The local equivariant Tamagawa number conjecture for almost abelian extensions
aa r X i v : . [ m a t h . N T ] J u l THE LOCAL EQUIVARIANT TAMAGAWA NUMBERCONJECTURE FOR ALMOST ABELIAN EXTENSIONS
JENNIFER JOHNSON-LEUNG
Abstract.
We prove the local equivariant Tamagawa numberconjecture for the motive of an abelian extension of an imaginaryquadratic field with the action of the Galois group ring for all splitprimes p = 2 , s < Introduction
Since Dirichlet’s remarkable proof of the analytic class number for-mula in the first half of the nineteenth century, conjectures on the rela-tionship between the values of L -functions and invariants of arithmeticobjects have motivated a great deal of research. The equivariant Tam-agawa number conjecture (ETNC) is a unifying statement concerningthe special values of motivic L -functions which encompasses both theBirch and Swinnerton-Dyer conjecture and the generalized Stark con-jectures. It is a deep and sweeping assertion which has yielded to proofin very few cases. The Tamagawa number conjecture builds on theconjectures of Beilinson [Bei], predicting that the L -values of smoothprojective varieties over Q are given by period integrals and regula-tor maps, up to a rational factor, q . Bloch and Kato [BK], furtherpredicted that the rational number q is given in terms of Tamagawanumbers and the order of a certain Tate-Shafarevic group. This con-jecture was reformulated by Fontaine and Perrin-Riou [FPR] in a lan-guage that was naturally extended to motives with extra symmetriesby Burns and Flach [BF2, BF3]. There are two equivalent formulationsof the conjecture. The first is a global formulation that concerns of thevanishing of a certain element in relative K -theory. The second is alocal formulation that concerns the equality of two lattices.In this paper, we study the conjecture for the motive of an abelianextension of an imaginary quadratic field. We call these almost abelianextensions because there are many similarities to the case of absolutely Date : November 7, 2018.2000
Mathematics Subject Classification.
Primary 11G40.The author was supported in part by an NSA Young Investigator’s Award. abelian extensions stemming from the fact that an imaginary quadraticfield has only one archimedean place. Notice that as this place is com-plex, the local conjecture at the prime 2 will be less complicated thanin the case of absolutely abelian extensions [F4]. However, we do notconsider the prime 2 in this paper. Our main result is a proof ofthe local ETNC at all split primes ℓ ∤ L -function. Bley also considers the case of abelian extensions ofimaginary quadratic fields for the L -value at 0 [Bl]. His proof has re-strictions similar to those in this work stemming from the vanishing ofthe µ -invariant of a certain Iwasawa module. It would be quite nice toprove compatibility of the conjecture with the functional equation forthis class of motives as well, as the combination of these results wouldgive the conjecture at any integer value of the L -function.The only completely proven case of the equivariant Tamagawa num-ber conjecture is the proof of Burns, Flach, and Greither for abelianextensions of Q [BG, BF4, F4]. Huber and Kings proved independentlya slightly weaker version of this cyclotomic case [HK2] which has sincebeen strengthened to a full proof by work of Witte [Wi]. Even partialresults are quite more sparse. Burns and Flach give a proof for aninfinite family of quaternion extensions [BF3], and Navilarekallu givesa method of proof for A extensions which he employs for a specificcase [Na]. There are also several theorems that are not equivariant.Gealy recently proved a weakened version of the Tamagawa numberconjecture for modular forms of weight greater than 1 [Ge]. Kings alsoproved a weakened version for elliptic curves with CM by an imagi-nary quadratic field of class number 1 [K2]. In both of these cases, theconjecture must be weakened because it is not known that the motiviccohomology groups are finitely generated. By working with the con-structible part of the group, however, a proof can be given. Bars buildson work of Kings to give some non-equivariant results for Hecke char-acters of imaginary quadratic fields [Ba]. The survey papers of Flach[F2, F3] include a nice formulation of the local version of the equivari-ant Tamagawa number conjecture for arbitrary motives over Q anddiscusses the proven cases. We strive here to keep notation consistentwith this overview.This paper is an improvement of the main result in the author’sthesis, and so many thanks are due her thesis advisor, Matthias Flach.She would also like to thank Werner Bley, Matthew Gealy, and GuidoKings for very helpful conversations and the referee for a careful readingof the manuscript. NTC FOR ALMOST ABELIAN EXTENSTIONS 3
Notation.
Let K be an imaginary quadratic field with ring ofintegers O K and let f be an integral ideal of O K . We will let K ( f ) denotethe ray class field of K of conductor f . By a CM pair of modulus f overa number field F , we mean a pair ( E, α ) where E is an elliptic curveover F with complex multiplication by O K and such that the inculsionof O K into F factors through End( E ), and α ∈ E ( C ) is a primitive f -division point. By [Ka, 15.3.1], there is a CM pair of modulus f over K ( f ) which is isomorphic to ( C / f , f ) over C . This pair isunique up to isomorphism, and whenever O × K → ( O K / f ) × is injectivethe isomorphism is unique. Denote ( C / f , f ) the canonical CMpair.We will make repeated use of the graded determinant functor Det ofKnudsen and Mumford [KM]. Let R be a commutative ring, and P aprojective R -module. The determinant of P is the invertible R -moduleDet R P := rk R P ^ R P. If C : · · · → P i − → P i → P i +1 → · · · is a perfect complex of projective R -modules, the determinant of the complex is defined to be the gradedinvertible R -module Det R C := O i ∈ Z Det ( − i R P i and depends only on the quasi-isomorphism class of C . Indeed, if thecohomology groups H i ( C ) are all perfect, one hasDet R C = O i ∈ Z Det ( − i R H i ( C ) . The Main Theorem
Let F be an abelian extension of K with Galois group G . We considerthe Chow motive M = h (Spec( F ))( j ) where j is a negative integer.2.1. The local statement of the ETNC.
We will formulate theequivariant Tamagawa number conjecture (ETNC) for this motive. M carries an action of the semisimple Q -algebra A = Q [ G ]. We study M via its realizations and the action of A on these spaces, focusing on theBetti realization M B = H (Spec( F )( C ) , Q ( j )) , which carries an action of complex conjugation, the de Rham realiza-tion M dR = H dR (Spec( F ) / Q )( j ) , JOHNSON-LEUNG with its Hodge filtration, and the ℓ -adic realization M ℓ = H et (Spec( F ) × Q ¯ Q , Q ℓ ( j ))) , which is a continuous representation of Gal( ¯ Q / Q ). The A -equivariant L -function of M is defined via an Euler product L ( A M, s ) = Y p Det A (1 − Frob − p · p − s | M I p ℓ ) − . The leading term of the Taylor expansion at s = 0 decomposes overthe characters of GL ∗ ( A M ) = ( L ′ ( η, j )) η ∈ ˆ G ∈ ( A ⊗ Q R ) × . We can now introduce one of the key objects in this formulation of theTamagawa number conjecture: the fundamental line is the A -moduleΞ( A M ) = Det A ( K − j ( O F ) ∗ ⊗ Z Q ) ⊗ A Det − A ( M + B ) , where + denotes the invariants under complex conjugation and K − j ( O F ) ∗ is the dual of the algebraic K -group K − j ( O F ) = K − j ( F ). This“line” is the tool which enables the comparison of the L -value withalgebraic invariants of the number field.Borel’s regulator [Bor], is an isomorphism K − j ( O F ) ⊗ Z R ρ ∞ −−→ M σ ∈T C / R · (2 πi ) − j · σ ! + where T = Hom( F, C ). Since j < K − j ( O F ) ≃ K − j ( F ). For an el-ement, P σ ∈T x σ · σ , the Galois group acts via g · (cid:0)P σ ∈T x · σ (cid:1) = P σ ∈T x · g − σ. With this action, ρ ∞ is A -equivariant just as in the case ofthe Dirichlet regulator. Now,the R -dual of (cid:0)L σ ∈T C / R · (2 πi ) − j · σ (cid:1) + is identified with M + B ⊗ Q R by taking invariants in the Gal( C / R )-equivariant perfect pairing M σ ∈T R · (2 πi ) j × M σ ∈T C / R · (2 πi ) − j → M σ ∈T C / πi · R Σ −→ R induced by multiplication. Hence, the dual of the Borel regulator in-duces an A -equivariant isomorphism ϑ ∞ : A ⊗ Q R → Ξ( A M ) ⊗ Q R . Note that the L ∗ ( A M ) lies in the domain of this isomorphism, andGross conjectured that its image lies in the rational space Ξ( A M ) ⊗ Q L -value is given by the Borel regulator and was proved by Deninger[D2] in his work on the Beilinson conjectures for Hecke characters ofimaginary quadratic fields. NTC FOR ALMOST ABELIAN EXTENSTIONS 5
Fix a prime number ℓ , let S be a set of primes containing ℓ , ∞ ,and the primes which ramify in F , and let A ℓ = A ⊗ Q Q ℓ . We nowconcern ourselves with the ℓ -part of the rational factor by consideringthe isomorphism induced by the Chern class map and the cycle classmap ϑ ℓ : Ξ( A M ) ⊗ Q A ℓ → Det A ℓ ( R Γ c ( Z [ 1 S ] , M ℓ )) , where the right hand side denotes the cohomology with compact sup-ports as defined by the mapping cone R Γ c ( Z [ 1 S ] , M ℓ ) → R Γ( Z [ 1 S ] , M ℓ ) → M p ∈ S R Γ( Q p , M ℓ ) , The conjecture then compares a natural lattice in the right hand side ofthis isomorphism to the lattice generated by the image of L ∗ ( A M ). Toconstruct the lattice, we choose the order Z [ G ] in A and the Gal( ¯ Q / Q )-stable projective Z ℓ [ G ]-lattice T ℓ = H (Spec( F ⊗ Q ¯ Q ) , Z ℓ ( j )) . Conjecture (Local ETNC) . There is an equality of lattices ϑ ℓ ϑ ∞ ( L ∗ ( A M ) − ) · Z ℓ [ G ] = Det Z ℓ [ G ] R Γ c ( Z [ 1 S ] , T ℓ ) . inside of Det A ℓ R Γ c ( Z [ S ] , M ℓ ) . The ETNC for number fields is equivalent to the statement that the lo-cal conjecture holds at every prime number ℓ . This determines L ∗ ( A M )up to a unit in Z [ G ]. Notice that the ETNC depends on the choice oforder but is independent of the choice of S and T ℓ [F1]. This indepenceof lattice is exploited to prove the main results of [JLK] which will bean important ingredient in the proof of our main theorem. Main Theorem.
