MATHEMATICS OF COMPUTATION
Volume 00, Number 0, Pages 000–000
S 0025-5718(XX)0000-0

THE BRUMER-STARK CONJECTURE IN SOME FAMILIES OF
EXTENSIONS OF SPECIFIED DEGREE
CORNELIUS GREITHER, XAVIER-FRANÇOIS ROBLOT, AND BRETT A. TANGEDAL

Abstract. As a starting point, an important link is established between
Brumer’s conjecture and the Brumer-Stark conjecture which allows one totranslate recent progress on the former into new results on the latter. For
example, if K/F is an abelian extension of relative degree 2p, p an odd prime,
we prove the l-part of the Brumer-Stark conjecture for all odd primes l 6= p
with F belonging to a wide class of base fields. In the same setting, we study
the 2-part and p-part of Brumer-Stark with no special restriction on F andare left with only two well-defined specific classes of extensions that elude
proof. Extensive computations were carried out within these two classes and acomplete numerical proof of the Brumer-Stark conjecture was obtained in allcases.

0. Overview and results
An important conjecture due to Brumer predicts that specific group ring elements constructed from the values of partial zeta-functions at s = 0 annihilate theideal class groups of certain number fields. Recent progress has been made on thisconjecture ([Gr1], [Wi]) and this will be used to obtain new results on the related
conjecture of Brumer-Stark where progress thus far has been more restricted (seeSection 1 of [RT] for the present status of the Brumer-Stark conjecture). The setting for these conjectures is the following: K/F is a relative Galois extension, with
G = Gal(K/F ) abelian, K totally complex, and F totally real. Both conjectures
predict that certain elements of ZZ[G] annihilate the ideal class group ClK of K.
Our results fall naturally into two parts. The first part (Sections 1 – 3) presents
our theoretical results, which we now briefly state:(1) (Probably well-known) There is the following localization principle: The
Brumer-Stark conjecture (BS) holds for K/F if and only if, for all prime numbersl, an “l-primary analog” (BS)l holds for K/F .
(2) (Semi-simple case) If F is an abelian extension of Q
I with the l-part ofGal(F/QI ) cyclic, K is a CM field, and l 6= 2 is prime to the order of G, then(BS)l holds. For example, given this set-up, if G is abelian of order 2p, p an odd
prime, then it suffices to prove (BS)2 and (BS)p in order to establish (BS) for K/F .
(The condition that F be abelian over Q
I can be relaxed somewhat, see Proposition
1.3 for details.)(3) If G is abelian of order 2p, p an odd prime, then (BS)p holds unless K/F isof type ] or [. Here ] means that K contains a primitive p-th root of unity ζp andReceived by the editor December ?, 2001 and, in revised form, September ?, 2002.
1991 Mathematics Subject Classification. Primary 11R42; Secondary 11R29, 11R80, 11Y40.
Key words and phrases. Algebraic Number Fields, Brumer-Stark Conjecture.
c 1997 American Mathematical Society

1

2

CORNELIUS GREITHER, XAVIER-FRANÇOIS ROBLOT, AND BRETT A. TANGEDAL

K/F is unramified outside the primes above p. The case [ is even more exceptional;
details will be given later. We shall see that, for instance, [ does not happen if p = 3
or if p = 5 and F is real quadratic.
(4) Suppose F is real quadratic and K/F is cyclic of degree 6. Then (BS) holds
in full, except perhaps in case ]. In particular, we prove (BS)2 without exception
in this setting.
The second part (Section 4) presents our computational work, where certain
situations not covered in full by our theoretical results are studied, for example:(5) In the setting of (4): The “first” 534 instances of case ] satisfy the full
Brumer-Stark conjecture as well. We will of course give details on what “first”
means. For the moment let us just say: A list of cases was generated, with various
cut-off criteria and elimination of uninteresting cases, for instance cases where theminus class number of K happened to not be divisible by 3, and consequently (BS)3 ,and (BS), are known to hold for simple reasons. This list, with elimination andsearch limits, came to 534 items, and (BS) was successfully verified in every case.
(6) We also carried out extensive computations involving cyclic extensions K/Fof degree 6 over real cubic fields F . Both ] and [ type extensions arise in this
scenario and we generated lists of non-trivial cases for both types using cut-off andelimination criteria. The Brumer-Stark conjecture was successfully verified in all
114 ] cases and all 145 [ cases considered.
We have one further result we would like to mention here: (BS) holds in fullwhen F is real quadratic and K/F is cyclic of degree 4. This was previously thesituation involving the lowest degree fields over Q
I still unproved (see [RT]). We areplanning to treat this situation and give further results in particular involving the2-primary part (BS)2 in a sequel to the present paper.
1. Statement of (BS) and its primary parts; the link with the Brumer
conjecture
For the first part of this section we assume that K/F is an abelian G-Galois extension with F totally real and K totally complex. (Towards the end of this section
and for the complete remainder of the paper we will make the extra assumptionthat K is a CM field.) For simplicity, we make the blanket assumption throughout
the paper that the base field F is not equal to Q
I , since the conjecture is known
to be true in that case based upon Stickelberger’s classic theorem concerning thefactorization of Gauss sums (see [Ta2], p. 109).
Given an extension K/F as above, let S(K/F ) denote the set of all primes ofF , finite and infinite, that ramify in K. With respect to a finite set S of primes
in F containing S(K/F ), we define a partial zeta-function corresponding to eachelement σ ∈ G in the usual way:
X 1
ζS (s, σ) =
Nas
σ =σ
a

where the sum is over all integral ideals a of F relatively prime to the finite primesin S and having the same Artin symbol (K/F, a) = σa = σ. The infinite sumon the right side converges only for <(s) > 1, but ζS (s, σ) has a meromorphic
continuation to all of C
I with exactly one (simple) pole at s = 1. In particular,
ζS (s, σ) is analytic at s = 0, and based upon the work of Klingen [Kl] and Siegel[Si] we know that ζS (0, σ) ∈ Q
I , and therefore the group ring element (referred toBRUMER-STARK CONJECTURE IN SOME FAMILIES OF EXTENSIONS

