L’INSTITUTFOURIER DE ANNALES

A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Andreas NICKEL

On non-abelian Stark-type conjectures Tome 61, n^o6 (2011), p. 2577-2608.

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Article mis en ligne dans le cadre du

ON NON-ABELIAN STARK-TYPE CONJECTURES

by Andreas NICKEL (*)

Abstract. — We introduce non-abelian generalizations of Brumer’s conjec- ture, the Brumer-Stark conjecture and the strong Brumer-Stark property attached to a Galois CM-extension of number fields. Moreover, we discuss how they are re- lated to the equivariant Tamagawa number conjecture, the strong Stark conjecture and a non-abelian generalization of Rubin’s conjecture due to D. Burns.

Résumé. — Nous présentons des généralisations non abéliennes de la conjecture de Brumer, de la conjecture de Brumer-Stark et de la propriété forte de Brumer- Stark, qui sont associées à une CM-extension galoisienne de corps de nombres. De plus, nous étudions les liens avec la conjecture équivariante sur les nombres de Tamagawa, la conjecture forte de Stark et la généralisation non abélienne d’une conjecture de Rubin due à D. Burns.

Let L/K be a finite Galois CM-extension of number fields with Galois groupG. To each finite setS of places ofK which contains all the infinite places, one can associate a so-called “Stickelberger element”θS(L/K) in the center of the group ring algebra CG. This Stickelberger element is defined viaL-values at zero ofS-truncated ArtinL-functions attached to the (complex) characters ofG. Let us denote the roots of unity ofLbyµL

and the class group ofLby clL. Assume thatScontains the setS_ram of all finite primes ofKwhich ramify inL/K. Then it was independently shown in [8], [13] and [1] that for abelianGone has

(0.1) Ann_ZG(µL)θS(L/K)⊂ZG.

Now Brumer’s conjecture asserts that Ann_ZG(µL)θS(L/K) annihilates clL. There is a large body of evidence in support of Brumer’s conjecture (cf.

the expository article [14]); in particular, C. Greither [15] has shown that the appropriate special case of the equivariant Tamagawa number conjec- ture (ETNC) as formulated by Burns and Flach [6] implies thep-part of Brumer’s conjecture for an odd primepif thep-part ofµL is a c.t. (short

Keywords:Stark conjectures,L-values, class groups.

Math. classification:11R42, 11R29.

(*) I acknowledge financial support provided by the DFG.

for cohomologically trivial)G-module. A similar result for arbitraryGwas recently proven by the author [19], improving an unconditional annihila- tion result due to D. Burns and H. Johnston [7]. Note that the assumptions made in loc.cit. are adapted to ensure the validity of the strong Stark con- jecture. These two results will provide some evidence for our conjecture.

Moreover, we will introduce a non-abelian generalization of the Brumer- Stark conjecture and of the strong Brumer-Stark property. The extension L/Kfulfills the latter if certain Stickelberger elements are contained in the (non-commutative) Fitting invariants of corresponding ray class groups;

but it does not hold in general, even if G is abelian, as follows from the results in [16]. But if this property happens to be true, this also implies the validity of the (non-abelian) Brumer-Stark conjecture and Brumer’s conjecture. We will show that thep-part of this property is implied by the ETNC if the ramification above the odd primepis at most tame.

D. Burns [3] has introduced a non-abelian analogue of a conjecture formu- lated by Rubin ([25], Conj. B). It is shown in loc.cit. that this conjecture is implied by the strong Stark conjecture, and we will show that Burns’

conjecture implies slightly weaker annihilation results as predicted by the (non-abelian) Brumer-Stark resp. Brumer’s conjecture.

1. Preliminaries

1.0.1. K-theory

Let Λ be a left noetherian ring with 1 and PMod(Λ) the category of all finitely generated projective Λ-modules. We writeK0(Λ) for the Grothen- dieck group of PMod(Λ), andK₁(Λ) for the Whitehead group of Λ which is the abelianized infinite general linear group. If S is a multiplicatively closed subset of the center of Λ which contains no zero divisors, 1 ∈ S, 0 6∈ S, we denote the Grothendieck group of the category of all finitely generatedS-torsion Λ-modules of finite projective dimension by K0S(Λ).

Writing Λ_S for the ring of quotients of Λ with denominators inS, we have the following Localization Sequence (cf.[12], p. 65)

(1.1) K1(Λ)→K1(ΛS)−→^∂ K0S(Λ)→K0(Λ)→K0(ΛS).

In the special case where Λ is ano-order over a commutative ring o and S is the set of all nonzerodivisors of o, we also write K₀T(Λ) instead of K0S(Λ). Moreover, we denote the relativeK-group corresponding to a ring

homomorphism Λ→Λ⁰byK₀(Λ,Λ⁰) (cf.[26]). Then we have a Localization Sequence (cf.[12], p. 72)

K1(Λ)→K1(Λ⁰)^∂−→^Λ,Λ0 K0(Λ,Λ⁰)→K0(Λ)→K0(Λ⁰).

It is also shown in [26] that there is an isomorphismK0(Λ,ΛS)'K0S(Λ).

