GENERALIZED STARK FORMULAE OVER FUNCTION FIELDS KI-SENG TAN 7 0 Abstract. We establishformulaeofStark type forthe Stickelbergerelements in 0 the functionfieldsetting. OurresultgeneralizesaworkofHayesandaconjecture 2 of Gross. It is used to deduce a p-adic version of Rubin-Stark Conjecture and n Burns Conjecture. a J 1. Introduction 2 ] In this paper, we study Stickelberger elements related to abelian extensions over T globalfunctionfields. OurmainresultisTheorem5.1,whichgeneralizesatheoremof N Hayes([Hay88]). InawaythetheoremputstogetherconjecturesofGross,Rubinand . h Stark. And we will show that it implies a p-adic version of Rubin-Stark conjecture t a (see Theorem 1.1 below). Furthermore, using the theorem, we are able to deduce a m p-adic version of a formula conjectured by Burns (see Theorem 1.2 below). [ For the purpose of having a better description of our work, we shall review in 1 the following paragraphs both Rubin-Stark conjecture and Burns conjecture. But v before we do so, let us first fix some notations. 1 6 From now on, K/k will be a finite abelian extension over a global function field of 0 characteristic p. We assume that the extension is unramified outside a given finite 1 set S of places of k. And we fix another finite non-empty set T of places of k such 0 7 that T ∩S = ∅. The notation K′ will be used to denote a subfield of K containing 0 k. Also, S(K′) (resp. T(K′)) will denote the set of places of K′ sitting over S (resp. / h T). Let F be the constant field of k, and put Γ = Gal(K/k), Γ′ = Gal(K′/k). q t a The analytical side of Rubin-Stark conjecture involves the equivariant L-function m which interpolates L-functions at each character. Recall that for each χ ∈ Γˆ the : v modified Artin L-function over k is defined as ([Gro88]) i X L (χ,s) = (1−χ([v])·N(v)1−s) (1−χ([v])·N(v)−s)−1. (1) S,T r a v∈T v6∈S Y Y Here [v] is the Frobenius element at v and N(v) = qdeg(v) is the norm of v. Because in a global field places of the same degree always form a finite set, one can easily deducethattheaboveinfiniteproductisexpandedinauniquewayasaformalpower series in q−s. It is well-known that this formal power series is in fact a polynomial 1991 Mathematics Subject Classification. 11S40 (primary), 11R42, 11R58 (secondary). Key words and phrases. Stickelberger element, special values of L-functions, Stark Conjecture, Conjecture of Gross, class numbers, local Leopoldt conjecture, Rubin’s conjecture, conjecture of Rubin and Burns, regulators. TheauthorwassupportedinpartbytheNationalScienceCouncilofTaiwan(R.O.C.),NSC91- 2115-M-002-001,NSC93-2115-M-002-007. 1 2 KI-SENGTAN in q−s ([Tat84]). Applying the theory of Fourier transforms, we see that there is a polynomial Θ (s) ∈ C[Γ][q−s] such that for every χ ∈ Γˆ Γ,S,T χ(Θ (s)) = L (χ,s). (2) Γ,S,T S,T This Θ (s) is called the modified equivariant L-function. From (1), we are able Γ,S,T to express Θ as an infinite product. Namely, Γ,S,T Θ (s) = (1−[v]·qdeg(v)(1−s)) (1−[v]·q−deg(v)s)−1, (3) Γ,S,T v∈T v6∈S Y Y In particular, this implies that Θ is actually an element in Z[Γ][q−s]. Γ,S,T Now let us start to describe the arithmetic side of Rubin-Stark conjecture. This will involve various regulator maps related to units groups. Consider OS(K′), the ring of S(K′)-integers of K′, and let O∗ be its units group. S(K′) Definition 1.1. Define U(K′) to be the kernel of the reduction modulo T(K′) O∗ −→ O∗/(1+π ·O ). S(K′) w w w w∈T(K′) M And define rK′ = #S(K′)−1. For simplicity, we denote U = U(K), r = rK. Note that U(K′) = UGal(K/K′) is a free abelian group of rank rK′ and there is an exact sequence ([Gro88]) 1 −→ U(K′) −→ OS∗(K′) −→ F∗w −→ Pic(OS(K′))T(K′) −→ Pic(OS(K′)) −→ 1. w∈T(K′) Y We recall that the (S(K′),T(K′))-class number of K′ is the group order hK′,S(K′),T(K′) = |Pic(OS(K′))T(K′)|. (4) To construct the regulator maps, we shall follow the notations and the methods used in [Rub96]. In particular, ifM isa finite Z[Γ]-modulethen QM denotes Q⊗M, and the dual module M∗ is defined as Hom (M,Z[Γ]) ⊂ Hom (QM,Q[Γ]). Also, Γ Γ if n is a non-negative integer, then ΛnM denotes the nth exterior power of M in the category of Z[Γ]-modules. We let ι denote the natural map ([Rub96], Sec.1.2) ι : Λn(M∗) −→ (ΛnM)∗, such that if φ ,...