Let F be an abelian extension of an imaginary qua-dratic field K with Galois group G . Then the local equivariant Tam-agawa number conjecture is valid for the motive h (Spec( F ))( j ) for j < at every rational prime p ∤ which splits in K .Remark . The restriction to split primes can be lifted whenever the µ -invariant of a certain Iwasawa module can be shown to vanish, asdiscussed in Section 4.2.2. Proof strategy.
We first reduce to the case that F = K ( m ) is theray class field of conductor m where the only root of unity in K whichis congruent to 1 modulo m is 1 by applying the general functorialityresult of Burns and Flach [BF2, Prop 4.1 b)]. Let G m denote theGalois group Gal( K ( m ) /K ). The conjecture asserts an equality of rank JOHNSON-LEUNG Z ℓ [ G m ]-modules inside of Det A ℓ R Γ c ( Z [ S ] , M ℓ ). Our strategy is tocompute a generator of each of these modules. By a rational characterof G m , we mean an Aut( C ) orbit of complex characters of G m . Thegroup ring A = Q [ G m ], splits as a product of number fields indexed bythese rational characters χ of G m , and thus A ℓ splits as well. Hence, itsuffices to compare our generators character by character. In section3 we compute the image of L ∗ ( A M ) under the composition ϑ ℓ ϑ ∞ . Insection 4 we compute a basis of Det Z ℓ [ G ] R Γ c ( Z [ S , T ℓ ) via descent fromthe Iwasawa main conjecture and show that it coincides with the imageof L ∗ ( A M ), completing the proof.3. The image of the L -value Let χ be a character of G m of conductor f χ . If f χ = m , then χ is induced from a character of G f χ . The Dirichlet L -function of χ differs from the Artin L -function of χ as a G m representation by afinite number of Euler factors L D ( χ, s ) = Y p | m ,p ∤ f χ (1 − χ ( p ) N p − s ) L ( χ, s ) . In [D1, D2] Deninger constructs elements in motivic cohomology in or-der to prove the Beilinson conjecture for Hecke characters of an imag-inary quadratic field. We will use these motivic elements to prove ourmain theorem, but as we seek a finer result about the value L ∗ ( A M ),we will have to revisit some details of the proof as well. We first trans-late our question to the setting of Hecke characters by twisting χ bythe norm character of K to obtain a Hecke character of weight 2, ϕ χ = χN K/ Q . Let E be an elliptic curve defined over K ( m ) with complex multipli-cation by O K where E has the additional property that the Serre-Tate character factors through the norm map from K ( m ) to K . Let A = R K ( m ) /K E be the Weil restriction of the elliptic curve. Then A is an abelian variety over K with CM by a semisimple K -algebra T and with Serre-Tate character ϕ A . Deninger proves that any Heckecharacter ϕ of weight w > w Y i =1 λ i ◦ ϕ A = w Y i =1 ϕ λ i where λ i ∈ Hom( T, C ), [D2, Prop 1.3.1]. We choose once and for all atype (1 ,
0) character ϕ with N K/ Q = ϕ ¯ ϕ and take m to be a multiple NTC FOR ALMOST ABELIAN EXTENSTIONS 7 of the conductor of ϕ . Then we have ϕ χ = ϕ λ ϕ λ and m is a multiple of the conductors of λ and λ .3.1. Torsion points.
Deninger computes the special values of the L -function of such Hecke characters in terms of torsion points on E . Let f be an ideal of O K and let ρ f ∈ A ∗ K be an id`ele with ideal f . Choosean approximation f f ∈ K ∗ with(3.1) v p ( f f ) ≤ p ∤ f v p ( f − f − ( ρ f ) − p ) ≥ p | f . We also fix an isomorphism θ E : O K ≃ End K ( m ) ( E )such that θ ∗ E ( k ) ω = kω for all ω ∈ H ( E, Ω E/K ( m ) ) and an embedding τ of K ( m ) into C such that j ( E ) = j ( O K ). Then we have a complexisomorphism E ( C ) ≃ C / Γ where Γ = Ω O K for some Ω ∈ C . This choice is non-canonical and determines a class inthe Betti cohomology of E . If z ∈ C , we let ([ z ]) denote the point on E under this isomorphism. Now f β = ([Ω f − f ]) is a point in E [ f ] whichis rational over K ( f ).Fix a set of ideals { b g ⊆ O K } g ∈ G m with Artin symbol ( b g , K ( m ) /K ) = g ∈ G m . For g ∈ G m , let g E be thecurve obtained by base change according to the diagram g E / / (cid:15) (cid:15) E (cid:15) (cid:15) Spec( K ( m )) g ∗ / / Spec( K ( m ))We denote the period lattice of g E by Γ g . Following Deninger, for anyideal a ⊆ O K which is prime to the conductors of ϕ λ and ϕ λ we defineΛ( a ) ∈ K ( m ) × by ϕ A ( a ) ∗ ω ( a ,K ( m ) /K ) = Λ( a ) ω where ω g ∈ H ( E g , Ω ) has period lattice Γ g and ϕ A ( a ) ∈ T × is viewedas an isogeny E → ( a ,K ( m ) /K ) E . Now we can consider a family of f -torsion points on the conjugates of E , f β g = ([Λ( b g Ω f − f )])with an action of G m given by h f β g = f β hg . JOHNSON-LEUNG
Proposition 3.1. [D2, (3.4)]
Let χ be a character of Gal( K ( m ) /K ) ofconductor f χ = f . The Artin L -series L ( χ, s ) has a first order zero forevery s = j < , and the special value is given by the formula L ′ ( χ, j ) = ( − j Φ( f )( − j )! Φ( m ) (cid:18) √ d K N f πi (cid:19) − j χ ( ρ f ) X g ∈ G m χ ( g ) A (Γ g ) − j X = γ ∈ Γ g ( f β g , γ ) g | γ | − j , where Φ is the totient function, d K is the discriminant of K , for any Z -basis of Γ g with Im( u/v ) > , A (Γ g ) = (¯ uv − ¯ vu ) / πi ) , and ( , ) g : C / Γ g × Γ g → U (1) given by ( z, γ ) g = exp ( A (Γ g ) − ( z ¯ γ − ¯ zγ )) is the Pontrjagin pairing. This formula is the essential first step to proving the ETNC as itdescribes the L -value in terms of points on an elliptic curve. Thesepoints can then be used to construct elements in motivic cohomology.Notice that the formula is unchanged if we consider χ to be a rationalcharacter of G m in the sense that it represents equality of tuples( L ′ ( η, j )) η ∈ χ = (cid:0) ( − j Φ( f )( − j )! Φ( m ) (cid:18) √ d K N f πi (cid:19) − j η ( ρ f ) X g ∈ G m η ( g ) A (Γ g ) − j X = γ ∈ Γ g ( f β g , γ ) g | γ | − j (cid:1) η ∈ χ , Where η is a complex character which is in the orbit represented bythe rational character χ .3.2. Eisenstein Symbol.
The Eisenstein symbol, originally constructedby Beilinson, is roughly a map from torsion points of an elliptic curveto the cohomology of a power of the curve. There are several variationsof Eisenstein symbol in the literature. In particular, Deninger uses avariation for which the domain is divisors of degree zero and defines adegree zero divisor f β ′ = f β + 1˜ N − j − − ˜ N − j ˜ N − j − X p ∈ E ( C )[ ˜ N ] ( p )to construct the motivic elements. Here ˜ N ≥ ℓ -adic regulator. Thus, we introduce our variation inthe following lemma and show that it is compatible with Deninger’s NTC FOR ALMOST ABELIAN EXTENSTIONS 9 construction. Note that this is used implicitly in work of Kings [K2],which we will discuss in section 3.4.
Lemma 3.2.
Let E be an elliptic curve. For any k > , there is avariation of the Eisenstein symbol E k M : Q [ E [ f ] \ → H k +1 M ( E k , k + 1) which is defined for divisors of any degree. Moreover, E k M ( f β ′ ) = E k M ( f β ) . Proof.