For any ring Λ we write ζ(Λ) for the subring of all elements which are central in Λ. Let L be a subfield of eitherC or Cp for some prime p and let G be a finite group. In the case where Λ⁰ is the group ring LG the reduced norm map nr_LG : K₁(LG) → ζ(LG)^× is always injective. If in additionL=R, there exists a canonical map ˆ∂G:ζ(RG)^× →K0(ZG,RG) such that the restriction of ˆ∂_G to the image of the reduced norm equals

∂_ZG,RG◦nr⁻¹_RG. This map is called the extended boundary homomorphism and was introduced by Burns and Flach [6].

1.0.2. χ-twists

We largely adopt the treatment of [3], § 1. Let Gbe a finite group and denote the set of all irreducible characters with values in C resp. Cp by Irr(G) resp. Irrp(G). Fix an irreducible character χ ∈ Irr(G) resp. χ ∈ Irrp(G) and letEχbe the minimal subfield ofCresp.Cpover whichχcan be realized and which is both, Galois and of finite degree overQresp.Qp. We put

pr_χ :=X

g∈G

χ(g⁻¹)g, e_χ := χ(1)

|G| pr_χ.

Henceeχis a central primitive idempotent ofEχGand pr_χis the associated projector. We writeoχfor the ring of integers ofEχ and choose a maximal oχ-orderMin E_χGwhich containsoχG. We fix an indecomposable idem- potentfχ ofeχMand define anoχ-torsionfree rightoχG-module by setting T_χ :=f_χM. Note that this slightly differs from the definition in [3], but follows the notation of [7] and [19].Tχ is (locally) free of rankχ(1) overoχ

and the associated rightE_χG-moduleV_χ:=E_χ⊗_oχT_χ has characterχ. For any leftG-moduleM we set M[χ] :=Tχ⊗_ZM resp.M[χ] :=Tχ⊗_ZpM, upon whichGacts on the left byt⊗m7→tg⁻¹⊗g(m) fort∈Tχ,m∈M andg∈G. For any integeriwe writeHⁱ(G, M) for the Tate cohomology in degreeiofM with respect toG. Moreover, we writeM^G resp.MGfor the maximal submodule resp. the maximal quotient module ofM upon which Gacts trivially. We obtain a left exact functorM 7→M^χ and a right exact functorM 7→ M_χ from the category of left G-modules to the category of oχ-modules by setting M^χ := M[χ]^G and Mχ := M[χ]G. The action of

N_G:=P

g∈Gg induces a homomorphismt(M, χ) :M_χ →M^χ with kernel H⁻¹(G, M[χ]) and cokernel H⁰(G, M[χ]). Thus Mχ ' M^χ whenever M and hence alsoM[χ] is a c.t.G-module.

1.0.3. Non-commutative Fitting invariants

For the following we refer the reader to [19]. We denote the set of all m×n matrices with entries in a ring R by M_m×n(R) and in the case m=nthe group of all invertible elements ofM_n×n(R) by Gln(R). Let A be a separableK-algebra and Λ be an o-order in A, finitely generated as o-module, where o is a complete commutative noetherian local ring with field of quotientsK. Moreover, we will assume that the integral closure of o in K is finitely generated as o-module. The group ring ZpG will serve as a standard example. Let N and M be two ζ(Λ)-submodules of an o- torsionfreeζ(Λ)-module. ThenNandMare called nr(Λ)-equivalentif there exists an integernand a matrixU ∈Gln(Λ) such thatN = nr(U)·M, where nr :A→ζ(A) denotes the reduced norm map which extends to matrix rings overAin the obvious way. We denote the corresponding equivalence class by [N]_nr(Λ). We say thatN is nr(Λ)-contained inM (and write [N]_nr(Λ)⊂ [M]nr(Λ)) if for all N⁰ ∈ [N]nr(Λ) there exists M⁰ ∈ [M]nr(Λ) such that N⁰ ⊂M⁰. Note that it suffices to check this property for oneN0∈[N]_nr(Λ). Moreover, we write [N]_nr(Λ) ⊂ M if N⁰ ⊂ M for all N⁰ ∈ [N]_nr(Λ). We will say thatxis contained in [N]_nr(Λ) (and writex∈[N]_nr(Λ)) if there is N₀∈[N]_nr(Λ)such thatx∈N₀.

Now letM be a finitely presented (left) Λ-module and let

(1.2) Λ^a −→^h Λ^bM

be a finite presentation ofM. We identify the homomorphismhwith the corresponding matrix inMa×b(Λ) and defineS(h) =Sb(h) to be the set of allb×bsubmatrices ofhifa>b. In the casea=bwe call (1.2) a quadratic presentation. The Fitting invariant ofhover Λ is defined to be

FittΛ(h) =

( [0]_nr(Λ) if a < b

hnr(H)|H ∈S(h)i_ζ(Λ)

nr(Λ) if a>b.

We call FittΛ(h) a Fitting invariant of M over Λ. One defines Fitt^max_Λ (M) to be the unique Fitting invariant ofM over Λ which is maximal among all Fitting invariants ofM with respect to the partial order “⊂”. IfM admits a quadratic presentationh, one also puts Fitt_Λ(M) := Fitt_Λ(h) which is independent of the chosen quadratic presentation (cf.also [22]). Finally, we

denote byI=I(Λ) theζ(Λ)-submodule ofζ(A) generated by the elements nr(H),H ∈M_b×b(Λ),b∈N.

