On the existence of special elements in odd K -theory groups
aa r X i v : . [ m a t h . N T ] J un On the existence of special elements in odd K -theory groups infinite abelian extensions of imaginary quadratic fields Jilali Assim and Saad El Boukhari Abstract
Let k be an imaginary quadratic number field, and F/k a finite abelian extension ofGalois group G . We investigate the relationship between the conjectural special elementsintroduced in [6] and ETNC in the semi-simple case. This provides a partial proof of theconjecture for F/k under certain conditions.
Key words and phrases: number field, algebraic K -theory, regulator, Artin L -function,Equivariant Tamagawa Number Conjecture. Let K be a number field and X a smooth projective variety over K . Let i and j denoteintegers in Z . We consider the motive M = h i ( X )( j ). For practical purposes, one needs onlyto understand the realisations of M : • For any rational prime p , the étale realisation M p := H i é t ( X × K K, Q p )( j ) which is acontinuous representation of the Galois group Gal( K/K ). • The Betti realisation M B := H i ( X ( C ) , Q )( j ) which is a finite dimentional space andcarries an action of complex conjugation. • The deRham realisation M dR := H idR ( X/ Q )( j ) which is a finite dimensional filteredspace. • We also attach to M the motivic cohomology spaces H M ( M ) and H M ( M ) which are(conjecturally) finite dimentional.For each such motive M , there is a dual motive M ∨ with dual realisations. The periodisomorphism M B ⊗ C ∼ = M dR ⊗ C induces a map α M : M + B ⊗ R → M dR / Fil M dR ⊗ R , and it is conjecured that we have the following Conjecture (c.f. [12]) There is an exact sequence0 → H M ( M ) ⊗ R → ker( α M ) ρ ∨ b → H M ( M ∨ (1)) ∨ ⊗ R → H M ( M ) ⊗ R ρ b → coker( α M ) → H M ( M ∨ (1)) ∨ ⊗ R → ρ b is the Beilinson regulator. 1 n the existence of special elements in odd K -theory groups L -function L ( M, s ) associated with the motive M , and defined for Re( s ) large enough, by the followingproduct over rational primes ℓL ( M, s ) := Y ℓ det(1 − Fr − ℓ T | M I ℓ p )where F r ℓ is the Frobenius element. In the most basic example, when M = h (Spec( K ))(0), H f ( M ∨ (1)) ≃ O × K ⊗ Q and M B ⊗ R ≃ L σ : K → C R . The Beilinson regulator is in this case theDirichlet regulator R DF defined as follows R DK : O × K ⊗ R → M σ : F → C R u Σ σ : F → C (log | σ ( u ) | ) σ where | σ ( u ) | := ( σ ( u ) σ ( u )) / . Suppose now that K is abelian and let G := Gal( K/ Q ). The L -function L ( M, s ) is a C [ G ]-valued function, and can be identified with L ( M, s ) = ( L ( χ, s )) χ ∈ Hom( G, C ) , where, L ( χ, s ) is the usual Dirichlet L -function associated with the character χ . If we denoteby L ∗ ( χ,
0) the special value of the L -function at s = 0, then L ∗ ( χ, ∈ R × We can make the regulator R DK map into R by slightly changing its definition: We canredefine R DK over ( O × K ) r + r , where r (resp. r ) is the number of real embeddings (resp.pairs of complex conjugate embeddings) of K , as the determinant in R of the real numbers(log( | σ i ( a j ) | )) ≤ i,j ≤ r + r , where ( a j ) ≤ j ≤ r + r ∈ ( O × K ) r + r .The question is then whether or not the special value L ∗ ( χ,
0) lies in the image of ( O × K ) r + r by R DK . The answer comes obviously, in this case, with the definition of cyclotomic units (e.g.[25], Chap. 8).Suppose now that F/k is a finite abelian extension of number fields with Galois group G , and M = h (Spec( F ))( j ) (The motive is over k ), with j = r <
0. The map ρ b is in this case theusual K -theory Beilinson regulator ρ b = ρ rF : K − r ( F ) ⊗ R → ( Y σ : F → C (2 πi ) − r R ) + . and the motivic L -function is identified with L ( M, s ) = ( L ( χ, s + r )) χ ∈ Hom( G, C ) , where L ( χ, s ) is the Artin L -function. Since L ∗ ( χ, r ) ∈ R × , we can ask again if one can mapa certain element in K − r ( F ) to a certain image involving L ∗ ( χ, r ) by the regulator ρ rF .In the case where k = Q , and F = Q ( ζ N ) ( N >
0) and ζ N = e πi/N , construction of suchelement ℓ ( w ) in K − r ( F ) ⊗ Q has been suggested by Bloch (for r = −
1) and Beilinson( r ≤− N th root of unity w , the image by Beilinson regulator of the element ℓ ( w ) ∈ K − r ( F ) ⊗ Q is computed in [1] and is given in ⊕ σ : F → C (2 πi ) − r R by ρ rF ( ℓ ( w )) = (Li − r ( σ ( w ))) σ : F → C where for any complex number s which verifies | s | <
1, Li − r ( s ) is the polylogarithm functiondefined as Li − r ( s ) = Σ ∞ i =1 s i i − r . n the existence of special elements in odd K -theory groups C − [1 , ∞ [. If the first derivative L ′ ( χ, r ) at s = r is non zero, one can constuct an element in K − r ( F ) ⊗ Z [ χ ] which maps to L ′ ( χ, r ) by ρ rF ,using Gross’s formula (c.f. [6], Proof of Thm. 3.1).In [6], the authors conjectured the existence of such elements for a wider choice of basenumber fields k . Unfortunately, the only available evidence for this conjecture is provided for k = Q .In the following work, we propose a proof of the conjecture when the base field is imaginaryquadratic given that some conditions are fulfilled. Our argument is based on the EquivariantTamagawa Number Conjecture, which is a generalization of both the analytic number formula,and the conjecutre of Birch and Swinnerton-Dyer. We adopt the formulation of Fontaine andPerrin-Riou ([12], [14]) which generalizes to motives with coefficients in an algebra other than Q . In this formulation, the ETNC links K -theory groups to motivic L -functions throughisomorphisms which implicitly involve the Beilinson regulator. This, along with the recentproof of the ETNC over an imaginary quadratic number field for strictly negative integersgiven in [17], will enable us to prove the following results: Theorem 1.1 (Thm. 4.4) Let
F/k be an abelian extension over the imaginary quadraticnumber field k . Suppose that E is a number field containing all values of characters χ of G := Gal( F/k ) and that r is a stricty negative integer. We make the identification e χ E ∼ = E .Then the following statements are equivalent1 Conjecture 4.3.1 holds for the pair ( E ( r ) F , E [ G ]) .2 For each character χ of G there exists an element ˜ ǫ χ ( F ) ∈ e χ ( K − r ( F ) ⊗ E ) whichverifies ρ rF (˜ ǫ χ ( F )) = L ′ ( r, χ − ) where the Beilinson Regulator ρ rF is defined here over K − r ( F ) ⊗ E by extension ofscalars. Theorem 1.2 (Thm. 4.8) Let
F/k be a finite abelian extension of number fields with k imaginary quadratic. Suppose that p is a rational prime which does not divide G , that E isa number field which contains all values of characters of G and let r denote a strictly negativeinteger. Let S be a finite set of places of k containing the infinite places, the p -places andthe places which ramify in F/k and S f the subset of finite places of S . Then the followingstatements are equivalent • Conjecture 4.3.4 holds for the pair ( E ( r ) F , O p [ G ]) . • For all p -adic characters χ of G one has ρ rF ( e χ ( K − r ( F ) /tors ⊗ O p )) = Y v ∈ S f (1 − N v − r χ − ( v )) . Fitt O p ( e χ ( H ( F, Q p / Z p (1 − r )) ⊗ O p ))Fitt − O p ( e χ ( K − r ( O F,S ) ⊗ O p )) L ′ ( r, χ − ) where O F,S refers to the ring of S -units of F , and ρ rF denotes the extension of scalars of theBeilinson regulator over O p . We also identified e χ O p [ G ] = e χ O p with O p . Let r denote a negetive integer.The Beilinson regulator defined over the K -group of the field of complex numbers is a map ρ r C : K − r ( C ) → H D (Spec( C ) , R (1 − r )) ∼ = (2 πi ) − r R n the existence of special elements in odd K -theory groups H D () is the first group of Deligne’s Cohomology.For a number field F we compose with the map K − r ( F ) → Y σ : F → C K − r ( C ) , and obtain ρ rF : K − r ( F ) → Y σ : F → C (2 πi ) − r R ≃ X F ⊗ (2 πi ) − r R , where X F := Z [Σ F ] and Σ F is the set of embeddings of F in C . Let τ denote the complexconjugation automorphism ∈ G( C / R ).The image of the Beilinson regulator over K n − ( F ) is invariant under the action of τ , hencewe have ρ rF : K − r ( F ) → ( Y σ : F → C (2 πi ) − r R ) + . Moreover, the image of K − r ( F ) by ρ rF is a complete lattice of the R -vector space ( Q σ : F → C (2 πi ) − r R ) + F/k
Let
F/k be a finite abelian extension of number fields and G = Gal( F/k ). Let χ be anirreducible (one dimensional) complex character of G , and S be a finite set of places of k containing the set of infinite places S ∞ .We regard the set Σ F of embeddings F → C as a left G × Gal( C / R )-module by setting( g × w )( σ ) = w ◦ σ ◦ g − , for all g ∈ G , w ∈ Gal( C / R ) and σ ∈ Σ F .Let Z [ χ ] denote the ring of values of the character χ , and suppose that r is a strictlynegative integer. We write ρ rF,χ : Z [ χ ] ⊗ K − r ( F ) → Z [ χ ] ⊗ ( Y σ : F → C (2 πi ) − r R ) + for the map induced from ρ rF by extension of scalars over Z [ χ ]. We denote by L ′ S ( r, χ ) thefirst derivative of the S -truncated Artin L -function L S ( s, χ ) of χ at r . Conjecture 2.2 ([6], Conjecture 1 .
