ON THE SEMI-SIMPLE CASE OF THE GALOISBRUMER-STARK CONJECTURE FOR MONOMIAL GROUPS
X.-F. ROBLOT

Abstract. In a previous work, we stated a conjecture, called the Galois
Brumer-Stark conjecture, that generalizes the (abelian) Brumer-Stark conjecture to Galois extensions. Other generalizations of the Brumer-Stark conjecture to non-abelian Galois extensions are due to Nickel. Nomura provedthat the Brumer-Stark conjecture implies the weak non-abelian Brumer-Stark
conjecture of Nickel when the group is monomial. In this paper, we use themethods of Nomura to prove that the Brumer-Stark conjecture implies the
Galois Brumer-Stark conjecture for monomial groups in the semi-simple case.

1. Introduction
Let K/k be an abelian extension of number fields. The Brumer-Stark conjecture[14] predicts that a group ring element, called the Brumer-Stickelberger element,
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 [5], we introduced a generalization of
the conjecture to Galois extensions, called the Galois Brumer-Stark conjecture.
Later, in [6], we introduced a refined version of the conjecture that focused on
the contribution of the non-linear irreducible characters. Since the new version
in [6] supersedes the version in [5], we will from now on refer to it as the Galois
Brumer-Stark conjecture (and not call it anymore the refined Galois Brumer-Stark
conjecture). Also, to avoid confusion, we call the original conjecture the abelian
Brumer-Stark conjecture.
In [10], Nickel introduced another generalization of the abelian Brumer-Stark
conjecture to Galois extensions and in [12], Nomura proved, among other things,
that the abelian Brumer-Stark conjecture implies the (weak) non-abelian BrumerStark conjecture of Nickel when the Galois group of K/k is monomial. In thispaper, we adapt the method used by Nomura to prove that the abelian BrumerStark conjecture implies the Galois Brumer-Stark conjecture in the semi-simple
case when the Galois group of K/k is monomial (see Theorem 3.1). Furthermore,using the fact that the local abelian Brumer-Stark conjecture is known to hold inseveral cases, we prove unconditionally some cases of the local Galois Brumer-Stark
conjecture (see Corollary 3.3).
2. The Galois Brumer-Stark conjecture
Before stating the Galois Brumer-Stark conjecture, we recall the statement of
the abelian Brumer-Stark conjecture, see [15, IV.§6] or [14]. Let K/k be an abelian
extension of number fields. Denote by G its Galois group. Let S be a finite set ofplaces of k containing the infinite places and the finite places ramified in K. To
1

2

X.-F. ROBLOT

simplify matters, we assume that the cardinality of S is at least 2. The interested
reader can refer to [15, Sec. IV.§6] for the statement of the conjecture when |S| = 1.
For χ ∈ Ĝ, where Ĝ denotes the group of irreducible characters of G, denote by
LK/k,S (s, χ) the Hecke L-function of the character χ with Euler factors associatedto prime ideals in S deleted. The Brumer-Stickelberger element associated to theextension K/k and the set S is defined byX
θK/k,S :=LK/k,S (0, χ) eχ̄
χ∈Ĝ

P
1

where eχ := |G|[4]) that(2.1)

σ∈G χ(σ)σ−1

is the idempotent of χ. It follows from [7] (see alsoξ θK/k,S ∈ Z[G]for any ξ ∈ AnnZ[G] (µK ), the annihilator in Z[G] of the group µK of roots of unity
in K. In particular, we have wK θK/k,S ∈ Z[G] where wK denotes the cardinality ofµK . We need one last notation before stating the abelian Brumer-Stark conjecture.
We say that a non-zero element α ∈ K is an anti-unit if all its conjugates haveabsolute value equal to 1. The group of anti-units of K is denoted by K ◦ .
Conjecture 2.1 (The abelian Brumer-Stark conjecture BS(K/k, S)). For any
fractional ideal A of K, the ideal AwK θK/k,S is principal and admits a generator
α ∈ K ◦ such that K(α1/wK )/k is abelian.
We refer to [5, §2] for a review of the current state of the abelian Brumer-Stark
conjecture. The following consequence of the abelian Brumer-Stark conjecture will
be useful later on. (The conclusion of the proposition is known as the Brumer
conjecture; thus, the proposition just states the well-known fact that the BrumerStark conjecture implies the Brumer conjecture.)
Proposition 2.2. Assume that the abelian Brumer-Stark conjecture BS(K/k, S)
holds. Let ξ ∈ AnnZ[G] (µK ). Then, ξ θK/k,S annihilates the class group ClK of K.
Proof. Under the assumption that BS(K/k, S) holds, there exists by [14, Proposition §2] a family (ai )i∈I of elements of AnnZ[G] (µK ), generating AnnZ[G] (µK ), andsuch that ai θK/k,S annihilates ClK for all i ∈ I. The result follows directly.

