ContemporaryMathematics Volume00,XXXX Rubin’s Integral Refinement of the Abelian Stark Conjecture Cristian D. Popescu ToHaroldStark,onhis65thbirthday. Abstract. This paper is a survey of results obtained by the present author towards proving Rubin’s integral version of Stark’s conjecture for abelian L– functions of arbitrary order of vanishing at the origin. Rubin’s conjecture is stated and its links to the classical integral Stark conjecture for L–functions oforderofvanishing1arediscussed. AweakerversionofRubin’sconjecture formulatedbytheauthorin[P4]isalsostatedanditslinkstoRubin’sconjec- turearediscussed. EvidenceinsupportofthevalidityofRubin’sconjectureis provided. AseriesofapplicationsofRubin’sconjecturetothetheoryofEuler Systems,groupsofspecialunitsandGras-typeconjecturesaregiven. Introduction In the 1970s and early 1980s, Stark [St] developed a remarkable Galois–equi- variant conjectural link between the values at s = 0 of the first non–vanishing derivatives of the Artin L–functions L (ρ,s) associated to a Galois extension K/k K/k of number fields and a certain Q[Gal(K/k)]–module invariant associated to thegroupofglobalunitsofK. Stark’sMainConjectureshouldbeviewedasavast Galois–equivariant generalization of the unrefined, rational version of Dirichlet’s class–number formula 1 lim ζ (s)∈Q×·R , s→0sr k k in which the zeta function ζ is replaced by a Galois–equivariant L–function k (cid:88) Θ (s)= L (ρ,s)·e , K/k,S K/k,S ρˇ ρ∈G(cid:98) with values in the center of the group–ring Z(C[Gal(K/k)]), the regulator R is k replacedbyaGalois–equivariantregulatorwithvaluesinZ(C[Gal(K/k)]), andthe 1991Mathematics Subject Classification. 11R42,11R58,11R27. Key words and phrases. globalL–functions,Units,ClassGroups,EulerSystems. TheauthorwassupportedinpartbyNSFGrantsDMS–0200543andDMS–0350441. (cid:176)cXXXXAmericanMathematicalSociety 1 2 CRISTIAN D. POPESCU rank r of the group of units in k is replaced by the local rank function of the (projective) Q[Gal(K/k)]–module QU of S–units in K. S In the 1970s and early 1980s, work of Stark, Tate, and Chinburg among others revealednotonlythedepthandimportanceofStark’sMainConjecturefornumber theory (e.g. Chinburg’s theory of multiplicative Galois Module Structure emerged in this context), but also the fact that an integral refinement of this statement, in the spirit of the integral Dirichlet class–number formula 1 h lim ζ (s)=− k ·R , s→0sr k wk k would have very far reaching applications to major unsolved problems in the field. In [St IV], Stark himself formulated such an integral refinement for abelian ex- tensions K/k andtheirassociatedimprimitiveL–functionsL (χ,s)oforder of K/k,S vanishingatmost1ats=0. Roughlyspeaking,thisintegralstatementpredictsthe existence of a special S–unit ε in K, which has remarkable arithmetic properties K and, if evaluated against the Galois–equivariant regulator, it produces the value at s = 0 of the first derivative of the Galois–equivariant L–function Θ (s). K/k,S Moreover, ε is unique and computable in terms of these special L–values. K In all the major instances in which this conjecture has been proved, the Stark units ε have turned out to be truly remarkable arithmetic objects. They are K cyclotomic S–units or Gauss sums if k = Q (see [St IV] and [Ta4]), elliptic units √ if k =Q( −d), with d∈Z (see [St IV] and [Ta4]), and norms of torsion points ≥1 of sign–normalized rank 1 Drinfeld modules if k is a function field (see [H1]). In eachof these instances the construction of Stark’s units ε is closely related to the K solution of