For N = N f ≥
3, let M be the modular curve parameterizingelliptic curves with full level N structure, and let E be the universalelliptic curve over M . Choose a level N structure on E , α : ( Z /N Z ) ∼ → E [ N ], rational over some extension K ′ of K ( m ). By the universality of E , we have the following diagram depending on the choice of level- N structure. E α ∗ / / (cid:15) (cid:15) E (cid:15) (cid:15) Spec( K ′ ) / / M Denote by ˜ E the fiber over the cusps of the connected component of thegeneralized elliptic curve over the compactification of M . Then we candefine Isom = Isom( G m , ˜ E ). Isom is a µ torsor over the subschemeof cusps, and we consider the subset Q [Isom] ( k ) ⊆ Q [Isom] where µ actsby ( − k . Define the horospherical map ̺ k : Q [ E [ N ]] → Q [Isom] ( k ) explicitly by ̺ k ( ψ )( g ) = N k k !( k + 2) X t ψ ( g − t ) B k +2 ( t N ) , where t = ( t , t ) ∈ ( Z /N Z ) and B k ( x ) is the k th Bernoulli polyno-mial. When k > ̺ is well-defined for divisors of any degree.For an elliptic curve over any base, Beilinson [Bei] constructs anEisenstein symbol E k M : Q [ E [ N ]] → H k +1 M ( E k , k +1) which is preservedunder base change. For the universal elliptic curve E , we also have aboundary map res k : H k+1 M ( E k , k + 1) → Q [Isom] (k) coming from the long exact cohomology sequence, and another mapalso called the Eisenstein symbolEis k : Q [Isom] (k) → H k+1 M ( E k , k + 1) with res k ◦ Eis k = id. The following diagram commutes when restrictingto degree zero divisors. Q [ E [ N ]] ̺ / / Q [Isom] ( k ) Eis / / H k +1 M ( E k , k + 1) α ∗ (cid:15) (cid:15) Q [ E [ N ]] E k M / / α O O H k +1 M ( E k , k + 1)Indeed, the horospherical map above was computed by Schappacherand Scholl to be the composition E k M ◦ res k [SS]. Combining this factwith base change, the diagram commutes, and we can compute theEisenstein symbol at torsion points on the elliptic curve. Moreover,this computation does not depend on the choice of full level structuresince the assignment of Eisenstein symbols commutes with the GL action on the torsion sections and is thus invariant under the trace Y ( N ) → Y ( N ) [K1, Lemma 3.1.2]To show that E k M ( f β ′ ) = E k M ( f β ), it suffices to show that f β ′ − f β ∈ ker ̺, As the action of G m preserves the identity section on the curve, ̺ − j (0)( g ) = N − j ( − j )!(2 − j ) B − j (0) . We compute ̺ − j X p ∈ E ( C )[ a ] ( p ) ( g ) = a − j ( − j )!(2 − j ) X ( p )=( t ,t ) ∈ ( Z /a Z ) B − j (cid:18) t a (cid:19) = a − j ( − j )!(2 − j ) a − X i =0 B − j (cid:18) ia (cid:19) . Moreover, the distribution relation B k ( X ) = a k − a − X i =0 B k (cid:18) X + ia (cid:19) implies that ̺ − j X p ∈ E ( C )[ a ] ( p ) ( g ) = 1 a − j ̺ − j (0)( g ) , which completes the proof of the lemma. (cid:3) NTC FOR ALMOST ABELIAN EXTENSTIONS 11
In order to study Hecke characters of K ( m ), we must consider theimage of the elements above in the cohomology of the number field.To this end, Deninger constructs the Kronecker map, K M , which is aprojector given by the composition H − j M ( E − j , − j ) ( id,θ E ( √ d K ) − j, ∗ / / K M * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ H − j M ( E − j , − j ) π − j, ∗ (cid:15) (cid:15) H M (Spec( K ( m )) , − j ) , where the map π − j ∗ is a proper push forward.3.3. Deninger’s Theorem.
We now have the tools to prove the fol-lowing adaptation of [D2, Theorem 3.1].
Theorem 3.3.
For every ideal f | m , there are motivic elements ξ f ( j ) ∈ H M ( K ( m ) , − j ) with the property that if χ is a rational character of G m of conductor f , then e χ ( ρ ∞ ( ξ f ( j ))) = (2 N f ) − (1+ j ) Φ( m )( − j ( − j )!Φ( f ) L ′ ( ¯ χ, j ) η Q . where η Q = e χ · τ is a basis of the χ -component of ( H B (Spec( F )( C ) , Q ( j )) + ) ∗ ,and Φ is Euler’s totient function. Moreover, the resulting elementsthe Betti cohomology form a norm-compatible system in the followingsense: If χ is a character of conductor f , then e χ ( ρ ∞ ( w f /w pf Tr K ( pf ) /K ( f ) ξ fp ( j )) = ( e χ ( ρ ∞ ( ξ f ( j )) p | f (1 − ¯ χ ( p ) N p − j ) e χ ( ρ ∞ ( ξ f ( j )) p ∤ f . Remark . The trace map in the above theorem should be understoodas corestriction. Moreover, this compatibility should already hold forthe elements ξ f ( j ), as under the ℓ -adic regulator these are the pullbackof the polylogarithm sheaf. In fact, Scholl proves some compatibilityfor the Eisenstein symbol on the universal elliptic curve in [Sch, A.2],but the above theorem is sufficient for our purposes. Proof.
Recall that we have fixed a choice of an embedding τ : K ֒ → C and of a uniformization E ≃ C / Ω O K . The torsion point f β is de-pendent on the choice of id`ele ρ f . By the main theorem of complexmultiplication, the Artin symbolArt( ρ f ) − : E → Art( ρ f ) E maps the pair ( E ( C ) , f β ) to ( C / Ω f , f ) since f β = Ω f − f and theideal f − f ( ρ f ) ≡ f . Indeed, the restrictions on the valuation of f f at each prime p | f in(3.1), give that ρ f , p /f f ∈ m ord p fp where m p is the maximal ideal in the local ring O K p . Moreover, onemay choose the id`eles ρ to be multiplicative in the sense that ρ fp = ρ f ρ p .Thus we define our motivic elements ξ f ( j ) := ( K M E − j (Art( ρ f ) − f β ) N f ≥ K M E − j (Art( ρ f ) − f β ′ ) N f ≤ f β ′ is the degree zero divisor used by Deninger. Notice that thecomputation of the ℓ -adic regulator is not possible for these divisors,but the norm compatibility allows us to bypass this difficulty.For an f -torsion point of E , we define the function M j ( x ) = X = γ ∈ Γ g ( x, γ ) g | γ | − j ) . In [D2, 3.2], Deninger computes that for an embedding τ of F into C , ρ ∞ ( K M E − j M ( f β ′ )) τ = − ( ˜ N N f ) − j A (Γ τ ) − j ( − j )! − j )! (2 p d K ) − j M j ( f β ′ τ ) . Deninger shows that we obtain a similar result when considering theoriginal torsion point f β . Indeed, by loc. cit. (2.6),˜ N − j M j ( f β ′ g ) = M j ( f β g ) . NTC FOR ALMOST ABELIAN EXTENSTIONS 13
For our purposes we must distinguish between the group G m and theprincipal homogeneous space of embeddings Hom K ( K ( m ) , C ), so ap-plying lemma 3.2 we compute ρ ∞ ( ξ f ( j ))= X τ ∈T (2 πi ) j (cid:18) − N f − j A (Γ τ ) − j ( − j )! − j )!(2 √ d K ) j M j (Art( ρ f ) − f β τ ) (cid:19) · τ = X g ∈ G m (2 πi ) j (cid:18) − N f − j − A (Γ g ) − j ( − j )! − j )!(2 √ d K ) j M j ( f β Art( ρ f ) − g ) (cid:19) · gτ = X g ∈ G m (2 πi ) j (cid:18) − N f − j − A (Γ g ) − j ( − j )! − j )!(2 √ d K ) j M j ( f β g ) (cid:19) · Art( ρ f ) gτ = X g ∈ G m g − Art( ρ f ) − (2 πi ) j (cid:18) − N f − j − A (Γ g ) − j ( − j )! − j )!(2 √ d K ) j M j ( f β g ) (cid:19) · τ . The analysis over Q [ G m ] is done character by character, so one projectsto the χ -isotypical component(3.2) e χ ( ρ ∞ ( ξ f ( j )) = X g ∈ G m − N f − j − A (Γ g ) − j ( − j )! − j )!