We now introduce the Galois Brumer-Stark conjecture (more precisely, as notedin the introduction, the refined version stated in [6]). Assume now that K/k is a
Galois extension of number fields. Denote by G its Galois group. Let S be a finite
set of places of k containing the infinite places and the finite places ramified in K.
Assume that the cardinality of S is at least 2. Denote by Ĝ(>1) the set of non-linear
irreducible characters of G and define the non-linear Brumer-Stickelberger element
by
X
(>1)
(2.2)
θK/k,S:=
LK/k,S (0, χ)eχ̄χ∈Ĝ(>1)

where, for χ ∈ Ĝ(>1) , LK/k,S (s, χ) denotes the Artin L-function of χ with Euler
factors associated to prime ideals in S deleted, andχ(1) Xeχ :=χ(σ)σ −1|G|σ∈GON THE GALOIS BRUMER-STARK CONJECTURE FOR MONOMIAL GROUPS

3

is the central idempotent of χ. It follows from the principal rank zero Stark conjecture, proved by Tate [15], that the non-linear Brumer-Stickelberger element lies
in Q[G]. Denote by [G, G] the commutator subgroup of G, i.e., the subgroupof G generated by the commutators [g1 , g2 ] := g1 g2 g1−1 g2−1 with g1 , g2 ∈ G. Let
Gab := G/[G, G] be the maximal abelian quotient of G and let K ab := K [G,G] be themaximal sub-extension of K/k that is abelian over k; we have Gal(K ab /k) = Gab .
Let sG denote the order of [G, G], let mG be the lcm of the cardinalities of the
conjugacy classes of G, and let dG be the lcm of mG and sG .
Conjecture 2.3 (The Galois Brumer-Stark conjecture BSGal (K/k, S)). Let K/kbe a Galois extension of number fields and let S be a finite set of places of k that
contains the infinite places and the finite places that ramify in K with |S| ≥ 2.
(>1)
Then, BS(K ab /k, S) holds, we have dG θK/k,S∈ Z[G] and, for any fractional idealdθ (>1)A of K, the ideal A G K/k,S is principal and admits a generator in K ◦ .
For p a prime, denote by ClK {p} the p-part of ClK , that is the subgroup of ClK
of classes of p-power order.
(p)

Conjecture 2.4 (The local Galois Brumer-Stark conjecture BSGal (K/k, S)). Let
K/k be a Galois extension of number fields and let S be a finite set of places of kthat contains the infinite places and the finite places that ramify in K with |S| ≥ 2.
Then, the local abelian Brumer-Stark conjecture at p for the extension K ab /k and(>1)
the set of places S holds, we have dG θK/k,S∈ Zp [G] and, for any fractional idealdθ (>1)A of K whose class lies in ClK {p}, the ideal A G K/k,S is principal and admits agenerator in K ◦ .
The statement of the local abelian Brumer-Stark conjecture at p is that, for all
ideals A whose class lies in ClK {p}, the ideal AwK θK/k,S is principal and admits agenerator α ∈ K ◦ such that K(α1/wK,p )/k is abelian where wK,p is the order ofthe p-part of the group of roots of unity in K, see [8]. One checks readily that
the Galois Brumer-Stark conjecture is equivalent to the local Galois Brumer-Stark
conjecture at p for all primes p. Some evidence for these conjectures is given in [6].
Relations between the Galois Brumer-Stark conjecture and the weak non-abelian
Brumer-Stark conjecture of Nickel are discussed in the appendix of [5]. To conclude
this section, we prove that the local versions of the weak non-abelian Brumer-Stark
conjecture and of the Galois Brumer-Stark conjecture are equivalent for primes p
not dividing wK |G|. First, we recall briefly the statement of the local weak nonabelian Brumer-Stark conjecture, see [10] for more details.
Let K/k be a Galois CM-extension with group G. Let S be a finite set of places ofk such that S contains the infinite places of k and the finite places of k that ramify
in K/k. Let Hyp(S) denote the set of finite sets T of places of k such that: S
and T are disjoint, and the group EK (S, T ) is torsion-free. Here, EK (S, T ) denotes
the group of (S, T )-units of K, that is the group of elements u ∈ K × such thatvP (u) = 0 for all prime ideals P of K such that (P ∩ k) 6∈ S and u ≡ 1 (mod Q)
for all prime ideals Q of K such that (Q ∩ k) ∈ T . For T ∈ Hyp(S), defineY−1δT := nr
1 − σPN (p)p∈T