3

as a “Brumer-Stickelberger element”)
θS (K/F ) =X

ζS (0, σ)σ −1

σ∈Glies in Q
I [G]. We generally take S = S(K/F ) and in that case we will usually omit
S from the notation and simply write θ(K/F ). If F , K, and S are all fixed in agiven context, we will sometimes just use θ to denote θS (K/F ).
Let w = wK denote the order of the group of roots of unity µK in K. A famous
result, due independently to Deligne and Ribet [DR], Barsky [Ba], and CassouNoguès [CN], states that wθS (K/F ) ∈ ZZ[G] and more generally ξθS (K/F ) ∈ ZZ[G]whenever ξ is an annihilator in ZZ[G] of µK . An element x ∈ K ∗ will be called
w-abelian if its w-th root generates an extension field of K which is abelian over F(not just over K). An anti-unit is an element x ∈ K ∗ which has absolute value 1
at all archimedean places of K (recall that by our assumptions all these places arecomplex). If K happens to be CM, with complex conjugation j ∈ G, then x is ananti-unit if and only if x1+j = 1. We may now state theBrumer-Stark Conjecture: For every nonzero fractional ideal a of K, the idealawθS (K/F ) is principal and has a generator x which is an anti-unit, and w-abelian.
The first part stating that awθS (K/F ) is always principal (i.e. that wθS (K/F )annihilates the class group ClK ) is due to Brumer (he predicts something evenstronger, see below). The generator x in the conjecture, if it exists, is unique up
to a root of unity in K, and the prediction that such an element in K exists withthe special properties above is due to Stark. The set of nonzero fractional ideals
a that satisfy the conjecture forms a group under multiplication which contains all
principal ideals (see [Ta1] for this and other basic properties). Thus one may speakof the subgroup in ClK made up by the classes [a] satisfying the BS property.
The (BS)l conjecture states, for any prime number l: For every nonzero fractional
ideal a whose class is in the l-primary part ClK {l} of ClK , the ideal awθS (K/F )is principal and has a generator x which is an anti-unit, and wl -abelian. Herewl = |M |, where M = µK {l}. The criterion that x be wl -abelian can be stated asfollows (see pp. 83–85 of [Ta2]): Let {σi }i∈I be a system of generators of G. Since
M is cyclic, there exists a system of integers {ni }i∈I such that ζ σi −ni = 1 for allζ ∈ M , i ∈ I. The element x ∈ K ∗ is wl -abelian if and only if there exists a system
σ −nof elements {αi }i∈I in K ∗ such that xσi −ni = αiwl for all i ∈ I and αi j j = αjσi −ni
for all i, j ∈ I. A weakened version of this criterion, sufficient for our purposes, will
be proved in the course of proving Proposition 1.2 below.
Remark: If l is odd, we may omit the condition that the generator x is an antiunit, by the following argument: For a given archimedean place v of F let jv be thenontrivial element of the decomposition group Gv ⊂ G. By property 3) on page24-02 of [Ta1], weQ have (1 + jv )θ = 0, and hence (1 − jv )θ = (2 − (1 + jv ))θ = 2θ.
On letting A = v (1 − jv ) (product over the archimedean places v of F ) we thus
obtain Aθ = 2a θ with a = [F : Q
I ]. Since squaring is a bijection on ClK {l}, we find2ab with a = (u)b for some u ∈ K ∗ , and hence awθ = (uwθ )bwθA . Thus, a generator
z for bwθ which is wl -abelian immediately gives a generator x = uwθ z A for awθ
which is again wl -abelian, and it is an anti-unit since 1 + jv annihilates both θ and
A for all archimedean places v of F .
Proposition 1.1. The Brumer-Stark conjecture holds for K/F if and only if (BS)lholds for every prime l.

4

CORNELIUS GREITHER, XAVIER-FRANÇOIS ROBLOT, AND BRETT A. TANGEDAL

Proof. It is obvious that (BS) implies (BS)l for all primes l. To prove the converse,
we have to show that the subgroup U of classes satisfying the BS property is thewhole of ClK . For this it suffices that U contains the l-primary part of ClK for
all l, so fix some l and take some [a] ∈ ClK {l}. Then (BS)l gives almost what wewant, the only problem being that the generator is only wl -abelian, and we want
it to be w-abelian. To mend this, write w = w0 wl with w0 prime to l and use asimilar trick as just before: exponentiation by w0 is bijective on ClK {l}. Thus we0find b and u ∈ K ∗ with a = (u)bw . Let bwθ = (z) with z a wl -abelian anti-unit.
0
Then awθ = (uwθ z w ), and the latter generator is now again an anti-unit, and wabelian.

Our next goal is to state and prove a result which links the Brumer conjecture
to the Brumer-Stark conjecture, and to explain which parts of the conjectures (the“semi-simple cases”) are already known in some generality. To this end, we recall
the Brumer conjecture. We keep our hypotheses on K, F , G, and S. The Brumer
conjecture (B) then affirms that IθS (K/F ) annihilates the class group ClK , where
I is the annihilator in ZZ[G] of µK . Just like the Brumer-Stark conjecture, (B) is
equivalent to the collection of all (B)l , l running over all prime numbers, where (B)lmeans that Il θS (K/F ) annihilates the l-primary part of ClK , with Il the annihilatorof M := µK {l} in ZZl [G], and ZZl denotes the l-adic integers.
C. Popescu has informed us that he has also obtained an essential part of thefollowing result:
Proposition 1.2. Fix an odd prime l and assume that, in the above notation, themodule M is G-cohomologically trivial. Then (B)l holds if and only if (BS)l holds.
Proof. We need two little preparations. Firstly we claim: whenever M is not zero,
the l-part G{l} of the (abelian) group G must be cyclic. Reason: G{l} must act
faithfully on M , otherwise a nontrivial l-subgroup H ⊂ G would act trivially on M ,and the cohomology of H with coefficients in M would be nontrivial. This easily
implies that G{l} is cyclic, since M is cyclic of l-power order.
Second, there is the following group-theoretical observation: Suppose G is abelianwith cyclic l-part and there is an extension of groups
1→C→Γ→G→1
with C an abelian l-group. Then Γ is abelian if and only if the action of G on C
via conjugation is the trivial action. (This is left to the reader. The main point isthat the extension splits if G has no l-part.)
Now assume that (B)l holds. Let wl = |M |; recall w = wK = |µK | and write
w = wl w0 with w0 prime to l. Since M is cyclic, we have a canonical isomorphism
M ∼= ZZl [G]/Il . The hypothesis on cohomological triviality yields that Il is again
cohomologically trivial; now Il has no ZZ-torsion, hence Il is ZZl [G]-projective, hence
free on one generator, let us say Il = (ξ). Since Il contains a suitable l-power, it isa routine matter to show that we may choose ξ ∈ ZZ[G]. Since wl ∈ Il and ξ is a
non zero divisor, we havewl = ξ · ν

for a unique ν ∈ ZZl [G].
0

Given an ideal a of K, we have to find a generator x of awl w θ which is wl -abelian.
(By a previous remark, we do not have to worry whether x is an anti-unit.) Let lN
be a high power of l (larger than wl times the l-class number of K), and ν 0 ∈ ZZ[G]0congruent to ν modulo lN . Thus, awl w θ is the product of a very high l-power ofBRUMER-STARK CONJECTURE IN SOME FAMILIES OF EXTENSIONS
0