Hilbert’s 12th problem for the respective base field k (i.e. the explicit generation of the abelian class–fields of the field k.) Tate [Ta4] showed that this is not at all coincidental – it turns out that, when non–trivial, the solution to Stark’s Integral Conjecture would lead to an explicit generation of the abelian class–fields of the base field k by exponentials of special values of L–functions. Inthelate1980sandearly1990s,theemergenceofKolyvagin’stheoryofEuler Systems revealed a new interpretation of the known Stark units ε – they provide K us with the only known non–trivial examples of Euler Systems of units: the Euler Systems of cyclotomic units, Gauss sums, elliptic units, and torsion points of rank 1 Drinfeld modules, leading to enlightening solutions of major problems in number theory – the Iwasawa Main Conjecture over Q and over quadratic imaginary fields, particularcasesoftheBirch–SwinnertonDyerconjectureetc. Rubin[Ru2]showed thatthisisnotacoincidenceeither–itturnsoutthat, whennon–trivial, theStark units ε for various extensions K/k give rise to Euler Systems over a fixed field k. K Unfortunately, since in most cases the order of vanishing at s=0 of the non– primitive L–functions L (χ,s) is strictly larger than 1, Stark’s integral refine- K/k,S mentofhisMainConjecturehasanon–trivialoutputε foraverylimitedclassof K abelian extensions K/k. This is why an integral refinement of Stark’s Main Con- jecture in its full generality is needed. Such a refinement is also expected to have the type of applications described in the previous paragraph for general extensions K/k. In 1994, Rubin [Ru3] formulated an integral refinement of the Main Con- jecture for abelian extensions K/k and their associated imprimitive L–functions L (χ,s) of arbitrary order of vanishing at s=0. K/k,S Themainobjectivesofthispaperareasfollows: afterintroducingthenecessary notations and definitions (§1 below), we state Rubin’s Conjecture and a related RUBIN’S INTEGRAL REFINEMENT OF THE ABELIAN STARK CONJECTURE 3 refinement of Stark’s Main Conjecture due to the present author and discuss their linkstotheclassicalIntegralStarkConjecture(§2);weprovideevidenceinsupport ofthevalidityofRubin’sConjecture(§3);finally,wediscussaseriesofapplications of Rubin’s Conjecture to the theory of Euler Systems, the construction of groups of special units and Gras–type conjectures (§4). We do not include applications of Rubin’s Conjecture to Hilbert’s 12th problem, nor do we discuss our recent (cid:96)–adic refinementsofRubin’sstatementandtheirlinkstowellknownconjecturesofGross [Gro1–3],postponingthetreatmentofthese(otherwise)importanttopicsforother occasions. This introduction would be incomplete without mentioning the recent remark- ableworkofBurns, Flach, andtheirstudentsandcollaboratorsontheEquivariant Tamagawa Number Conjecture (ETNC), which has brought a wide variety of new andexcitingideas,techniques,andinterpretationstothesubjectofStark’sConjec- tures. Building upon earlier work of Bloch–Kato and Fontaine–Perrin-Riou, Burns and Flach [BF1–3] have formulated the ETNC for L–functions associated to mo- tives with not necessarily abelian coefficients. If restricted to Artin motives, this statement can also be viewed as an integral refinement of Stark’s Main Conjecture for general (not necessarily abelian) extensions K/k. In the more restrictive case of Dirichlet motives, the ETNC is closely related to Rubin’s Conjecture, as shown by Burns in [Bu1]. We refer the reader to Flach’s contribution to this volume and Burns’s upcoming papers for more