For anyZpG-moduleM we denote the Pontryagin dual Hom(M,Qp/Zp) of M by M^∨ which is equipped with the natural G-action (gf)(m) = f(g⁻¹m) forf ∈M^∨,g∈Gandm∈M. IfM is finite, we have

(1.3) Ann_ZpG(M^∨) = Ann_ZpG(M)^],

where we denote by^] :QpG→QpGthe involution induced by g 7→g⁻¹. We will frequently make use of the following proposition.

Proposition 1.1. — Let M, M⁰ and M⁰⁰ be finitely presentedΛ-mo- dules. Then it holds:

(1) IfM M⁰ is an epimorphism, thenFitt^max_Λ (M)⊂Fitt^max_Λ (M⁰).

(2) IfM⁰→M M⁰⁰is an exact sequence ofΛ-modules, then Fitt^max_Λ (M⁰)·Fitt^max_Λ (M⁰⁰)⊂Fitt^max_Λ (M).

(3) Ifθ∈Fitt^max_Λ (M)andλ∈ I, then alsoλ·θ∈Fitt^max_Λ (M).

(4) IfMadmits a quadratic presentation, thenFitt^max_Λ (M) =I·FittΛ(M).

(5) IfΛ =ZpGandM C→C⁰M⁰ is an exact sequence of finite Λ-modules, whereC andC⁰ are c.t., then we have an equality

Fitt^max_Λ (M^∨)^]·FittΛ(C⁰) = Fitt^max_Λ (M⁰)·FittΛ(C).

Proof. — For (1), (2) and (5) see [19], Prop. 3.5 (i), (iii) and Prop. 5.3 (ii). For (3) let h be a finite presentation of M such that Fitt_Λ(h) = Fitt^max_Λ (M). Then θ = P

Hz_Hnr(H), z_H ∈ ζ(Λ), where the sum runs through all the submatrices H ∈ Sb(h). Hence it suffices to show that λ·nr(H)∈Fitt^max_Λ (M) for anyH∈S_b(h). We may assume thatλ= nr(H⁰) withH⁰∈Mb⁰×b⁰(Λ), and by adding an appropriate identity matrix onH⁰ resp.hwe may also assume thatb=b⁰. Consider the diagram

Λ^{b H◦H}

0//

H⁰

Λ^b ////cok(H◦H⁰)

Λ^{b H} //Λ^b ////cok(H).

Now (1) implies nr(H) nr(H⁰)∈Fitt^max_Λ (cok(H◦H⁰))⊂Fitt^max_Λ (cok(H)), and since there is an epimorphism cok(H)M, also Fitt^max_Λ (cok(H))⊂ Fitt^max_Λ (M). This shows (3) and the inclusionI ·Fitt_Λ(M)⊂Fitt^max_Λ (M) of (4). Now letψ be a quadratic presentation and hbe an arbitrary pre- sentation ofM. Then it follows from [19], Th. 3.2 resp. its proof that we may assume thath= (ψ|0)◦X, whereX ∈Gla(Λ) for somea∈N. Since

ψis quadratic, eachH ∈S(h) is the productψ◦X˜ for some submatrix ˜X ofX and thus nr(H) = nr( ˜X)·nr(ψ)∈ I ·FittΛ(M).

Assume now thatois an integrally closed commutative noetherian ring, but not necessarily complete or local. We choose a maximal order Λ⁰ con- taining Λ. We may decompose the separableK-algebra A into its simple components

A=A₁⊕ · · · ⊕A_t,

i.e., each Ai is a simple K-algebra and Ai = Aei = eiA with central primitive idempotentse_i, 16 i6 t. For any matrix H ∈M_b×b(Λ) there is a unique matrixH^∗∈Mb×b(Λ⁰) such thatH^∗H =HH^∗= nr(H)·1b×b

andH^∗ei = 0 whenever nr(H)ei = 0 (cf. [19], Lemma 4.1; the additional assumption onoto be complete local is not necessary). If ˜H ∈Mb×b(Λ) is a second matrix, then (HH˜)^∗= ˜H^∗H^∗. We define

H=H(Λ) :={x∈ζ(Λ)|xH^∗∈M_b×b(Λ)∀b∈N∀H∈M_b×b(Λ)}. Sincex·nr(H) =xH^∗H, we have in particular

(1.4) H · I=H ⊂ζ(Λ),

whereI is defined as before even ifois not complete and local. The impor- tance of theζ(Λ)-moduleH will become clear by means of the following result which is [19], Th. 4.2.

Theorem 1.2. — Ifois an integrally closed complete commutative noe- therian local ring andM is a finitely presentedΛ-module, then

H ·Fitt^max_Λ (M)⊂AnnΛ(M).

Now we specialize to Λ = oG, whereo is either Zor Zp. As before, let Λ⁰ be a maximal order containing Λ. The central conductor of Λ⁰ over Λ is defined to beF =F(Λ) :={x∈ζ(Λ⁰) :xΛ⁰ ⊂Λ}and is explicitly given by (cf.[11], Th. 27.13)

(1.5) F=M

χ

|G|

χ(1) D⁻¹(K(χ)/K),

whereD⁻¹(K(χ)/K) denotes the inverse different of the extensionK(χ) :=

K(χ(g) : g ∈ G) over K = Quot(o) and the sum runs through all the irreducible characters with values inCresp.Cp modulo Galois action.