0

5

0

an ideal with the ideal aξν w θ . By hypothesis, aξw θ is principal, with generator y,0say. What remains is to show that x = y ν has the “abelian property”: L := K(ρ)
is abelian over F with ρwl = x. We show that L is normal over K: the elementx̄ ∈ K ∗ /K ∗wl is annihilated by ξ, hence by the whole of Il . Thus, for σ ∈ G, wefind some integer a with σ − a ∈ Il (this is the cyclicity of M over ZZ), and σ(x̄) is
just the a-th power of x̄. Thus, all conjugates of L/K over K coincide with L.
Now we show L is abelian over F . If wl = 1, there is nothing to show, so we
may assume wl > 1, which implies, as mentioned above, that the l-part of G is
cyclic. By virtue of the second preparatory remark, it suffices to show that theaction by conjugation of G on Gal(L/K) is the trivial one. Let σ be any elementof G and lift it to an automorphism σ̃ of L over F . Then we just have to show thatσ̃ commutes with every automorphism τ : ρ 7→ ζρ of L/K (ζ ∈ M ). If a ∈ ZZ suchthat σ − a ∈ Il , then as seen above, σ̃(ρ) must have the form sρa with s ∈ K. We
getσ̃τ (ρ) = σ(ζ)σ̃(ρ) = σ(ζ)sρa ,andτ σ̃(ρ) = τ (sρa ) = ζ a sρa .
This is the same result as in the previous equation since σ(ζ) = ζ a , thus Gal(L/F )is indeed abelian.
Remark: This last part of the argument is standard and well known. It actuallyshows that the property “x is wl -abelian” is equivalent in the present setting to thestatement that x̄ is annihilated by Il , since Il is generated by terms of the form
σ − a.
We now prove the reverse implication, so assume (BS)l . Let J be the groupof non-zero fractional ideals in K. This is a torsion-free ZZ-module, containing
the submodule P of principal ideals. It is convenient to calculate with the l-adic
0
completions Jl and Pl . Let [b] ∈ ClK {l}. Then bwl w θ = (x), and x satisfies theabelian condition, that is: the class of x modulo wl -th powers is annihilated by Il .
Thus xξ can be written as y wl . In Pl we then have equations
ξ(x)ξ = (y)wl = (y)ν ;since ξ divides wl in ZZl [G] andPl has no torsion, we may cancel ξ and obtain00(x) = (y)ν . Now (x) is bwl w θ = bξw θν. With the same argument we may cancel

w0 ξθ

the exponent ν and find b= (y) in Jl . Since [bξθ ] ∈ ClK {l}, we may concludeξθthat b is principal. Since ξ generates Il , we have proved the l-part of the Brumer
conjecture.
This proposition will be important in the proof of the next result.
We will continue to keep all of the assumptions made at the beginning of thissection, but from now onward we impose the extra hypothesis that K is a CM field.
Thus, K is a quadratic extension of a totally real subfield K + . Necessarily, thetotally real field F lies in K + . The nontrivial element j of Gal(K/K + ) coincides
with all jv in earlier notation; it is “the” complex conjugation in G = Gal(K/F ).
Proposition 1.3. If the relative degree [K : F ] is prime to l (which still is an odd
prime), then the l-part of the Brumer conjecture is true for K/F under either ofthe following two conditions:(a) F/Q
I is abelian and the l-part of Gal(F/Q
I ) is cyclic, or6

CORNELIUS GREITHER, XAVIER-FRANÇOIS ROBLOT, AND BRETT A. TANGEDAL(b) F is cubic over Q
I , and l2 does not divide the discriminant of F . (Remark:We do not strive for maximum generality here but rather wish to provide
the results we need later in the paper.)
Proof. We use character decompositions: for every l-adic character χ of G, there
is an exact functor M → Mχ on ZZl [G]-modules, defined by base change: Mχ =M ⊗ZZl [G] ZZl [χ]. The functor M → Mχ is naturally isomorphic to the functor
M → M χ in [Wi]. This property, and the exactness, are due to the circumstancethat l does not divide the order of G. Every module M decomposes as a direct sumof Mχ , with χ ranging over the Q
I l -conjugacy classes of all characters of G. The
l-part of the Brumer conjecture for K/F is, in the present situation, equivalent tothe conjunction of all the χ-parts of this conjecture, in an obvious sense. Note thatsince l 6= 2 and F 6= Q
I , the Brumer-Stickelberger element has zero plus part and it
suffices to consider the Brumer conjecture on A(K), which denotes the l-part of theminus class group of K. In particular we may restrict attention to odd charactersin what follows.
Now we begin with case (a). Standard module theory over discrete valuation
rings shows that the χ-part of the l-part of the Brumer conjecture is a consequenceof the following statement à la Leopoldt-Iwasawa:|A(K)χ | divides [ZZl (χ) : (χ(θ(K/F )))] · |(µK {l})χ |,(∗)

for any odd character χ of G. This formula follows directly from the statementof Theorem 3 in [Wi]; note that the equality given there becomes a divisibility
statement since there may be primes ramifying in K/F but not in the conductorof χ, which gives extra Euler factors. This theorem is stated and proved under the
hypothesis that a certain set Sχ,l is empty. So if this set is empty, we are done. If
it isn’t empty, then by the very definition of Sχ,l , the character χ must be inflated
from a character χ0 of F cl (ζl )/F , where the superscript cl means taking the normal
closure over Q
I . Of course we may omit this superscript here. Write K0 for F (ζl ).
Then we have a canonical isomorphism A(K)χ ∼= A(K0 )χ0 , due to the fact thatl does not divide [K : K0 ]. On the other hand, χ(θ(K/F )) equals χ0 (θ(K0 /F ))times a product of Euler factors, by standard properties of L-functions.
This means that we are left with proving the above divisibility statement underthe extra hypothesis that K = F (ζl ) and thus K is absolutely abelian (we now write
again K for K0 and χ for χ0 ). Again, this can be extracted from the literature:
we will use a result of Solomon [So]. Let Γ = Gal(K/QI ) and Γ0 the non-l-part of0
Γ. Then G = Gal(K/F ) is a subgroup of Γ , and A(K)χ is the direct sum of all
A(K)η with η running over all characters up to conjugacy of Γ0 that restrict to χ
on G. We use the formula for A(K)η at the top of p.6 of [Gr2], which is based on[So], and we take the product over all η up to conjugacy. (The required condition
that the l-part of Γ is cyclic does hold, thanks to the corresponding condition we
imposed on F , and also thanks to K = F (ζl ).) This gives the following formulain which ψ runs over all characters of the whole group Γ whose restriction to G isconjugate to χ:
YY|A(K)χ | =l δη ·
B1,ψ−1 l .
η|G∼χvl (b)

ψ|G∼χ

Here |b|l = l, where vl : Q
I l → Q
I is the extension of the l-adic valuation, sovl (l) = 1. The quantity δη is defined as follows: it is 1 unless η is the Teichmüller character, and in that case it equals r with lr = |µK {l}|. Since B1,ψ−1 =BRUMER-STARK CONJECTURE IN SOME FAMILIES OF EXTENSIONS

7−L(0, ψ −1 ), and since L-functions are fully inductive, we getYY
B1,ψ−1 l =LF (0, χ−1Zl (χ) : (LF (0, χ−1 ))].
1 ) l = [Z
ψ|G∼χ

χ1 ∼χ

On the other)) is a multiple of LF (0, χ−1 ). Moreover it is easy to
Q hand, δχ(θ(K/Fη
check that η|G∼χ l is exactly the order of the χ-part of µK {l}. This concludes
the verification of statement (*).
It remains to handle case (b). We may assume F is not abelian over Q
I . Again
we must show that for all characters χ of G the set Sχ,l is empty, so supposeq ∈ Sχ,l . Then we have by the definition in [Wi]: q divides l; χ factors throughGal(F cl (ζl )/F ); χ(q) = 1 and the order of χ (m, say) is at least 3. We show thatsuch a setup cannot exist.
Indeed the condition that l2 does not divide d = disc(F ) gives that the ramicl
fication degree√ of q | l is at most 2. Since F /F is composed from the quadratic
extension Q
I ( d)/QI , χ actually factors through√ a faithful character of Gal(M F/F )with M a cyclic degree m subextension of Q
I ( d, ζl ). Since m > 2, the ramification
exponent of l in M is m, or possibly m/2 if m/2 is odd. Therefore all primes of
M F above l have ramification degree at least 3. On the other hand q does not
ramify in M F/F since χ(q) = 1. This is a contradiction.