2) Assume that r < L ′ S ( r, χ − ) = 0. Then foreach σ ∈ Σ F there exists an element ǫ σ ( χ, S ) ∈ Z [ χ ] ⊗ K − r ( F ) such that(2 πi ) r ρ rF,χ ( ǫ σ ( χ, S )) = w − r ( F ker( χ ) ) L ′ S ( r, χ − )( c χσ,σ ′ ) σ ′ : F → C where F ker is the fixed field of χ , w − r ( F ker( χ ) ) := | H (Gal( Q /F ker( χ ) ) , Q / Z (1 − r )) | , andif we write τ σ for the generator of the decomposition subgroup of G of the place of F thatcorresponds to σ , the elements c χσ,σ ′ are given for each σ ′ ∈ Σ F by c χσ,σ ′ = χ − ( g ) + ( − − r χ − ( gτ σ ) if σ ( k ) ⊂ R and σ ′ ◦ g = σ f or some g ∈ G,χ − ( g ) if σ ( k ) R and σ ′ ◦ g = σ f or some g ∈ G, ( − − r χ − ( g ) if σ ( k ) R and σ ′ ◦ τ ◦ g = σ f or some g ∈ G, otherwise. Remarks k = Q . n the existence of special elements in odd K -theory groups
52 If we write ǫ σ ( χ, S ) to denote the image of ǫ σ ( χ, S ) by the surjetive map K − r ( F ) ⊗ Z [ χ ] → ( K − r ( F )) /tors ⊗ Z [ χ ]where ( K − r ( F )) /tors is the torsion-free quotient of K − r ( F ) (i.e. the quotient of K − r ( F ) by its Z -torsion submodule), then, since ρ rF is injective over ( K − r ( F )) /tors and Z [ χ ] is a flat Z -module, the element ǫ σ ( χ, S ) is unique in ( K − r ( F )) /tors ⊗ Z [ χ ]. In this section, we assume that the base field k is imaginary quadratic. Recall that we suppose r to be a strictly negative integer.In this case L S ( r, χ ) = 0 and L ′ S ( r, χ ) = 0 and the elements c χσ,σ ′ are given for each σ ′ ∈ Σ F by c χσ,σ ′ = ( χ − ( g ) if σ ′ ◦ g = σ f or some g ∈ G, ( − − r χ − ( g ) if σ ′ ◦ τ ◦ g = σ f or some g ∈ G. We also have the following
Lemma 3.1
Let the number field k be imaginary quadratic. Then there is a canonical iso-morphism of Z [ G ] -modules ι : ( Y σ : F → C (2 πi ) − r R ) + ∼ = R [ G ] Proof If k is imaginary quadratic, then for each embedding ˆ σ ∈ Hom( F, C ), either ˆ σ or τ ˆ σ identifies with an automorphism σ ∈ G .The isomorphism ι is explicitly given by mapping, for each such embedding ˆ σ ∈ Hom( F, C ),the element (0 , .., (2 πi ) − r a ˆ σ , .., (2 πi ) − r a τ ˆ σ , ..,
0) (with a σ = ± a τσ ) to a ˆ σ σ − if ˆ σ = σ and to a τ ˆ σ σ − if τ ˆ σ = σ . (cid:3) Let E be the field extension generated over Q by the values of all characters χ ∈ Hom( G, C )and O be its ring of integers.We extend the Beilinson regulator to the following by composing with the isomorphism ι andtensoring with O ρ rF : K − r ( F ) ⊗ O −→ R [ G ] ⊗ O. Explicitly if a ∈ K − r ( F ) and x ∈ Oρ rF ( a ⊗ x ) = ι ( ρ rF ( a )) ⊗ x. Remark
Let d − r := rk Z ( K − r ( F )). Since the image of K − r ( F ) by the Beilinson regulatoris a Z -lattice of Z -rank d − r , and O is a flat Z module, the image of K − r ( F ) ⊗ O by ρ rF isa free O -module of rank d − r over O .We need the following proposition Proposition 3.2
Suppose that the base field k is imaginary quadratic and σ ∈ G . Let S beany finite set of places of k containing the set of infinite places. The image of the element ǫ σ ( χ, S ) ∈ K − r ( F ) ⊗ O (where σ is also viewed as an emebedding in Σ F ) by ρ rF is given by ρ rF ( ǫ σ ( χ, S )) = w − r ( F ker( χ ) ) L ′ S ( r, χ − ) χ − ( σ ) | G | e χ , where e χ := | G | Σ g ∈ G χ − ( g ) g . n the existence of special elements in odd K -theory groups Proof
Since k R Conjecture 2.2 states that ρ rF ( ǫ σ ( χ, S )) = (2 πi ) − r w − r ( F ker( χ ) ) L ′ S ( r, χ − )(( − a χ − ( hσ ′ − σ )) σ ′ : F → C where a = 0 and h = Id if σ ′ identifies with an automorphism of G and a = − r and h = τ otherwise.If r is even, ( − a = 1 and the extension of scalars of the isomorphism ι over O sends theelement ((2 πi ) − r χ − ( hσ ′ − )) σ ′ : F → C to Σ σ ∈ G χ − ( σ − ) σ − = | G | e χ .The case of r odd is proved in a similar way. (cid:3) Let p be a rational prime, and let O p := O ⊗ Z p . We want to formulate a p -adic analog ofconjecture 2.2.First, the isomorphism ι in Lemma 3.1, is also an isomorphism of R -vector spaces. Sincethe image by the Beilinson regulator of K − r ( F ) is a complete Z -lattice in the R -vectorspace ( Q σ : F → C (2 πi ) − r R ) + , it is also a complete Z -lattice T in R [ G ] once we compose withthe isomorphism ι . We can, then, tensor T by the ring O p , and define a p -adic regulator byextension of scalars over O p ρ rF,p : K − r ( F ) ⊗ Z O p → T ⊗ Z O p Note that
T ⊗ Z O p ≃ O rk Z K − r ( F ) p = O | G | p .By Proposition 3.2, we can reformulate conjecture 2.2 for the extension F/k as follows
Conjecture 3 ( p -adic reformulation of conjecture 2.2) Let k be an imaginary quadraticnumber field, and F/k a finite abelian extension. Suppose that r is a strictly negative integerand that p is a rational prime. Let S be a finite set of places of k containing the set of infiniteplaces.Then for each σ ∈ G := Gal( F/k ) and each one dimensional character χ of G the followingholds1 The element w − r ( F ker( χ ) ) L ′ S ( r, χ − ) χ − ( σ ) | G | e χ of the R -vector R [ G ] = T ⊗ R belongs to the E -vector space T ⊗ E (recall that E is the field generated over Q by allvalues of characters of G ).2 Suppose that the previous condition is fulfilled. Then, there exists an element ǫ σ ( χ, S, p ) ∈ K − r ( F ) ⊗ O p such that ρ rF,p ( ǫ σ ( χ, S, p )) = w − r ( F ker( χ ) ) L ′ S ( r, χ − ) χ − ( σ ) | G | e χ , where e χ := | G | Σ g ∈ G χ − ( g ) g . Remarks ρ rF,p ( K − r ( F ) ⊗ O p ) = T ⊗ O p , the second statement in conjecture 3 is equivalent tothe assumption that w − r ( F ker( χ ) ) L ′ S ( r, χ − ) χ − ( σ ) | G | e χ ∈ T ⊗ O p F/k and a character χ of G , itsuffices to prove the existence of the element ǫ σ ( χ, S, p ) for one given choice of automorphism σ ∈ G . In fact if σ ′ ∈ G is another automorphism, then by Proposition 3.2 one has (modulotosion) ǫ σ ′ ( χ, S, p ) = χ − ( σ ′ σ − ) ǫ σ ( χ, S, p ) . n the existence of special elements in odd K -theory groups
73 Suppose that p doesn’t divide | G | . The image ǫ σ ( χ, S, p ) of the element ǫ σ ( χ, S, p ) in( K − r ( F )) /tors ⊗ O p lies in e χ (( K − r ( F )) /tors ⊗ O p ).This is because ρ rF,p is injective over ( K − r ( F )) /tors ⊗ O p (since O p is a flat Z module), and ρ rF,p ( ǫ σ ( χ, S, p )) = e χ ρ rF,p ( ǫ σ ( χ, S, p ))= ρ rF,p ( e χ ǫ σ ( χ, S, p ))Suppose, as before, that r <
0, and that S is a finite set of places of the totally imaginaryquadratic field k containing the infinite places. Fix a character χ of G . Let E be a number fieldwhich contains all values of χ , and O the ring of integers of E . We will say that Conjecture2.2 holds for the set of data ( F/k, r, S, χ, O ), if the elements ǫ σ ( χ, S ) of Conjecture 2.2 existwithin K − r ⊗ O for all σ ∈ G . For example, if Conjecture 2.2 holds exactly as stated abovefor all σ ∈ G , we will say that it holds for the set of data ( F/k, r, S, χ, Z [ χ ]).If E contains values of all characters of G , and if Conjecture 2.2 holds for all sets of data( F/k, r, S, χ, O ) for all characters χ of G , then we simply say that Conjecture 2.1 holds forthe set ( F/k, r, S, O ).Similarly, we will say that conjecture 3 holds for the set of data ( F/k, r, S, O p ), if it does holdfor the rational prime p and all characters χ of G . Proposition 3.3
If conjecture 2.2 holds for the set of data ( F/k, r, S, χ, O ) , then it holds for ( F/k, r, S, χ, O + [ χ ]) , where O + is the ring of integers of the maximal real subfield E + of E . Proof