details in this direction. We dedicate this paper to Harold Stark, on the occasion of his 65th birthday, withadmirationandgratitudeforhisoutstandingwork,whichhasalreadyinspired several generations of number theorists and will continue to do so for many more generations to come. 1. Notations and definitions 1.1. Notations. Let K/k be a finite, abelian extension of global fields of arbitrary characteristic and of Galois group G := Gal(K/k). We denote by G(cid:98) the group of irreducible complex valued characters of G. Let µ be the group of roots K of unity in K, w := card(µ ), and S and T two finite, nonempty sets of primes K K in k. For a finite extension K(cid:48)/k, S and T will denote the sets of primes in K(cid:48) K(cid:48) K(cid:48) dividing primes in S and T, respectively . For the moment, we require that the sets S and T satisfy the following set of hypotheses. Hypotheses (H ) 0 1. S contains all the primes which ramify in K/k, and all the infinite primes of k in the case where k is a number field. 2. T ∩S =∅. 3. Therearenonontrivialelementsinµ whicharecongruentto1moduloall K the primes w in T . K The reader will note that the last hypothesis above is automatically satisfied in the function field case. In the number field case, it is satisfied if, for example, T contains either at least two primes of different residual characteristic or a prime whose corresponding residue field is large compared to the size of µ . K ForafiniteextensionK(cid:48)/k,O willdenoteitsringofS –integers,U := K(cid:48),S K(cid:48) K(cid:48),S O× is the group of S –units in K(cid:48), and A the ideal–class group of O . K(cid:48),S K(cid:48) K(cid:48),S K(cid:48),S ForanysuchK(cid:48),wealsodefinethe(S,T)–modifiedgroupofunitsandrespectively 4 CRISTIAN D. POPESCU ideal class–group as follows. U :={x∈U |x≡1 mod w, ∀w ∈T } . K(cid:48),S,T K(cid:48),S K(cid:48) {fractional ideals of O coprime to T } A := K(cid:48),S K(cid:48) . K(cid:48),S,T {x·O |x≡1 mod w, ∀w ∈T } K(cid:48),S K(cid:48) For simplicity, we will set U := U , A := A , U := U , and A := S K,S S K,S S,T K,S,T S,T A . Since S and T are G–invariant, these groups are endowed with natural K,S,T K K Z[G]–module structures. 1.2. The G–equivariant L–function. ForK/k,S,andT asabove,andany χ in G(cid:98), let L (χ,s) denote, as usual, the L–function associated to χ with Euler S factorsatprimesinS removed,ofthecomplexvariables. Thisisacomplexvalued function,holomorphiceverywhereifχisnon–trivial,andholomorphicoutsides=1, with a pole of order 1 at s=1 if χ is the trivial character. With the help of these L–functions, one can define Θ , Θ :C−→C[G], S S,T (cid:88) (cid:89) Θ (s):= L (χ,s)·e , Θ := (1−σ−1·(Nv)1−s)·Θ (s), S S χ−1 S,T v S χ∈G(cid:98) v∈T where σ and Nv denote the Frobenius morphism in G and the cardinality of the v residue field associated to v, and (cid:88) e :=1/|G| χ(σ)·σ χ−1 σ∈G istheidempotentelementassociatedtoχ−1inC[G]. ThefunctionsΘ andΘ are S S,T the so–called S–modified respectively (S,T)–modified G–equivariant L–function. If the group–ring C[G] is viewed in the obvious manner as a direct product of |G| copies of C, then the projections of Θ and Θ onto the various components of S S,T C[G] with respect to this product decomposition are holomorphic away from s=1 (holomorphiceverywhere,respectively)asfunctionsofthecomplexvariables. The values of Θ at non-positive integers satisfy the following remarkable integrality S,T property. Theorem 1.2.1. Under hypotheses (H ), one has 0 Θ (1−n)∈Z[G], S,T for all integers n≥1. Inthenumberfieldcase,Theorem1.2.1wasindependentlyprovedbyDeligne–Ribet [DR], P. Cassou-Nogues [CN], and D. Barski [Bar1]. In fact, in [DR] it is proved that one has (cid:88) Ann (µ⊗n)·( L (χ,1−n)·e )⊆Z[G], for all n∈Z . Z[G] K S χ−1 ≥1 χ∈G(cid:98) However, the reader will notice right away that hypothesis (H )3 is equivalent to 0 the statement (cid:89) (1−σ−1·(Nv)n)∈Ann (µ⊗n), for all n∈Z . v Z[G] K ≥1 v∈T RUBIN’S INTEGRAL REFINEMENT OF THE ABELIAN STARK CONJECTURE 5 In fact, the following lemma, whose proof in characteristic 0 can be found in [Co], holds true in both characteristics 0 and p. Lemma 1.2.2. Assumethatthesetofdata(K/k,S )satisfieshypothesis(H )– 0 0 1. Then, for all n ∈ Z , Ann (µ⊗n) is generated as a Z[G]–module by the ≥1 Z[G] K elements (cid:89) δ (n):= (1−σ−1·(Nv)n), T v v∈T whereT runsthroughallthefinitesetsofprimesink,suchthat(K/k,S,T)satisfies hypotheses (H ). 0 In the function field case, Theorem 1.2.1 is a direct consequence of Weil’s theorem expressing the L–functions as alternating products of characteristic polynomials of the action of a geometric Frobenius morphism on various G–eigenspaces of (cid:96)–adic ´etale cohomology groups of the smooth projective curve associated to the top field K. By using this interpretation, one can show that, if q is the cardinality of the field of constants for the base field k, then there exists a polynomial P (X) in S,T Z[G][X], such that Θ (s)=P (q−s) S,T S,T (see [Ta4] and §3.1 below.) Theorem 1.2.1 follows immediately from this equality. In what follows, in order to simplify notations we will set δ :=δ (1), T T for any set T as in Lemma 1.2.2. In the context described above, Stark’s Conjecture provides a link between the leadtermintheTaylorexpansionats=0oftheG–equivariantL–functionΘ (s) S,T and certain arithmetic invariants of the abelian extension K/k. In order to make this link precise, we will need to impose additional conditions on the sets of primes S and T. Let us fix an integer r ≥ 0. We associate to r the following (extended) set of hypotheses to be satisfied by the set of data (K/k,S,T,r). Hypotheses (H ) r 1. S contains all the primes which ramify in K/k, and all the infinite primes of k in the case where k is a number field. 2. card(S)≥r+1. 3. S contains at least r distinct primes which split completely in K/k. 4. T ∩S =∅. 5. Therearenonontrivialelementsinµ whicharecongruentto1moduloall K the primes w in T . K The following Lemma shows how hypotheses (H ) control the order of vanishing r ord at s=0 of the associated G–equivariant L–functions. s=0 Lemma 1.2.3. If (K/k,S,T) satisfies hypotheses (H ), then ord L (χ,s)≥ r s=0 S r, ∀χ∈G(cid:98) and, consequently, ord Θ (s)=ord Θ (s)≥r. s=0 S s=0 S,T 6 CRISTIAN D. POPESCU Proof. Thisisadirectconsequenceofthefollowingequalityprovedin[Ta4]. (cid:189) card{v ∈S|χ| =1 }, for χ(cid:54)=1 (1) ord L (χ,s)= Gv Gv G , s=0 S card(S)−1, for χ=1 G where G is the decomposition group of v in K/k and 1 denotes the trivial char- v H acter of a group H. (cid:164) If the set of data (K/k,S,T) satisfies hypotheses (H ), we let r 1 Θ(r) (0):= lim Θ (s) S,T s→0sr S,T denote the coefficient of sr in the Taylor expansion of Θ (s) at s=0. S,T 1.3. The G–equivariant regulator maps. The link between the analytic aspects of the picture (represented by the G–equivariant L–function described in the previous section) and its arithmetic aspects (represented by the Z[G]–modules ofunitsU andidealclassesA )predictedbyStark’sConjectureisachievedvia S,T S,T certainG–equivariantregulatormaps. Thesewillbedefinedinthepresentsection. Throughout this