Corollary 1.4. Assume F is a field satisfying the conditions in either part (a) or(b) of Prop. 1.3. If K/F is as above (K is a CM field) with [K : F ] prime to l (lan odd prime), then the l-part of the Brumer-Stark conjecture is true for K/F .
Proof. The module M = µK {l} is G-cohomologically trivial for the simple reason
that its cardinality is coprime to the order of G; we may thus invoke Prop. 1.2 and
Prop. 1.3.

2. The p-primary part of (BS) in case [K : F ] = 2p
We assume in this section that the abelian group G = Gal(K/F ) is of order 2p,
p an odd prime number, with F totally real and K totally complex. The field K is
automatically CM since each jv in earlier notation is equal to the unique element
j of order 2 in the cyclic group G. In this setting, Corollary 1.4 gives a widely
applicable result covering (BS)l with l 6= p an odd prime. We study (BS)p in thissection and (BS)2 in section 3. Some extra notation:
E denotes the quadratic extension of F sitting in K;
H = Gal(K/E), which we will identify with Gal(K + /F ), hence H has order p;
σ is a fixed generator of H;j ∈ G is complex conjugation.
Thus G is generated by jσ. Just for this section we will use the following nota−tion for a CM field L: AL = Cl−
L {p}, the p-part of ClL . Our approach is now to
distinguish some cases and subcases, and in most of them we show how the conjecture can be deduced from the analytic class number formula with some algebraic
tricks. Two classes of exceptional cases will remain where these arguments fail: thefirst one (called type [) given by the condition that
ζp 6∈ K and primes splitting in E/F do not ramify in K/E and K cl ⊂ (K cl )+ (ζp ),8

CORNELIUS GREITHER, XAVIER-FRANÇOIS ROBLOT, AND BRETT A. TANGEDAL

where the exponent cl means: taking normal closure over Q
I ; the second (type ]) is
given by the conditionζp ∈ K and K/E is at most ramified at primes over p.
It will soon be discussed how frequently these exceptions occur.
We distinguish, first of all, the two major cases I and II: ζp is not, respectively
is, contained in E. Note that ζp ∈ E if and only if ζp ∈ K.
We further distinguish Cases I and II in two subcases: (a) some prime p in E
which is split in E/F ramifies in K/E, and subcase (b), the logical complement of(a). In Case II a simpler, equivalent, formulation of (b) will be given.
Proposition 2.1. (BS)p holds in Case I with the possible exception of [.
Proof. We first do the subcase (a). By Lemma 2.5 (below), the order of thesubgroup AH
K fixed under H in AK is strictly larger than the order of AE . This
will be important later.
In what follows, we always consider projections to minus parts. That is, θ(K/F )will be considered as an element of Q
I [H] (originally it is in Q
I [G] and G = H × {j};we send complex conjugation j to −1.) In the same vein, θ(E/F ) will be considered
as an element of Q
I . Now θ(K/F ) · N has an expression as  · N · θ(E/F ) whereN = 1 + σ + . . . + σ p−1 is the norm element attached to H and  is a product ofEuler factors attached to the primes that ramify in K but not in E. The hypothesisof case (a) is just chosen to ensure that (the minus projection of)  is zero. In otherwords, θ(K/F ) is annihilated by N , and we’ll make the most of that: we write
θ(K/F ) = (σ − 1)α for some α ∈ Q
I [H].
Now we consider the analytic class number formula which gives, thanks to ζp 6∈ E
(overbar is the canonical map ZZp [H] → ZZp [ζp ]):|AK | = [ZZp [ζp ] : (θ̄(K/F ))][ZZp : (θ(E/F ))]= p · [ZZp [ζp ] : (ᾱ)][ZZp : (θ(E/F ))];
|AE | = [ZZp : (θ(E/F ))].
Some remarks concerning integrality: Both |AE | and |AK |/(p|AE |) are integers (thelatter since |AE | strictly divides |AH
K | and this divides |AK |). Therefore θ(E/F ) ∈ZZp and ᾱ ∈ ZZp [ζp ], and this shows that α can also be taken in ZZp [H].
We can now show that θ(K/F ) kills AK as follows: The module (σ − 1)AK is,
via multiplication by σ − 1, an epimorphic image of AK /AH
K . This latter quotient
has cardinal strictly less than |AK |/|AE |, so|(σ − 1)AK | ≤1|AK |/|AE | ≤ [ZZp [ζp ] : (ᾱ)].
p

Now we argue over the discrete valuation ring R = ZZp [ζp ]. The R-module (σ−1)AK
is annihilated by α as a consequence of the last equation; hence AK is annihilated by
θ(K/F ) = (σ − 1)α. Since K has no nontrivial p-th roots of unity, this annihilation
statement already proves (BS)p by virtue of Prop. 1.2.
It remains to treat case (b). Again by Prop. 1.2, we just want to see that thep-part of the Brumer conjecture (minus Stark) holds for K/F . Here we rely on thepaper [Gr1]. Theorems 4.10 and 4.11 of that paper show the p-part of the Brumer
conjecture under the condition that K/F is “nice”. Those theorems were not stated “prime by prime”, and for this reason, one does not need the full “niceness”

BRUMER-STARK CONJECTURE IN SOME FAMILIES OF EXTENSIONS

9hypothesis for the p-part of the conjecture. An extension was called nice if a certain condition CP holds, and for all odd l, the module µK {l} is G-cohomologicallytrivial. This latter condition is clearly satisfied in our situation for l = p. The pointis that we only need a part of condition CP which originally stated that no criticalprime of F splits from K + to K. There are two kinds of critical primes: those pthat ramify in K/F (first kind), and those q which lie over a rational prime q suchthat K cl ⊂ (K cl )+ (ζq ) (second kind). It is easy to see from [Gr1] that the q with
q 6= p play no role for the proof of the p-part. One has only to look at §4 for this,since critical primes of the second kind never make an appearance in the earlier
sections. Let’s sum up. To ensure the validity of (B)p it is enough to check twothings:(CP1) no critical prime of the first kind splits from K + to K;(CP2) there exist no critical primes of the second kind that divide p.
The first condition is assured by the hypotheses of case (b). The second condition is equivalent to saying that we are not in the exceptional case [, by the very
definition of exceptional primes of the second kind.
The subdivision of Case II (ζp ∈ E) is the same as in Case I: with case II.a wemean that some prime p in F splits in E and ramifies in K; case II.b is then what’sleft. In contrast to Case I, we can make a much sharper statement concerning
subcase II.b:
Remark: In subcase II.b, only primes over p can ramify in K/E (or equivalently,
in K + /F ); in other words, subcase II.b is identical with the Exceptional Case ]defined above. Suppose the prime p of E does not divide p and is ramified in K.
Then the prime p+ below p in E + = F ramifies in K + , and the absolute norm of
p+ is congruent to 1 modulo p (see [FT], Theorem 28(ii) on p. 145). But then,since E = F (ζp ), p+ must split in E (see [Ja], Proposition 3.1 on p. 131), whichputs us in case II.a.
Proposition 2.2. (BS)p holds in case II.a; in other words, (BS)p holds in case II
with the possible exception of ].
Proof. This is very similar to the first proposition. Lemma 2.5 gives that AH
K has
order strictly larger than AE . We now again look at the analytic class number
formula. Let us first assume that ζp2 is not in K. Then
|AK | = p · [ZZp [ζp ] : (θ̄(K/F ))][ZZp : (θ(E/F ))]= p2 · [ZZp [ζp ] : (ᾱ)][ZZp : (θ(E/F ))];
|AE | = p · [ZZp : (θ(E/F ))].
Here α is defined exactly as in the proof of Prop. 2.1. The module (σ − 1)AKis an epimorphic image of AK /AH
K which in turn has cardinality smaller than|AK |/|AE | = p · [ZZp [ζp ] : (ᾱ)]. Therefore (σ − 1)AK is annihilated by α, so AK isannihilated by θ(K/F ). The punchline is that we still have a margin: the conjecture (BS)p talks about the exponent pθ(K/F ), so (BS)p is a direct consequence of
the annihilation statement.
We still have to consider the case that K contains a higher p-power root of unity.
No new ideas come in. The final step just becomes a little technical. So assume|µp∞ (K)| = pe with e ≥ 2. Then |µp∞ (E)| is at least pe−1 . If it is pe , the proof
runs as before, so let us assume it is pe−1 . If one suitably replaces the factors