Let σ be an automorphism of G and let χ be a one dimensional character of G .Suppose there exists an element ǫ σ ( χ, S ) ′ ∈ K − r ( F ) ⊗ O such that ρ rF ( ǫ σ ( χ, S ) ′ ) = w − r ( F ker( χ ) ) L ′ S ( r, χ − ) χ − ( σ ) | G | e χ . This means that x := w − r ( F ker( χ ) ) L ′ S ( r, χ − ) χ − ( σ ) | G | e χ ∈ T ⊗ O Let g := | G | and let ( a i ) i =1 ,...,g be a Z -basis of T . Since x ∈ T ⊗ O , there exist elements x , ..., x g in O such that x = Σ gi =1 x i a i (1)Consider now the complete Z -lattice T ′ of the R -vector space R [ G ] generated by the basis( b h := L ′ S ( r, χ − ) h ) h ∈ G (it is implied that L ′ S ( r, χ − ) = 0). The element x is written in thebasis ( b h ) h ∈ G as a sum x = Σ h ∈ G x ′ h b h , where, for each h ∈ G , x ′ h = w − r ( F ker( χ ) ) χ − ( σh ) ∈ Z [ χ ]. Since R T = R T ′ (= R [ G ]), bychange of basis we get again x = Σ gi =1 y i a i (2)with y i ∈ R ( χ ), for each i = 1 , ..., g . We compare now (1) and (2). For this we should firststudy the linear dependency of the family of vectors ( a i ) i =1 ,...,g over C .The Z -basis ( a i ) i =1 ,...,g is also a basis of the vector space R T = R [ G ]. If, the family of vectors( a i ) i =1 ,...,g is linearly dependent over the field of complex numbers, then the C -vector spaceΣ gi =1 C a i has necessarily dimension strictly less than g . However, since Σ gi =1 R a i = R [ G ], weget Σ gi =1 C a i = C [ G ]which shows indeeed that ( a i ) i =1 ,...,g are linearly independent over C . Therefore for each i = 1 , ..., g , we have x i = y i ∈ O ∩ R ( χ ) ⊆ O + [ χ ]. (cid:3) n the existence of special elements in odd K -theory groups Proposition 3.4
If Conjecture 3 holds for all sets of data ( F/k, r, S, O p ) for all p -adic inte-gers p , then Conjecture 2.2 holds for ( F/k, r, S, O ) .If Conjecture 3 holds for ( F/k, r, S, O p ) for all but a finite set of rational primes { p , ..., p n } ,then Conjecure 2.2 holds for ( F/k, r, S, O [1 /m ]) , where m = Q ni =1 p i ∈ Z . Proof
Fix a choice of an abelian extension
F/k with k imaginary quadratic, an integer r < S of places of k containing the set of infinite places.Let σ ∈ G and χ be a character of G . Suppose that conjecture 3 holds for ( F/k, r, S, O p ) forall primes p . In particular, for each rational prime p , there exists an element ǫ σ ( χ, S, p ) ∈ K − r ⊗ O p which verifies the two statements of conjecture 3.The first statement of Conjecture 3 ensures that x := w − r ( F ker( χ ) ) L ′ S ( r, χ − ) χ − ( σ ) | G | e χ ∈ T ⊗ E (recall that E is the field of fractions of O ). Let g := | G | and ( a i ) i =1 ,...,g be a basis of the Z -lattice T . By the above, there exist coefficients x i ∈ E , for i = 1 , ..., g such that x = Σ gi =1 x i a i Yet, the second statement of Conjecture 3 shows that x ∈ T ⊗ O p for all rational primes p which means that x i ∈ O p for all primes p . Fix an integer i ∈ { , ..., g } . Since E is the field offractions of O , there exist b i and c i relatively prime in O , such that x i = b i /c i . As, for each p , x i ∈ O p , the denominator c i is necessarily not divisible by p . Hence, c i is not divisible byall primes p and is therefore a unit in O . which gives x i ∈ O and the first claim ensues.Similarly, if x i ∈ O p for all but a finite set of primes { p , ..., p n } , then c i is only divisible byprimes p ∈ { p , ..., p n } . Thus x i ∈ O [ p p ...p n ]. (cid:3) The functor
DetLet R be a ring and P a finitely generated projective R -module. For every prime ideal p ∈ Spec( R ), the localization P p is a finitely generated free R p -module (since R p is local). Definition
The rank of a finitely generated projective R -module at a prime ideal p of R isthe integer rk p P = rk R p P p .Note that rk p P is well defined (unique) since local fields have IBN (invariant basis number).One can also verify that alternativelyrk p P = rk κ p ( P ⊗ R κ p )where κ p = R p / p R p is the residue field at p .For a fixed finitely generated projective R -module P , one can define a function Spec( R ) → Z ,which maps any prime p ∈ Spec( R ) to rk p P . We recall the following ([3], Chapter II, §5.3,Theorem 1) Property
The function rk P : Spec( R ) → Z is locally constant (and hence continuous withrespect to the Zariski topology on Spec( R ) and the discrete topology on Z ) and bounded. n the existence of special elements in odd K -theory groups R -module P has constant rank over R , if the asso-ciated function rk P : Spec( R ) → Z is constant. We write in this case rk R ( P ) = rk P (Spec( R )). Corollary If Spec( R ) is connected (which is equivalent to R having no nontrivial idempo-tents), then every finitely generated projective module over R has constant rank. Example If R is an integral domain then every finitely generated projective R -module hasconstant rank. Definition
Let A • denote a complex of R -modules. The complex A • is said to be perfect ifthere exists a bounded complex P • of finitely generated projective R -modules and a morphismof complexes π : P • → A • which induces isomorphisms for all iH i ( P • ) ∼ −−→ H i ( A • ) . (In this case we say that π is a quasi-isomorphism). Remark An R -module is said to be perfect if it is perfect when considered as the complex → A → . We denote by D p ( R ) the category of perfect complexes of R -modules.Let P be a finitely generated projective R -module. Let R = ⊕ si =1 R i be a decomposition of R as a direct finite sum of rings R i with connected spectra. For each i one has R i = e i .R , where e i is an indecomposable idempotent (e.g. [23], §1.3). We have a direct sum decomposition P = ⊕ si =1 P i , where P i = e i P is a projective R i -module of constant rank r i .The Knudsen-Mumford determinant functor [19] of P over R , is then the graded invertible(projective of constant rank 1) R -moduleDet R P := ⊕ si =1 r i ^ R i P i Let A • be an object of D p ( R ) and suppose that A • is quasi-isomorphic to the boundedcomplex of finitely generated projective R -modules P • . The Knudsen-Mumford determinantfunctor of A • over R depends only on the quasi-isomorphism class of A • and is defined asDet R A • = Det R P • := ⊗ i ∈ Z Det ( − i R P i . Remark
We used in the defintion of Det R A • the parity convention of [19] rather than thatof [9]. However, this will not affect our results and the formulation of ETNC with thisconvention is equivalent to the formulation given in [9].We recall some properties of the Knudsen-Mumford determinant functor (e.g. [9], §2) Properties of the functor
Det1. Det − R A • = Hom R (Det R A • , R ).2. If the cohomology groups H i ( A • ) are all perfect thenDet R A • = ⊗ i ∈ Z Det ( − i R H i ( A • ) n the existence of special elements in odd K -theory groups C • → C • → C • is a distinguished triangle (e.g. [22], 1.1.3) of perfect complexes of R -modules then Det R C • ≃ Det R C • ⊗ Det R C •
4. If A is a finitely generated torsion R -module which has a projective dimension at most1, then A is perfect and Det R A = (Fitt R A ) − . The fundamental line
Let
F/k be an abelian extension of number fields, E a number field. We suppose in the restof the article that r is a strictly negative integer . We set E ( r ) F := h (Spec( F )) E ( r ) . for the motive of F over k with coefficients in E and twist r . Since r < Y r ( F ) = Y σ ∈ Σ F (2 πi ) − r Z . where Σ F denotes again the set of embeddings of F in the field of complex numbers.For a motive M of F over k with coefficients in E one has a dual motive M ∨ with dualrealisations. In general, if X is a smooth projective variety of dimenion d , and M = h i ( X )( j ),then M ∨ identifies with h d − i ( X )( d − j ) (e.g. [11], Part 1, §2).For M = E ( r ) F , r <