where P is a fixed prime ideal of K above p, σP is the Frobenius element of P in
G, and nr : Q[G] → Z(Q[G]) is the reduced norm (see [13, §9]). Let Λ0 denote a

4X.-F. ROBLOT

fixed maximal order of Q[G] containing Z[G] and denote by F(G) := {x ∈ Z(Λ0 ) :xΛ0 ⊂ Z[G]} the central conductor of Λ0 over Z[G].
Conjecture 2.5 (The local weak non-abelian Brumer-Stark conjecture [10]).
Let wK := nr(wK ). Then wK θK/k,S ∈ Z(Λ0 ) ⊗ Zp . Furthermore, for any fractional
ideal A of K whose class lies in ClK {p} and for each x ∈ F(G), there exists ananti-unit αx ∈ K ◦ such that
AxwK θK/k,S = αx OK
and, for any set of places T ∈ Hyp(S ∪ Sαx ), there exists αx,T ∈ EK (Sαx , T ) suchthat, for all z ∈ F(G)zwK
αxzδT = αx,T
where Sαx is the set of prime ideals p of k such that vp (NK/k (αx )) 6= 0.
Observe that the conjecture stated above is slightly different from the original
conjecture given by Nickel in [10]. Indeed, Nickel does not state explicitly the
local conjecture in this paper but writes instead that one should restrict to ideals
whose class lies in ClK {p} in the global conjecture to get the local conjecture at
p. In particular, the local conjecture does not have an specific statement on wherewK θK/k,S should lie. However, it seems reasonable to only asks for wK θK/k,S to
be in Z(Λ0 ) ⊗ Zp in this case.
Theorem 2.6. Let K/k be a Galois CM-extension of number fields with Galois
group G and let S be a finite set of places of k that contains the infinite places
and the finite places that ramify in K with |S| ≥ 2. Let p be a prime number not(p)
dividing wK |G|. Then, the local Galois Brumer-Stark conjecture BSGal (K/k, S)
is equivalent to the local weak non-abelian Brumer-Stark conjecture at p for theextension K/k and the set of prime ideals S.
Proof. We will use the following fact several times whose proof is direct and left tothe reader: let t be an integer not divisible by p and let H be a group of fractional
ideals containing the principal ideals and all the ideals Bt where B runs throughthe fractional ideals of K whose class lies in ClK {p}. Then, H is the group of
fractional ideals whose class lies in ClK {p}.
(p)
Assume that BSGal (K/k, S) holds. Then, θK/k,S ∈ Z(Zp [G]) and therefore wehave wK θK/k,S ∈ Z(Λ0 ) ⊗ Zp = Z(Zp [G]). Let B be a fractional ideal of K whoseclass lies in ClK {p}. By [5, Prop. A.1], for any x ∈ F(G), there exists βx ∈ K ◦ suchthat
BdG xwK θK/k,S = βx OK
wK
and, for any set of places T ∈ Hyp(S ∪ Sβx ), there exists βx,T ∈ K × with βx,T∈EK (Sβx , T ) such that, for all z ∈ F(G)zwK
βxzδT = βx,T.

Observe that the proof of [5, Prop. A.1] uses the original formulation of the (global)
Galois Brumer-Stark conjecture but that is not a concern since the refined version
that we use now implies the original conjecture; furthermore, one can check readily
that the local version of the conjecture is enough for the proof of the result in thiswKcase. Let A := BdG wK . We set αx := βxwK and αx,T := βx,T
∈ EK (Sβx , T ) =EK (Sαx , T ) for all T ∈ Hyp(S ∪ Sβx ) = Hyp(S ∪ Sαx ). Then, it is direct to checkthat these elements satisfy the required properties for the statement of the local