10

CORNELIUS GREITHER, XAVIER-FRANÇOIS ROBLOT, AND BRETT A. TANGEDAL

immediately after the equality signs in the formulas for |AK | and |AE | displayed
above, one ends up with the statement that the order of (σ − 1)AK is strictly lessthan p2 [ZZp [ζp ] : (ᾱ)] (so the factor p has become p2 ). This means that (σ − 1)AKis annihilated by (σ − 1)α, and AK by (σ − 1)θ. We must show that this is strong
enough to imply that pe θ is a Brumer-Stark annihilator.
Now the action of σ on µp∞ (K) is exponentiation by g, with g − 1 exactly
divisible by pe−1 , and x ∈ K is pe -abelian if and only if xσ−g is a pe -th power in
K. For this it is sufficient that x be of the form z γ , γ = σ p−1 + gσ p−2 + . . . + g p−1 ,since (σ − g)γ = 1 − g p , which is exactly divisible by pe . It is thus sufficient toshow that the equation pe θ = γα(σ − 1)θ has a solution α in ZZp [G], for theneevery ideal ap θ has a generator which is a γ-th power. Since θ is killed by thenorm element, we may just solve the equation with ζp substituted for σ, thus:
pe = (1 − g p ) · (ζp − g)−1 · α · (ζp − 1). This equation is indeed solvable, since 1 − g p
is exactly divisible by pe and since ζp − g is associated to ζp − 1.

We now discuss the “range” of the exceptional cases. If F/Q
I is normal and E/QI
is unramified over p, then [ is impossible. In case ] we must have E = F (ζp );in particular, since E/F is quadratic, the absolute degree of F must be at least(p − 1)/2. If F is real quadratic and p > 5, case ] is therefore
ruled out; if F is real√
quadratic and p = 5, case ] is only possible if F = Q
I ( 5) and E = Q
I (ζ5 ). With aview towards calculations, we also show:
Lemma 2.3. For p = 3 and p = 5, case [ does not happen if F is real quadratic.
Proof. Under the assumption ζp 6∈ E, we prove that the inclusion K cl ⊂ (K cl )+ (ζp )cannot hold. Indeed, K cl is the compositum M M 0 , with M quadratic or biquadraticover F , and M 0 abelian of degree p or p2 over F . If the inclusion were true, we
could pass to the maximal fields of exponent 2 over F , which would giveM ⊂ M + (ζp ).
Note that ζp has 2-power degree over Q
I , and over F . The above inclusion would
have to be an equality since M is not real. Therefore ζp would lie in M . We showthat this implies ζp ∈ E which finishes the proof. Indeed, ζp and F generate anabelian extension of Q
I . If E is normal over Q
I , then E = M and we are done. If
not, M is dihedral of degree 8 over Q
I and M = EE 0 where E 0 /F is the unique
conjugate of E/F over F . Then the maximal subfield of M abelian over Q
I is E 00 ,0
the third relatively quadratic subfield of the biquadratic extension EE /F , and ζpmust be in E 00 . On the other hand, E/F and E 0 /F are both totally complex, so
E 00 would have to be totally real, a contradiction.

From 2.1, 2.2, and 2.3 we obtain the following which we point out as a corollary
for later reference:
Corollary 2.4. Let K/F be a relative abelian extension with K totally complex,
F totally real, and [K : F ] = 2p, p an odd prime. Then (BS)p is proved in generaloutside of cases ] and [. If F is real quadratic and p = 3 or 5, then (BS)p is proved
except in case ].
It remains to state and prove the lemma used in the proof of Proposition 2.1.
Until the end of this section, we consider a cyclic Galois extension K/E of CM
fields of relative degree p an odd prime and Galois group H. One defines the total
ramification index e(K/E) as the product over all ramification indices ep with p

BRUMER-STARK CONJECTURE IN SOME FAMILIES OF EXTENSIONS

11

running over the primesof E that ramify in K. There is also the index e− (K/E)
+
defined as e(K/E) e(K /E + ). The index e− (K/E) is equal to pt with t the
number of primes of E + that split in E and ramify in K. We recall that AK means
the p-part of the minus part of ClK .
Lemma 2.5. Suppose H is cyclic of odd prime order p. Then−|AH
K | = |AE | · e (K/E).

Proof. From Lemma 4.1 on p. 307 of [La] we have the formula
|ClH
K| =

|ClE | e(K/E).
[K : E][OE ∗ : NK/E K ∗ ∩ OE ∗ ]An analogous formula holds with K/E replaced by K + /E + . We now divide theformer by the latter and extract the maximal p-power. On the left hand, we do
end up with |AH
K |. On the right hand, the quotient of |ClE | by |ClE + | contributes|AE |. The quotient of the e-terms gives the e− term. The term [K : E] simply
cancels against [K + : E + ], and what is most important, the quotient of the twonorm indices gives[µE : NK/E K ∗ ∩ µE ],by the theory of units in CM fields: the minus part of the multiplicative group ofunits is the group of roots of unity, and we may decompose into plus and minus
parts with no worries since we are only interested in p-primary torsion.
We have to show that the above norm index is 1. Indeed, let η ∈ µE . We have toshow it is in NK/E K ∗ . By the Hasse norm theorem, this reduces to showing that ηis locally a norm in K/E. Now K/E is induced from the cyclic extension K + /E +by composing with E. Thus, locally, η is a norm from K if and only if NE/E + η
is a norm from K + . But luckily the latter norm NE/E + η is 1 since Gal(E/E + ) is
generated by complex conjugation.