0, we define the fundamental line in terms of motivic cohomology to beΞ( M ) := Det E [ G ] ( H M ( M )) ⊗ Det E [ G ] (( H M ( M ∨ (1)) ∨ ) ⊗ Det − E [ G ] ( M + B )where ( H M ( M ∨ (1)) ∨ = Hom E ( H M ( M ∨ (1) , E ), and M B denotes the Betti realization of M .For M = E ( r ) F , one has H M ( E ( r ) F ) = 0 ( r < H M ( E ( r ) ∨ F (1)) = K − r ( O F ) ⊗ E and( E ( r ) F ) B ∼ = Y r ( F ) ∨ ⊗ E .Let the map x x denote the Z -linear involution of the group ring Z [ G ] which satisfies g = g − for each g ∈ G . If X is any (complex of) Z [ G ]-module(s), then we write X forthe scalar extension of X with respect to the morphism x x .If X is an object of the category D p ( E [ G ]) of perfect complexes of E [ G ]-modules we denoteby X ∨ the complex X ∨ = RHom E ( X, E ) which is endowed with the contragredient G -actionand is also an object of D p ( E [ G ]).Recall that we have a canonical isomorphism M ∨ ∼ = Hom E [ G ] ( M, E [ G ]) for any E [ G ]-module M (e.g. [23], Remark 1.1.1), which induces for each object X in the category ofperfect complexes of E [ G ]-modules the following canonical isomorphism (e.g. [9], §2 (afterLemma 2.1)) Det E [ G ] X ∨ ∼ = Det − E [ G ] X In fact, since the functor Det E [ G ] only depends on the quasi-isomorphism class of X , it sufficesto show the result for a bounded complex of finitely generated projective E [ G ]-modules P • .One has Det E [ G ] ( P • ) ∨ := Det E [ G ] RHom E ( P • , E ) n the existence of special elements in odd K -theory groups E ( P • , E ) is quasi-iomorphic to the bounded complex of finitely generated projec-tive E [ G ]-modules whose ( − n )-th entry is Hom E ( P n , E ). HenceDet E [ G ] ( P • ) ∨ = ⊗ i ∈ Z Det ( − i E [ G ] ( P i ) ∨ ∼ = ⊗ i ∈ Z Det ( − i E [ G ] Hom E [ G ] ( P i , E [ G ]) Clearly, we have Hom E [ G ] ( P i , E [ G ]) = Hom E [ G ] (( P i ) , E [ G ]). ThusDet E [ G ] ( P • ) ∨ ∼ = ⊗ i ∈ Z Det ( − i E [ G ] Hom E [ G ] (( P i ) , E [ G ]) ∼ = ⊗ i ∈ Z Det ( − i +1 E [ G ] ( P i ) (use e . g . [23] , Remark 1 . . − E [ G ] ( P • ) We apply this result to the fundamental line and get the followingΞ( E ( r ) F ) = Det − E [ G ] ( K − r ( F ) ⊗ E ) ⊗ Det E [ G ] ( Y r ( F ) + ⊗ E ) , This is the definition of the fundamental line as given in ([9], §3.1), but with inverted signsof Det since we adopted here the parity convention of [19]. f -cohomology For any commutative ring Z and any étale sheaf F , we abbreviate in the sequel R Γ(Spec( Z ) é t , F )and H i ( R Γ(Spec( Z ) é t , F )) to R Γ( Z, F ) and H i ( Z, F ) respectively.Let p be a rational prime and let us denote by M p := H t (Spec( F ) × k k, E p ( r )) (with E p := E ⊗ Q p ) the étale realization of the motive E ( r ) F . Following Fontaine [12], for everyfinite place v of k define the local unramified cohomology of M p to be the complex R Γ f ( k v , M p ) = M I v p − Frob − v → M I v p v ∤ p ( B cris ⊗ M p ) G kv − Frob − → ( B cris ⊗ M p ) G kv ⊕ ( B dR / Fil ⊗ M p ) G kv v | p where I v is the inertia group at v , and B cris ⊂ B dR are Fontaine’s rings of p -adic periods (cf.[13]). We also write (Fil i ) i ∈ Z = (Fil i ( B dR )) i ∈ Z for the decreasing filtration associated to B dR .Note that the first cohomology group H f ( k v , M p ) of the complex RΓ f ( k v , M p ) is Bloch-Kato’slocal condition ([2], §3, (3.8.2)) or Mazur-Rubin’s finite condition ([20], Def. 1.1.6) definedby H f ( k v , M p ) = ( ker( H ( k v , M p ) → H ( I v , M p )) v ∤ p ker( H ( k v , M p ) → H ( k v , M p ⊗ B cris )) v | p It is also worth noting that the tangent space t v ( M ) := ( B dR / Fil ⊗ M p ) G kv is trivial for ourmotive. In fact by ([8], §1.4) one has an isomorphism ⊕ v ∈ S p t v ( M ) ∼ = ( H dR ( M ) / Fil ( H dR ( M )) ⊗ E p where E p := E ⊗ Q p , and S p is the set of places of k above p . However for M = E ( r ) F , wehave H dR ( M ) = Fil ( H dR ( M ) = F (e.g. [8], §1.3) or [8], §1.1, p. 70).If the place v is archimedean, we define the complex R Γ f ( k v , M p ) of f -cohomology as R Γ f ( k v , M p ) := R Γ( k v , M p ) v | ∞ n the existence of special elements in odd K -theory groups R Γ /f ( k v , M p ) := Cone( R Γ f ( k v , M p ) → R Γ( k v , M p ))where Cone() is the cone functor (e.g. [22], (1.1.2)).Note that when v | ∞ , the complex R Γ /f ( k v , M p ) is acyclic.For any finite set S of places of k containing the set of p -places, and the set S ∞ of infiniteplaces, we define the global unramified cohomology as R Γ f ( O k,S , M p ) := Cone( R Γ( O k,S , M p ) → M v ∈ S R Γ /f ( k v , M p ))[ − O k,S denotes the ring of S -integers of k . Compact Support cohomology
Let T p be any Galois stable lattice of M p , and S a finite set of places of k containing theset S p of places of k above p and the set S ∞ of infinite places of k . The cohomology withcompact support (e.g. [22], 5.3.1) is defined so as to lie in a canonical distinguished triangle R Γ c ( O k,S , T p ) → R Γ( O k,S , T p ) → ⊕ v ∈ S R Γ( k v , T p ) , where, we write R Γ( O k,S , T p ) for the complex of étale cohomology R Γ et (Spec O k,S , T p ). Remark
Let E p := E ⊗ Q p and O p := O ⊗ Z p , where O denotes the ring of integers of E .Let S be a finite et of place of k containing the set S p of p -places of k .1 The complex R Γ f ( k v , M p ) is perfect over E p [ G ] (e.g. [4], Lemma 12.2.1).2 Let T p denote any Glois-stable lattice inside M p . The complexes R Γ c ( O k,S , T p ) is perfectover O p [ G ] (e.g. [8], Proposition 1.20).3 Since E p [ G ] is semisimple, by ([22], 4.2.8), a complexe of E p [ G ]-modules is perfect ifand only if it has bounded finitely generated cohomology. If k is totally imaginary, thecomplexes R Γ( O k,S , M p ) and R Γ( k v , M p ) are then perfect.4 The complexes R Γ /f ( k v , M p ) and R Γ f ( O k,S , M p ) are also perfect over E p [ G ]. Thisensues from their respective definitions and the points 1 and 3 made above.5 If p doesn’t divide G , the previous observations can be also used to show for examplethat the complexe R Γ( O k,S , T p ) is perfect over O p [ G ]. ϑ rF,S p and ϑ rF, ∞ ϑ rF,S p Lemma 4.1
For every sequence of morphisms of complexes X f −→ Y g −→ Z one has a distinguished triangle Cone( f ) −→ Cone( g ◦ f ) −→ Cone( g ) . Proof e.g. [21], Chap. II, Proposition 0.10. (cid:3) n the existence of special elements in odd K -theory groups S p denote the set of primes of k above p . In the sequel, we always abbreviate R Γ ? ( O k,S p ∪ S ∞ , .. )as R Γ ? ( O k,S p , .. ), and H i ? ( O k,S p ∪ S ∞ , .. ) as H i ? ( O k,S p , .. ). We claim the following: Proposition 4.2
There is a distinguished triangle R Γ c ( O k,S p , M p ) → R Γ f ( O k,S p , M p ) → ⊕ v ∈ S p R Γ f ( k v , M p ) ⊕ ⊕ v ∈ S ∞ R Γ( k v , M p ) Proof
As shown above we have a distinguished triangle for every finite place v of kR Γ f ( k v , M p ) → R Γ( k v , M p ) g → R Γ /f ( k v , M p ) . Since R Γ /f ( k v , M p ) = 0 when v is infinite, we consider the following (non exact) sequence ofcomplexes R Γ( O k,S p , M p ) f → ⊕ v ∈ S p ∪ S ∞ R Γ( k v , M p ) g → ⊕ v ∈ S p R Γ /f ( k v , M p ) . Lemma 4.1 gives a distinguished triangle which we tranlate by -1
