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The local Tamagawa number conjecture for Hecke characters, II Francesc Bars ∗ Abstract 7 Inthispaperweprovetheweak local Tamagawa numberconjecture fortheremaining 0 non-critical cases for the motives associated to Hecke characters ψθ : AK → K∗ of [1], 0 where K is an imaginary quadratic field with cl(K)=1, undercertain restrictions which 2 originate mainly from the Iwasawa theory of imaginary quadratic fields. n a 1. Introduction J Let E be a CM elliptic curve defined over an imaginary quadratic field K of 3 cl(K)= 1, with CM given by the ring of integers, D the discriminant of K. 2 K K O Wecanconsiderϕthe HeckecharacterassociatedtoE. ConsidertheChowmotive ] h1(E),andalso wh1(E). TheChowmotive wh1(E)hasadecompositioninpure T ⊗ ⊗ motivesM whereeveryoneofthemisrelatedwithaHeckecharacterψ . Observe N θ θ that every Hecke character from A the adèle group to K correspond to some K ∗ . h concrete ψθ. In [1] we study the Tamagawa number conjecture for Mθ(w+l+1) t associated to a Hecke character ψ with l 0. a θ ≥ m ThevalueoftheL-functionatzeroforMθ(w+l+1)isrelatedwiththefirstnon- zero coefficient of the Taylor development at l of the L-function associated to ψ [ − θ the complex conjugation from the Hecke character ψ by the use of the functional θ 1 equation of L-functions. Observe that l is a non-critical value associated to the v − Hecke character. The non-critical values associated to the Hecke character ψ (we 4 θ restrict to the situation a b (mod )) are the integers l such that l 3 θ 6≡ θ |OK∗ | − ≤ 6 min(aθ,bθ) where aθ,bθ are associated to the Hecke character ψθ (see [3, Theorem 1 1.4.1]) 0 The generalformulationof the Tamagawanumber conjecture atthe non-critical 7 values following [5] assumes w+l+1 > w [5, Conjecture 2.2.7] because then one 0 avoidsthe poles in the badEuler factors,andtherefore the assumptionl 0. But, h/ for M (w+l+1), there are no poles in the bad Euler factors, see [1, rem≥ark 2.6]. θ t a Thus, it is natural to study the Tamagawa number conjecture for l <0. m InthispaperweconstructelementsinK-theoryforM (w+l+1)with0< l θ − ≤ : min(aθ,bθ) and obtain the image of these by the Beilinson regulator map and the v Soulé regulator map, obtaining the weak local Tamagawa number conjecture. i X Deninger[3,pp.142-144]alreadyconstructedelementsinK-theoryforthemotive r Mθ(w+l+1)withl<0non-criticalandtheimageoftheseelementsbytheBeilinson a regulatormapsatisfytheexpectedproperties,provingtheBeilinsonconjecture. He constructedtheseelementsinK-theorybyuseofaprojectormap withoutusing KM complex multiplication. The problem of his construction is that the Weil pairing appearing in [1, 5] to a E[pr]-torsion point t , γ(t ) =< t ,t > is trivial and r r r r § the arguments through [1, 5] does not generalize in order to construct an Euler § e e e e system to control the image by the Soulé regulator map. We modify Deninger’s projector map by (we use now complex multiplication), and we construct the ′ elements in K-theoKrMy using and we reobtain Beilinson’s conjecture. With this ′ modificationtheargumentsiKnMthep-partoftheweakTamagawanumberconjecture, i.e. the image by the Soulé regulator map of these K-theory elements [1, 4, 5], § § apply straightforwardobtaining the result. ∗MSC(2000): Primary: 11G40;Secondary: 11G55, 11R42,14G10,19F27. 