3. Completing the proof for sextic extensions
We suppose now that F is totally real, and K is an abelian sextic CM extensionof F , with Galois group G. From Proposition 1.1 and Corollary 1.4 we know that
if F is a real quadratic field then (BS) holds for K/F if and only if the local
counterparts (BS)l hold for l = 2 and l = 3. Moreover, the case l = 3 was provedin the preceding section (Cor. 2.4) for F real quadratic, with the exception of one
particular class of cases, called ]. The point now is that we can prove the case l = 2
without exception, even over an arbitrary totally real field F . Actually the proof
is quite similar in spirit to Sands’ [Sa1] treatment of the relative quadratic case,adorned with some slight algebraic complications. We do not push for maximum
generality here: certainly other abelian extensions could be treated as well.
Let G0 be the subgroup of order 3 of G and let K0 be the relatively quadratic
subfield of K/F . (This was E in the last section.) The group ring ZZ2 [G] contains
two orthogonal idempotents e = NG0 and e0 = 1 − e; we have eZZ2 [G] ∼=LZZ02 [j]and e0 ZZ2 [G] ∼e M.
= ZZ2 [ζ3 ][j]. All ZZ2 [G]-modules M decompose as M = eM
Actually eM is the part of M belonging to the trivial character of G0 , and e0 M
belongs to either of the two nontrivial characters. Let A (resp. A0 ) be the 2-part ofthe class group of K (of K0 respectively); let B (resp. B0 ) be the factor group of A
(A0 ) modulo the group of ambiguous ideals in the extension K/K + (resp. K0 /F ).

12

CORNELIUS GREITHER, XAVIER-FRANÇOIS ROBLOT, AND BRETT A. TANGEDAL

Since θ(K/F ) maps to a multiple of θ(K0 /F ), and since (BS)2 holds for the
relative quadratic extension K0 /F by [Sa1], we know right away that A0 = eA (thegroup of ideal classes coming from K0 ) is contained in the subgroup of A of classes
satisfying the (BS)2 property. Thus we need only show the same thing for e0 A.
This takes some preparation.
Let d denote the number of finite primes of K + that ramify in K, and let d0 be
defined accordingly for K0 /F . Then the analytic class number formula yields (cf.
[Sa1] Prop. 3.2):+

wK θS(K/K + ) (K/K + ) ∼ 2[K :QI]+d−2 |B|(1 − j),where ∼ means that the two quantities differ by a unit factor in ZZ2 ; and a similar
formula results on replacing K by K0 , K + by F , B by B0 , and d by d0 . (Bibliographical note: the factor wK is erroneously missing in the quoted proposition;
however, it is present, as it should, at the beginning of the proof of Prop. 5.2 1) of
the same paper.) Let ξ be the nontrivial character on {1, j}, and let χ0 , χ1 , χ2 be
the three odd characters of G, numbered so that χ0 is trivial on G0 . We recall ournotational convention that θ(K/F ) = θS(K/F ) (K/F ), with S(K/F ) precisely theset of places that ramify in K/F . We need a lemma:
Lemma 3.1. ξ(θ(K/K + )) = ξ(θ(K0 /F )) · χ1 (θ(K/F )) · χ2 (θ(K/F )).
Proof. From well known inductive properties of L-functions we have
LS (K/K + , ξ, s) = LS (K/F, χ0 , s)LS (K/F, χ1 , s)LS (K/F, χ2 , s)with S = S(K/F ) throughout. The first factor on the right is LS (K0 /F,
Qξ, s),and this may be expressed as LS(K0 /F ) (K0 /F, ξ, s) times U , with U = (1 −ξ(p)N p−s ) and p running over the primes of F that ramify in K but not in K0 .
The expression on the left of the = sign is LS(K/K + ) (K/K + , ξ, s) times V withQ
V = (1 − ξ(P)N P−s ) and P running over the primes of K + that are unramified
in K but ramified over F . One sees easily that U = V , which gives, by cancellation:
LS(K/K + ) (K/K + , ξ, s) = LS(K0 /F ) (K0 /F, ξ, s)LS (K/F, χ1 , s)LS (K/F, χ2 , s).
It now suffices to set s = 0 and recall the construction of the Brumer-Stickelberger
elements.

From Sands’ formula (see above) we get+|B| = wK ξ(θ(K/K + ) · 2−[K :QI]−d+1(note 1 − j became a factor 2 upon applying ξ), and a similar formula for |B0 |.
Since |e0 B| = |B|/|B0 |, we obtain by division, noting wK = wK0 :+|e0 B| = χ1 (θ(K/F )) · χ2 (θ(K/F )) · 2−[K :QI]−d+[F :QI]+d0 .
Now e0 B is a module over the PID R = ZZ2 [χ1 ] = ZZ2 [ζ3 ]. We use α to denoteχ1 (θ(K/F )) ∈ R. The exponent of 2 in the last formula is −4 or even morenegative, so |e0 B| divides the norm of α/22 from R to ZZ2 . This means that α/22
annihilates e0 B.
The step from this last statement to the desired statement that every class in
e0 A has the BS property is exactly the same as in [Sa1], proof of (5.2) 1). (In ourcase, we have more factors of 2 than we actually need.)
We sum up what we have proved here:

BRUMER-STARK CONJECTURE IN SOME FAMILIES OF EXTENSIONS

13

Theorem 3.2. For K/F sextic abelian with F totally real and K totally complex(automatically CM), (BS)2 holds.
Applying Prop. 1.1, Cor. 1.4, Cor. 2.4, and Theorem 3.2, we finally obtain:
Theorem 3.3. Let K/F be a sextic abelian extension with K totally complex and F
either a real quadratic or a real cubic field. If F is real quadratic, the Brumer-Stark
conjecture holds in full unless K/F falls under case ]. If F is real cubic, abelianover Q
I or such that no square of a prime other than 2 or 3 divides its discriminant,then the Brumer-Stark conjecture holds in full unless K/F falls under case ] orcase [. In all of these cases only (BS)3 remains to be verified.
4. Computations
The Brumer-Stark conjecture has been proved in general when G = Gal(K/F )is cyclic of order 2 [Ta1] or bicyclic of order 4 [Sa1]. When G is cyclic of order 4
and F is real quadratic, Corollary 1.4 shows that only (BS)2 requires verification.
One reason we directed our efforts towards relative abelian sextic extensions was theintention of studying a situation where (BS)3 is of paramount importance. Theorem
3.3 delineates for us exactly those extensions over real quadratic or real cubic basefields where (BS)3 seems most difficult to prove. Our goal in this section is to give acomplete numerical verification of the Brumer-Stark conjecture for a large numberof examples within each of the three families of extensions where Theorem 3.3 falls
short of a complete proof: ] over real quadratic fields, ] over real cubic fields, and[ over real cubic fields.
Our method of constructing the examples in this section is of interest in itself. In each case we start with a real quadratic or real cubic field F satisfying
the conditions in Theorem 3.3. We then need to construct a field K + that is a
cyclic cubic extension of F (note that K + is totally real as well) and has restricted
ramification (for example, we only want ramification above the prime 3 with ] extensions). A particularly attractive way to generate such extensions is offered by arelated conjecture of Stark and we refer to [Co] or [Ro] for a detailed explanationof this procedure. Even if we generate K + based upon an unproved conjecture, an
independent check can be made to ensure that a root of the irreducible cubic polynomial in F [x] produced by Stark’s conjecture truly does generate a cyclic cubic
extension K + /F of the desired type. In the ] cases, K is obtained from K + simply
by adjoining ζ3 . We need to be slightly more clever to obtain [ examples but K +is obtained there as well using Stark’s related conjecture. Given an extension K/Ffalling into one of our three categories, we still face the task of computing θS (K/F ).
In every case, S was taken to be the minimal set S(K/F ) which is precisely the setof primes in F that ramify in K. Throughout this section we simply write θ in placeof θS(K/F ) (K/F ). A detailed explanation of how to compute θ over any totally realbase field F was given in [RT] and this method was used in all of our computations.
Further details on our method of verifying the Brumer-Stark conjecture itself may
be found in [RT] as well.
All computations were carried out using the PARI/GP package [PARI].
4.1. The real quadratic case. We generated 534 examples of type ] over real
quadratic fields using the search constraints listed below on F and K + . We have
K = K + (ζ3 ) in each case.
(1) F is real quadratic,