φ ∈ M∗ and m ,...,m ∈ M, then 1 n 1 n ι(φ ∧···∧φ )(m ∧···∧m ) = det(φ (m )). (5) 1 n 1 n i j And following [Rub96], we define ΛnM := {m ∈ QΛnM | ι(φ ∧···∧φ )(m) ∈ Z[Γ] for every φ ,...,φ ∈ M∗}. 0 1 n 1 n Now, we start to define the regulator maps. First, let Y(K′) = Z · w, w∈S(K′) and L X(K′) = { a ·w ∈ Y(K′)| a = 0}. w w w∈S(K′) w∈S(K′) X X For each place w of K, let deg be the local degree map w deg : K∗ −→ K∗/O∗ −→ Z, w w w w GENERALIZED STARK FORMULAE OVER FUNCTION FIELDS 3 such that if | | is the normalized absolute value associated to w, then w log(|x| ) = −deg (x)·log(q). w w Compose this local degree map with the natural embedding U −→ K∗ to form w λ : U −→ K∗ −de→gw Z. w w And we define λ : U(K) −→ X(K) to be the Γ-equivariant homomorphism such that λ(u) = λ (u)·w for every u ∈ U(K). Write λ(n) : ΛnU −→ ΛnX(K) w∈S(K) w (resp. i(n) : ΛPnX(K) −i(→n) ΛnY(K)) for the map induced by the map λ (resp. the inclusion i : X(K) −→ Y(K)). Then the regulator map R : QΛnU −→ Q[Γ] (6) Ψ associated to a Ψ ∈ ΛnY(K)∗ is defined as the one that linearly extends the map ΛnU −λ(→n) ΛnX(K) −i(→n) ΛnY(K) −ι(→Ψ) Z[Γ]. It is easy to see that if ψ ∈ Y(K)∗, then R ∈ U∗. Also, if Ψ = ψ ∧···∧ψ , i ψi 1 n then we have R = ι(R ∧···∧R ). (7) Ψ ψ1 ψn Consequently, we have R (ΛnU) ⊂ Z[Γ] for every Ψ ∈ ΛnY(K)∗. In this paper, Ψ 0 some special elements in Y(K)∗ of the form w∗ will be used. Recall that for each place w over K, the element w∗ ∈ Y(K)∗ is defined ([Rub96]) such that for every place w′ w∗(w′) = γ. γw=w′ X What we called Rubin-Stark Conjecture is the one proposed by Rubin in [Rub96] (Conjecture B′), because it can be viewed as an integral version of Stark’s conjec- ture ([San87, Stk71, Stk75, Stk76, Stk80]). In a way, the conjecture relates some derivative of the equivariant L-function to certain exterior product of units arising form regular representations of Γ. It is easy to see that the units group U contains a regular representation of Γ if and only if some place in S splits completely over K. Thus, for the purpose of having an interesting theory, we need to assume the following: Assumption 1.1. From now on, we assume that there exist n, n ≥ 1, different places S = {v ,...,v } ( S such that every place in S splits completely over K, 0 1 n 0 Definition 1.2. Let v ,...,v be as in Assumption 1.1 and let w ,...,w be a fixed 1 n 1 n set of places of K such that each w is sitting over the place v ∈ S. And let i i η = w∗ ∧···∧w∗. 1 n First we note that Assumption 1.1 (together with the class number formula at s = 0) implies Θ = a (q−s −1)n +··· ∈ (q−s −1)·Z[Γ][q−s]. Γ,S,T n 4 KI-SENGTAN And the coefficient a ∈ Z[Γ] will be denoted as Θ(n) (0). For each χ ∈ Γˆ let e n Γ,S,T χ be the associated idempotent element in the group ring C[Γ], and let r denote the χ C-dimension of the χ-eigenspace of C⊗Z U. Define Λn = {u ∈ ΛnU|e (u) = 0, for every χ ∈ Γˆ such that r > n}. S,T 0 χ χ Then in our settings Rubin-Stark conjecture reads as following. Conjecture 1.1. (Rubin,[Rub96], Conjecture B′) There exists an ǫ ∈ Λn such S,T that (n) R (ǫ) = Θ (0). (8) η Γ,S,T We will show in Section 5.1 that our main result implies the following p-adic