On non-abelian Brumer and Brumer–Stark conjecture 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" θ 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 D4p, Q 2 n+2 and A4 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 D4p, Q 2 n+2 or A4 ×Z/2Z. 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 Q(ζp) in the dihedral case and in Q(ζ3) 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.

Conjectures of Brumer, Gross and Stark

“…In certain situations, we can also remove the condition that S p ⊆ S. To illustrate this, we conclude with the following two results (the first is [23, Thm. 10.10], whereas the second is due to Nomura [31,32]). Theorem 4.28.…”

Annihilation of Selmer groups for monomial CM-extensions

“…Next we consider the case G = D 4p with odd prime p. Each of the irreducible characters of D 4p is 1-dimensional or 2-dimensional, and as in the proof of [Nom,Lemma 2.1] (also see [JN,Example 6.22…”

“…The above theorem is an exact analogue of [Nom,Theorem 3.3] In the following, we assume F/k is a CM-extension with the unique complex conjugation j. For each G-module M and α ∈ {±1}, we set M α := {m ∈ M | jm = αm}, and…”

“…(See[Nom, Lemma 2.1].) If an irreducible character ψ of G is induced by an irreducible character of a subgroup H of G, then we have…”

A Note on the Galois Brumer-Stark Conjecture

“…The keys of the proofs of these theorems are [11,Propositions 5.3 and 5.8] (in fact we need slight refinements of [11,Propositions 5.3 and 5.8], which are Propositions 5.2 and 5.6 in § §5.1 and 5.2, respectively). These propositions were also needed to prove the "weak non-abelian Brumer-Stark conjecture" by Nickel under the same assumption as above [11,Theorems 5.1 and 5.6]. Therefore the author thinks it is a very interesting problem to find a relation between two conjectures, however, we do not study the problem in this paper.…”