(2 √ d K ) j M j ( f β g ) ¯ χ ( g ) ¯ χ ( ρ f ) ! · η Q where η Q = e χ · (2 πi ) j τ is the basis of e χ (cid:0) M + ∗ B (cid:1) determined by thechoice of embedding. Comparing equation (3.2) with the formula inproposition 3.1 we have the first part of the theorem. A careful readerwill note that there is a difference of a factor of N f between our formulaand Deninger’s formula. This is due to the fact that we scale ρ ∞ bythat factor in have agreement with [HK1].To deduce the norm compatibility, we first note that corestrictioncommutes with the regulator map, so it suffices to study the elements w f /w pf Tr K ( pf ) /K ( f ) ρ ∞ ( ξ fp ( j )). By the computation in (3.2) we have thatTr K ( pf ) /K ( f ) ρ ∞ ( ξ fp ( j ))= Tr K ( pf ) /K ( f ) X g ∈ G m − g − Art( ρ fp ) − (2 πi ) j N fp − j A (Γ g ) − j ( − j )! − j )!(2 √ d K ) j M j ( fp β g ) · τ = X g ∈ G m − g − Art( ρ fp ) − (2 πi ) j N fp − j A (Γ g ) − j ( − j )! − j )!(2 √ d K ) j Tr K ( pf ) /K ( f ) M j ( fp β g ) · τ . Focusing on M j , we proceed by taking first the case of p | f .Tr K ( pf ) /K ( f ) Art( ρ p ) − M j ( fp β g ) =Tr K ( pf ) /K ( f ) Art( ρ p ) − M j ( fp β g )=Tr K ( pf ) /K ( f ) M j ( Frob p fp β g )= w pf /w f X u ∈ Frob − , ∗ p f β g M j ( f β g + u )= w pf /w f N p j M j ( f β g )(3.3)Here the u are the primitive p th roots of p β g resulting from pullingback by the isogeny E Frob − p → Frob p E , and the equality in (3.3) followsfrom a formula in the proof of [D2, prop. 2.6]. Now in the case that p ∤ f , there is a unique point u which is not a primitive root and hencein Frob − ∗ p f β g but not a conjugate of any primitive root. We add andsubtract this point from the trace to conclude thatArt( ρ p ) − Tr K ( pf ) /K ( f ) M j ( fp β g ) = (1 − Frob − p ) w pf /w f N p j M j ( f β g ) . Thus, we have shown that for a character χ of conductor f e χ ( ρ ∞ ( w f /w pf Tr K ( pf ) /K ( f ) ξ fp ( j )) = R f ,j L ′ ( ¯ χ, j ) η Q p | f (1 − ¯ χ ( p ) N p − j ) R f ,j L ′ ( ¯ χ, j ) η Q p ∤ f , where R f ,j = ( − N f ) − − j Φ( m )( − j )!Φ( f ) . (cid:3) NTC FOR ALMOST ABELIAN EXTENSTIONS 15 ℓ -adic regulator. We now study the image of the elements ξ f ( j )under the ´etale Chern class map, which can be considered an ℓ -adicregulator. We begin this section with a brief review of the Euler systemof elliptic units. We use Kato’s description of these elements, and referto [F3] for a comparison with more classical constructions. Lemma 3.5. ( [Ka, 15.4.4] Let E be an elliptic curve over a field F with complex multiplication O K ∼ = End F ( E ) and let a be an ideal of O K prime to . Then there is a unique function a Θ E ∈ Γ( E \ a E, O × ) satisfying (i) div ( a Θ E ) = N a · (0) − E a , where E a denotes the a -torsion pointsof E . (ii) For any b ∈ Z prime to a we have N b ( a Θ E ) = a Θ E where N b isthe norm map associated to the finite flat morphism E \ E b a → E \ E a given by multiplication with b .Moreover, for any isogeny φ : E → E ′ of CM elliptic curves where End F ( E ′ ) = O K , we have φ ∗ ( a Θ E ) = a Θ E ′ , in particular property (ii)also holds with b ∈ O K prime to a . Given f = 1 and any (auxiliary) a which is prime to 6 f we define ananalog of the cyclotomic unit 1 − ζ f by a z f = a Θ C / f (1)and for f = 1 we define a family of elements indexed by all ideals a of K by u ( a ) = ∆( O K )∆( a − ) . where ∆( τ ) = q τ Q (1 − q nτ ) for q τ = e πiτ is the Ramanujan ∆-function. Lemma 3.6. [F3, Lemma 2.2]
The complex numbers a z f and u ( a ) sat-isfy the following properties a) (Integrality) a z f ∈ ( O × K ( f ) f divisible by primes p = q O × K ( f ) , { v | f } f = p n for some prime p u ( a ) · O K (1) = a − O K (1) b) (Galois action) For ( c , fa ) = 1 with Artin symbol Art( c ) ∈ Gal( K ( f ) /K ) we have a z Art( c ) f = a z c − f ; u ( a ) Art( c ) = u ( ac ) /u ( c ) . This implies as in 3.5 a z N c − Art( c ) f = c z N a − Art( a ) f ; u ( a ) − Art( c ) = u ( c ) − Art( a ) . c) (Norm compatibility) For a prime ideal p one has N K ( pf ) /K ( f ) ( a z pf ) w f /w pf = a z f p | f = 1 a z − Frob − p f p ∤ f = 1 u ( p ) (Art( a ) − N a ) / f = 1e) (Kronecker limit formula). Let η be a complex character of G f .If f = 1 and η = 1 choose any ideal a so that η ( a ) = 1 . Then L ( η,
0) = ζ K (0) = − hw R η = 1 dds L ( s, η ) | s =0 = − − η ( a ) 112 w X σ ∈ G log | σ ( u ( a )) | η ( σ ) η = 1 , f = 1 dds L ( s, η ) | s =0 = − N a − η ( a ) 1 w f X σ ∈ G f log | σ ( a z f ) | η ( σ ) f = 1 . Remark . (i) The relations in b) show the auxiliary nature of a . (ii)The Galois action in b) together with the invariance under homothetyshows that the Galois conjugates of a z f are the numbers a Θ E ( α ) where( E, α ) runs through all pairs with E/ C an elliptic curve and α ∈ E ( C )a primitive f -division point.We compute the image of ξ f ( j ) under the ´etale Chern class map ρ et in terms of elliptic units. Theorem 3.8.
For all = f | m , we have that ρ et ( ξ f ( j )) = N f − − j w f ( N a − Art( a ))( − j )! Q l | ℓ (1 − Frob − l ) · (cid:0) Tr K ( ℓ n f ) /K ( f ) a z ℓ n f ζ ⊗− jℓ n (cid:1) n up to a sign, where a ∤ ℓ f is an auxilliary ideal and the a z ℓ n f are ellipticunits.Proof. As E is an abelian variety, the Todd classes vanish and thefollowing diagram commutes. H − j M ( E − j , − j ) ρ et / / K M (cid:15) (cid:15) H − jet ( E − j , Q ℓ (1 − j )) K ℓ (cid:15) (cid:15) H M (Spec( K ( m )) , − j ) ρ et / / H et ( K ( m ) , Q ℓ (1 − j )) NTC FOR ALMOST ABELIAN EXTENSTIONS 17
By [HK1, Theorem 2.2.4], the ´etale realization of Eisenstein symbolcan be computed in terms of the pullback of the elliptic polylogarithmsheaf along torsion sections. Indeed, ρ et ( ξ f ( j )) = K ℓ ( ρ et ( E − j ( ρ f · f β ))) = Art( ρ f ) − · N f − j − K ℓ ( f β ∗ P ol Q ℓ ) − . Happily, Kings computes this pullback up to a sign for an elliptic curveover any base [K2, Theorem 4.2.9] using the geometric elliptic polylogunder the assumption that ℓ ∤ f . So we can now consider the action ofthe Kronecker map K ℓ on(3.4)( f β ∗ P ol Q ℓ ) − j = ±N f j N a ([ a ] − j N a − − j )! δ X [ ℓ n ] t n = f β a Θ E ( − t n )( ˜ t n ) ⊗− j n which is an element of H − jet ( E − j , Q ℓ (1 − j )). Here, δ is the con-necting homomorphism in a Kummer sequence, a ⊂ O K is chosenprime to ℓ f , [ a ] is the corresponding isogeny, and ˜ t n is a projection of t n ∈ E [ f ℓ n ] = E [ f ] ⊕ E [ ℓ n ] to E [ ℓ n ]. Kings gives the projection map asa composition E [ f ℓ n ] Art( ρ f ) − −→ Art( ρ f ) E [ ℓ n ] Art( ρ f ) −→ E [ ℓ n ] , which accounts for the multiplication of his result by N f j above.For a point t ∈ E [ ℓ n ], we define γ ( t ) k := < t, √ d K t > ⊗ k where <, > is the Weil pairing. Following section 5.1.1 of [K2] we have that K ℓ (˜ t ⊗− jn ) = γ (˜ t n ) − j = ζ ⊗− jℓ n and K ℓ ([ a ] − j ) = N a − j , where ζ nℓ = e πiℓn is the canonical ℓ n th root of unity. Note also thatthe Artin automorphism Art( a ) acts on the space H ( K ( m ) , Q ℓ ) viamultiplication by N a , and thus on the space H ( K ( m ) , Q ℓ (1 − j )) by N a − j . We conclude that ρ et ( ξ f ( j )) = Art( ρ f ) − N a − σ ( a ) · N f − − j δ X [ ℓ n ] t n = f β a Θ E ( − t n ) ζ ⊗− jℓ n n . Now, it follows from the proof of lemma 3.10 below that the Kummermap δ gives an isomorphism O K ( m ) [ m ℓ ] × ⊗ Q ℓ ( − j ) ≃ H ( K ( m ) , Q ℓ (1 − j )). In the sequel, we will drop the map δ from our formulas andconsider the equalities as occurring inside of the unit group. As weconsider imaginary quadratic fields with any class number, we needthe following analog of [K2, Lemma 5.1.2]. Lemma 3.9.