Cone ( f )[ − → Cone ( g ◦ f )[ − → Cone ( g )[ − R Γ c ( O k,S p , M p ) → R Γ f ( O k,S p , M p ) → ⊕ v ∈ S p R Γ f ( k v , M p ) ⊕ ⊕ v ∈ S ∞ R Γ( k v , M p ) (cid:3) We denote By E p the field E p := E ⊗ Q p . Proposition 4.3
Suppose that p is odd or that k is totally imaginary if p = 2 . Let r < .There exists an E p [ G ] -equivariant isomorphism ϑ rF,S p : Ξ( E ( r ) F ) ⊗ E p ∼ −−→ Det E p [ G ] RΓ c ( O k,S p , (( E ( r ) F )) p ) Proof
The fundamental line is given byΞ( M ) := Det E [ G ] ( H M ( M )) ⊗ Det E [ G ] ( H M ( M ∨ (1)) ∨ ) ⊗ Det − E [ G ] ( M + B )We are interested in the motive M = E ( r ) F . In that case H M ( M ) = 0, H M ( M ) = 0 and wehave the following isomorphisms • M + B ⊗ E p α ∼ −−→ ( ⊕ v ∈ S ∞ H ( k v , M p )) • The Chern map isomorphism H M ( M ∨ (1)) ⊗ E p ch ∼ −−→ H f ( O k,S p , M ∨ p (1)) . These two isomorphisms induce a third one(Det E [ G ] ( H M (( M ) ∨ (1)) ∨ ) ⊗ Det − E [ G ] ( M + B ) ) ⊗ E p Det( ch ∨ ) ⊗ Det − ( α ) ∼ −−−−−−→ Det E p [ G ] ( H f ( O k,S p , M ∨ p (1)) ∨ ) ⊗ Det − E p [ G ] ( ⊕ v ∈ S ∞ R Γ( k v , M p ))By Artin-Verdier duality we have H if ( O k,S p , M p ) ≃ H − if ( O k,S p , M ∨ p (1)) ∨ n the existence of special elements in odd K -theory groups R Γ f ( O k,S p , M p ) in terms of one degree (since degrees0 and 1 are null in our case) and we haveDet E p [ G ] R Γ f ( O k,S p , M p ) = Det E p [ G ] ( H f ( O k,S p , M ∨ p (1)) ∨ )Let us write J S p : Det E p [ G ] R Γ f ( O k,S p , M p ) ⊗ Det − E p [ G ] ( ⊕ v ∈ S p R Γ f ( k v , M p ) ⊕⊕ v ∈ S ∞ R Γ( k v , M p )) ∼ −−→ Det E p [ G ] R Γ c ( O k,S p , M p )for the isomorphism obtained from the distinguished triangle of Proposition 4.2. Since r < R Γ f ( k v , M p ) is acyclic for all finite places (e.g. [7], §2, after Lemma 1), andDet − E p [ G ] ( ⊕ v ∈ S p R Γ f ( k v , M p ) = ⊗ v ∈ S p Det − E p [ G ] R Γ f ( k v , M p ) maps to E p [ G ] in the followingfashion (e.g. [7], §2):Write V v → V v for the complex R Γ f ( k v , M p ), then Det − E p [ G ] R Γ f ( k v , M p ) maps to E p [ G ] viaId V v ,triv : det − E p [ G ] V v ⊗ det E p [ G ] V v ∼ −−→ E p [ G ] . and ϑ rF,S p := J S p ◦ (Det( ch ∨ ) ⊗ Det − ( α )) . There is another way to map Ξ( E ( r ) F ) ⊗ E p to Det E p [ G ] RΓ c ( O k,S p , (( E ( r ) F )) p ) by mappingrespective cohomology groups one by one (e.g. [8], §1.4, the remark after (1.16). This can beachieved as the following :Since R Γ f ( k v , M p ) is acyclic and the complex R Γ f ( O k,S p , M p ) is acyclic outside degree 2( r < ( H c ( O k,S p , M p ) ≃ ⊕ v ∈ S ∞ H ( k v , M p ) H c ( O k,S p , M p ) ≃ H f ( O k,S p , M p )which implies an isomorphism˜ J S p : Det E p [ G ] R Γ f ( O k,S p , M p ) ⊗ Det − E p [ G ] ( ⊕ v ∈ S ∞ R Γ( k v , M p )) ∼ −−→ Det E p [ G ] R Γ c ( O k,S p , M p )and this provides us with a new isomorphic map˜ ϑ rF,S p : Ξ( E ( r ) F ) ⊗ E p ∼ −−→ Det E p [ G ] RΓ c ( O k,S p , M p )where ˜ ϑ rF,S p := ˜ J S p ◦ (Det( ch ∨ ) ⊗ Det − ( α )) , in which we implicitly use the mapdet − E p [ G ] H f ( k v , M p ) ⊗ det E p [ G ] H f ( k v , M p ) f → E p [ G ]to get rid of f -cohomology.Note that the maps ϑ rF,S p and ˜ ϑ rF,S p only differ by the way ⊗ v ∈ S p Det − E p [ G ] R Γ f ( k v , M p ) mapsto E p [ G ]. In fact one has ([7] , (11), (12))˜ ϑ rF,S p = ε S p ( r ) .ϑ rF,S p and ε S p ( r ) = Y v ∈ S p (1 − N v − r Frob v e I v ) ∈ ( E [ G ]) × . where Frob v is any representative in G of the Frobenius map associated to the finite place v ,and e I v is the idempotent of Q [ G ] which corresponds to the inertia subgroup I v .Note that ε S p ( r ) is the inverse of the factor used in [7], since we adopted a dual formulationof the ETNC compared to Loc. cit. (cid:3) n the existence of special elements in odd K -theory groups ν be any integer. We define the lattice I pF,S p ( r ) := Det O p [ G ] RΓ c ( O k,S p , p ν T p ) ⊂ Det E p [ G ] RΓ c ( O k,S p , M p ) . The definition of I pF,S p ( r ) doesn’t depend on the choice of ν (e.g. [7], Prop. 1.20) and we setΞ( E ( r ) F ) O := ∩ p (Ξ( E ( r ) F ) ∩ ( ϑ rF,S p ) − ( I pF,S p ( r ))) . ϑ rF, ∞ Definition
For any isomorphism of finitely generated R -modules φ : V ∼ −−→ W , we let φ triv : Det − R ( V ) ⊗ Det R ( W ) ∼ −−→ R, obtained by the following composition of isomorphismsDet − R ( V ) ⊗ Det R ( W ) Det − R ( φ ) ⊗ id ∼ −−−−→ Det − R ( W ) ⊗ Det R ( W ) ˜ → R. The Beilinson regulator map induces an isomorphism ρ rF : K − r ( F ) ⊗ R ∼ −−→ Y r ( F ) + ⊗ R , which defines the isomorphism ϑ rF, ∞ := (( ρ rF ) ) triv : Ξ( Q ( r ) F ) ⊗ R ∼ −−→ R [ G ] . For the motive E ( r ) F , extension of scalars gives ϑ rF, ∞ : Ξ( E ( r ) F ) ⊗ R ∼ −−→ R ⊗ E [ G ] . Let S be any finite set of places of k containing the set S ∞ . We write L S ( E ( r ) F , s ) for the S -truncated C [ G ]-valued L -function of the motive E ( r ) F . Then with respect to the canonicalidentification C [ G ] = Q ˆ G C we have L S ( E ( r ) F , s ) = ( L S ( s + r, χ )) χ ∈ ˆ G . Remark L ∗ ( E ( r ) F ,
0) := L ∗ S ∞ ( E ( r ) F , ∈ R [ G ] × The statement of the Equivariant Tamagawa Number conjecture for the motive E ( r ) F isgiven as follows Conjecture 4.3 (ETNC)
One has
Conjecture 4.3.1 (Rationality Conjecture) ( ϑ rF, ∞ ) − ( L ∗ ( E ( r ) F , − ) .E [ G ] ⊇ Ξ( E ( r ) F )Further, each of the following equivalent conjectures is true n the existence of special elements in odd K -theory groups Conjecture 4.3.2 O p [ G ] ϑ rF,S p ◦ ( ϑ rF, ∞ ) − ( L ∗ ( E ( r ) F , − ) = I pF,S ( r ) Conjecture 4.3.3 O [ G ]( ϑ rF, ∞ ) − ( L ∗ ( E ( r ) F , − ) = Ξ( E ( r ) F ) O Conjecture 4.3.4 O p [ G ] ϑ rF,S p ◦ ( ϑ rF, ∞ ) − ( L ∗ ( E ( r ) F , − ) = Det O p [ G ] RΓ c ( O k,S p , T p ) Remarks S p to any finite set of primes containingthe infinite primes (e.g. [18], Lemma 2.3).3 Conjecture 4.3.1 is equivalent to Stark’s conjecture for r = 0 and E = Q , it is also equivalentto the central conjecture formulated by Gross in [15].4 Each of the equalities in Conjectures 4.3.2, 4.3.3 and 4.3.4 is equivalent to the "Lifted RootNumber Conjecture" formulated by Gruenberg & al [16] but only when E = Q . Let us work first with the non integral case (Conjecture 4.3.1). We have the following
Theorem 4.4
Let
F/k be an abelian extension over the imaginary quadratic number field k .Suppose that E is a number field containing all values of characters χ of G := Gal( F/k ) andthat r is a stricty negative integer. We make the identification e χ E ∼ = E .Then the following statements are equivalent1 Conjecture 4.3.1 holds for the pair ( E ( r ) F , E [ G ]) .2 For each character χ of G there exists an element ˜ ǫ χ ( F ) ∈ e χ ( K − r ( F ) ⊗ E ) whichverifies ρ rF (˜ ǫ χ ( F )) = L ′ ( r, χ − ) where the Beilinson Regulator ρ rF is defined here over K − r ( F ) ⊗ E by extension ofscalars. Proof
Conjecture 4.3.1 reads( ϑ rF, ∞ ) − ( L ∗ ( E ( r ) F , − ) .E [ G ] ⊇ Ξ( E ( r ) F )where Ξ( E ( r ) F = Det − E [ G ] ( K − r ( F ) ⊗ E ) ⊗ Det E [ G ] ( Y r ( F ) + ⊗ E ) . By § 4.2.2 this is equivalent to L ∗ ( E ( r ) F , − .E [ G ] ⊇ (( ρ rF ) ) triv (Det − E [ G ] ( K − r ( F ) ⊗ E ) ⊗ Det E [ G ] ( Y r ( F ) + ⊗ E ) ) n the existence of special elements in odd K -theory groups k is totally imaginary. Then L ∗ ( E ( r ) F ,
0) = L ′ ( E ( r ) F ,