1 2 2. The motives associated to Hecke characters The elliptic curve E has conductor f an ideal of which coincides with the K O conductor of the Hecke character ϕ. Let us introduce the motives M . Consider the category of Chow motives θ (K) over K with morphisms induced by graded correspondences in Chow the- Q M ory tensored with Q. Then, the motive of the elliptic curve E has a canonical decomposition h(E) = h0(E) h1(E) h2(E) . The motive h1(E) has a Q Q Q Q Q ⊕ ⊕ multiplicationbyK [2, 1.3]. Letusconsiderthemotive wh1(E) ,forwastrictly § ⊗Q Q positive integer, which has multiplication by T := wK. Observe that T has a w ⊗Q w decomposition T as a product of fields T , where θ runs through the Aut(C)- θ θ θ orbitsofHomQQ(Tw,C), andθ canbe consideredasasubsetofHomQ(Tw,C). This decompositiondefinessomeidempotentse andgivesadecompositionofthemotive θ wh1E and its realizations. We have that T =K. The idempotent e belongs to ⊗ θ ∼ θ [1/D ], then K K O M :=e ( wh1(E) [1/D ]), θ θ ⊗ ⊗OK OK K is an integral Chow motive with coefficients in [1/D ], that is an element in K K O the category (K) constructed like (K) tensoring the correspondences with Q M M [1/D ] istead of tensoring by Q. We have that it has multiplication by e⊗O(KwOK [1/DK ]), recall that h1(E) ha⊗sZmultiplication by . θ K K K ⊗ O O Let us fix once and for all an immersion λ : K C as in [3, p.135]. The → L-function associated to the motive e ( wh1(E) ) corresponds to the L-function θ Q ⊗ associated to ψ =e ( wϕ):A K ([3, 1.3.1]) a CM-character, which is the θ θ ⊗ ∗K → ∗ § complex Hecke character that we also denote by ψ :A /K C from our fixed θ ∗K ∗ → ∗ embeddingλ. Letusrememberforreader’sconveniencethatwehavealreadythree equivalentnotions ofHeckecharacters,one ofthemis the CM-character[1,remark 2.2]. The CM-character is equivalent to a map named also ψθ : A∗K/K∗ → IK∗ where IK∗ is the idèle group of K, [1, remark 2.2]. A complex Hecke character is takingthe valueatsomearchimedianplaceofthis lastnotionofaHeckecharacter. The CM-character corresponds to ϕaθϕbθ, where aθ,bθ 0 are non-negative ≥ integerssuchthatw =a +b . Thepair(a ,b )istheinfinitetypeforψ . Thereare θ θ θ θ θ different θ with the same infinite type. Everyθ has two elements of Hom (T ,C), Q w oneisgivenbythe infinite typeϑ θ Hom (T ,C)andthe otheris ϑcomposed K w ∈ ∩ with the complex conjugation. The motivic cohomologygroupHw+1(M ,Q(w+l+1))is the K-theorygroup θQ M H :=K (M )(w+l+1) Q 2(w+l) w+1 θ M − ⊗ where the K-groups are the Quillen K-groups and the superscript indicates the Adam’s filtrationonthem. LetS be a finite setofprimesofK,whichcontainsthe primes above pf , where f is the conductor of the Hecke character ψ . Denote for θ θ θ simplicity M (w+l+1):=Hw(M K,Z (w+l+1)) [1/D ], θZp e´t θ×K p ⊗OK OK K and M (w+l+1):=Hw(M K,Q (w+l+1)). θQp e´t θ×K p We impose that w 2(w+l+1) 3. We have a Beilinson regulator map, − ≤− r :H R Hw(M ,Q(w+l)) R, D M⊗ → B θC ⊗ where the cohomology group on the right is the Betti realization for our motive, which coincides with e ( wH1(E(C),Q(1))(l). We have in this Q-vector space a θ ⊗Q Z[1/D ]-lattice givenbyH :=e ( wH1(E(C),Z(1)) [1/D ])(l), K h,Z[1/DK] θ ⊗Z B ⊗OKOK K which is an -module of rank 1. θ O 3 We have also, for every prime number p, the Soulé regulator map: r :H Q H1( [1/S],M (w+l+1)). p M⊗ p → e´t OK θQp The L-function associated to the motive M is defined by θ L (M ,s)=L (M ,s):= P (M ,s) for Re(s)>>0, S θ S θQp p θQp pY/S ∈ whereM =Hw(M K,Q )andthelocalEulerfactorsP (M ,s)aregiven θQp et θ×K p p θQp by P (M ,s):=det (1 Fr Np s MIp )=(1 ψ (p)Np s)(1 ψ (p)Np s) p θQp Qp − p − | θQp − θ − − θ − whereFristhegeometricFrobeniusatpandI istheinertiagroupatp[1,Lemma p 2.5]. 