For any rational prime ℓ , Y l | ℓ (1 − Frob − l ) − X [ ℓ n ] t n =Ω f − f a Θ E ( − t n ) ⊗ ζ ⊗− jℓ n n = w f (cid:0) Tr K ( ℓ n f ) /K ( f ) a Θ E ( − s n ) ⊗ ζ ⊗− jℓ n (cid:1) n , where s n is a primitive ℓ n th root of f β .Proof. Let l be a prime of K and ν = ord l ( f ). Define t r via the maintheorem of CM so that Frob − r l t r = Ω f − f and write t r = (˜ t r , t r, ) ∈ E [ l r + ν ] ⊕ E [ f ] = E [ l r f ]. We also abuse notation and write l i t r forFrob − i l t r . Define a filtration F • on the set H l r = { l r t r = Ω f − f } by F ir := { t r = (˜ t r , t r, ) ∈ H l r,t : l r + ν − i ˜ t r = 0 } . The Frobenius at l acts via (Frob − l ) ζ ⊗ kℓ r = ζ ⊗ kℓ r − and fixes Tr K ( l r f ) /K ( l r − i f ) a Θ E ( − s r )as the Galois group is generated by the frobenius. Thus, we computeFrob − i l Tr K ( l r f ) /K ( l r − i f ) a Θ E ( − s r ) ⊗ ζ ⊗− jℓ r = Tr K ( l r f ) /K ( l r − i f ) a Θ E ( − (˜ s r , s r, )) ⊗ ζ ⊗− jℓ r − i = a Θ E ( − (˜ s r − i , s r − i, )) ⊗ ζ ⊗− jℓ r − i . The second equality follows from the distribution relation for ellipticunits in lemma 3.6. Notice that the elliptic function a Θ E does notchange in the distribution relation even though the curve does becausethe lattices are homothetic.The Galois group Gal( K ( l r − i f ) /K ( f )) acts transitively on F ir \ F i +1 r with each conjugate appearing w f times. Hence we can writeFrob − i l Tr K ( l r f ) /K ( f ) a Θ E ( − s r ) ⊗ ζ ⊗− jℓ r = 1 w f X t r − i ∈ F ir \ F i +1 r a Θ E ( − (˜ t r − i , t r − i, )) ⊗ ζ ⊗− jℓ r − i These elements are annihilated by l r , so summing over i we can takethe limit as r → ∞ to get (cid:0) X l r t r = f β a Θ E ( − t r ) ⊗ ζ ⊗− jℓ r (cid:1) r = w f r X i =1 (Frob − l ) i Tr K ( l r f ) /K ( f ) a Θ E ( − s r ) ⊗ ζ ⊗− jℓ r ! r = w f (1 − Frob − l ) − (cid:0) Tr K ( l r f ) /K ( f ) a Θ E ( − s r ) ⊗ ζ ⊗− jℓ r (cid:1) r . For ℓ inert in K , the lemma is proved, and for ℓ split or ramified in K we apply the results to Tr K ( ℓ n f ) /K ( f ) = Tr K ( ℓ n f ) /K ( l n f ) Tr K ( l n f ) /K ( f ) . (cid:3) NTC FOR ALMOST ABELIAN EXTENSTIONS 19
Again, by the main theorem of complex multiplication, Art( ρ f ) − · s n gives a primitive torsion point of 1 mod f on the curve Art( ρ f ) E with C / Ω f ≃ Art( ρ f ) E ( C ). Therefore, we effectively undo our choice of f β viathe identity Art( ρ f ) − a Θ E ( − s n ) = a z f ℓ n . In particular, we have shown that the χ component is given by(3.5) e χ · ρ et ( ξ f ( j )) = Y l | ℓ (1 − χ ( l ) N l − j ) − N f − − j w f ( − j )! (cid:0) Tr K ( ℓ n f ) /K ( f ) z ℓ n f ζ − jℓ n (cid:1) n . where we follow Kato to set z ℓ n f = ( N a − σ ( a )) − a z ℓ n f . This completesthe proof of theorem 3.8. (cid:3) Putting it all together.
We can now compute the image of thespecial value L ∗ ( A M ) under the composition ϑ ℓ ϑ ∞ . We begin by study-ing the complex which computes the cohomology groups of interest. Let µ = ord ℓ m be compound notation denoting the following ℓ µ = l µ l µ ℓ = l l split l µ ℓ = l ramified ℓ µ ℓ inert , where µ , µ ∈ Z , and we write m = m ℓ µ . We choose a projective G Q -stable Z ℓ [ G m ] lattice T ′ ℓ = H et (Spec( K ( m ) ⊗ K K ) , Z ℓ ) = T ℓ ( − j )in the ℓ -adic realization, M ℓ ( − j ) = H et (Spec( K ( m ) ⊗ K K ) , Q ℓ ) , define a perfect complex of Z ℓ [ G m ]-modules,(3.6) ∆( K ( m )) := R Γ( O K [ 1 m ℓ ] , T ′ ℓ (1))For any finite set of places S of K ( m ), the Z [ G m ] module X S isdefined to be the kernel of the sum map0 → X S ( K ( m )) → Y S ( K ( m )) → Z → Y S ( K ( m )) := L v ∈ S Z . When there is no confusion we will sup-press the field. Lemma 3.10.
The cohomology of ∆( K ( m )) is given by a canonicalisomorphism, H (∆( K ( m )) ≃ H ( O K ( m ) [ 1 m ℓ ] , Z ℓ (1)) ≃ O K ( m ) [ 1 m ℓ ] × ⊗ Z Z ℓ , a short exact sequence, → Pic( O K ( m ) [ 1 m ℓ ]) ⊗ Z Z ℓ → H (∆( K ( m ))) → X { v | m ℓ ∞} ⊗ Z Z ℓ → , and H i (∆( K ( m ))) = 0 for i = 1 , .Proof. By Shapiro’s lemma, R Γ( O K [ 1 m ℓ ] , T ′ ℓ (1)) ≃ R Γ( O K ( m ) [ 1 m ℓ ] , Z ℓ (1)) . The Kummer sequence0 → µ ℓ n → G m ℓ n → G m → ℓ n → H i ( O K ( m ) [ 1 m ℓ ] , G m ) → H i +1 ( O K ( m ) [ 1 m ℓ ] , µ ℓ n ) → H i +1 ( O K ( m ) [ 1 m ℓ ] , G m ) ℓ n → . The Galois cohomology is then computed by the short exact sequences0 → H ( O K ( m ) [ 1 m ℓ ] , G m ) /ℓ n → H ( O K ( m ) [ 1 m ℓ ] , µ ℓ n ) → H ( O K ( m ) [ 1 m ℓ ] , G m )[ ℓ n ] → → H ( O K ( m ) [ 1 m ℓ ] , G m ) /ℓ n → H ( O K ( m ) [ 1 m ℓ ] , µ ℓ n ) → H ( O K ( m ) [ 1 m ℓ ] , G m )[ ℓ n ] → H ( O K ( m ) [ 1 m ℓ ] , µ ℓ n ) ≃ H ( O K ( m ) [ 1 m ℓ ] , G m )[ ℓ n ] . The Galois cohomology of O K ( m ) [ m ℓ ] × is given by [NSW, Prop. 8.3.10] H i ( O K ( m ) [ 1 m ℓ ] , G m ) = ( O K ( m ) [ m ℓ ] × i = 0Pic( O K ( m ) [ m ℓ ]) i = 1 , and H ( O K ( m ) [ m ℓ ] , G m )( ℓ ) = 0. Moreover, H ( O K ( m ) [ m ℓ ] , G m ) is ℓ -divisible (loc. cit. Corollary 8.3.11). Hence, taking inverse limits wecompute that H i ( O K ( m ) [ 1 m ℓ ] , Z ℓ (1)) = O K ( m ) [ m ℓ ] × ⊗ Z Z ℓ i = 1Pic( O K ( m ) [ m ℓ ]) ⊗ Z Z ℓ i = 20 otherwise. (cid:3) NTC FOR ALMOST ABELIAN EXTENSTIONS 21
Remark . A similar computation is given in [BF2, Prop 3.3] un-der the condition that the S -restricted class group is trivial. In ourcase, since K is a totally imaginary field, the Tate cohomology is justthe usual cohomology with compact supports. Thus the definition of∆( K ( m )) is simplified since Artin-Verdier duality gives(3.7) R Hom Z ℓ ( R Γ c ( O K ( m ) [ 1 m ℓ ] , Z ℓ ) , Z ℓ )[ − ≃ R Γ( O K ( m ) [ 1 m ℓ ] , Z ℓ (1)) . For invertible Z ℓ [ G m ]-modules, the dual of the inverse module (orvice versa) is isomorphic to the original module with the action of G m twisted by the automorphism g g − . We denote the twisted actionwith a Z ℓ [ G m ] ∆( K ( m )) ≃ Det Z ℓ [ G m ] R Γ c ( O K [ 1 m ℓ ] , T ′ ℓ ) . Theorem 3.12.