ver- sionoftheconjecture. Here”p-adic”meanstensoring thingswithZ . Inparticular, (p) Z Λ = Z ⊗Λ ⊂ QΛ . (p) S,T (p) S,T S,T Theorem 1.1. There exists an ǫ ∈ Z Λn such that in Z [Γ] (p) S,T (p) (n) R (ǫ) = Θ (0). (9) η Γ,S,T A different proof of the theorem can be found in [Pop05], and a proof for the l-adic (l 6= p) version of Rubin-Stark conjecture is given in [Bun04]. In view of this, over function fields, Rubin-Stark conjecture actually holds. Now we review the conjecture of Burns. We will follow the construction described in [Bun02, Haw04]. The conjecture involves regulators of another type, and we are going to define them in the follow paragraphs. First we note that if M is a Z[Γ]-module, then for each φ ∈ M∗ there is a unique φ(id) ∈ HomZ(M,Z) such that for x ∈ M φ(x) = φ(id)(γ−1x)γ. (10) γ∈Γ X For a place v over k, define λ¯ : U(k) −→ k∗ −→ Γ ֒→ Γ, (11) v,Γ v v where the first and the last arrows are natural embeddings and the second is the norm residue map in the local class field theory. Let u ,...,u be a Z-basis for U(k), 1 rk v ,...,v be distinct places in S \ S and φ ,...,φ ∈ U∗. Consider the matrix n+1 rk 0 1 n A = (a ) with ij 1≤i,j≤rk (id) φ (u ),if 1 ≤ i ≤ n a = i j ij ¯ (λvi,Γ(uj)−1,if n+1 ≤ i ≤ rk. For each pair i,j the entry a is an element in Z[Γ]. And it is obvious that the ij determinant det(A) is in Irk−n where I is the augmentation ideal of Z[Γ]. Up to ±1, the residue class of det(A) modulo Irk−n+1 depends on neither the ordering of v ,...,v nor the choice of the basis u ,...,u . We assume that the ordering 1 rk 1 rk of v ,...,v is fixed and the basis u ,...,u are ordered in a way such that the 1 rk 1 rk classical regulator formed by them is positive. On the other hand, the residue class of det(A) actually depends on the exterior product Φ = ι(φ ∧···∧φ ) ∈ ι(ΛnU∗), 1 n GENERALIZED STARK FORMULAE OVER FUNCTION FIELDS 5 and therefore we will denote it as RegΦ. On top of Conjecture 1.1, Burns [Bun02] Γ proposes the following strengthened conjecture. Conjecture 1.2. Assume that Conjecture 1.1 holds, so that for every Φ ∈ ι(ΛnU∗), we have Φ(ǫ) ∈ Z[Γ]. Then this element satisfies Φ(ǫ) ≡ h RegΦ (mod Irk−n+1). k,S,T Γ For more material related to this conjecture, see for instance [Bun02, Bun04, Haw04, Pop99a, Pop99b, Pop02, Rub96]. The l-adic version (for l 6= p) of the conjecture is proved in [Bun04], but it seems the technique used in the proof can not be applied to cover the following p-adic version, which will be proved in Section 5.2. Let I be the augmentation ideal of Z [Γ]. p p Theorem 1.2. Let notations be as those in Theorem 1.1. Then for every Φ ∈ ι(ΛnU∗), we have Φ(ǫ) ∈ Z [Γ] and this element satisfies p Φ(ǫ) ≡ h RegΦ (mod Irk−n+1). (12) k,S,T Γ p Now we begin to describe Theorem 5.1, our main result. In short, it is a p-adic refinement of Theorem 1.1. The method for making this kind of refinement comes from [Gro88, Gro90], and the main idea is to replace Z by certain Galois groups in p order to construct refinements of both side of the equality (9). To explain it, let us start with those degree maps deg which play important roles in the construction w of the regulator maps. These local degree maps together form the global degree map deg : A∗K −→ Z defined on the ideles group A∗K. Let L0 = KFqp∞ be the constant Z -extension over K. If we view Z as the Galois group Gal(L /K) and p p 0 compose the map deg with the embedding Z −→ Z which sends 1 to the Frobenius p in Gal(L /K), then we get the norm residue map A∗ −→ Gal(L /K), and the local 0 K 0 degree map is just the composite K∗ −→ A∗ −→ Gal(L /K). From this we see w K 0 that the field extension L /K and the related norm residue maps are implicitly used 0 in the construction of the previous