0) and we canoni-cally identify Y r ( F ) + ⊗ E ∼ = E [ G ] . This identification sends for each σ ∈ G , seen as an embedding σ : F → C , the element(0 , .., (2 πi ) − r a σ , .., (2 πi ) − r a τσ , ..,
0) (with a σ = ± a τσ ) to a σ σ − . HenceDet E [ G ] ( Y r ( F ) + ⊗ E ) = E [ G ]Therefore, Conjecture 4.3.1 is equivalent to( L ∗ ( E ( r ) F , − ) E [ G ] ⊇ ( ρ rF ) triv (Det − E [ G ] ( K − r ( F ) ⊗ E ) ⊗ E [ G ])Let χ ∈ Hom( G, C ) be a character of G and e χ the corresponding idempotent. Since E contains all values of such characters of G , we get e χ E [ G ] = e χ E . In the following we usethe identification e χ E ∼ = E .By § 4.3 one can also identify L ′ ( E ( r ) F ,
0) = Σ χ ∈ Hom( G, C L ′ ( r, χ ) e χ . Conjecture 4.3.1 is thenequivalent to the following being true for all characters of GL ′ ( r, χ − ) − E ⊇ ( ρ rF ) triv (Det − E ( e χ ( K − r ( F ) ⊗ E )) ⊗ E )The number fields k is totally imaginary, thus by Beilinson regulator( K − r ( F ) ⊗ E ) ⊗ R ≃ E [ G ] ⊗ R The E -vector space e χ ( K − r ( F ) ⊗ E ) is then necessarily of dimension 1 and the restrictionof ( ρ rF ) triv to the χ -eigenspaces is defined as follows( ρ rF ) triv : Det − E ( e χ ( K − r ( F ) ⊗ E )) ⊗ E → E : Hom( e χ ( K − r ( F ) ⊗ E ) , E ) ⊗ E (( ρ rF ) ∨ ) − ⊗ Id E → E It follows that Conjecture 4.3.1 is then equivalent to the following being true for all charactersof
G L ′ ( r, χ − ) E ⊆ ρ rF ( e χ ( K − r ( F ) ⊗ E ))which means there exists for each character χ of G an element ˜ ǫ χ ( F ) ∈ e χ ( K − r ( F ) ⊗ E )which verifies ρ rF (˜ ǫ χ ( F )) = L ′ ( r, χ − ) . (cid:3) Corollary 4.5
Let
F/k be a finite abelian extension with k imaginary quadratic. Supposethat r < and that S is a finite set of places of k containing the set of infinites places. Let E be a number field which contains all values of characters χ of G , and O be its ring of integers.If Conjecture 4.3.1 holds for the pair ( E ( r ) F , E [ G ]) , then, the first statement of Conjecure 3holds for the set of data ( F/k, r, S, O, p ) , for all rational primes p . Proof
Recall that T denotes the complete Z -lattice in R [ G ], which is axactly the image of K − r ( F ) by the Beilinson regulator.If Conjecture 4.3.1 holds for the pair ( E ( r ) F , E [ G ]), then, Theorem 4.4, shows that theelement w − r ( F ker( χ ) ) L ′ ( r, χ − ) χ − ( σ ) | G | e χ belongs to T ⊗ E .Consequently, the first statement of Conjecure 3 holds for the set of data ( F/k, r, S ∞ , O, p ),for all rational primes p . But, this also means that the first statement of Conjecure 3.1 holds n the existence of special elements in odd K -theory groups F/k, r, S, O, p ), for all rational primes p and any finite set of places of k containing the infinite places since L ′ S ( r, χ ) = Y v ∈ S \ S ∞ (1 − N v − r χ − ( v )) L ′ S ∞ ( r, χ ) = Y v ∈ S \ S ∞ (1 − N v − r χ − ( v )) L ′ ( − r, χ )and Q v ∈ S \ S ∞ (1 − N v − r χ − ( v )) ∈ O . (cid:3) Let us return now to the integral case using conjecture 4.3.4 for more results. We need tointroduce some results first.In the rest of the paper we suppose that the integer r is always strictly negative and we fix T p = O p [ G ]( r ). Lemma 4.6
Let p be a rational prime and S a finite set of places of k containing the infiniteplaces, the p -places and the places which ramify in F/k .Suppose that either p is odd or k is totally imaginary. There is a distinguished triangle R Γ c ( O k,S , T p ) → R Γ( O k,S , T ∨ p (1)) ∨ [ − → ( Y σ : k → C T p ) + [0] (3) Proof
The distinguished triangle 3 is the same as the one given in ([9], §3.4). It arises inthe following way: The exact sequence (Poitou-Tate global duality, e.g.[21])0 / / H ( O k,S , T p ) / / ⊕ v ∈ S f H ( k v , T p ) ⊕ ⊕ v ∈ S ∞ ˆ H ( k v , T p ) / / H ( O k,S , T ∗ p (1)) ∗ (cid:15) (cid:15) H ( O k,S , T ∗ p (1)) ∗ (cid:15) (cid:15) ⊕ v ∈ S H ( k v , T p ) o o H ( O k,S , T p ) o o H ( O k,S , T p ) / / ⊕ v ∈ S H ( k v , T p ) / / H ( O k,S , T ∗ p (1)) ∗ / / ∗ denotes the Pontryagin dual, and S f the subset of finite places inside S .The sequence above gives a distinguished triangle whenever p is odd or k is totally imaginary R Γ( O k,S , T p ) → ⊕ v ∈ S f R Γ( k v , T p ) → R Γ( O k,S , T ∗ p (1)) ∗ [ − R Γ( O k,S , T ∗ p (1)) ∗ ∼ = R Γ( O k,S , T ∨ p (1)) ∨ this along with the application of Lemma 4.1 to the (non distinguished) triangle R Γ( O k,S , T p ) f → ⊕ v ∈ S R Γ( k v , T p ) g → ⊕ v ∈ S f R Γ( k v , T p )gives the triangle 3. (cid:3) By Lemma 4.6 we get an isomorphismDet O p [ G ] R Γ( O k,S , T ∨ p (1)) ∨ [ − ⊗ Det − O p [ G ] ( Y σ : k → C T p ) + [0] ℑ → Det O p [ G ] R Γ c ( O k,S , T p ) (4) n the existence of special elements in odd K -theory groups ℑ ⊗ Id E p explicitly maps the following cohomology groups ( H c ( O k,S , M p ) ≃ ⊕ v ∈ S ∞ H ( k v , M p ) H c ( O k,S , M p ) ≃ H ( O k,S , M ∨ p (1)) ∨ Let us go even further: If we apply Lemma 4.1 to the (non exact) triangle ⊕ v ∈ S f R Γ f ( k v , M p ) ⊕ ⊕ v ∈ S ∞ R Γ( k v , M p ) f → ⊕ v ∈ S ∞ R Γ( k v , M p ) g → R Γ c ( O k,S , M p )[1]we get a distinguished triangle ⊕ v ∈ S f R Γ f ( k v , M p ) → R Γ f ( O k,S , M p ) → R Γ( O k,S , M ∨ p (1)) ∨ [ − R Γ( O k,S , M ∨ p (1)) ∨ [ −
3] identifies with f -cohomologygiven that R Γ f ( k v , M p ) is acyclic H ( O k,S , M ∨ p (1)) ∨ ≃ H f ( O k,S , M p ) (5)A second isomorphism then ensues Det E p [ G ] R Γ( O k,S , M ∨ p (1)) ∨ [ − ⊗ Det − E p [ G ] ( Y σ : k → C M p ) + [0] ℧ → Det E p [ G ] R Γ f ( O k,S , M p ) ⊗⊗ v ∈ S ∞ (Det − E p [ G ] R Γ( k v , M p ))(6) To take the enlarged set of primes S into account in the formulation of the ETNC we recallthe distinguished triangle (e.g. [18], Lemma 2.3) R Γ c ( O k,S , M p ) → R Γ c ( O k,S p , M p ) → ⊕ v ∈ S f \ S p R Γ f ( k v , M p ) (7)One gets then an isomorphismDet E p [ G ] R Γ c ( O k,S , M p ) ˜ ∇ → Det E p [ G ] R Γ c ( O k,S p , M p ) (8)We use the results ((3),..,(8)) above to define the diagram in the proposition below: Proposition 4.7
The following diagram is commutative (all arrows are isomorphisms)
Det E p [ G ] R Γ( O k,S , M ∨ p (1)) ∨ [ − ⊗ Det − E p [ G ] ( Q σ : k → C M p ) + [0] ℧ (cid:15) (cid:15) ℑ⊗ Id Ep / / Det E p [ G ] R Γ c ( O k,S , M p ) ˜ ∇ (cid:15) (cid:15) Det E p [ G ] R Γ f ( O k,S , M p ) ⊗ ⊗ v ∈ S ∞ (Det − E p [ G ] R Γ( k v , M p )) (cid:15) (cid:15) Det E p [ G ] R Γ c ( O k,S p , M p ) ( ˜ ϑ rF,Sp ) − (cid:15) (cid:15) Det E p [ G ] R Γ f ( O k,S p , M p ) ⊗ ⊗ v ∈ S ∞ (Det − E p [ G ] R Γ( k v , M p )) H / / Ξ( E ( r ) F ) ⊗ E p where identifies cohomology of R Γ f ( O k,S , M p ) with the cohomology of R Γ f ( O k,S p , M p ) and H = ( ˜ ϑ rF,S p ) − ◦ ˜ J S p where the isomorphisms ˜ ϑ rF,S p and ˜ J S p are explained in the proof of Proposition 4.3. n the existence of special elements in odd K -theory groups Proof