The element A ϑ ℓ ( A ϑ ∞ ( L ∗ ( A M, − )) of Det A ℓ ∆( K ( m )) = Y χ ∈ ˆ G (Det Q ℓ ( χ ) ∆( K ( m )) ⊗ Q ℓ ( χ )) has χ component given by R χ,j (cid:0) Tr K ( f χ, ℓ n ) /K ( f χ ) ( a z f χ, ℓ n ζ ⊗− jℓ n ) (cid:1) − n ⊗ ζ ⊗− jℓ ∞ · e χ τ , where R χ,j = Q p | m (1 − χ ( p ) N p − j ) − K ( m ): K ( f χ )]( − j ( N a − χ ( a ) N a − j ) and ζ ℓ ∞ = ( ζ ℓ n ) n .Proof. The dual of the regulator isomorphism ρ ∨∞ : H B ( K ( m )( C ) , Q ( j )) + ⊗ Q R ∼ → K − j ( O K ( m ) ) ∗ ⊗ Z R induces an isomorphism of rank 1 A ⊗ R -modules A ϑ ∞ : A ⊗ Q R → Ξ( A M ) ⊗ Q R , where we recall thatΞ( A M ) = ( K − j ( O K ( m ) ) ∗ ⊗ Z Q ) ⊗ H B ( K ( m )( C ) , Q ( j )) + . In Theorem 3.3 we proved that for f χ = 1, e χ ( ρ ∞ ( ξ f χ ( j ))) = ( − N f ) − − j Φ( m )( − j )!Φ( f ) L ′ ( ¯ χ, j ) η Q , where η Q is a basis of e χ ( M + ∗ B ). Moreover, for f χ = 1, we have acomputation via trace maps e χ · Tr K ( q ) /K (1) ρ ∞ ( w K (1 − Frob − q ) − ξ q ( j )) = ( − − − j Φ( m )( − j )! L ′ ( ¯ χ, j ) η Q , where we take the primitive L -function for χ . As corestriction com-mutes with the regulator maps, we will sometimes abuse notation atwrite ξ ( j ) for a choice of Tr K ( q ) /K (1) q ξ ( j ).Since both H B ( K ( m )( C ) , Q ( j )) + and K − j ( O K ( m ) ) ⊗ Z Q are invert-ible A -modules duality manifests in terms of the twist g g − accord-ing to the computationΞ( A M ) = ( K − j ( O K ( m ) ) ∗ ⊗ Z Q ) ⊗ ( H B ( K ( m )( C ) , Q ( j )) + , − ) = ( K − j ( O K ( m ) ) ⊗ Z Q ) − ⊗ ( H B ( K ( m )( C ) , Q ( j )) ∗ ) + = ( K − j ( O K ( m ) ) ⊗ Z Q ) − ⊗ Y ( − j ) , (3.8)where for v a place of K ( m ) Y ( − j ) := M v |∞ Q · (2 πi ) − j . The Gal( C / R )-equivariant perfect pairing M τ ∈T R · (2 πi ) j × M τ ∈T C / R · (2 πi ) − j → M τ ∈T C / πi · R Σ → R for T = Hom( K ( m ) , C ) identifies the Q -dual of H B ( K ( m )( C ) , Q ( j ))with L τ ∈T Q · (2 πi ) − j . Taking invariants under complex conjugationgives the equality in (3.8). We compute that the χ components of A ϑ ∞ ( L ∗ ( A M, − ) = ( L ∗ ( A M, − ) A ϑ ∞ (1) are given by( A ϑ ∞ ( L ∗ ( A M, − )) χ = ( − N f ) − − j Φ( m )( − j )!Φ( f ) [ ξ f χ ( j )] − ⊗ (2 πi ) − j e χ τ . Denote by ∆( K ( m )) j the “twist” of the Z ℓ [ G m ]-module ∆( K ( m )). Namely,∆( K ( m )) j := ( R Γ( O K [ 1 m ℓ ] , T ℓ ) . The natural isomorphismDet Z ℓ [ G ] ∆( K ( m )) = (Det Z ℓ [ G ] R Γ c ( O K [ 1 m ℓ ] , T ′ ℓ ) ∗ ) − ≃ Det Z ℓ [ G ] R Γ c ( O K [ 1 m ℓ ] , T ′ ℓ ) induces Det Z ℓ [ G ] ∆( K ( m )) j ≃ Det Z ℓ [ G ] R Γ c ( O K [ 1 m ℓ ] , T ℓ ) . NTC FOR ALMOST ABELIAN EXTENSTIONS 23
By lemma 3.10 there are isomorphisms in cohomology H (∆( K ( m )) j ) ⊗ Z ℓ Q ℓ ≃ H ( O K ( m ) [ 1 m ℓ ] , Q ℓ (1 − j )) H (∆( K ( m )) j ) ⊗ Z ℓ Q ℓ ≃ M τ ∈T Q ℓ ( − j ) ! + with H i (∆( K ( m )) j ) = 0 for i = 1 , f χ = 1,( A ϑ ℓ ◦ A ϑ ∞ ( L ∗ ( A M, − )) χ = Y p | m ℓ (1 − χ ( p ) N p − j ) − ( − N f ) − − j Φ( m )( − j )!Φ( f ) ρ et ( ξ f χ ( j )) − ⊗ ζ ⊗− jℓ ∞ · σ, and for f χ = 1, we choose a q | m to show(3.9) ( A ϑ ℓ ◦ A ϑ ∞ ( L ∗ ( A M, − )) χ = Y q = p | m ℓ (1 − χ ( p ) N p − j ) − ( − j Φ( m ) ρ et ( w K ξ ( j )) − ⊗ ζ ⊗− jℓ ∞ · σ. Theorem 3.8 states that for any 1 = f | m , ρ et ( ξ f ( j )) = N f − − j w f ( N a − σ ( a )) Q l | ℓ ( − j )!(1 − Frob − l ) · (cid:0) Tr K ( ℓ n f ) /K ( f ) a z ℓ n f ζ ⊗− jℓ n (cid:1) n . We recall that [ K ( f ) : K (1)] = Φ( f ) w f /w K where w K ∈ { , , } isthe number of roots of unity in the imaginary quadratic field K , and w f is the number of roots of unity in K which are congruent to 1modulo f . For f large enough (at least bigger than 2) this number is1. Recall that we have chosen m so that w m = 1, so we have thatΦ( m ) / Φ( f χ ) = [ K ( m ) : K ( f χ )] w f χ . What’s more, if ( ℓ, f ) = 1, then bylemma 3.6, (cid:0) Tr K ( ℓ n f ) /K ( f ) a z ℓ n f ζ ⊗− jℓ n (cid:1) n = (cid:0) Tr K ( ℓ n f ) /K ( f ) (cid:0) Tr K ( ℓ n + µ f ) /K ( ℓ n f ) a z ℓ n f (cid:1) ζ ⊗− jℓ n (cid:1) n = (cid:0) Tr K ( ℓ n f ) /K ( f ) a z ℓ n f ζ ⊗− jℓ n (cid:1) n where µ denotes the compound notation discussed above. Thus for f χ = 1 we have computed the χ -component of the image of the L -value. When f χ = 1, choose q so that w q = 1 and compute ρ et ( w K Tr K ( q ) /K (1) ξ q ( j ))= w K N a − Art( a )) Q l | ℓ (1 − Frob − l ) · (Tr K ( ℓ n q ) /K (1) a z ℓ n q ζ ⊗− jℓ n ) n = (1 − Frob − q )( N a − Art( a )) Q l | ℓ (1 − Frob − l ) · (Tr K ( ℓ n ) /K (1) a z ℓ n ζ ⊗− jℓ n ) n . Substituting the formulas for ρ et completes the proof of the theorem. (cid:3) Descent from the main conjecture of Iwasawa theory
Formulation of the conjecture.
We first formulate the 2-variablemain conjecture by considering the tower of ray class fields over K ( m )unramified outside of the primes above ℓ . The Iwasawa algebraΛ := lim ←− n Z ℓ [ G m ℓ n ] ≃ Z ℓ [ G tor m ℓ ∞ ][[ S, T ]]is a finite product of complete local 3-dimensional Cohen-Macaulayrings, where G tor m ℓ ∞ is the torsion subgroup of G m ℓ ∞ = lim ←− n G m ℓ n . Λ isregular if and only if ℓ ∤ G tor m ℓ ∞ . In general, this torsion subgroup isnot G m ℓ where m is the prime to ℓ part of m . (Consider the case that ℓ | h K .) The elements S, T ∈ Λ depend on the choice of a complement F ≃ Z ℓ of the torsion subgroup in G m ℓ ∞ as well as the choice topologicalgenerators γ , γ of F . The cohomology of the perfect complex of Λmodules, ∆ ∞ = lim ←− n ∆( K ( m ℓ n ))is computed by functoriality. By Lemma 3.10, H i (∆ ∞ ) = 0 for i = 1 , H (∆ ∞ ) ≃ U ∞{ v | m ℓ } := lim ←− n O K ( m ℓ n ) [ 1 m ℓ ] × ⊗ Z Z ℓ , and a short exact sequence,0 → P ∞{ v | m ℓ } → H (∆ ∞ ) → X ∞{ v | m ℓ ∞} → , NTC FOR ALMOST ABELIAN EXTENSTIONS 25 where P ∞{ v | m ℓ } := lim ←− n Pic( O K ( m ℓ n ) [ 1 m ℓ ]) ⊗ Z Z ℓ X ∞{ v | m ℓ ∞} := lim ←− n X { v | m ℓ ∞} ( K ( m ℓ n )) ⊗ Z Z ℓ . The limits are taken with respect to the Norm maps, which on themodule Y S is the map sending a place to its restriction. We also consider K ( m ℓ n ) as a subfield of C and denote the corresponding archimedeanplace by σ m ℓ n . Notice that for f | m , the elliptic units a z f ℓ n discussedin section 3.4 form a Norm-compatible system of units. We set a η f := ( a z f ℓ n ) n>> ∈ U ∞{ v | m ℓ } σ := ( σ m ℓ n ) n>> ∈ Y ∞{ v | m ℓ ∞} We fix an embedding ¯ Q ℓ → C and identify ˆ G with the set of ¯ Q ℓ -valuedcharacters. The total ring of fractions(4.1) Q (Λ) ∼ = Y ψ ∈ ( ˆ G tor m ℓ ∞ ) Q ℓ Q ( ψ )of Λ is a product of fields indexed by the Q ℓ -rational characters of G tor m ℓ ∞ . Since for any place w of K , the Z [ G m ℓ n ]-module Y { v | w } ( K ( m ℓ n ))is induced from the trivial module Z on the decomposition group D w ⊆ G m ℓ n , and for w = ∞ (resp. nonarchimedean w ) we have [ G m ℓ n : D w ] =[ K ( m ℓ n ) : K ] (resp. the index [ G m ℓ n : D w ] is bounded as n → ∞ ), onecomputes easily(4.2) dim Q ( ψ ) ( Y ∞{ v | mℓ ∞} ⊗ Λ Q ( ψ )) = 1for all characters ψ . Note that the inclusion X ∞{ v | m ℓ ∞} ⊆ Y ∞{ v | m ℓ ∞} be-comes an isomorphism after tensoring with Q ( ψ ), and thus by the unittheorem(4.3) dim Q ( ψ ) ( U ∞{ v | m ℓ } ⊗ Λ Q ( ψ )) = 1 . So we have that e ψ ( a η − m ⊗ σ ) is a Q ( ψ )-basis ofDet − Q ( ψ ) ( U ∞{ v | m ℓ } ⊗ Λ Q ( ψ )) ⊗ Det Q ( ψ ) ( X ∞{ v | m ℓ ∞} ⊗ Λ Q ( ψ ) ∼ =Det Q ( ψ ) (∆ ∞ ⊗ Λ Q ( ψ )) . The last isomorphism follows from the fact that the class group, P ∞{ v | m ℓ } is a torsion Λ-module. Hence we obtain an element L := ( N a − Art( a )) a η − m ⊗ σ ∈ Det Q (Λ) (∆ ∞ ⊗ Λ Q (Λ)) . Iwasawa Main Conjecture.