Lemma 1.3. — LetΛ =oG, where oisZorZp. Then it holds:

(1) F ⊂ H.

(2) IfGis abelian or ifΛ =ZpGandp-|G|, thenH=ζ(Λ).

Proof. — (1) and the casep-|G|of (2) are clear from the observations above. IfGis abelian, we may choose H^∗ to be the adjoined matrix of H

and we get (2).

In the sequel we will use the following notation:F(G) =F(ZG),Fp(G) = F(ZpG) and similarly for Hand I. We denote the normalized valuation at a primeP by vP and forx= P

g∈Gxgg ∈ZG resp.x ∈ZpGwe put v_p(x) := min_g∈Gv_p(x_g).

Lemma 1.4. — Letpbe a prime and letGbe a finite group. ThenH(G) is dense inHp(G)with respect to thep-adic topology.

Proof. — Letx∈ Hp(G) and choosex⁰ ∈ζ(ZG) close tox. Then for any H∈M_n×n(ZpG) we have

x⁰H^∗=xH^∗+ (x⁰−x)H^∗∈Mn×n(ZpG)

ifvp(x⁰−x)>n, where|G|=m·pⁿ withp-m. Thusx⁰ ∈ Hp(G). Now letN ∈ Nbe large; since pdoes not divide m, we can choose a multiple m⁰ of m such that m⁰ ≡ 1 modp^N. Then m⁰x⁰ is also close to x, since vp(x−m⁰x⁰)>min{vp(x−x⁰), vp((1−m⁰)x⁰)}.But since m|m⁰, we have m⁰x⁰ ∈ Hq(G) for all primesq. Now letH ∈Mn×n(ZG). Then

m⁰x⁰H^∗∈\

q

M_n×n(ZqG)∩M_n×n(QG) =M_n×n(ZG)

and hencem⁰x⁰∈ H(G).

1.0.4. EquivariantL-values

Let us fix a finite Galois extension L/K of number fields with Galois group G. For any prime p of K we fix a prime Pof L above p and write GP resp. IP for the decomposition group resp. inertia subgroup ofL/K atP. Moreover, we denote the residual group atPbyG_P =G_P/I_P and choose a liftφP∈GPof the Frobenius automorphism atP.

IfSis a finite set of places ofKcontaining the setS_∞of all infinite places ofK, andχis a (complex) character ofG, we denote theS-truncated Artin L-function attached toχ andS byLS(s, χ) and defineL^∗_S(0, χ) to be the leading coefficient of the Taylor expansion of L_S(s, χ) at s = 0. Recall that there is a canonical isomorphismζ(CG) =Q

χ∈Irr(G)C. We define the equivariant ArtinL-function to be the meromorphicζ(CG)-valued function

LS(s) := (LS(s, χ))_χ∈Irr(G).

We putL^∗_S(0) = (L^∗_S(0, χ))_χ∈Irr(G)and abbreviateL_S∞(s) byL(s). IfT is a second finite set of places ofK sucht thatS∩T =∅, we defineδT(s) :=

(δT(s, χ))_χ∈Irr(G), where δT(s, χ) = Q

p∈Tdet(1−N(p)^1−sφ⁻¹_P |Vχ^IP), and put

ΘS,T(s) :=δT(s)·LS(s)^].

These functions are the so-called (S, T)-modifiedG-equivariantL-functions and we define Stickelberger elements

θ^T_S := ΘS,T(0)∈ζ(CG).

IfT is empty, we abbreviateθ^T_S byθ_S. Note that theχ-part ofθ^T_S vanishes for a non-trivial characterχif there is an (infinite) primep∈S such that Vχ^GP 6= 0. This is the main reason why we will assume henceforth that L/Kis a CM-extension,i.e.,Lis a CM-field,Kis totally real and complex conjugation induces an unique automorphismjofLwhich lies in the center ofG. IfRis a subring of eitherCorCpfor a primepsuch that 2 is invertible overR, we putRG₋:=RG/(1 +j) which is a ring, since the idempotent

1−j

2 lies inRG. For anyRG-moduleM we defineM⁻ =RG₋⊗RGM which is an exact functor since 2∈R^×. IfM is aZG-module, we defineM⁻ to beZ[¹₂]G/(1 +j)⊗_ZGM. This notation is nonstandard, but practical, since taking minus parts is again an exact functor. Now Stark’s conjecture (which is a theorem for odd characters, see [28], Th. 1.2, p. 70) implies

(1.6) θ^T_S ∈ζ(QG₋).

Note that we actually have to exclude the special case|S∞(L)|= 1 (cf.the proof of [20], Prop. 3, where (1.6) is shown in the relevant caseS=S_∞and T =∅), but in this situation the extension L/K is abelian. Let us fix an embeddingι:CCp; then the image ofθ^T_S inζ(QpG₋) via the canonical embedding

ζ(QG₋)ζ(QpG₋) = M

χ∈Irrp(G)/∼

χ odd

Qp(χ),

is given byP

χ(δT(0, χ^ι−1)·LS(0,χˇ^ι−1))^ι. Here the sum runs over allCp- valued irreducible odd characters ofGmodulo Galois action. Note that we will frequently dropι andι⁻¹ from the notation.