regulator maps. For the refinements we are going to use various Galois groups of the form H := Gal(L/K) where L/K is a pro-p abelian extension such that L/k is also abelian and unramified outside S (such extension is called admissible, see Definition 2.1). We let H play the role of Z = Gal(L /K) and use the related norm residue maps p 0 to construct, for each Ψ, the associated refined regulator map R (Definition Ψ,H 4.2) which has values in the nth relative augmentation quotient associated to H (Definition 2.3). To see that R actually refines R , we only need to take L = L , Ψ,H Ψ 0 because in this situation R can be recovered from R (Lemma 4.2). We would Ψ Ψ,H like to emphasize that H, the Galois group of the maximal admissible extension, is a direct product of countable infinite many copies of Z (Lemma 2.1). And, p in a way, Lemma 2.5 together with the isomorphism (15) says that an element in the nth relative augmentation quotient associated to H can be identified as a Z [Γ]-coefficient n’th degree homogeneous ”polynomial in countable infinite many p variables”. Furthermore, under this identification, if the Z -basis of H is suitably p (n) arranged, then for each ǫ ∈ Z Λ the value R (ǫ) is just the coefficient of certain (p) S,T Ψ monomial in RΨ,H(ǫ). In particular, it is fair to say that the map Rη,H, where η 6 KI-SENGTAN is the one in Definition 1.2, carries a rich amount of information about the units group. In fact, the universal property studied in Section 4.4 (see Corollary 4.2) tells us that most of the important information about the integer structure of the n’th exterior product of the units group can be obtained from Rη,H. The refinement of the equivariant L-function Θ turns out to be the Stickel- Γ,S,T berger element θ (see Definition 3.1) where G = Gal(L/k). For its reason, please G see Lemma 3.1. It is somewhat a surprise since the Stickelberger element only inter- polates special values of L-functions while the equivariant L-function interpolates the complete L-functions. Lemma 3.1 also tells us that in the case where L = L , 0 (n) the ”nth derivative” Θ (0) can be recovered from the residue class [θ ] of Γ,S,T G (n,H) (n) θ in the nth relative augmentation quotient. That there is a unique ǫ in Z Λ G (p) S,T such that [θ ] = R (ǫ) G (n,H) η,H for every admissible H is exactly the content of Theorem 5.1. In the case where H = H we have an equality between two ”polynomials in infinite variables” while (9) in Theorem 1.1 is an equality between the corresponding coefficients of certain ”monomial”. And this is the reason why Theorem 5.1 implies Theorem 1.1. The proof of Theorem 1.2 involves a refined class number formula proposed by Gross[Gro88](see Conjecture 5.1). What we actuallyuse isits p-adicversion proved in [Tan95] (see Theorem 5.2) in which the congruence (38) relates the Stickelberger element θ with the product of h and a regulator det defined by Gross. In G k,S,T G contrast to this, Theorem 5.1 relates θ with the refined regulator R (ǫ) which is G η,H the left-hand side of the congruence (12). The main step for proving Theorem 1.2 is to use the aforementioned universal property to relate the right-hand side of (12) to the product h det . k,S,T G Finally, let us have some words about the proof of Theorem 5.1. In brief, it is based on two observations. First, we find that, via Fourier transform, the theorem is equivalent to its twisted version, Theorem 5.3 in which the main part is the congruence (43). And we have discovered that both the left-hand and right-hand sides of (43) can be found as factors of the corresponding left-hand and right-hand sides of the congruence (38) in Theorem 5.2. Furthermore, the two sides of (38) are indeed products of these kind of factors (indexed by characters, see Proposition 6.1 and Proposition 7.1). To use (38) to prove (43), we apply Fourier transforms, the universal property and the result of Hayes for the case n = 1. This manuscript has evolved through several versions, since 1996. It is a great pleasuretothankDavidBurns, Wen-ChengChi, BenedictGross,Po-YiHuang,King F. Lai, Cristian Popescu, Karl Rubin and John Tate for stimulating discussions. 