There is an equality of invertible Λ -submodules Λ · L = Det Λ ∆ ∞ of Det Q (Λ) (∆ ∞ ⊗ Λ Q (Λ)) . Theorem 4.1. [JLK, Theorem 5.7]
The Iwasawa main conjecture holds,for all prime number ℓ ∤ which are split in K .Remark . i) In order to prove the theorem, it is necessary to showthat the µ -invariant of a certain Iwasawa module vanishes. This followsfrom a result of Gillard [Gi, 3.4] when ℓ ∤ K . If one were toprove that the µ -invariant vanishes for non-split ℓ , the Iwasawa mainconjecture would follow immediately.ii) The statement of the theorem in [JLK] can be rewritten to coincidewith the conjecture above when one notes that∆ ∞ = R Γ( O K [ 1 m ℓ ] , Λ(1)) . Descent and proof of the main theorem.Theorem 4.3.
The Iwasawa main conjecture implies the local equi-variant Tamagawa number conjecture for the pair ( K ( m ) , G m ) when j < for every prime p = 2 . To prove this theorem we will show that the equality of Λ-modulesin the Iwasawa main conjecture descends to A ϑ ℓ ◦ A ϑ ∞ ( L ∗ ( A M, − ) · Z ℓ [ G m ] = Det Z ℓ [ G m ] ∆( K ( m ))in Det Q ℓ [ G m ] (∆( K ( m )) ⊗ Z ℓ Q ℓ ). We begin by proving a twisting lemma.For j ∈ Z we denote by κ j : G m ℓ ∞ → Λ × the character g χ cyclo ( g ) j g as well as the induced ring automorphism κ j : Λ → Λ. If there is no riskof confusion we also denote by κ j : Λ → Z ℓ [ G m ] ⊆ A ℓ the composite of κ j and the natural projection to Z ℓ [ G m ] or A ℓ . Lemma 4.4. a) For j ∈ Z there is a natural isomorphism ∆ ∞ ⊗ L Λ ,κ j Z ℓ [ G m ] → ∆( K ( m )) j . b) On the cohomology groups, the map H i (∆ ∞ ) → H i (∆ ∞ j ) induces u ( u n ∪ ζ ⊗− jℓ n ) n>> and s ( s n ∪ ζ ⊗− jℓ n ) n ≥ NTC FOR ALMOST ABELIAN EXTENSTIONS 27 where u = ( u n ) n ≥ ∈ lim ←− n H ( O K ( m ℓ n ) [ 1 m ℓ ] , Z /ℓ n Z (1)) ≃ U ∞{ v | m ℓ } = H (∆ ∞ ) and s = ( s n ) n ≥ ∈ lim ←− Z /ℓ n Z [ G m ℓ n ] · σ = Y ∞{ v |∞} Proof. (As in [F2, Lemma 5.1.3]) The automorphism κ j can be viewedas the inverse limit of similarly defined automorphisms κ j of the ringsΛ n := Z /ℓ n Z [ G m ℓ n ]. Let f n : Spec( O K ( m ℓ n ) [ m ℓ ]) → Spec( O K ( m ) ) bethe natural map. The sheaf F n := f n, ∗ f ∗ n Z /ℓ n Z is free of rank oneover Λ n with π (Spec( O K ( m ) ))-action given by the natural projection G Q → G m ℓ n , twisted by the automorphism g g − . There is a Λ n - κ − j -semilinear isomorphism tw j : F n → F n ( j ) so that Shapiro’s lemmagives a commutative diagram of isomorphisms(4.4) R Γ c ( O K ( m ) , F n ) tw j −−−→ R Γ c ( O K ( m ) , F n ( j )) y y R Γ c ( O K ( m ℓ n ) [ m ℓ ] , Z /ℓ n Z ) ∪ ζ ⊗ jℓn −−−→ R Γ c ( O K m ℓn [ m ℓ ] , Z /ℓ n Z ( j )) , with the horizontal arrows Λ n - κ − j -semilinear. Taking the Z /ℓ n Z -dualof the lower row (with contragredient G m ℓ n -action), we obtain a ◦ κ − j ◦ κ j -semilinear isomorphism R Γ( O K ( m ℓ n ) [ 1 m ℓ ] , Z /ℓ n Z ( j )) → R Γ( O K ( m ℓ n ) [ 1 m ℓ ] , Z /ℓ n Z ) . After passage to the limit this gives a κ j -semilinear isomorphism ∆ ∞ ≃ ∆ ∞ j , i.e. a Λ-linear isomorphism ∆ ∞ ⊗ Λ ,κ j Λ ≃ ∆ ∞ j . The part a) followsby tensoring over Λ with Z ℓ [ G m ]. For b), consider the inverse map ofthe lower row of (4.4) on the degree two cohomology given by H c ( O K ( m ℓ n ) [ 1 m ℓ ] , Z /ℓ n Z ) ∪ ζ ⊗ jℓn ← H c ( O K ( m ℓ n ) [ 1 m ℓ ] , Z /ℓ n Z ( j )) . Artin-Verdier duality says that H ic ( O K ( m ℓ n ) [ 1 m ℓ ] , Z /ℓ n Z ( j )) ∨ = H − i ( O K ( m ℓ n ) [ 1 m ℓ ] , Z /ℓ n Z (1 − j )) . Thus we have a dual map which is a κ j semi-linear isomorphism. H ( O K ( m ℓ n ) [ 1 m ℓ ] , Z /ℓ n Z (1)) ∪ ζ ⊗ jℓn → H ( O K ( m ℓ n ) [ 1 m ℓ ] , Z /ℓ n Z (1 − j )) . Moreover, we have a similar diagram to (4.4) on the level of sheaveswhere the lower row is obtained by taking invariants under complexconjugation(4.5) F n tw j −−−→ F n ( j ) y y H ( K ( m ℓ n ) ⊗ R , Z /ℓ n Z ) ∪ ζ ⊗ jℓn −−−→ H ( K ( m ℓ n ) ⊗ R , Z /ℓ n Z ( j )) . Again using the inverse map and taking the Z /ℓ n Z dual, we again havea κ j -semilinear isomorphism given by the cup product with ζ ⊗− jℓ n Λ n · σ Λ n · σ ∪ ζ ⊗− jℓ n . Taking inverse limits, we have part b). (cid:3)
We follow with the proof of Theorem 4.3.
Proof.
As ∆( K ( m )) j is a rank 1 Z ℓ [ G m ]-module, the image of L ⊗ A ϑ ℓ ◦ A ϑ ∞ ( L ∗ ( A M, − ) inside of the rational space ∆( K ( m )) ⊗ Z ℓ Q ℓ whichis a rank one module over A ℓ and thus splits over the Q ℓ -rational char-acters χ of G m . Thus, it suffices to show that( A ϑ ℓ ◦ A ϑ ∞ ( L ∗ ( A M, χ = ( L m ,j ) χ . for every Q ℓ -rational character χ of G m , where ( L m ,j ) χ is the image of L in ∆( K ( m )) j ⊗ Q ℓ ( χ ). Let q = q χ,j be the height 2 prime of Λ givenby the kernel of the composite ring homomorphism χκ j : Λ κ j → Λ → Z ℓ [ G ( m )] ⊆ A ℓ → Q ℓ ( χ ) .R := Λ q is a regular local ring of dimension 2 with residue field k := Q ℓ ( χ ). Let ∆ be the module ∆ ∞ q over the localized ring R . To indicatethe ℓ -divisibility of m and f χ , we continue with the compound notationabove m = m ℓ µ and f χ = f χ, ℓ µ ′ , where ( m , ℓ ) = ( f χ, , ℓ ) = 1. For ℓ = l l split, ℓ µ = l µ l µ , and for ℓ = l ramified, ℓ µ = l µ where µ and µ are integers. By the IwasawaMain Conjecture, we can consider L to be a basis of the R -module(Det Λ ∆ ∞ ) q which is isomorphic to Det R ∆ since localization is exact.Lemma 4.4 gives the following isomorphism of complexes of R -modules,∆ ⊗ L R k ≃ → ∆( K ( m )) j ⊗ Z ℓ [ G m ] k. Lemma 4.5. H i (∆ ⊗ L R k ) ≃ H i (∆) ⊗ R k. NTC FOR ALMOST ABELIAN EXTENSTIONS 29
Proof.