1.0.5. Ray class groups

For any set S of places of K, we write S(L) for the set of places of L which lie above those in S. Now let T and S be as above. We write cl^T_L for the ray class group ofL to the rayM_T :=Q

P∈T(L)P ando_S for the ring ofS(L)-integers of L. LetSf be the set of all finite primes in S(L);

then there is a natural map ZSf → cl^T_L which sends each prime P ∈ Sf

to the corresponding class [P]∈cl^T_L. We denote the cokernel of this map by cl^T_S(L) =: cl^T_S. Further, we denote theS(L)-units ofLbyES and define E^T_S :={x∈E_S :x≡1 modM_T}. All these modules are equipped with a naturalG-action and we have the following exact sequences ofG-modules (1.7) E_S^T_∞E_S^T −→^v ZSf →cl^T_Lcl^T_S,

wherev(x) =P

P∈SfvP(x)Pforx∈E_S^T, and

(1.8) E_S^T E_S →(o_S/M_T)^× −→^ν cl^T_S cl_S,

where the mapν lifts an elementx∈(oS/MT)^× tox∈oS and sends it to the ideal class [(x)]∈cl^T_S of the principal ideal (x). Note that theG-module (o_S/M_T)^× is c.t. if no prime inT ramifies inL/K. We define

A^T_S := (cl^T_S)⁻.

IfS = S_∞, we also write A^T_L and E_L^T instead of A^T_S∞ and E_S^T_∞. Finally, we suppress the superscriptT from the notation ifT is empty. If M is a finitely generatedZ-module andpis a prime, we putM(p) :=Zp⊗_ZM. In particular,AL(p) is thep-part of the minus class group ifpis odd.

2. Statement of the conjectures

Let L/K be a Galois CM-extension with Galois groupG. Let S and T be two finite sets of places ofKsuch that

• S contains all the infinite places of K and all the places which ramify inL/K,i.e.,S⊃S_ram∪S_∞.

• S∩T =∅.

• E^T_S is torsionfree.

We refer to the above hypotheses as Hyp(S, T). We put Λ =ZGand choose a maximal order Λ⁰ containing Λ. For a fixed setS we defineAS to be the ζ(Λ)-submodule of ζ(Λ⁰) generated by the elements δ_T(0), whereT runs through the finite sets of places ofKsuch that Hyp(S, T) is satisfied. The following is a non-abelian generalization of Brumer’s conjecture.

Conjecture 2.1 (B(L/K, S)). — LetS be a finite set of places ofK containingS_ram∪S∞. ThenASθS ⊂ I(G)and for eachx∈ H(G)we have

x·ASθS ⊂AnnΛ(clL).

Before we make some clarifying remarks, we state the following lemma.

Lemma 2.2. — LetSbe a finite set of places ofKcontainingS_ram∪S∞. Then the elementsφP−N(p), wherepruns through all the finite places ofK such that the setsSandT_p={p}satisfyHyp(S, T_p), generateAnn_ZG(µ_L).

Moreover, if we restrict to the primes p such that φP = 1, the greatest common divisor of the integers1−N(p)is|µL|.

Proof. — The proof of [28], Lemma 1.1, p. 82 (whereGis assumed to be

abelian) remains unchanged.

Remark 2.3. — (1) Since AS is generated by the elements δT(0) such that Hyp(S, T) holds, Conjecture 2.1 is equivalent to the as- sertion that for all these setsT the Stickelberger elementθ_S^T lies in I(G) andxθ^T_S annihilates the class group for each x∈ H(G). Note thatθ^T_S ∈ I(G) impliesxθ^T_S ∈ζ(Λ).

(2) Lemma 2.2 implies that in fact [AS(p)]_nr(ZpG)is a Fitting invariant ofµ_L(p) overZpG. Moreover, we have

Ip(G)·[AS(p)]_nr(ZpG)⊂Fitt^max_Z

pG(µL(p))

by Proposition 1.1. So it is reasonable to ask if this inclusion might be an equality (at least ifS=S_ram∪S_∞).

(3) If G is abelian, Lemma 2.2 implies that the module AS equals Ann_ZG(µ_L). In this case the inclusion A_Sθ_S ⊂ I(G) = Λ follows from (0.1), and since H(G) = Λ by Lemma 1.3, Conjecture 2.1 recovers Brumer’s conjecture.

Since H(G) always contains the central conductor F(G), we can state the following weaker form of Conjecture 2.1.

Conjecture 2.4(Bw(L/K, S)). — LetS be a finite set of places ofK containingS_ram∪S_∞. ThenASθS ⊂ζ(Λ⁰)and for eachx∈ F(G)we have

x·ASθS ⊂Ann_Λ(clL).

Lemma 2.5. — LetSbe a finite set of places ofKcontainingS_ram∪S∞. Then

B(L/K, S) =⇒ B_w(L/K, S).

IfS ⊂S⁰, then

B(L/K, S) =⇒ B(L/K, S⁰), Bw(L/K, S) =⇒ Bw(L/K, S⁰).

Proof. — The first assertion is obvious. Now assume that B(L/K, S) resp. Bw(L/K, S) holds. Since θS⁰ = nr(λ)θS, where λ = Q

p∈S⁰rS(1− φ⁻¹_P ) ∈Λ, we see that also AS⁰θS⁰ ⊂ASnr(λ)θS lies in I(G) resp. ζ(Λ⁰).