section we assume that the data (K/k,S,T,r) satisfies hy- potheses(H ), forafixedr ∈Z . Wefixanr–tupleV :=(v ,...,v )ofr distinct r ≥0 1 r primesinS whichsplitcompletelyinK/k,andprimesw inK,withw dividingv , i i i for all i=1,...,r. Let W :=(w ,...,w ). For all primes w in K, let | · | denote 1 r w theirassociatedmetrics, canonicallynormalizedsothattheproductformulaholds. This means that for all x∈K×, we let (cid:189) (Nw)−ordw(x), if w is finite |x| := , w |σ (x)|, if w is infinite w where σ denotes the unique embedding of K into C associated to w and | · | is w the usual complex absolute value. Throughout this paper, if M is a Z[G]–module and R is a commutative ring with 1, then RM denotes the tensor product R ⊗ M endowed with the usual Z R[G]–module structure, M(cid:102)denotes the image of M via the canonical morphism M −→QM, and M∗ :=Hom (M,Z[G]) is the dual of M in the category of Z[G]–modules. Z[G] Definition 1.3.1. The G–equivariant regulator map associated to W is the unique Q[G]–linear morphism r R :Q ∧ U −→C[G], W S,T Z[G] such that, for all u ,...,u in U , we have 1 r S,T (cid:88) R (u ∧···∧u ):=det(− log|uσ−1| ·σ), W 1 r i wj σ∈G where the determinant is taken over C[G], and i,j =1,...,r. RUBIN’S INTEGRAL REFINEMENT OF THE ABELIAN STARK CONJECTURE 7 Remark1. IfextendedbyC–linearitytheregulatorR definedaboveinduces W a C[G]–isomorphism (see [Ru3] for a proof) r ∼ R :C ∧ U −→C[G], W S,T Z[G] also denoted by R in what follows. By definition, the special value Θ(r) (0) W S,T belongs to the C[G]–submodule of C[G], consisting of all elements x∈C[G] which satisfy e ·x=0, χ for all χ in G(cid:98), such that ord L (χ,s) > r. Since R is a C[G]–isomorphism, s=0 S W this implies that there exists a unique element r ε ∈C ∧ U , S,T S,T Z[G] such that the following equalities hold. 1. R (ε )=Θ(r) (0). W S,T S,T 2. e ·ε =0 in C ∧r U , for all χ in G(cid:98), such that ord L (χ,s)>r. χ S,T S,T s=0 S Z[G] The above remark prompts us to give the following definition. Definition1.3.2. Assumethatthesetofdata(K/k,S,T,r)satisfieshypothe- ses (H ). Let M be a Z[G]–module. We define r M(cid:102) :={x∈M(cid:102)|e ·x=0 in CM, for all χ∈G(cid:98) such that ord L (χ,s)>r}. r,S χ s=0 S 2. The Conjectures In this section, we state Stark’s Conjecture (“over Q”) in the abelian situation outlined in §1, as well as its integral refinements (“over Z”) due to Rubin and the present author. 2.1. Stark’s Conjecture “over Q”. CONJECTURE A(K/k,S,T,r) (Stark). If the set of data (K/k,S,T,r) satisfies hypotheses (H ), then there exists a unique element r r ε ∈Q ∧ U , S,T S,T Z[G] such that the following equalities hold. 1. R (ε )=Θ(r) (0). W S,T S,T 2. e ·ε =0 in C ∧r U , for all χ in G(cid:98), such that ord L (χ,s)>r. χ S,T S,T s=0 S Z[G] A few remarks concerning this statement are in order. Remark1. Theuniquenesspartofthestatementaboveisadirectconsequence of Remark 1, §1.3. The (highly non–trivial) conjectural part is the statement that r the unique element ε ∈ (C ∧ U ) satisfying the regulator condition (1) S,T S,T r,S Z[G] and the vanishing condition (2) in Remark 1, §1.3, belongs in fact to the Q[G]– r submodule (Q ∧ U ) . S,T r,S Z[G] 8 CRISTIAN D. POPESCU Remark 2. No doubt, the reader has noticed that the notation used for ε S,T does not capture its dependence of our choice of r–tuples V := (v ,...,v ) and 1 r W = (w ,...,w ) of primes in S and S , as in §1.3. It is indeed true that ε 1 r K S,T depends on these choices. However, it turns out that this dependence is simple and, most importantly, if Conjecture A is true for one choice of V and W, then it is true for any other choice. To show this, let us first assume that V(cid:48) and W(cid:48) differ from V and W by a permutation τ ∈ Sym and respectively an r–tuple r σ := (σ ,...,σ ) ∈ Gr, i.e. V(cid:48) = (v ,...,v ) and W(cid:48) = (vσ1 ,...,vσr ). 