ON THE GALOIS BRUMER-STARK CONJECTURE FOR MONOMIAL GROUPS

5

weak non-abelian conjecture to be satisfied for the ideal A. Since it is proved in[10] that the set of ideals satisfying the properties of the weak non-abelian BrumerStark conjecture is a group containing the principal ideals, it follows by the above
remark that the local weak non-abelian Brumer-Stark conjecture holds at p for theextension K/k and the set S.
Reciprocally, assume that the local weak non-abelian Brumer-Stark conjectureholds at p for the extension K/k and the set S. We first prove that this implies thatthe local abelian Brumer-Stark conjecture holds at p for the extension K ab /k andthe set S. Let b be a fractional ideal of K ab whose class lies in ClK ab {p}, thus theclass of bOK is in ClK {p}. Thanks to [13, Th. 41.1], we can take x = |G| ∈ F(G) inthe local weak non-abelian Brumer-Stark conjecture, and thus there exists β0 ∈ K ◦such that (bOK )|G|wK θK/k,S = β0 OK . Taking norms down to K ab and using theproperties of the Brumer-Stickelberger element, we deduce thatbsG |G|wK θK ab /k,S = awK θK ab /k,S = α0 OK ab
where a := bsG |G| and α0 := NK/K ab (β0 ) ∈ (K ab )◦ . Now, since p does not divide
1/w

wK , we have wK,p = 1 and thus K ab (α0 K,p ) = K ab is abelian over k. The set ofideals that satisfy the local abelian Brumer-Stark conjecture is a group containing
the principal ideals therefore, by the remark at the start of the proof, the local
abelian Brumer-Stark conjecture holds at p for the extension K ab /k and the set
S. To prove the part of the statement of the local Galois Brumer-Stark conjecture
concerning the non-linear Brumer-Stickelberger element, we proceed in a similar
way. Observe that it follows from [5, Eq. (12)] that one can write θK/k,S = θ0 +
(>1)
(>1)
θK/k,S
with sG wK θ0 ∈ Z[G]. In particular, we have θ0 ∈ Zp [G] and thus θK/k,S∈
Zp [G]. Now, let B be a fractional ideal of K whose class is in ClK {p}. Let ` be
the maximum of the χ(1)’s for χ ∈ Ĝ. Let
X`−χ(1)
x = |G|2
wKeχ ∈ |G|Z[G].
χ∈Ĝ(>1)

As noted above |G| ∈ F(G), therefore x ∈ F(G) and there exists α ∈ K ◦ such that
BxwK θK/k,S = αOK . Observe that` (>1)
xwK θK/k,S = |G|2 wK
θK/k,S .
2`d θ (>1)

Let A := B|G| wK /dG . Then, we have A G K/k,S = αOK . Since the set of ideals
that satisfy the statement of the local Galois Brumer-Stark conjecture is groupcontaining the principal ideals, it follows that the local Galois Brumer-Stark conjecture holds at p for the extension K/k and the set of places S. This concludes
the proof.

3. The semi-simple case for monomial group
In this section, we prove the main result of this paper.
Theorem 3.1. Let K/k be a Galois extension of number fields with Galois group
G and let S be a finite set of places of k that contains the infinite places and thefinite places that ramify in K with |S| ≥ 2. Assume that the group G is monomial
and that the abelian Brumer-Stark conjecture BS(E/F, SF ) holds for any abelian
extension E/F contained in K/k where SF denotes the set of places of F above

6

X.-F. ROBLOT

the places in S. Let p be a prime number such that p - |G|. Then, the local Galois(p)
Brumer-Stark conjecture BSGal (K/k, S) holds.
Proof. In order to prove Theorem 3.1, it is enough to prove the following two facts:(>1)(1) θK/k,S∈ Zp [G],(>1)(2) θK/k,Sannihilates ClK {p}.
(>1)∈
Indeed, dG is a divisor of G and thus it is invertible in Zp [G], therefore dG θK/k,S
(>1)
(>1)
Zp [G] if and only if θK/k,S∈ Zp [G]. Now, assume that θK/k,Sannihilates ClK {p}(>1)
then, by the previous remark, that means that dG θK/k,Salso annihilates ClK {p}.
The only thing remaining to prove is that, for any ideal A of K whose class lies ind θ (>1)
ClK {p}, one can find a generator of A G K/k,S that is an anti-unit. But, since p is
odd (otherwise the conjecture is trivially true, see Remark 2.2 of [6]), this is always
possible using the trick explained on page 299 of [8].
We now prove the two assertions. As we mentioned in the introduction, themethod we use is a direct adaptation of the method used by Nomura in [12]. Since
the result is trivial if G is abelian, we assume from now on that G is non-abelian.
Let ν be a character defined on some subgroup Hν of G. (From now on, we will
always use the notation Hν to denote the subgroup of G on which the character ν
is defined.) For g ∈ G, define the character ν[g] of g −1 Hν g by ν[g](x) := ν(gxg −1 )for all x ∈ g −1 Hν g. Note that χ[g] = χ[g 0 ] if g and g 0 are in the same right cosetof G modulo Hν . Observe that ν G = (ν[g])G for all g ∈ G, where we denote by ν G
the induced character of ν on G (see [9, Chapter 5]). In particular, for χ ∈ Ĝ(>1) ,the group G acts on the set of linear characters ν defined on some subgroup Hν of
G and such that ν G = χ. (This is a non-empty set by hypothesis.) We denote byΩ(χ) a fixed orbit of this set under the action of G. Then, we have
X
(3.3)
χ=
ν̇