Moreover ˜x:=x·nr(λ) lies in H(G) resp. F(G) if xdoes. Hence we find thatxAS⁰θS⁰ ⊂˜xASθS annihilates clL. Replacing the class group clLby itsp-parts clL(p) for each rational prime p, Conjecture B(L/K, S) resp. Conjecture Bw(L/K, S) naturally decom- poses into local conjectures B(L/K, S, p) resp. B_w(L/K, S, p). Note that it is possible to replaceH(G) by Hp(G) by means of Lemma 1.4. Taking Lemma 1.3 into account, a similar proof shows the following lemma.

Lemma 2.6. — LetS be a finite set of places ofKcontainingS_ram∪S_∞ and letpbe a prime. Then

B(L/K, S, p) =⇒ Bw(L/K, S, p).

Ifp-|G|then

B(L/K, S, p) ⇐⇒ B_w(L/K, S, p).

IfS ⊂S⁰, then

B(L/K, S, p) =⇒ B(L/K, S⁰, p), Bw(L/K, S, p) =⇒ Bw(L/K, S⁰, p).

Forα∈L^× we define

Sα:={pfinite prime ofK:p|N_L/K(α)}

and we callαan anti-unitifα^1+j = 1. Let ωL := nr(|µL|). The following is a non-abelian generalization of the Brumer-Stark conjecture.

Conjecture 2.7(BS(L/K, S)). — LetSbe a finite set of places ofK containingS_ram ∪S_∞. Then ωL·θS ∈ I(G) and for each x ∈ H(G) and each fractional idealaof L, there is an anti-unitα=α(x,a, S)∈L^× such that

a^x·ωL·θS = (α)

and for each finite setT of primes ofKsuch thatHyp(S∪Sα, T)is satisfied there is anα_T ∈E_S^T

α such that

(2.1) α^z·δT(0) =α^z·ω_T ^L for eachz∈ H(G).

Remark 2.8. — IfGis abelian, we have I(G) =H(G) =ZGandωL=

|µL|. Hence it suffices to treat the case x= z = 1. Then [28], Prop. 1.2, p. 83 states that the condition (2.1) on the anti-unitαis equivalent to the assertion that the extensionL(α^1/ωL)/K is abelian.

As above we can state the following weaker conjecture.

Conjecture 2.9 (BS_w(L/K, S)). — Let S be a finite set of places of KcontainingS_ram∪S_∞. ThenωL·θS ∈ζ(Λ⁰)and for eachx∈ F(G)and each fractional idealaof L, there is an anti-unitα=α(x,a, S)∈L^× such that

a^x·ωL·θS = (α)

and for each finite set T of primes of K such that Hyp(S∪Sα, T) holds there is anα_T ∈E_S^T

α such that

(2.2) α^z·δT(0) =α^z·ω_T ^L for eachz∈ F(G).

Remark 2.10. — (1) Since E_S^T

α is torsionfree, we may replace the equalities (2.1) and (2.2) by the equalityα^δT(0)=α^ω_T^L inQ⊗E_S^T

α. (2) If a and b are two fractional ideals of L for which Conjecture BS(L/K, S) resp.BSw(L/K, S) holds, then it is easy to see that this conjecture is also true for the producta·b. Since it is also true for principal ideals, it suffices to verify these conjectures for totally decomposed primes, as these primes generate the class group.

(3) If we restrict ConjectureBS(L/K, S) resp.BS_w(L/K, S) to ideals whose class in clL has p-power order, we get local conjectures BS(L/K, S, p) resp.BS_w(L/K, S, p).

(4) If the primepis odd, we may omit the condition that the generator αis an anti-unit by the following reason (cf.[17], remark preceding Prop. 1.1): Let a be a fractional ideal whose class in clL has p- power order. Since squaring is invertible on clL(p) we findb such thata= (u)b²for someu∈L^×. Now letx∈ H(G) resp.x∈ F(G) and assume thatb^x·ωLθS is generated by β ∈ L^× such that (2.1) holds (with αreplaced byβ). But since (1−j)θ_S = 2θ_S, we have a^x·ωLθS = (u^x·ωLθS)b^x·ωL2θS = (u^x·ωLθS ·β^1−j) and this generator is an appropriate anti-unit.

(5) Burns [5] has meanwhile formulated a new conjecture which gener- alizes many refined Stark conjectures to the non-abelian situation.

In particular, it implies our generalization of Brumer’s conjecture (cf.loc.cit., Prop. 3.5.1). Since it also implies the Brumer-Stark con- jecture (cf. loc.cit., Remark 3.5.2) in the abelian case, the author expects that it also implies ConjectureBS(L/K, S). If this is true, loc.cit., Th. 4.1.1 would give a different proof of Theorem 5.1 below.

We have the following implications which are either obvious or which are proved by a similar argument as in Lemma 2.5.

Lemma 2.11. — LetSbe a finite set of places ofKcontainingS_ram∪S∞

and letpbe a prime. Then

BS(L/K, S) =⇒ BS_w(L/K, S), BS(L/K, S, p) =⇒ BS_w(L/K, S, p).

Ifp-|G|, then

BS(L/K, S, p) ⇐⇒ BSw(L/K, S, p).