3. Modification of Deninger’s projector map. Beilinson conjecture revisited. Let us fix w 1 and l < 0 such that w 2l 3 with 0 < l min(a ,b ) θ θ ≥ − − ≤ − − ≤ and let us consider the motive M (w + l + 1). With the fixed embedding we θ have ϑ = (λ ,...,λ ) θ and set I = iλ Hom (K,C) and I = iλ / 1 w 1 i K 2 i ∈ { | ∈ } { | ∈ Hom (K,C) and we have now that 0 < l #I = a and 0 < l #I = K 1 θ 2 } | | ≤ | | ≤ b . Denote by ∆ = id1 id2 : E E E the diagonal map and by ∆ = θ CM id1,CM id2,CM : E ×E E giv→en by×e (e,(√d )e) where we understand K × → × 7→ √d End(E). Let us choose exactly l elements in the sets I and I denote it K 1 2 ∈ | | in increasing order i ,...,i I and j ,...,j I . Let us define the projector 1 l 1 1 l 2 map pr :Ew+l Ew+2l by||th∈e projection of th|e| ∈first w+2l-components of Ew+l → and let us define (id ∆l) : Ew+l Ew (which it depends of the choice in the || × → sets I and I ) by (e ,...,e ,e ,...,e ) (e ,...,e ) where e is 1 2 1 w+2l w+2l+1 w+l 7→ α1 αw αs defined as follows: ifα appearsinonecomponentofthesetoftuplesL:= (i ,j ),...,(i ,j ) s 1 1 l l • then { || || } id1(e ) if α =i e := w+2l+m s m , αs (cid:26) id2(ew+2l+m) if αs =jm in the other case, then it is defined by e := e with 1 ny w+2l • αs ny ≤ ≤ such that α = ny+ 1 where the sum runs the naturals that appear in s some component of thPe elements of L and which are lower than αs. We define also the map (id×∆|Cl|M) similarly as the map (id×∆|l|) but inter- changing idi by idi,CM. We define the projector map by ′ H2l+w+1(Sym2l+wh1E,Q(w+2KlM+1)) −pr→∗ H2l+w+1(E2l+w+|l|,Q(2l+w+1)) M M K′ ↓ ↓(id×∆|l| ) M CM ∗ HMw+1(MθQ,Q(w+l+1)) ←eθ− HMw+1(h1(E)⊗w,Q(l+w+1)) Deninger defines a projector map with a similar diagram as our by ′ replacing the map (id×∆|l|)∗ insteadKoMf the map (id×∆|Cl|M)∗. KM Define an element in H by M Υθ := ′ 2l+w(NK(E[f])/K((Ωf−1))), KMEM where 2l+w is the Eisenstein symbol, f a generator of f , Ω the period of E and θ (Ωf 1)EMmeans the divisor in Z[E[f ] 0]. − θ \ The next result is a modification of Deninger’s result [3, pp.143-145]. 4 Theorem 3.1. Suppose a b mod with a ,b 0, l <0 w =a +b , with θ 6≡ θ |OK∗ | θ θ ≥ θ θ w 2l 3. Define, up to sign, − − ≤− (√d )2l(2l+w)!L (ψ , l) 1Φ(f ) K p θ − θ ξ := − Υ θ,l 2 1N flψ (ρ )Φ(f) θ − K/Q θ θ θ which belongs to Hw+1(M ,Q(w+l+1)) where L (ψ , l) means the product of θ p θ − the Euler factors ofMthe primes above p of K at l (is well defined by [1, remark − 2.7]), and ρ is the idèle of K such that v (ρ 1 f 1) 0 for q f and v (ρ )=0 θ q −q − − ≥ | θ q q in the other primes q. Then rD(ξθ,l)=L∗S(ψθ,−l)ηθ, where S are the set of primes of K that divide f p, and η means an -basis for θ θ K O Hh,Z and L∗S(ψθ,−l)=lims+l→0LSs(ψ+θl,s). Proof. We will follow closely Deninger’s papers [2] and [3], we follow also in this proofhis notationwhere his nis ourw+2l. Deningerdefines the elementξ from θ,l 2l+w(N ((Ωf 1))) instead of Υ . We modify only the calculation in K(E[f])/K − θ [K3M,(2E.M13)Lemma]for insteadof ,andforourchosenorderforthefactorsof ′ the map (id ∆|l| )KwMe have to conKtrMolsome sign that we do not do. One obtains × CM (up to sign) 1 ˜ (ξ˜) dz(ε) = (2πi)w ZEwKD′ ∧ 1 BεpdK|l|(cid:18) n+|ln|−|ε| (cid:19)− A(Γ)n+|l|cn+|l|−|ε| by it calculation at the top of [2, p.63]. Then the argument [3, p.143-144] applies in our situation obtaining, r (Υ )=t L (ψ , l)η D θ θ ∗ θ − θ where t is given by (up to sign) 2−1NK/Qflθψθ(ρθ)Φ(f). By [1, remark 2.7] we can θ (√dK)2l(2l+w)!