtained. In [2], we introduced a generalization of the conjecture to Galois extensions, called the Galois Brumer-Stark conjecture. To avoid confusion, we call the original conjecture the abelian Brumer-Stark conjecture.…”

Conjectures of Brumer, Gross and Stark

“…The following conjecture was first formulated in [27] and is a non-abelian generalisation of Brumer's Conjecture 2.1. [14]. ♦…”