IfS ⊂S⁰, then

BS(L/K, S) =⇒ BS(L/K, S⁰), BS(L/K, S, p) =⇒ BS(L/K, S⁰, p), BS_w(L/K, S) =⇒ BS_w(L/K, S⁰), BS_w(L/K, S, p) =⇒BS_w(L/K, S⁰, p).

We have the following relation to the non-abelian Brumer Conjectures:

Lemma 2.12. — LetSbe a finite set of places ofKcontainingS_ram∪S_∞ and letpbe a prime. Then

BS(L/K, S) =⇒ B(L/K, S), BS(L/K, S, p) =⇒ B(L/K, S, p), BS_w(L/K, S) =⇒ B_w(L/K, S), BS_w(L/K, S, p) =⇒ B_w(L/K, S, p).

Proof. — Letabe a fractional ideal ofLwhose class in cl_Lis assumed to havep-power order if we are in the local case. Letx∈ H(G) resp.x∈ F(G).

Thena^x·ωLθS = (α) and (α)^z·δT(0)= (α_T)^z·ωL for allz∈ F(G). Hence (2.3) a^{x·z·ωL·θST} = (α)^z·δT(0) = (αT)^z·ωL.

SinceωL ∈ζ(QG)^×, we findN ∈Nsuch thatN·ω_L⁻¹ ∈ζ(Λ). Moreover,

|G| ·ζ(Λ) ⊂ F(G) such that we may choose z = |G| ·N ·ω⁻¹_L . But the group of fractional ideals has noZ-torsion such that equation (2.3) implies

a^x·θTS = (αT).

3. Burns’ Conjecture

and the strong Brumer-Stark property

We first recall a non-abelian generalization of the Rubin-Stark conjecture due to D. Burns [3]. Note that our slightly different definition ofχ-twist will lead to some minor changes. LetL/Kbe an extension of number fields with Galois groupGand fix a non-trivial irreducible complex characterχ of G. For any finite non-empty set S of places of K we write Y_S for the free abelian group onS(L) andXS for the kernel of the augmentation map

Y_S → Z which sends each element of S(L) to 1. If S contains S_∞, the Dirichlet map

λS :R⊗_ZES →R⊗_ZXS, ε7→ − X

P∈S(L)

log|ε|pP

is an isomorphism of RG-modules. The Noether-Deuring Theorem com- bines with the fact that XS is torsionfree to imply the existence of G- invariant embeddingsφ:XS ES. For any suchφwe set

R^S_φ(χ) := det(λ_S◦φ|C⊗_Eχ (V_χˇ⊗_ZGX_S))∈C^×,

where ˇχdenotes the character contragredient toχ. Then Stark’s conjecture (as interpreted in [28], Conj. 5.1,p. 27, but see [3], § 2) states that for each α∈Aut(C) one has

L^∗_S(0, χ^α)

R^S_φ(χ^α) =α L^∗_S(0, χ) R^S_φ(χ)

! , whereχ^α:=α◦χ. We put

rS=rS(χ) :=X

p∈S

dimE_χ(Vχ^GP).

Then, sinceχ is non-trivial, one has

r_S = dim_Eχ(V_χˇ⊗_ZGX_S) = dim_Eχ(E_χ⊗_oχ X_S,ˇχ)

and the functionLS(s, χ) vanishes to orderrS at s= 0 by [28], Prop. 3.4, p. 24. Further, if we denote by

λ^(χ)_S :C⊗E_χ(∧^r_E^S

χ(Vχˇ⊗_ZGES))−→^∼ C⊗E_χ(∧^r_E^S

χ(Vχˇ⊗_ZGXS)) the isomorphism ofC-spaces induced byλ_S, one finds that Stark’s conjec- ture implies

(3.1) L^∗_S(0, χ)· ∧^r_E^S

χ(Vχˇ⊗_ZGXS)) =λ^(χ)_S (∧^r_E^S

χ(Vχˇ⊗_ZGES)).

LetL^∗_S,T(0, χ) :=δ_T(0,χ)ˇ ·L^∗_S(0, χ) ifS andT satisfy Hyp(S, T). For any G-module resp. oχ-module M we writeMtor for the Z-torsion submodule ofM and setM :=M/M_tor which we identify as a submodule ofQ⊗_ZM resp.Eχ⊗o_χM in the natural way. Now Burns’ conjecture ([3], Conj. 2.1) asserts the following refinement of (3.1):

Conjecture 3.1 (RS(L/K, S, T, χ) (Burns)). — Let S and T be two finite sets of places of K such that Hyp(S, T) is satisfied and let χ be a non-trivial irreducible complex character. Then Stark’s conjecture holds for χand inC⊗E_χ(∧^r_E^S

χ(Vχˇ⊗_ZGXS))one has

|G|^rSL^∗_S,T(0, χ)· ∧^r_o^S_χXS,χˇ⊂Fitto_χ(H⁻¹(G, XS[ ˇχ]))·λ^(χ)_S (∧^r_o^S_χE_S^{T ,ˇχ}).

Moreover, we will say thatRS(L/K, S, χ) holds ifRS(L/K, S, T, χ) holds for all finite sets of placesT such that Hyp(S, T) is satisfied.

Remark 3.2. — It is reasonable to expect that the inclusion in Conjec- ture 3.1 is an equality for sufficiently largeS (cf.[3], Prop. 4.8).