ν∈Ω(χ)

where ν̇ denotes the function of G obtained by setting ν̇(x) := ν(x) if x ∈ Hν andν̇(x) := 0 otherwise. In particular, it follows that
X
(3.4)
eχ =
eν .
ν∈Ω(χ)For ν ∈ Ω(χ), we denote by πν : Hν → Hν / Ker(ν) the canonical surjection andby ν̂ the unique linear character of Hν / Ker(ν) such that ν = ν̂ ◦ πν . We also
associate to ν two extensions: Eν := K Ker(ν) and Fν := K Hν . Thus, Eν /Fν is a
cyclic extension with Galois group isomorphic to Hν / Ker(ν). Finally, we define Sν
to be the set of places of Fν above the places in S. From the properties of Artin
L-functions, we see that(>1)¯ 0) = ν̂(θE /F ,S ).
χ(θK/k,S) = LK/k,S (χ̄, 0) = LK/Fν ,Sν (ν̄, 0) = LEν /Fν ,Sν (ν̂,νννSLet Ω := χ∈Ĝ(>1) Ω(χ) (note that it is a disjoint union). Combining the previous
equalities with (2.2) and (3.4), we obtain the following identity (which is the nonlinear equivalent of [12, Lemma 4.4])
X
(>1)
(3.5)
θK/k,S=ν̂(θEν /Fν ,Sν ) eν .
ν∈ΩThe following result plays a crucial role in the proof.

ON THE GALOIS BRUMER-STARK CONJECTURE FOR MONOMIAL GROUPS

7

Lemma 3.2. Let ν ∈ Ω. Define Tν := πν (Hν ∩ [G, G]) andXAν :=(1 − c).
c∈Tν

Then, the element Aν annihilates the group of roots of unity of Eν and we haveν̂(Aν ) = tν where tν := |Tν |.
Proof of the lemma. The first assertion follows from the fact that elements of Tν areimage of elements of [G, G], thus they act trivially on roots of unity. For the second
assertion, fix g ∈ G \ Hν (g exists since χ is non-linear). Since ν G is irreducible, itfollows from Mackey’s irreducibility criterion that the restriction to Hν ∩ g −1 Hν gof ν and ν[g] do not have a common irreducible constituent. Since the characters νand ν[g] are linear characters, this implies that there exists h ∈ Hν ∩ g −1 Hν g suchthat ν[g](h) 6= ν(h), i.e., ν(ghg −1 h−1 ) 6= 1. Therefore, Hν ∩ [G, G] is not containedin the kernel of ν and Tν is non-trivial. It follows thatX
ν̂(Aν ) = tν −ν̂(c) = tνc∈Tνand the lemma is proved.