2. Admissible extensions and Augmentation Quotients Inthischapter,westudyadmissibleextensionsandthepropertiesoftheassociated augmentation quotients. 2.1. Admissible extensions. GENERALIZED STARK FORMULAE OVER FUNCTION FIELDS 7 Definition 2.1. An abelian extension L/K and its Galois group H = Gal(L/K) are admissible if the followings are satisfied: (1) The extension L/k is abelian and is unramified outside S. (2) The extension L/K is a pro-p extension. Throughout this paper, we will fix an admissible extension L/K, and we will also fix the notations: G = Gal(L/k), H = Gal(L/K), Γ = G/H. Also, a subgroup of G denoted as H′ always contains H, and we always denote K′ = LH′ and Γ′ = G/H′. 2.2. The maximal admissible extension. Although there are infinitely many different admissible extensions, the theory in this paper can be summed up to a theory for a single extension, that is, the maximal admissible extension with respect to K/k and S. We will denote the associated Galois group by H and will first study its structure. Lemma 2.1. The maximaladmissibleGalois group H is a directproduct of countable infinite many copies of Z . p Before we prove the lemma, let us recall some known results related to the local Leopoldt conjecture (see [Kis93, Tan95]). Lemma 2.2. Suppose that K is a global function field of characteristic p and v is a place over K. If an element u ∈ K∗ is divisible by p in K∗, then it is divisible by p v in K∗. As a consequence, we have the following . Lemma 2.3. ([Kis93]) Suppose that K is a global function field of characteristic p and S is a finite set of places of K. Then the Galois group of the maximal pro-p abelian extension over K unramified outside S is a direct product of countable infinite many copies of Z . p Proof. (of Lemma 2.1) Let Γ = Γ ⊕Γ be the natural decomposition of Γ into the p-part, Γ , and the p 0 p non-p-part, Γ . Suppose that G is the Galois group over k of the maximal pro- 0 p abelian extension unramified outside S. Then G is an extension of Γ which is p viewed as a quotient group of Γ. Let H = ker(G −→ Γ ) be the kernel of the natural p quotient map. Then H is isomorphic to H. By Lemma 2.3, G is a direct product of countable infinite many copies of Z , and p so is H. (cid:3) 2.3. Group rings and augmentation ideals. For the rest of this chapter, we will study group rings with various coefficient rings together with two types of augmen- tation ideals and the associated augmentation quotients. Let R be an integral domain finite over Z or Z . If C is the fraction field of R p and M is an R-module, then we use CM to denote C ⊗ M. R Definition 2.2. For a pro-finite group H, let R[H] be the projective limit of R[∆], where ∆ runs through all the finite quotient groups of H. Also, for every positive integer n, let I (H)n be the projective limit of I (∆)n, where I (∆)n is the nth power R R R 8 KI-SENGTAN of the augmentation ideal I (∆). We call respectively I (H)n and I (H)n/I (H)n+1 R R R R the nth augmentation ideal and the nth augmentation quotient of R[H]. For the rest of the paper, if Ξ : H −→ H is a group homomorphism, then we 1 2 will also use Ξ to denote the induced homomorphisms on the group rings and the augmentation quotients. Definition 2.3. If H′ is finite, let