If Stark’s conjecture holds, the quotient L^∗_S(0, χ)/R^S_φ(χ) belongs toE_χ. The strong Stark conjecture as formulated by T. Chinburg ([9], Conj. 2.2) further predicts that

(3.2) L^∗_S(0, χ)

R^S_φ(χ) oχ =q(ψ_χˇ)⁻¹,

whereψχ denotes the composite homomorphism ofoχ-modules XS,χ

φχ //ES,χ t(E_S,χ)

//E^χ_S,

and for each irreducible characterχ we use the following general notion:

iff : M → N is a homomorphism of finitely generated o_χ-modules with finite kernel and finite cokernel, then q(f) denotes the frational oχ-ideal Fitto_χ(cok(f))·Fitto_χ(ker(f))⁻¹.

Theorem 3.3 (([3], Th. 4.1)). — If the strong Stark conjecture (3.2) holds for the character χ, then RS(L/K, S, χ) is valid for all admissible setsS.

We will need the following result which is [3], Prop. 3.2. We setcS(χ) :=

Fitt_oχ(H⁻¹(G, X_S[ ˇχ])).

Proposition 3.4. — Assume that rS = 1 and |S| > 1. Let p1 be the unique place in S with Vχ^GP1 6= 0 and set S₁ := S_∞ ∪ {p₁}. If RS(L/K, S, T, χ) is valid, then for any element d of cS(χ)⁻¹D(Eχ/Q)⁻¹ there exists an elementu(d)∈E_S^T

1 which at each placePofL satisfies

−log|u(d)|P=

_{0, ifP}

-p₁

P

γ∈Gal(Eχ/Q)

P

h∈GP1

γ(d)χ^γ(gh)L^∗_S,T(0, χ^γ), ifP=P^g₁, g∈G.

Theorem 3.5. — LetSbe a finite set of places ofK containingS_ram∪ S_∞. If RS(L/K, S∪ {q}, χ)is valid for all characters χ such thatrS = 0 and all primes q which are totally split in L/K, then BS_w(L/K, S) and Bw(L/K, S)hold. In particular, if the strong Stark conjecture (3.2) holds for these characters, thenBS_w(L/K, S) andB_w(L/K, S) hold for all ad- missible setsS.

Proof. — By means of Lemma 2.12 and Theorem 3.3, it suffices to show that the relevant case of Burns’ conjecture implies BSw(L/K, S). Since

e_χ·θ_S^T = 0 ifr_S >0, we only have to treat charactersχwithr_S = 0. LetT be a finite set of primes ofK such that Hyp(S, T) is satisfied and letqbe a totally decomposed prime not inS∪T. We claim that ¹₂ ∈c_S∪{q}(χ)⁻¹. Taking this for granted for the moment, letx∈ F(G) and write

1

2 ·x·θ^T_S =X

χ

X

γ∈Gal(Q(χ)/Q)

γ1 2x_χ

L^∗_S,T(0, χ^γ)·pr_χˇγ,

where the first sum runs over all irreducible characters with r_S(χ) = 0 modulo Galois action, and wherexχ ∈ D⁻¹(Q(χ)/Q) according to the de- scription (1.5) of the central conductor. SinceL^∗_S,T(0, χ) =L^∗_S∪{q},T(0, χ) and the trace mapsD⁻¹(E_χ/Q) ontoD⁻¹(Q(χ)/Q), we can apply Propo- sition 3.4 to the setS∪ {q} and obtain

Q^12xθTS = (α_T), where α_T ∈ E^T_S

∞∪{q} and Q is a prime inL above q. Since the ray class group cl^T_L is generated by totally decomposed primes, we have for any fractional ideala ofL, coprime to the ideals inT that

(3.3) a^12xθTS = (αT(a))

with α_T(a) ∈ E_S(a)^T , where S(a) contains all the primes of K below the primes dividinga. Now letpbe a prime and leta be a fractional ideal of Lsuch that its class [a]∈clLhasp-power order . Then Lemma 2.2 implies that there is a totally decomposed primep0such that|µL|= (1−N(p0))·c, wherec ∈Qwithvp(c) = 0. We may assume thatp0 is coprime toa and that Hyp(S,{p0}) is satisfied. The observations above imply that for any x∈ F(G) we have

a^xωLθS =a^12xωL2θS =a^12xnr(c)(1−j)θ^{p_S^o} = (α⁰)^(1−j)= (α)

for an appropriateα⁰ ∈L^× and an anti-unit α=α^0(1−j). Note that there is a natural numberN withvp(N) = 0 such thatN ·xnr(c)∈ F(G) and raising to theN^th power is a bijection on cl_L(p). Moreover, ifT is a finite set of primes such that Hyp(S∪Sα, T) holds, then (3.3) implies that for anyz∈ F(G) we have

(α)^z·δT(0)=a^(1−j)z·

1 2xωLθ_S^T

= (α⁰_T(a)^(1−j))^z·ωL, whereα⁰_T(a)∈E^T_S

α andαT :=α⁰_T(a)^(1−j)is an anti-unit. Henceα^z·δT(0)= u_T ·α^z·ω_T ^L, whereu_T is a unit and an anti-unit, thus a root of unity. But uT is also congruent 1 modulo MT and thereforeuT = 1.