(>1)
We prove the first assertion, that is θK/k,S∈ Zp [G]. Let ν ∈ Ω. First, notethat tν ν̂(θEν /Fν ,Sν ) = ν̂(Aν θEν /Fν ,Sν ) is an algebraic integer. Indeed, since theextension Eν /Fν is abelian, it follows from (2.1) and Lemma 3.2 that Aν θEν /Fν ,Sνlies in Z[Hν / Ker(ν)]. But, the integer tν divides |G| and therefore |G| ν̂(θEν /Fν ,Sν )is an algebraic integer for all ν ∈ Ω. Since |G|eν is also an algebraic integer for all(>1)
ν ∈ Ω, we deduce using (3.5) that the coefficients of |G|2 θK/k,S
are all algebraic
integers and, since it is rational, it lies in Z[G]. Finally, since p does not divide |G|,(>1)
we get that θK/k,S∈ Zp [G].
(>1)
We prove the second assertion, i.e., θK/k,Sannihilates ClK {p}. Let ν be acharacter of a subgroup Hν of G and let σ ∈ Γ := Gal(Q̄/Q). We denote by ν σ thecharacter of Hν defined by ν σ (x) := σ(ν(x)) for all x ∈ Hν . The group Γ acts on
ν via its quotient Γ(ν) := Γ/ StabΓ (ν) which is also the Galois group of Q(ν)/Q
where Q(ν) is the extension of Q generated by the values of ν. Assume now thatν ∈ Ω(χ) with χ ∈ Ĝ(>1) . We see that Γ(ν) = Γ(ν̂), but Γ(χ) is a quotient of Γ(ν)
where χ := ν G (although we will not use this fact). Observe that, for σ ∈ Γ(ν), we
have Hν σ = Hν , Ker(ν σ ) = Ker(ν), πν σ = πν , Tν σ = Tν , Aν σ = Aν , Eν σ = Eν ,Fν σ = Fν , and Sν σ = Sν . Let Ω0 be a set of representatives of Ω under the actionof Gal(Q̄/Q). Then, we can rewrite equation (3.5) as
X X
(>1)
(3.6)
θK/k,S=ν̂ σ (θEν /Fν ,Sν ) eν σ .

ν∈Ω0 σ∈Γ(ν)

Now, for ν ∈ Ω0 and σ ∈ Γ(ν), one checks readily that eν σ = ιν σ NKer(ν) wherePιν ∈ Q̄[Hν ] is such that πν (ιν σ ) = eν̂ σ and NKer(ν) := x∈Ker(ν) x. Thus, we have XXν̂ σ (θEν /Fν ,Sν ) eν σ =ν̂ σ (θEν /Fν ,Sν ) ιν σ NKer(ν) .
σ∈Γ(ν)σ∈Γ(ν)

8

X.-F. ROBLOTLet C be a class in ClK of p-power order. We compute
 X

X
tν
ν̂ σ (θEν /Fν ,Sν ) eν σ C =tν ν̂ σ (θEν /Fν ,Sν ) ιν σ NKer(ν) Cσ∈Γ(ν)σ∈Γ(ν)= X

ν̂ (Aν θEν /Fν ,Sν ) ιν σ NK/Eν (C)σσ∈Γ(ν)

= πν

 X


ν̂ σ (Aν θEν /Fν ,Sν ) ιν σ NK/Eν (C)σ∈Γ(ν)

=

 X



σ

ν̂ (Aν θEν /Fν ,Sν ) eν̂ σ NK/Eν (C)σ∈Γ(ν)= X

Aν θEν /Fν ,Sν eν̂ σ NK/Eν (C)σ∈Γ(ν)= Aν θEν /Fν ,Sν X

eν̂ σ NK/Eν (C).

σ∈Γ(ν)PσThe element σ∈Γ(ν) eν̂ σ is p-integral and thusσ∈Γ(ν) eν̂ NK/Eν (C) is welldefined and is a class in ClEν {p}. But, by Lemma 3.2 and Proposition
P2.2, Aν θEν /Fν ,Sνannihilates ClEν . Since tν is prime to p, we deduce that the element σ∈Γ(ν) ν̂ σ (θEν /Fν ,Sν ) eν̂ σ(>1)
kills ClK {p}. This is true for all ν ∈ Ω0 , hence we get by (3.6) that θK/k,Sannihilates ClK {p}. This concludes the proof of Theorem 3.1.
P