1 r τ(1) τ(r) τ(1) τ(r) Then one can define a unique Z[G]–linear isomorphism r ∼ r Φ : ∧ U −→ ∧ U , τ,σ S,T S,T Z[G] Z[G] σ−1 σ−1 bysettingΦ (u ∧···∧u ):=u 1 ∧···∧u r ,forallu ,...,u inU . LetΦ τ,σ 1 r τ(1) τ(r) 1 r S,T τ,σ r denote the unique C-linear extension of the above map to C ∧ U . If ε and S,T S,T Z[G] r ε(cid:48) aretheuniqueelementsinC ∧ U satisfying(1)and(2)inRemark1,§1.3, S,T S,T Z[G] for R and R respectively, then one can easily show that ε(cid:48) = Φ (ε ). W W(cid:48) S,T τ,σ S,T r r This shows that if ε ∈ Q ∧ U , then ε(cid:48) ∈ Q ∧ U and vice versa. S,T S,T S,T S,T Z[G] Z[G] Secondly,letusassumethatS containsmorethanr primeswhichsplitcompletely, i.e. thereareatleasttwochoicesV andV(cid:48) whichdonotdifferfromoneanothervia a permutation τ ∈Sym . If card(S)>r+1, then Lemma 1.2.3 above implies that r Θ(r) (0)=0andthereforeConjectureAistriviallytrueforε =0,foranychoice S,T S,T of V and W. Now, let us assume that S ={v ,...,v }, and v splits completely 1 r+1 i for all i=1,...,r+1. Then Lemma 1.2.3 combined with the (S,T)–class–number formula of [Ru3] implies that Θ(r) (0)=−card(A )·R ·e , S,T k,S,T k,S,T 1G where 1 denotes the trivial character of G and R is the Dirichlet regulator G k,S,T associated to the group of (S,T)–units U . The equalities above imply right k,S,T away that in this case Conjecture A is true with card(A ) ε :=± k,S,T u ∧···∧u , S,T |G|r 1 r where u ,...,u is a Z–basis for U and the sign is uniquely determined by 1 r k,S,T the choice of V and W. Based on these considerations and in order to simplify notations, wehavedroppedthedependenceofε onthechoiceofV andW from S,T the notation in the statement of Conjecture A and throughout this paper. Remark 3. The reader familiar with [St] will notice that Stark’s original for- mulationofhisconjecture“overQ”isquitedifferentfromours. Themaindifference stemsfromthefactthatwhilethestatementaboveisgiveninaGalois–equivariant form, Stark’s original conjecture was formulated in a character–by–character man- ner(i.e. foroneL–functionatatime). Yetanotherdifferenceismarkedbythefact that Stark deals with all the characters χ∈G(cid:98), while Conjecture A only deals with those characters χ whose associated L–functions have minimal order of vanishing r at s=0. The statement presented here is essentially due to Tate [Ta4] and Ru- bin [Ru3] and, under the present hypotheses (H ), is equivalent to Stark’s original r RUBIN’S INTEGRAL REFINEMENT OF THE ABELIAN STARK CONJECTURE 9 conjecture for L–functions of minimal order of vanishing r at s=0 (see [Ru3] for a proof). 2.2. Rubin’s integral refinement of Conjecture A. In this section, we state Rubin’s integral refinement of Conjecture A. The main idea behind any inte- gral refinement of Conjecture A is to construct an arithmetically meaningful Z[G]– submodule (i.e. a G–equivariant lattice of rank which is not necessarily maximal) r of the Q[G]–module Q ∧ U , which contains the element ε . We will first de- S,T S,T Z[G] scribeRubin’sconstructionofsuchalattice. Wearestillworkingunderhypotheses (H ) for the set of data (K/k,S,T,r). For any (r−1)–tuple r Φ:=(φ ,...,φ )∈(U∗ )r−1, 1 r−1 S,T there exists a unique Q[G]–linear morphism Φ(cid:101) :Q ∧r U −→QU , S,T S,T Z[G] such that, for all u ,...,u in U , we have 1 r S,T (cid:88)r Φ(cid:101)(u ∧···∧u )= (−1)kdet(φ (u )) ·u . 