14

CORNELIUS GREITHER, XAVIER-FRANÇOIS ROBLOT, AND BRETT A. TANGEDAL(2) K + /F is a cyclic cubic extension,(3) K + /F is unramified outside 3,(4) dK + ≤ 1012 ,(5) the minus class number of K is divisible by 3.
Note that if condition (5) does not hold, then (BS)3 follows easily. Of the 534 extensions K/F satisfying conditions (1)-(5), there are 91 where K is abelian over Q
I .
When K/Q
I is abelian, Sands [Sa2] and Hayes [Ha] have proved the (BS)conjectureQ
for K/F with respect to the Brumer-Stickelberger element θSH = θ · (1 − σp−1 ),with the product taken over all primes p in F ramified in F/Q
I but not ramified
in K/F (this product always includes at least one factor in each of the examples
in our family). If all of the additional factors in θSH are 3-adic units in the minus
part of the group ring, then the result of Sands-Hayes applies to prove (BS) withrespect to θ as well and this phenomenon occurred in 44 of our examples. For theremaining 490 examples, we computationally verified (BS)3 to be true and therefore
the full (BS) conjecture follows.
Proposition 4.1. The Brumer-Stark conjecture is true for all 534 extensions K/Fsatisfying conditions (1)-(5) above.
√
We now give an example of such a verification. Let F = Q
I ( 69), let K + be theray class field of conductor 3, and K = K + (ζ3 ). The extension K/F is a type ]extension satisfying conditions (1)-(5) above. The field K is generated over Q
I by
an algebraic integer α satisfying
α12 − 6α11 + 21α10 − 44α9 + 63α8 − 72α7 + 103α6 − 174α5 + 51α4 +192α3 − 63α2 − 180α + 144 = 0.
Let σ be the generator of Gal(K/F ) defined by1805α11 − 3995α10 + 12649α9 − 20944α8 + 23354α7 −39384
18508α6 + 39459α5 − 68889α4 − 68817α3 + 160020α2 + 14454α − 121896 .

σ(α) =The element θ was computed to be equal to1θ = (4 + 2σ − 2σ 2 − 4σ 3 − 2σ 4 + 2σ 5 ).
3
The class group of K is of order 3 and K + is a class number one field. The following
ideal is a generator of ClK :a = 2OK + (1 + α4 + α5 + α8 )OK .
We computed a3θ and its class in the class group. We found that it is principal andgenerated byz=

1− 289089 + 234063α − 529186α2 − 807811α3 − 35091α4 + 125033α5 −256
222916α6 − 47864α7 + 38987α8 − 106532α9 + 55649α10 − 602512α11 .

We now look for a generator of a3θ that is also an anti-unit. It is not necessary
to find a generator that is also an anti-unit in order to prove (BS)3 by the remark
preceding Prop. 1.1. But since this gives a neat way to find also a generator whichis 3-abelian (if (BS)3 is true!), this seems to be the easiest way to proceed froma computational point of view. Here, it turns out that the generator z is actually

BRUMER-STARK CONJECTURE IN SOME FAMILIES OF EXTENSIONS

15

already an anti-unit (this was also true in almost all examples; this can be explainedby the fact that PARI tries to return a generator as “small” as possible). Finally,K(z 1/3 ) is abelian over F if and only if z N −σ is a cube in K, where N is an integer
such that σ(ζ3 ) = ζ3N ; in this setting, it is easy to see that one can always takeN = −1. We findh 1
z −1−σ =− 15453α11 + 73560α10 − 240326α9 + 414447α8 −630144568768α7 + 595508α6 − 1125963α5 + 1585780α4 + 586818α3 − i31319271α2 − 654420α + 1880208 ,and therefore (BS)3 is true for this extension.
Listed below is a table of the structure of the minus 3-class groups occurring in
the 490 examples we computationally verified. Each entry consists of two parts.
The first part gives the invariant factor decomposition of an abelian group A in theform n1 , . . . , nr where 3 | nj for all j and nj+1 | nj for 1 ≤ j < r. The group A has
structure Cn1 × · · · × Cnr , where Cn denotes the cyclic group of n elements. The
second part gives the number of minus 3-class groups isomorphic to A.
3 258
9
79
27 10
3, 3 33
9, 3 76

9, 9
2
27, 9
5
9, 3, 3 20
27, 3, 3 4
27, 9, 3 3√
Remark: For F real quadratic and p = 5, case ] is only possible if F = Q
I ( 5)and E = Q
I (ζ5 ) since E/F must be a quadratic extension. To be in case ] meansthat K + /F is cyclic of degree 5 and unramified outside 5. A short calculation in
local class field theory shows that K + is then uniquely determined: it is the degree
10 field over Q
I of conductor 25. The field K = Q
I (ζ25 ) has class number one andthus the full (BS) conjecture is true for K/F by [Ta1].
4.2. The real cubic case. In this subsection we consider both type ] and [ extensions over real cubic fields. For ] type extensions, we proceed similarly to thereal quadratic case and consider all extensions K/F with K = K + (ζ3 ), and F andK + satisfying the following constraints:(1) F is real cubic,(2) F/Q
I abelian or (q prime and q 2 | dF ⇒ q = 2 or 3),+(3) K /F is cyclic cubic,(4) K + /F is unramified outside 3,(5) dK + ≤ 1015 ,(6) the minus class number of K is divisible by 3.
Condition (2) is necessary to ensure that Theorem 3.3 applies and condition (4)ensures that K/F is a type ] extension. There are 114 extensions satisfying these
conditions and 15 of these are such that K is abelian over Q
I . The result of SandsHayes ([Sa2], [Ha]) did not apply in any of these abelian cases to prove the (BS)conjecture with respect to θ and therefore we computationally verified (BS)3 in all
114 examples using the same method as in the real quadratic case.