IR,H′ be the kernel of the ring homomorphism R[G] −→ R[Γ′] induced from the natural quotient map G −→ Γ′. In general, for every positive integer n, let In be the projective limit of In , where N runs R,H′ R,H′/N through the family of all open subgroups of H′ contained in H. We call respectively In and In /In+1 the nth relative H′-augmentation ideal and the nth relative R,H′ R,H′ R,H′ H′-augmentation quotient of R[G] For simplicity, we let I(H′), Ip(H′), IH′ and Ip,H′ denote respectively IZ(H′), IZp(H′), IZ,H′ and IZp,H′. Definition 2.4. For ξ ∈ I (H′)n ⊂ R[H′], let [ξ] be its residue class in the R (n) augmentation quotient IR(H′)n/IR(H′)n+1. Also, for ξ ∈ IRn,H′ ⊂ R[G], let [ξ](n,H′) be its residue class in In /In+1. R,H′ R,H′ Definition 2.5. Let F be a fixed number field containing all the values of characters of Γ and let O = O be its ring of integers. Let O be the completion of O at a fixed F p place sitting over p and let F be its field of fractions. Define the group rings over p F, F and the corresponding augmentation ideals as follow. Let F[H′] = FO[H′] p and F [H′] = F O [H′]. Also, for every positive integer n, let I (H′)n = FI (H′)n, p p p F O I (H′)n = F I (H′)n, In = FIn , and In = F In . Fp p Op F,H′ O,H′ Fp,H′ p Op,H′ In many situations, the structure of the augmentation quotients can be explicitly expressed. First of all, we have the following isomorphism ([Gro88]) δH′ : H′ −→ I(H′)/I(H′)2 (13) which sends h ∈ H′ to h − 1 (mod I(H′)2). Also, if H ≃ Zd for some d and R p is either Z or O , then the graded ring formed by augmentation quotients can p p be identified with a polynomial ring. To see this, let R[[s ,...,s ]] be the ring of 1 d formal power series in d variables. If E = {σ ,...,σ } is a basis of H over Z and 1 d p x = σ −1 ∈ R[H], i = 1,...,d, then the map R[H] −→ R[[s ,...,s ]], x 7→ s is an i i 1 d i i isomorphism. Consequently, for every positive integer n, I (H)n = (x ,...,x )n ≃ (s ,...,s )n, (14) R 1 d 1 d and the augmentation quotient I (H)n/I (H)n+1 is isomorphic to the R-module of R R nth degree homogeneous polynomials in s ,...,s . This induces an isomorphism 1 d ∞ d : I (H)n/I (H)n+1−→R[s ,...,s ]. (15) E,R R R 1 d n=0 M Since F [H] = F ⊗ O [H], tensoring with F , We get the induced ring homomor- p p Op p p phism ∞ d : I (H)n/I (H)n+1−→F [s ,...,s ]. (16) E,Fp Fp Fp p 1 d n=0 M GENERALIZED STARK FORMULAE OVER FUNCTION FIELDS 9 Lemma 2.4. (a) For each non-negative integer n we have I (H)n ∩ Z[H] = p I(H)n, I (H)n ∩O[H] = I (H)n and I (H)n ∩F[H] = I (H)n. Op O Fp F (b) Suppose that either H is finite free over Z or H = H. For i = 1,2, let A p i be one of the rings Z, Z , O, O , F and F . If A ⊂ A , then for each n, p p p 1 2 I (H)n ∩A [H] = I (H)n. A2 1 A1 (c) If H is finite free over Z or H = H, then the natural map p i : I(H)n/I(H)n −→ I (H)n/I (H)n+1 p p is an isomorphism. Proof. In Part (a), the third equality is from the second. If H is finite, then the proof of the first equality can be found in [Tan95], Lemma 2.5. The second equality can be proved in a similar way. If H = H, we prove them by taking projective limits. To prove Part (b), we first assume that H is finite free over Z . If we are in the p special case where {A ,A } ⊂ {Z ,O ,F }, then Part (b) is proved by using (14). 