The commutativity of the diagram above can be shown easily when passing to coho-mology. Note that since E p [ G ] is semi-simple, all the involved (finitely generated) cohomologygroups are perfect.By definition of the isomorphism H one has a commutative diagramDet E p [ G ] R Γ c ( O k,S p , M p ) ( ˜ ϑ rF,Sp ) − (cid:15) (cid:15) Det E p [ G ] R Γ f ( O k,S p , M p ) ⊗ ⊗ v ∈ S ∞ (Det − E p [ G ] R Γ( k v , M p )) ˜ J Sp ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ H / / Ξ( E ( r ) F ) ⊗ E p Next we shall define the isomorphism . By the proof of proposition 4.3, the isomorphism˜ J S p arises from the distinguished triangle R Γ c ( O k,S p , M p ) → R Γ f ( O k,S p , M p ) → ⊕ v ∈ S p R Γ f ( k v , M p ) ⊕ ⊕ v ∈ S ∞ R Γ( k v , M p )Passing to cohomology, ˜ J S p links the following cohomology groups ( H c ( O k,S p , M p ) ≃ ⊕ v ∈ S ∞ H ( k v , M p ) H c ( O k,S p , M p ) ≃ H f ( O k,S p , M p )We can use the same arguments of proposition 4.3 to get a distinguished triangle R Γ c ( O k,S , M p ) → R Γ f ( O k,S , M p ) → ⊕ v ∈ S f R Γ f ( k v , M p ) ⊕ ⊕ v ∈ S ∞ R Γ( k v , M p )where S f is the set of finite places of S . This gives us an isomorphism˜ J S : Det E p [ G ] R Γ f ( O k,S , M p ) ⊗ Det − E p [ G ] ( ⊕ v ∈ S ∞ R Γ( k v , M p )) ∼ −−→ Det E p [ G ] R Γ c ( O k,S p , M p )which explicitly maps the following cohomology groups ( H c ( O k,S , M p ) ≃ ⊕ v ∈ S ∞ H ( k v , M p ) H c ( O k,S , M p ) ≃ H f ( O k,S , M p )By the distinguished triangle (7), we have an isomorphism (this isomorphism induces theisomorphism ˜ ∇ ) H c ( O k,S , M p ) ≃ H c ( O k,S p , M p )hence H f ( O k,S , M p ) ≃ H f ( O k,S p , M p )The isomorphism is induced by this isomorphism and the identity on ⊗ v ∈ S ∞ Det − E p [ G ] R Γ( k v , M p ).Therefore, we have (by definition of ) the following commutative diagramDet E p [ G ] R Γ c ( O k,S , M p ) ˜ ∇ (cid:15) (cid:15) Det E p [ G ] R Γ f ( O k,S , M p ) ⊗ ⊗ v ∈ S ∞ (Det − E p [ G ] R Γ( k v , M p )) ˜ J S ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ (cid:15) (cid:15) Det E p [ G ] R Γ c ( O k,S p , M p )Det E p [ G ] R Γ f ( O k,S p , M p ) ⊗ ⊗ v ∈ S ∞ (Det − E p [ G ] R Γ( k v , M p )) ˜ J Sp ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ n the existence of special elements in odd K -theory groups E p [ G ] R Γ( O k,S , M ∨ p (1)) ∨ [ − ⊗ Det − E p [ G ] ( Q σ : k → C M p ) + [0] ℧ (cid:15) (cid:15) ℑ⊗ Id Ep / / Det E p [ G ] R Γ c ( O k,S , M p )Det E p [ G ] R Γ f ( O k,S , M p ) ⊗ ⊗ v ∈ S ∞ (Det − E p [ G ] R Γ( k v , M p )) ˜ J S ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ for, this we recall again that ˜ J S maps the cohomology groups ( H c ( O k,S , M p ) ≃ ⊕ v ∈ S ∞ H ( k v , M p ) H c ( O k,S , M p ) ≃ H f ( O k,S , M p )while, as stated above, ℑ ⊗ Id E p maps the following cohomology groups ( H c ( O k,S , M p ) ≃ ⊕ v ∈ S ∞ H ( k v , M p ) H c ( O k,S , M p ) ≃ H ( O k,S , M ∨ p (1)) ∨ This induces an isomorphism H ( O k,S , M ∨ p (1)) ∨ ≃ H f ( O k,S , M p )which is exactly the isomorphism (5) used to induce the isomorphism ℧ in (6). Thus ℧ = ˜ J S ◦ ( ℑ ⊗ Id E p ) − (cid:3) For later computations, we need a detailed description of how the isomorphism H − worksin more general terms.In the proof of proposition 4.3 we mapped motivic cohomology to f -cohomolgy provided thatthe complex R Γ f ( O k,S p , M p ) has non-trivial cohomolgy only in degree 2 (that is the group H f ( O k,S p , M p ) ∼ = H f ( O k,S p , M ∨ p (1)) ∨ . The other two degrees isomorphisms are • The cycle class map isomorphism H M ( M ) ⊗ E p cyc ∼ −−→ H f ( O k,S p , M p ) • The chern class map isomorphism H M ( M ) ⊗ E p ch ∼ −−→ H f ( O k,S p , M p )(remember that H M ( M ) = 0 and H M ( M ) = 0 for our motive, but we need to computethe above isomorphisms nonetheless to take account of the finite groups which appearin the integral case).Hence, more accurately H − := Det( ch ∨ ) ⊗ Det − ( ch ) ⊗ Det( cyc ) ⊗ Det − ( cyc ∨ ) ⊗ Det − ( α ) n the existence of special elements in odd K -theory groups − E p [ G ] ( H M ( M ∨ (1)) ∨ ⊗ E p ) Det − ( cyc ∨ ) (cid:15) (cid:15) ⊗ Det E p [ G ] ( H M ( M ) ⊗ E p ) Det( cyc ) (cid:15) (cid:15) ⊗ Det − E p [ G ] ( H M ( M ) ⊗ E p ) Det − ( ch ) (cid:15) (cid:15) Det − E p [ G ] H f ( O k,S p , M ∨ p (1)) ∨ ) ⊗ Det E p [ G ] H f ( O k,S p , M p ) ⊗ Det − E p [ G ] H f ( O k,S p , M p )and Det E p [ G ] ( H M (( M ) ∨ (1)) ∨ ) Det( ch ∨ ) (cid:15) (cid:15) ⊗ Det − E p [ G ] ( M + B ⊗ E p ) Det − ( α ) (cid:15) (cid:15) Det E p [ G ] H f ( O k,S p , M p ) Det − E p [ G ] ( ⊕ σ : k → C M p ) + Theorem 4.8
Let
F/k be a finite abelian extension of number fields with k imaginaryquadratic. Suppose that p is a rational prime which does not divide G , that E is a numberfield which contains all values of characters of G and let r denote a strictly negative integer.Let S be a finite set of places of k containing the infinite places, the p -places and the placeswhich ramify in F/k and S f the subset of finite places of S . Then the following statementsare equivalent • Conjecture 4.3.4 holds for the pair ( E ( r ) F , O p [ G ]) . • For all p -adic characters χ of G one has ρ rF ( e χ ( K − r ( F ) /tors ⊗ O p )) = Y v ∈ S f (1 − N v − r χ − ( v )) . Fitt O p ( e χ ( H ( F, Q p / Z p (1 − r )) ⊗ O p ))Fitt − O p ( e χ ( K − r ( O F,S ) ⊗ O p )) L ′ ( r, χ − ) where O F,S refers to the ring of S -units of F , and ρ rF denotes the extension of scalars of theBeilinson regulator over O p . We also identified e χ O p [ G ] = e χ O p with O p . Proof
Let p be a rational prime and S a finite set of places of k containing the infinite places,the p -places and the places which ramify in F/k . We denote by S f the subset of finite placesinside S .Recall that O denotes the ring of integers of E , and O p = O ⊗ Z p .We also fix T p = O p [ G ]( r ).If we denote by ϑ rF,S the compositum map ϑ rF,S := ∇ − ◦ ϑ rF,S p where ∇ : Det E p [ G ] R Γ c ( O k,S , M p ) f → Det E p [ G ] R Γ c ( O k,S p , M p ) is the isomorphism obtainedfrom the distinguished triangle (7) by mapping ⊗ v ∈ S f \ S p Det E p [ G ] R Γ f ( k v , M p ) to E p [ G ] via ⊗ v ∈ S f \ S p Id V v ,triv (as explained in the proof of proposition 4.3), then the statement of theETNC for the enlarged set of primes S reads O p [ G ] ϑ rF,S ◦ ( ϑ rF, ∞ ) − ( L ∗ ( E ( r ) F , − ) = Det O p [ G ] R Γ c ( O k,S , T p )For any subset of finite places S ′ of S we write ε S ′ ( r ) := Y v ∈ S ′ (1 − N v − r Frob v e I v ) n the existence of special elements in odd K -theory groups v is any representative in G of the Frobenius map associated to the finite place v ,and e I v := Σ δ ∈ Iv δ I v is the idempotent of Q [ G ] which corresponds to the inertia subgroup I v .In the proof of Proposition 4.3 we mentioned the equality ([7] , (11), (12))˜ ϑ rF,S p = ε S p ( r ) ϑ rF,S p )Again, by the same reasoning, we also have ∇ = ε S f \ S p ( r ) . ˜ ∇ Then ( ϑ rF,S ) − Det O p [ G ] R Γ c ( O k,S , T p ) = (( ϑ rF,S p ) − ◦ ∇ )(Det O p [ G ] R Γ c ( O k,S , T p ))=( ε S p ( r ) . ( ˜ ϑ rF,S p ) − ◦ ε S f \ S p ( r ) . ˜ ∇ )Det O p [ G ] R Γ c ( O k,S , T p ))= ε S f ( r )(( ˜ ϑ rF,S p ) − ◦ ˜ ∇ )Det O p [ G ] R Γ c ( O k,S , T p ))= ε S f ( r )( H ◦ ◦ ℧ )(Det O p [ G ] ( R Γ( O k,S , T ∨ p (1)) ∨ [ − ⊗ Det − O p [ G ] ( Y σ : k → C T p ) + ))= ε S f ( r )(Det( ch ∨ ) ⊗ Det − ( ch ) ⊗ Det − ( cyc ∨ ) ⊗ Det( cyc ) ⊗ Det − ( α )) − (Det O p [ G ] ( R Γ( O k,S , T ∨ p (1)) ∨ [ − ⊗ Det − O p [ G ] ( Y σ : k → C T p ) + )) In the last equality, the maps Det( ch ∨ ), Det − ( ch ), Det − ( cyc ∨ ) and Det( cyc ) are the in-duced maps which apply to Det O p [ G ] ( R Γ( O k,S , T ∨ p (1)) ∨ [ − ◦ ℧ .Note that, since we suppose k to be imaginary quadratic, we have ( Q σ : k → C T p ) + = T p = O p [ G ]( r ).Thus ( ϑ rF,S ) − Det O p [ G ] R Γ c ( O k,S , T p ) == ε S f ( r )(Det( ch ∨ ) ⊗ Det − ( ch ) ⊗ Det − ( cyc ∨ ) ⊗ Det( cyc ) ⊗ Det − ( α )) − (Det O p [ G ] ( R Γ( O k,S , T ∨ p (1)) ⊗ Det − O p [ G ] T p )= ε S f ( r ) (cid:18)(cid:16) (Det( ch ∨ ) ⊗ Det − ( ch ) ⊗ Det − ( cyc ∨ ) ⊗ Det( cyc )) − Det O p [ G ] ( R Γ( O k,S , T ∨ p (1)) (cid:17) ⊗ Det − O p [ G ] ( α − ( T p )) (cid:19) which is equivalent to ( ϑ rF,S ) − , Det O p [ G ] R Γ c ( O k,S , T p ) == ε S f ( r ) (cid:18)(cid:16) (Det( ch ∨ ) ⊗ Det − ( ch ) ⊗ Det − ( cyc ∨ ) ⊗ Det( cyc )) − , Det O p [ G ] ( R Γ( O k,S , T ∨ p (1)) (cid:17) ⊗ Det O p [ G ] ( α − ( T p )) ∨ (cid:19) yet we have • H ( O k,S , T ∨ p (1)) = 0 since r < H ( O k,S , T ∨ p (1)) = 0 . • Assuming that the Chern maps are isomorphisms on the integral level [24], we write ch O p r,k : K k − r ( O F,S ) ⊗ O p ∼ −−→ H − k ( O k,S , O p (1 − r ))for the isomorphisms induced by tensoring with Id O p for k = 0 , ch ∨ ) ⊗ Det − ( ch )) − , (Det − O p [ G ] H ( O k,S , T ∨ p (1)) ⊗ Det O p [ G ] H ( O F,S , T ∨ p (1))=(Det − (( ch O p r, ) − ) ⊗ Det(( ch O p r, ) − ))(Det − O p [ G ] H ( O F,S , O p (1 − r )) ⊗ Det O p [ G ] H ( O F,S , O p (1 − r )))=Det − O p [ G ] (( ch O p r, ) − H ( O F,S , O p (1 − r ))) ⊗ Det O p [ G ] (( ch O p r, ) − H ( O F,S , O p (1 − r )))=Det − O p [ G ] ( K − r ( O F,S ) ⊗ O p ) ⊗ Det O p [ G ] ( K − r ( O F,S ) ⊗ O p ) n the existence of special elements in odd K -theory groups • One also has Det O p [ G ] ( α − ( T p )) ∨ ∼ = Det O p [ G ] ( O p [ G ])= O p [ G ] . Conjecture 4.3.4 is equivalent to ( L ∗ ( E ( r ) F , − ) = ε S f ( r ) . ( ϑ rF, ∞ ) (cid:18)(cid:16) Det − O p [ G ] ( K − r ( O F,S ) ⊗ O p ) ⊗ Det O p [ G ] ( K − r ( O F,S ) ⊗ O p ) (cid:17)(cid:19) = Y v ∈ S f (1 − N v − r Frob − v e I v ) . ( ϑ rF, ∞ ) (cid:18)(cid:16) Det − O p [ G ] ( K − r ( O F,S ) ⊗ O p ) ⊗ Det O p [ G ] ( K − r ( O F,S ) ⊗ O p ) (cid:17)(cid:19) and since p does not divide | G | and O p contains all values of characters of G , the latteris equivalent to the following being true for all characters χ of GL ′ ( r, χ − ) − = Y v ∈ S f (1 − N v − r χ − ( v )) . ( ϑ rF, ∞ ) (cid:18)(cid:16) Det − O p ( e χ ( K − r ( O F,S ) ⊗ O p )) ⊗ Det O p ( e χ ( K − r ( O F,S ) ⊗ O p )) (cid:17)(cid:19) The ring O p is a product of discrete valuation rings, which means that every finite module isof projective dimension less or equal to 1 over O p , henceDet O p ( e χ ( K − r ( F ) ⊗ O p )) = Det O p ( e χ ( K − r ( F ) ⊗ O p ) /tors )) ⊗ Det O p ( e χ ( tors O p ( K − r ( F ) ⊗ O p )))= Det O p ( e χ ( K − r ( F ) ⊗ O p ) /tors )) ⊗ Fitt − O p ( e χ ( H ( F, Q p / Z p (1 − r )) ⊗ O p )) . The map ( ϑ rF, ∞ ) applies to Det O p ( e χ ( K − r ( O F,S ) ⊗ O p )) and Det O p ( e χ ( tors O p ( K − r ( F ) ⊗ O p ))) via identity since K − r ( O F,S ) ⊗ E and tors O ( K − r ( F ) ⊗ O ) ⊗ E are trivial.It also maps (Det − O p ( e χ ( K − r ( O F,S ) ⊗ O p ) /tors ) by definition to (Det − O p ρ rF ( e χ ( K − r ( O F,S ) ⊗ O p ) /tors ) (as explained in the proof of Theorem 4.4).We then get ρ rF ( e χ ( K − r ( F ) /tors ⊗ O p )) = Y v ∈ S f (1 − N v − r χ − ( v )) . Fitt O p ( e χ ( H ( F, Q p / Z p (1 − r )) ⊗ O p ))Fitt − O p ( e χ ( K − r ( O F,S ) ⊗ O p )) L ′ ( r, χ − ) (cid:3) In [17], the author suggests a proof of the ETNC for abelian extensions over an imaginaryquadratic field for odd primes whenever a certain condition is fulfilled ([17], Theorem 1.1).This motivates the following corollary
Corollary 4.9
Let
F/k be a finite abelian extension of number fields with k imaginaryquadratic. Suppose that p is a rational prime which is split and does not divide . G andthat r < . Then ρ rF ( e χ ( K − r ( F ) /tors ⊗ O p )) = Y v ∈ S f (1 − N v − r χ − ( v )) . Fitt O p ( e χ ( H ( F, Q p / Z p (1 − r )) ⊗ O p ))Fitt − O p ( e χ ( K − r ( O F,S ) ⊗ O p )) L ′ ( r, χ − ) where O F,S refers to the ring of S -units of F , and ρ rF denotes the extension of scalars of the Beilinsonregulator over O p . n the existence of special elements in odd K -theory groups Proof
This follows from Theorem 4.8 and ([17], Corollary 1.2). (cid:3)
This implies the following
Corollary 4.10
Let
F/k be a finite abelian extension of number fields with k imaginaryquadratic, and r < . Suppose that O is the ring of integers of the extension generated over Q by all values of characters of G := Gal( F/k ) , and let S be any finite set of places containingthe infinite places and the places which ramify in F/k .Then Conjecture 3 holds for the set of data ( F/k, r, S, O p ) , for all rational primes p which aresplit and such that p ∤ G . Proof
Let p be a rational prime which is split and doesn’t divide 6 . G . Corollary 4.5 ensuresthat the first statement of Conjecture 3 is fulfilled for ( F/k, r, S, O p ).By Corollary 4.9 there exists at least one element ǫ ∈ K − r ⊗ O p such that ρ rF,p ( ǫ ) ∈ Y v ∈ S f ∪ S p (1 − N v − r χ − ( v )) . Fitt O p ( e χ ( H ( F, Q p / Z p (1 − r )) ⊗ O p )) L ′ ( r, χ − )where S f is the subset of finite places in S , and S p is the set of p places of k . Since e χ ( H ( F, Q p / Z p (1 − r )) ⊗ O p ) is invariant under the action of Gal( F/F ker( χ ) ), we get e χ ( H ( F, Q p / Z p (1 − r )) ⊗ O p ) ⊂ ( H ( F, Q p / Z p (1 − r )) ⊗ O p ) Gal(
F/F ker( χ ) ) = H ( F ker( χ ) , Q p / Z p (1 − r )) ⊗ O p Since the two modules e χ ( H ( F, Q p / Z p (1 − r )) ⊗ O p ) and H ( F ker( χ ) , Q p / Z p (1 − r )) ⊗ O p arefinite and O p is the direct sum of principal rings we getFitt O p ( e χ ( H ( F, Q p / Z p (1 − r )) ⊗ O p )) | Fitt O p ( H ( F ker( χ ) , Q p / Z p (1 − r )) ⊗ O p ) = w − r ( F ker( χ ) ) O p This means that there exists elements a and b in O p , such that Fitt O p ( e χ ( H ( F, Q p / Z p (1 − r )) ⊗ O p )) = aO p , and w − r ( F ker( χ ) ) = ab .We have then ρ rF,p ( bǫ ) ∈ Y v ∈ S f ∪ S p (1 − N v − r χ − ( v )) .w − r ( F ker( χ ) ) L ′ ( r, χ − ) O p Further, since Y v ∈ S p (1 − N v − r χ − ( v )) ≡ p, the element Q v ∈ S p (1 − N v − r χ − ( v )) is a unit in O p , and if we write ǫ ′ = bǫ , we also have ρ rF,p ( ǫ ′ ) ∈ Y v ∈ S f (1 − N v − r χ − ( v )) .w − r ( F ker( χ ) ) L ′ ( r, χ − ) O p . Yet, Y v ∈ S f (1 − N v − r χ − ( v )) L ′ ( r, χ − ) = L ′ S ( r, χ − )Hence ρ rF,p ( ǫ ′ ) ∈ w − r ( F ker( χ ) ) L ′ S ( r, χ − ) O p By Theorem 4.8, we can choose the element ǫ ′ , so as to exactly have the following ρ rF,p ( ǫ ′ ) = w − r ( F ker( χ ) ) L ′ S ( r, χ − ) | G | n the existence of special elements in odd K -theory groups e χ O p = O p , the exact statementis ρ rF,p ( ǫ ′ ) = w − r ( F ker( χ ) ) L ′ S ( r, χ − ) | G | e χ The result ensues. (cid:3)
Acknowledgement
In ([10], Theorem 6.3) the second author introduced an element Q (0) := Y ℓ | N,ℓ = p (1 − Fr − ℓ ℓ m − )where, for a finite abelian extension F/ Q , we denote by N the conductor of F and by m anodd integer such that m ≥
3. We write Fr ℓ for the Frobenius automorphism at the prime ℓ .However, in the latter expression, the eulerian factors 1 − Fr − ℓ ℓ m − should be replaced by(1 − Fr − ℓ ℓ m − ) e I ℓ ∈ Z [ G ]. Here I ℓ is the inertia subgroup of Gal( F/ Q ) corresponding tothe prime ℓ , e I ℓ := ( I ℓ ) − Σ g ∈ I ℓ g and Fr ℓ denotes a representative of the Frobenius map in G := Gal( F/ Q ).This corrects the definition given in [10] and has been suggested by the authors in [5] whowe particularily thank for their observation. References [1] A. Beilinson
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E-mail: [email protected] S. El Boukhari, Moulay Ismail University of Meknès, Department of Math., B.P. 11201Zitoune, Meknès, Morocco.