1 r i j j(cid:54)=k j k=1 In the last equality, the determinant in the k–th term of the sum is taken with respecttoalli=1,...,r−1andallj =1,...,r, suchthatj (cid:54)=k. Pleasenotethat sinceU hasnoZ–torsion(seehypothesis(H )5),U canbenaturallyviewedas S,T r S,T a Z[G]–submodule of QU . Consequently, the k–term sum above can be viewed S,T without any ambiguity inside QU . S,T r Definition2.2.1. Rubin’slatticeΛ consistsofallelements(cid:178)inQ ∧ U , S,T S,T Z[G] which satisfy the following properties. 1. Φ˜((cid:178))∈U , for all Φ:=(φ ,...,φ )∈(U∗ )r−1. S,T 1 r−1 S,T 2. e ·(cid:178)=0 in C[G], for all χ∈G(cid:98), such that ord L (χ,s)>r. χ s=0 S Remark 1. It is immediate from Definition 2.2.1 that for r =0,1, we have (cid:40) (U(cid:93)) , if r =1 Λ = S,T 1,S S,T Z[G] , if r =0. 0,S For a general r ≥1, we have inclusions r(cid:94) |G|n·Λ ⊆( ∧ U ) ⊆Λ , S,T S,T r,S S,T Z[G] for sufficiently large positive integers n. It is not difficult to show that if U has S,T finiteprojectivedimensionoverZ[G],thenthesecondinclusionaboveisanequality. However, as Rubin shows in [Ru3], the second inclusion above is in general strict. After tensoring with Z[1/|G|], we have an equality r(cid:94) Z[1/|G|]Λ =(Z[1/|G|] ∧ U ) , S,T S,T r,S Z[G] 10 CRISTIAN D. POPESCU which follows directly from the sequence of inclusions above. Also, since U sits S,T inside U with a finite index, we have equalities S r r QΛ =(Q ∧ U ) =(Q ∧ U ) . S,T S,T r,S S r,S Z[G] Z[G] CONJECTURE B(K/k,S,T,r) (Rubin). If the set of data (K/k,S,T,r) satisfies hypotheses (H ), then there exists a unique element ε ∈Λ , such that r S,T S,T R (ε )=Θ(r) (0). W S,T S,T Remark 2. In the case r = 0 Rubin’s conjecture states that Θ (0) ∈ S,T Z[G] . This statement is true and follows immediately if one sets n=1 in Theo- 0,S rem 1.2.1 and applies equalities (1) above. Remark 3. In the case r = 1, Conjecture B is equivalent to Stark’s integral refinementofConjectureAforL–functionsoforderofvanishing1ats=0, formu- lated in [St IV]. A proof of this equivalence can be found in [Ta4] and [Ru3] and it is a direct consequence of Remark 1 for r =1 and Lemma 2.2.3 below. At times, it is convenient to drop dependence on the auxiliary set T in conjec- ture B formulated above. This is why we will sometimes work with the following statement instead. CONJECTUREB(K/k,S,r). Ifthesetofdata(K/k,S,r)satisfieshypothe- ses (H )1–3, then for all sets T such that the set of data (K/k,S,T,r) satisfies r hypotheses (H ), there exists a unique element ε ∈Λ , such that r S,T S,T R (ε )=Θ(r) (0). W S,T S,T The following result proved in [P4] (see Proposition 5.3.1) shows that in the above conjecture it is sufficient to work with minimal sets T. Theorem 2.2.2. Let (K/k,S,T,r) and (K/k,S,T(cid:48),r) be two sets of data sat- isfying hypotheses (H ), such that T ⊆T(cid:48). Then r B(K/k,S,T,r)=⇒B(K/k,S,T(cid:48),r). In particular, this theorem shows that, in the case where char(k) = p > 0, it suffices to prove conjecture B(K/k,S,T,r) for sets T of cardinality 1. Remark 4. In what follows, we will give a somewhat detailed description of theconnectionbetweenRubin’sConjectureandtheclassicalconjecturesofBrumer and Brumer–Stark. In the 1970s, Brumer stated the following conjecture, as a natural extension of the classical Theorem of Stickelberger for abelian extensions of Q (see [Co]) to abelian extensions of general number fields. The conjecture was later extended to global fields of arbitrary characteristic by Mazur and Tate (see [Ta4]).