1 2 p p p In general, put A′i = (Ai)p, for i = 1,2. Then we have IA′(H)n ∩A′1[H] = IA′(H)n. 2 1 Also, part (a) implies that IA′i(H)n ∩Ai[H] = IAi(H)n. These imply Part (b). The case H = H is proved by taking projective limits. To prove Part (c), we note that by Part (b), the map i is injective. First assume that H is finite free over Z . Then by Equation (14), the homomorphism φ sending p h to the residue class of h − 1 is an isomorphism from H to I (H)/I (H)2. It is p p obvious that (see (13)) φ = i◦δ . Since δ is an isomorphism, so is i. This proves H H the lemma for n = 1. For n > 1, we observe that the multiplication map I /I2 ×In−1/In −→ In/In+1 p p p p p P maybe not surjective but its image generates the whole group (as an abelian group). Then the surjectivity of i is proved by induction. Again, the H = H case can be proved by taking projective limits. (cid:3) Lemma 2.5. For g ∈ G, let γ ∈ Γ be its residue class modulo H. Then for a g nonnegative n, the homomorphism £ : R[Γ]⊗ I (H)n/I (H)n+1−→In /In+1, n R R R R,H R,H which sends the residue class of γ ⊗(h −1)·····(h −1) to that of g 1 n g(h −1)·····(h −1), h ,...,h ∈ H, is an isomorphism. 1 n 1 n Proof. That the homomorphism £ is well defined is due to the simple fact that if n h ,h ∈ H, g ∈ G, then gh ≡ gh (mod I ). The rest is obvious for the case where 1 2 1 2 H H is finite. In general, it is proved through projective limits. (cid:3) The lemma shows that the structure of the augmentation quotient In /In+1 R,H R,H actually depends only on the structures of Γ and H. 2.4. Numerical extensions. The relative H-augmentation quotients can be easily expressed in the following situation. Definition 2.6. The extension L/K and its Galois group H are called numerical if H ≃ Z . p 10 KI-SENGTAN In particular, L/K is numerical, if it is the constant Z -extension. If H is numer- p ical and σ is a Z -generator of it, then the isomorphisms in the previous sections p together form the following isomorphism Val = Val : In/In+1 −£→−n1 Z[Γ]⊗I(H)n/I(H)n+1 d−σ→,Zp Z [Γ]. σ,n σ,n,G/H H H p HereweidentifyZ[Γ]⊗I(H)n/I(H)n+1 withZ[Γ]⊗I (H)n/I (H)n+1,andweidentify p p an one variable homogeneous polynomial with its coefficient. If σ′ = uσ, u ∈ Z∗, is p another generator, then Valσ,n = un ·Valσ′,n. IfH isnumerical, thenGcanbeidentifiedwithΓ′×H′ forsomesubgroupH′ ≃ Z p containing H. In this case, Γ = Γ′ ×H′/H. We will relate the group ring and the ˘ H-augmentation quotients of the group G to those of the direct product G = Γ×H. To do so, we let ̟ = |H′/H| and define U : G = Γ′ ×H′ −→ Γ′ ×H′/H ×H = Γ×H (17) ′ (γ′,h′) 7→ (γ′,h,̟h′) BothGand G˘ are extensions ofΓ by H, andby Lemma 2.5 we have theassociated isomorphisms £ : Z[Γ]⊗I(H)n/I(H)n+1 −→ In/In+1 and n,G H H £ : Z[Γ] ⊗ I(H)n/I(H)n+1 −→ I˘n/I˘n+1, where I˘n denotes the nth relative H- n,G˘ H H H augmentation ideal of Z[G˘]. The following lemma is obvious. Lemma 2.6. Let n be a nonnegative integer. An element ξ ∈ Z[G] is in In, if and H only if U(ξ) is in I˘n. In fact, if as in Lemma 2.5 we associate the isomorphisms H £ and £ respectively to the groups G and G˘, then n,G n,G˘ U◦£ = ̟n£ , (18) n,G n,G˘ and Val ◦U◦£ = ̟nVal ◦£ . (19) σ,n,G/H n,G σ,n,G˘/H n,G˘ 3. Stickelberger elements as refinements of the equivariant L-functions In this chapter, we review the definition of Stickelberger elements and show that they can be viewed as refinements of the equivariant L-functions. 3.1. The Stickelberger elements. Definition 3.1. ([Gro88],[Tat84]) The Stickelberger element θH′ = θL/K′ associated to the extension L/K′ is the unique element of Z[H′] such that for each continuous character ψ of H′, ψ(θH′) = LS(K′),T(K′)(ψ,0). (20) For the existence of the Stickelberger element, see [Gro88, Tat84]. In the case where L/K is the constant Z -extension, we can relate θ to Θ in p G Γ,S,T the following way. First we note that the Galois group of the constant Z -extension p over k can be identified with some H′ such that G can be identified with Γ′ ×H′. This is actually the situation discussed in Section 2.4. We recall the notations used there and in particular, we have ̟ = |H′/H|. Let σ′ be the Frobenius of H′. Then