The local abelian Brumer-Stark conjecture is known to hold unconditionally inmany cases. Combining Theorem 3.1 and several results in [1] by Burns and Flach,and in [2] and [3] by Burns, Kurihara, and Sano, we can thus deduce cases where
the local Galois Brumer-Stark conjecture is satisfied.
Corollary 3.3. Let K/k be a Galois CM-extension of number fields and let S bea finite set of places of k that contains the infinite places and the finite places thatramify in K with |S| ≥ 2. Assume that Gal(K/k) is monomial. Then, for any odd
prime p such that p does not divide [K : k] and at least one of the two following
condition is satisfied: (1) p is unramified in K/Q, or (2) at most one prime ideal of(p)k above p splits in K/K + , the local Galois Brumer-Stark conjecture BSGal (K/k, S)
holds.
Proof. Let E/F be an abelian CM-extension of number fields. It is known thatthe abelian Brumer-Stark conjecture for the extension E/F follows from the equivariant Tamagawa number conjecture [1] for the pair (h0 (Spec(E)), Z[H]) whereH := Gal(E/F ). For example, using the results of [3], we get that this special caseof the equivariant Tamagawa number conjecture is equivalent to Conjecture 3.1 ofibid by Remark 3.2 of ibid, which is in turn equivalent to the ‘leading term conjecture’ (Conjecture 3.6 of ibid ) and ‘the leading term conjecture’ implies the abelian
Brumer-Stark conjecture by Remark 1.11(i) of ibid. More precisely, the equivariant
Tamagawa number conjecture for (h0 (Spec(E)), Zp [H]− ) implies the local abelian
Brumer-Stark conjecture at p for E/F . (Here, Zp [H]− := Zp [H]/(1 + τ ) whereτ is the complex conjugation in H.) Therefore, cases where this special case of
the equivariant Tamagawa number conjecture is proved together with Theorem 3.1

ON THE GALOIS BRUMER-STARK CONJECTURE FOR MONOMIAL GROUPS

9yield cases where the local Galois Brumer-Stark conjecture holds unconditionally.
Case (1) follows from Theorem 4 of [11]. Indeed, for any CM-subextension E/Fof K/k, the prime p is unramified in E and thus the conditions of the theorem aresatisfied by the remark just before the theorem; since p - [E : F ], the condition on
the vanishing of the Iwasawa µ-invariant is not necessary (see Remark 6 of ibid ).
For case (2), we use [2, Cor. 1.2] since, in every abelian subextension E/F of K/k,
there is at most one prime ideal of F above p that splits in E/E + ; for the same
reasons as in case (1), the condition on the vanishing of the Iwasawa µ-invariant is
not necessary.

References[1] D. Burns and M. Flach. Tamagawa numbers for motives with (non-commutative) coefficients.
Doc. Math., 6:501–570 (electronic), 2001.
[2] D. Burns, M. Kurihara, and T. Sano. Iwasawa theory and zeta elements for Gm . ArXiv
e-prints, ArXiv:1506.07935, 2015.
[3] D. Burns, M. Kurihara, and T. Sano. On zeta elements for Gm . Doc. Math., 21:555–626,
2016.
[4] Pi. Cassou-Noguès. Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta p-adiques.
Invent. Math., 51(1):29–59, 1979.
[5] G. Dejou and X.-F. Roblot. A Brumer-Stark conjecture for non-abelian Galois extensions. J.
Number Theory, 142:51–88, 2014.
[6] G. Dejou and X.-F. Roblot. The Galois Brumer-Stark conjecture for SL2 (F3 )-extensions. Int.
J. Number Theory, 12(1):165–188, 2016.
[7] P. Deligne and K. Ribet. Values of abelian L-functions at negative integers over totally real
fields. Invent. Math., 59(3):227–286, 1980.
[8] C. Greither, X.-F. Roblot, and B. Tangedal. The Brumer-Stark conjecture in some familiesof extensions of specified degree. Math. Comp., 73(245):297–315, 2004.
[9] I. M. Isaacs. Character theory of finite groups. Dover Publications, Inc., New York, 1994.
Corrected reprint of the 1976 original.
[10] A. Nickel. On non-abelian Stark-type conjectures. Ann. Inst. Fourier (Grenoble), 61(6):2577–
2608, 2011.
[11] A. Nickel. On the equivariant Tamagawa number conjecture in tame CM-extensions. Math.
Z., 268(1-2):1–35, 2011.
[12] J. Nomura. On non-abelian Brumer and Brumer-Stark conjecture for monomial CMextensions. Int. J. Number Theory, 10(4):817–848, 2014.
[13] I. Reiner. Maximal orders, volume 28 of London Mathematical Society Monographs. New
Series. The Clarendon Press Oxford University Press, Oxford, 2003.
[14] J. Tate. Brumer-Stark-Stickelberger. In Seminar on Number Theory, 1980–1981 (Talence,
1980–1981 ), pages 16, Exp. No. 24. Univ. Bordeaux I, Talence, 1981.
[15] J. Tate. Les conjectures de Stark sur les fonctions L d’Artin en s = 0, volume 47 of Progressin Mathematics. Birkhäuser Boston Inc., Boston, MA, 1984.