16

CORNELIUS GREITHER, XAVIER-FRANÇOIS ROBLOT, AND BRETT A. TANGEDAL

Proposition 4.2. The Brumer-Stark conjecture is true for all 114 extensions K/Fsatisfying conditions (1)-(6) above.
The following table gives the structure of the minus 3-class groups of these 114
fields:3 47
9, 9 5
9 11
3, 3, 3 4
27 1
9, 3, 3 2
3, 3 199, 9, 3 3
9, 3 22
In order to construct type [ extensions, we use the following tactic. Assume thatthe discriminant of F has the form dF = 3dh2 with d, h ≥ 1, d square-freeand√3 - d (F is not abelian over Q
I since dF is not a square). Set E = F ( −d), and let
K + be a (totally real) cyclic cubic extension of F such that all the primes ramified√+
in K + /F are either ramified or inert in E/F√
. Now,
let K = KE√
= K + ( −d).
√√
It is clear that ζ3 6∈ K, but since E cl = F ( −d, dF ) = F ( −d, −3) ⊂ K cl ,we see that ζ3 ∈ K cl and actually K cl = (K cl )+ (ζ3 ). Therefore K/F is a type [extension. Finally, in order to ensure that Theorem 3.3 applies we must restrict
the allowed values of h to 1, 2, 3, 6, 9, or 18 (23 and 35 are the highest powers of 2
and 3 respectively that can divide dF ). We thus consider all extensions K/F with
F , K + , and K satisfying the following constraints:(1) F is real cubic,(2) dF = 3dh2 with d square-free, 3 - d, and h = 1, 2, 3, 6, 9, or 18,(3) K + /F is cyclic cubic,√(4) p ramified in K + /F ⇒ p ramified or inert in F ( −d)/F ,(5) dK + ≤ 1015 ,√(6) the minus class number of K = K + ( −d) is divisible by 3.
There are 145 such extensions. For each of these we have computationally proved(BS)3 . However, we did not verify (BS)3 directly. Indeed, the way these extensions
are constructed implies that the discriminants of the fields K are much larger
than the ones we found for ] type extensions over real cubic fields (the average
discriminant of K in our [ examples is about 1040 as compared with about 1032 forthe ] examples). Finding generators of the ideals awK θ in the [ case would require atremendous amount of computer work. Fortunately, we can take advantage of thefact that ζ3 6∈ K and deduce from Prop. 1.2 that (BS)3 is equivalent to (B)3 which
is a lot easier to check computationally.
Proposition 4.3. The Brumer-Stark conjecture is true for all 145 extensions K/Fsatisfying conditions (1)-(6) above.
We now give an example of how (B)3 was checked in this setting. Let F = Q
I (β)
with β 3 −√15β − 11 = 0 and dF = 33 · 379. If K + is the ray class field modulo 3 andK = K + ( −379), then K/F is a type [ extension satisfying the conditions above.
The field K is generated over Q
I by an algebraic integer α satisfying
α18 − 18α16 − 27α15 + 135α14 + 108α13 + 345α12 + 243α11 + 7065α10 −26730α9 + 47304α8 − 183465α7 + 212529α6 − 337041α5 +1982187α4 − 319113α3 + 4234869α2 − 1074978α + 2138797 = 0.

BRUMER-STARK CONJECTURE IN SOME FAMILIES OF EXTENSIONS

17

Let σ be the generator of Gal(K/F ) defined by
σ(α) =

1
25642504251998015091363136823852538505350
16+ 472143852594195097517708119595806967α14− 4879316343979543590590886861288093147α

40102332900383807248126529375112654α
40556171859845878461143042418164629α− 25269652050923497929212771120026077α

+

−

13

+

12

− 472973717538113819727856636934455350α

10− 175034778726883761526482554920769518064α +2143997949648729840779430938111638246α
12874314031468850703781916134606295925α

15

17

11

+
9

8

7

937707788369275078182681363965190126961α − 2147256501457017200394102724004023763715α +656063904950213330802965554608991729941733α − 12246526239717823142768415853103938661601α +4

314923188674625299628566095708659108616482α − 60032163251501835275377729140669965121226α +286713900497115826322509607741480020610254α − 84338008294198391998255893276165939205806α+
136920780019591940272486568579277920858014 .

We computed θ to be equal toθ = 60 − 60σ − 84σ 2 − 60σ 3 + 60σ 4 + 84σ 5 = 12 1 − σ 3 5 − 5σ − 7σ 2 .

The field K + has class number one and the minus 3-class group of K is isomorphic to C27 × C9 × C3 . Each cyclic component is generated respectively by theclasses of the ideals a1 , a2 and a3 where
a1 = 1346743OK + α17 + 113401α16 + 164111α15 − 6120α14 + 60471α13 +192014α12 + 588905α11 + 301501α10 + 227250α9 + 176104α8 −486356α7 + 187668α6 + 392157α5 + 2273α4 − 544623α3 +334792α2 + 485257α + 14022 OK ,a2 = 21701161OK + α17 + 5967947α16 + 6035814α15 + 901785α14 +5050230α13 − 2257687α12 + 5599077α11 + 10036919α10 + 3808177α9 +9283078α8 − 797894α7 − 5933662α6 + 595599α5 − 9968717α4 + 4645261α3 −267451α2 − 5650796α + 2159654 OK ,a3 = 31538539OK + α17 + 11072706α16 + 7219250α15 − 9719308α14 −14233721α13 − 11309241α12 + 9919126α11 + 13593987α10 − 1474838α9 +3019847α8 + 13482312α7 + 2534813α6 + 5113039α5 + 4458431α4 −14063623α3 − 864926α2 + 8475998α + 1373957 .

We use the notation (b) to denote the class in the minus class group of the ideal b.
Since ζ3 6∈ K, the annihilator of µK {3} is I3 = ZZ[G], so to verify the conjecture, itsuffices to prove that aθi is principal for i = 1, 2, 3. We use the factorized form of θθto simplify the computations. Since a3 is of order 3, we have that a12
3 , hence a3 , is33
principal. The factor 1 − σ won’t contribute to the annihilation. Indeed, σ acts3
as complex conjugation and K + is a class number one field so that (bσ ) = (b)−1 .
3Therefore (b1−σ ) = (b)2 and 1 − σ 3 induces an isomorphism of the minus 3-classgroup. We now look at the action of the factor 5 − 5σ − 7σ 2 :2
a5−5σ−7σ= (a32 · a23 ),12
a5−5σ−7σ= (a62 ),2

18

CORNELIUS GREITHER, XAVIER-FRANÇOIS ROBLOT, AND BRETT A. TANGEDAL
12(5−5σ−7σ 2 )12(5−5σ−7σ 2 )so a1
and a2are both principal, and hence so are aθ1 and aθ2 .
Thus (B)3 , and therefore (BS)3 , is true for the extension K/F .
The following table gives the structure of the minus 3-class groups of these 145
fields.
27, 27 1
96481, 3, 3
1
81, 3 327 15
9, 3, 3, 3 1
81, 9 49, 3 827, 9, 3, 3 1
243, 3 1
9, 9 927, 9, 9, 3 2
3, 3, 3 527, 3 1481, 9, 3, 3 1
27, 3, 3 4
27, 9 581, 27, 3, 3 1
27, 9, 3 5
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BRUMER-STARK CONJECTURE IN SOME FAMILIES OF EXTENSIONS

19

Institut für theoretische Informatik und Mathematik, Fakultät für Informatik, Universität der Bundeswehr München, 85577 Neubiberg, F.R.Germany
E-mail address: greither@informatik.unibw-muenchen.deInstitut Girard Desargues, Université Claude Bernard (Lyon I), 69622 Villeurbanne,
FranceE-mail address: roblot@euler.univ-lyon1.fr
Department of Mathematics, College of Charleston, Charleston, SC 29424-0001
E-mail address: tangedal@math.cofc.edu