Indeed, if ( x, y ) is a regular sequence for R , then the Koszulcomplex is the resolution0 → R ( x y ) → R ⊕ R ( y, − x ) → R → k → . Thus, the homological spectral sequence for Tor degenerates to give anisomorphism H (∆ ⊗ L R k ) ≃ H (∆) ⊗ k and in degree 1 an exact sequence0 → Tor ( H (∆) , k ) → H (∆) ⊗ k → H (∆ ⊗ L R k ) → Tor ( H (∆) , k ) → . Now, the second degree cohomology is given by an exact sequence wherethe quotient is a free module (lemma 3.10)0 → P ∞{ v | m ℓ } → H (∆ ∞ ) → X ∞{ v | m ℓ ∞} → . Again, localization is exact, so we must show that the higher torsiongroups of the localized class groups are zero. As R is a 2-dimensionallocal ring, the localization R π at a height 1 prime is a DVR, and theimage of a η f χ, in H (∆) π is non-zero because of its relationship to thenon-vanishing L -value. Then, by [JLK, Section 5.5] , the fitting idealof ( P ∞ q ) π vanishes, and so by Nakayama’s lemma does P ∞ q . (cid:3) By lemma 4.5 the isomorphism of determinants φ : Det k (∆ ⊗ L R k ) → Det k (∆( K ( m )) j ⊗ Z ℓ [ G m ] k )can be computed as a map on the cohomology groups φ : O i =1 H i (∆) ⊗ k → O i =1 H i (∆ ⊗ L R k ) → O i =1 H i (∆( K ( m )) j ) ⊗ Q ℓ [ G ] k. To compute φ ( L ⊗ a η m and σ indepen-dently. Recall that for an ideal d | m N d := X τ ∈ Gal( K ( m ) /K ( d )) τ. When f χ, | d , N d is invertible is the ring R since χ ( N d ) = [ K ( m ) : K ( d )]. Thus, in the localized module ∆, the norm compatibility prop-erties of the elliptic units give the equality a η m = N − f χ, N f χ, a η m (4.6) = N − f χ, Y p | m , p ∤ f χ, (1 − Frob − p )( w m /w f χ, ) a η f χ, = ( w m /w f χ, ) X τ ∈ Gal( K ( m ) /K ( m ℓ µ ′ )) τ X τ ∈ Gal( K ( m ) /K ( m ℓ µ ′ )) τ − N − f χ, Y p | m , p ∤ f χ, (1 − Frob − p ) a η f χ, = (cid:0) w m ℓ µ ′ w f χ [ K ( m ) : K ( f χ )] (cid:1) Tr K ( m ) /K ( m ℓ µ ′ ) Y p | m , p ∤ f χ, (1 − Frob − p ) a η f χ, . The last equality in (4.6) can be deduced from the diagram of fieldsbelow. K ( m ℓ µ ′ ) w m ℓµ ′ w f χ, w m w f χ K ( m ) K ( f χ ) ♣♣♣♣♣♣♣♣♣♣♣ ▼▼▼▼▼▼▼▼▼▼ K ( m ) ◆◆◆◆◆◆◆◆◆◆◆ K ( f χ ) qqqqqqqqqq K ( f χ, )Thus, by Lemma 4.4 φ ( a η m ) =( w m ℓ µ ′ /w f χ )[ K ( m ) : K ( f χ )] − Y p | m , p ∤ f χ, (1 − χ ( p ) N p − j ) · Tr K ( m ) /K ( m ℓ µ ′ ) (Tr K ( m ℓ n ) /K ( m ) a z f χ, ℓ n ⊗ ζ ⊗− jℓ n ) n =[ K ( m ) : K ( f χ )] − Y p | m , p ∤ f χ, (1 − χ ( p ) N p − j )(Tr K ( f χ, ℓ n ) /K ( f χ ) a z f χ, ℓ n ⊗ ζ ⊗− jℓ n ) n . NTC FOR ALMOST ABELIAN EXTENSTIONS 31
The second equality follows from a similar diagram of fields K ( m ℓ n ) w f χw m ℓµ ′ K ( m ℓ µ ′ ) K ( f χ, ℓ n ) ♠♠♠♠♠♠♠♠♠♠♠♠ ◗◗◗◗◗◗◗◗◗◗◗◗◗ K ( m ℓ µ ′ ) ◗◗◗◗◗◗◗◗◗◗◗◗◗◗ K ( f χ, ℓ n ) ♠♠♠♠♠♠♠♠♠♠♠♠♠♠ K ( f χ )where we recall that we take m and n to be large enough that w m = 1and w f χ, ℓ n = 1. For the second degree cohomology, the situation issomewhat more simple. Indeed, by lemma 4.4, φ ( σ ) = e χ ( σ m ⊗ ζ ⊗− jℓ n ) n = e χ σ m ⊗ ζ ⊗− jℓ ∞ . Recalling that σ m was our fixed choice of embedding τ and multiplyingby N a − σ ( a ), we see that in fact(4.7) φ ( L ) = [ K ( m ) : K ( f χ )] Y p | m , p ∤ f χ, (1 − χ ( p ) N p − j ) − (Tr K ( f χ, ℓ n ) /K ( f χ ) z f χ, ℓ n ⊗ ζ ⊗− jℓ n ) − n ⊗ ζ ⊗− jℓ ∞ · e χ τ . Since χ ( p ) = 0 for p | f χ and 2 is a unit Λ q we have proved that φ ( L ⊗
1) = ( A ϑ ℓ ◦ A ϑ ∞ ( L ∗ ( A M, − )) . By its relation to the L -valueestablished in theorem 3.12, the image of ( N a − σ ( a )) − a η m does notvanish and thus is a basis of H (∆( K ( m )) j ) ⊗ Q ℓ ( χ ). Further, theimage of σ is a basis of H (∆( K ( m )) j ) ⊗ Q ℓ ( χ ), completing the proofof the theorem. (cid:3) References [Ba] F. Bars
On the Tamagawa number conjecture for Hecke characters , Math.Nach. (2011), 608–628[Bei] A. A. Beilinson
Higher regulators of modular curves , Contemp.Math. (1986), 1–34.[Bl] W. Bley On the equivariant Tamagawa number conjecture for abelianextensions of a quadratic imaginary field
Documenta Mathematics (2006), 73–118. [BK] S. Bloch and K. Kato L-functions and Tamagawa numbers of motives ,The Grothendieck Festschrift I, Profress in Math, Birkh¨auser (1990),333–400.[Bor] A. Borel
Stable real cohomology of Arithmetic Groups , Ann. Sci ENS (1974), 235–272.[BF1] D. Burns and M. Flach On Galois structure invariants associated to Tatemodules , Amer. J. Math. (1998), 1343–1397.[BF2] D. Burns and M. Flach
Tamagawa numbers for motives with (non-commutative) coefficients , Documenta Mathematica (2001), 501–569.[BF3] D. Burns and M. Flach Tamagawa numbers for motives with (non-commutative) coefficients II , Amer. J. Math. (2003), 475–512.[BF4] D. Burns and M. Flach
On the equivariant Tamagawa number conjecturefor Tate motives. II
Doc. Math. Extra Vol. (2006), 133163 .[BG] D. Burns and C. Greither
On the equivariant Tamagawa Number Conjec-ture for Tate Motives , Invent. Math.
53 (2003), 303–359.[D1] C. Deninger
Higher regulators and Hecke L -series of imaginary quadraticfields I , Invent. Math. (1989), 1–69.[D2] C. Deninger Higher regulators and Hecke L -series of imaginary quadraticfields II , Annals of Math. (1990), 131–158.[F1] M. Flach Euler characteristics in relative K -groups , Bull. London Math.Soc. (2000), 272–284.[F2] M. Flach The equivariant Tamagawa number conjecture: a survey , Stark’sconjectures: Recent work and new directions, Contemp. Math., vol 358,AMS, 2004, 79–125.[F3] M. Flach
Iwasawa theory and motivic L-functions , Pure and AppliedMath. Quarterly (2009) 255–294.[F4] M. Flach On the cyclotomic main conjecture for the prime 2 , J. ReineAngew. Math. (2011), 1–36[FPR] J.-M. Fontaine and B. Perrin-Riou
Atour des conjectures de Bloch et Ka-toL cohomologie Galoisienne et valeurs de fonctions L
Motives (Seattle),Proc. Symp. Pure Math. (1994) 599–706.[Ge] M. Gealy
On the Tamagawa number conjecture for motives attached tomodular forms
PhD Thesis, California Institute of Technology (2006).[Gi] R. Gillard
Fonctions L p-adiques deq corps quadratiques imaginaires et deleurs extensions ab´eliennes , J. Reine Agnew. Math. (1985), 76–91.[HK1] A. Huber and G. Kings
Degeneration of l-adic Eisenstein classes and ofthe elliptic polylog , Invent. Math. (1999), 545–594.[HK2] A. Huber and G. Kings
Bloch-Kato conjecture and main conjecture ofIwasawa theory for Dirichlet character
Duke Math. J. (2003) 393–464.[JLK] J. Johnson-Leung and G. Kings
On the equivariant main conjecture forimaginary quadratic fields , J. Reine Angew. Math. (2011), 75–114.[Ka] K. Kato p-adic Hodge theory and values of zeta functions of modularforms , Ast´erisque (2004), 117–290.[K1] G. Kings
K-theory for the polylogarithm of abelian schemes,
J. ReineAngew. Math. (1999), 103–116.[K2] G. Kings
The Tamagawa number conjecture for CM elliptic curves , Invent.Math. (2001), 571–627.
NTC FOR ALMOST ABELIAN EXTENSTIONS 33 [KM] F. Knudsen and D. Mumford
The projectivity of the moduli space of stablecurves I: Preliminaries on det and div , Math. Scand. (1976), 19–55.[Na] T. Navilarekallu On the equivariant Tamagawa number conjecture for A extensions of number fields J. of Number Theory (2006) 67–89.[N] J. Neukirch
The Beilinson conjecture for algebraic number fields
Beilin-son’s conjectures on special values of L -functions, Perspect. Math, vol 4,Academic Press, 1988, 193–247.[NSW] J. Neukirch, A. Schmidt, K. Winberg Cohomology of Number Fields ,Springer (2000).[SS] N. Schappacher and A. J. Scholl
The boundary of the Eisenstein symbol ,Math. Ann. (1991), 303–321.[Sch] A. J. Scholl
An introduction to Kato’s Euler systems , Galois representa-tions in arithmetic algebraic geometry (Durham, 1996), London Math.Soc. Lecture Note Ser., vol. 254, Cambridge Univ. Press (1998), 379–460.[Wi] M. Witte
On the equivariant main conjecture of Iwasawa theory
ActaArith. (2006), 275-296.
Department of Mathematics, University of Idaho, Moscow, Idaho
E-mail address ::