Φ(fθ) introduce the Euler factors above p in the constant and introduce the function L S for the motive, obtaining the result. (cid:3) 4. The weak Tamagawa number conjecture for l<0 Following [1, 3] we define for l<0 the constructible space by § :=ξ , θ θ,l K R O with ξ given by the theorem 3.1. Let us observe that with this notation we can θ,l follow straightforward all the results and proofs of [1, 3] and [1, 4]. In [1, 5] we § § § needtocompute w+2l(N (Ωf 1)). Denote bye=(t ) andelement ′ K(E[f])/K − r r of the Tate moduKleMT◦EEMwhere t E[pr] a pr-torsion point for E. p r ∈ e Lemma 4.1. The realization oneGalois cohomology of the projector map ′ has the property, ′ (tr⊗2l+w)=eθ( wtr) γ(tr)l where γ(tr)=<tr,√dKtr >KMwhich is the Tate twKisMt. ⊗ ⊗ e e e e e e Proof. Observe first that the projector map is the composite of = e (id ′ θ ∆δ∗|Cl|=M)p∗r◦∗p◦r(∗i.dL×et∆u|Csl|Mta)k∗eisδ∗in:=th(iedd×efi∆n|Cilt|Mio)n∗o◦nprK∗Mand=oebθse◦rvδe∗ twKhiaMtthitls>tra0◦nsgpivoes×ne at [6] and therefore we know that it has the claimed property of this lemma [6]. We need only to study these projector maps on the Galois cohomology. Denote by the étale realizationof h1(E)(1) and observe that there is an isomorphism HQp 5 HQ∗p(1) ∼= HQp, from (h1(E)(1))∗ = h1(E)(−1) ∼= h1(E)(1)(−1) = h1(E) and therefore (h1(E)(1)) (1)=h1(E)(1). The map δ is given by ∗ ∼ ∗ H1( ,Sym2l+w( )(1)) H1( ,Symw( )(l+1)), OS HQp → OS HQp and because the map δ is the transpose for the map δ up to Tate twist by w+l ∗ ∗ it is represented by global Tate duality by, H1(OS,Symw(HQp)∗(−l−1)(1))→H1(OS,Sym2l+w(HQp)∗(−1)(1)). Is known [6] that δ (lim( 2l+wt ))=lim(( wt )γ(t )l) ∗ r r r ←− ⊗ ←− ⊗ r r writeitalsobyδ ( w+2le)=( we)γe(e)l(**). TakeenowethedualmapbyHom(,Z ) ∗ p ⊗ ⊗ and do the identification T E =Hom(T E,Z (1)), to obtain p ∼ p p ( we( 1))γ(e) l w+2le( 1) − ⊗ − 7→⊗ − twisting now by w+l we arrive to the definition for δ and, ∗ δ ( we) ( w+2le)γ(e) l − ∗ ⊗ 7→ ⊗ and taking this equality at level r we finish. (cid:3) After the lemma 4.1 all the results of [1, 5] and the proofs of [1, 5] follow § § straightforward up to a power of 2 and D , (the reader could make these modifi- K cationswhichfollowonlyfromourdefinitionof ). Thereforeweobtainthe weak θ R local Tamagawa number conjecture with K-coefficients and Q-coefficients, under standard hypothesis from Iwasawa theory for imaginary quadratic fields: (*)Letpbe afix prime differentfrom2and3 (hence, inparticular,p∤# ), |OK∗ | and p ∤ N f. Suppose that ψ has infinity type (a ,b ) with a ,b non-negative K/Q θ θ θ θ θ integers, such that a b mod and w = a +b 1 verifies w 2l 3 θ 6≡ θ |OK∗ | θ θ ≥ − − ≤ − with l < 0. Suppose that ( /f ) is injective. Suppose moreover that OK∗ → OK θ ∗ the representationχ ofGal(K(E[p])/K)in Hom (Hw(M K,Z (w+l)), ) Op θ×K p Op is a non-trivial representation that if p inert we impose moreover that as a Z - p representation is irreducible. Suppose moreover χ is not the cyclotomic character and it satisfies two results on isomorphism between Iwasawa modules: (i) Iwasawa main conjecture for imaginary quadratic fields is true [6, Rubin Theorem2.1.3]for χ and the elliptic units of [1, definition 4.1] and (ii) the conclusion of [6, Corollary 2.2.11]is satisfied. See [1, remark4.4]for results abouthow this conditions impose restrictions on M (w+l+1). θ Theorem4.2. Assumehypotheses(*). IfwedenotebyM (w+m)=Hw(M θZp θ×K K,Z (w+m)), (by hypothesis p∤D ) then, there is an -submodule H p K K θ O R ⊂ M of rank 1 such that: (1) det (r ( [1/D ]))= OK[1/DK] D Rθ ⊗OK OK K ∼ L (ψ , l)det (Hw(M ,Z(w+l)) [1/D ]) ∗S θ − OK[1/DK] B θC ⊗OK OK K in det (Hw(M ,Z(w+l)) [1/D ] R). OK[1/DK]⊗R B θC ⊗OK OK K ⊗ (2) The map r induces an isomorphism p detOK⊗Zp(Rθ)∼=detOK⊗Zp(RΓ(OK[1/S],MθZp(w+l+1))−1. Here L (ψ ,s) L (ψ , l)= lim S θ , ∗S θ − s l s+l →− and S is the set of primes of K dividing p and the ones dividing f . θ 6 Moreover, if r is injective on , the second part can be written as p θ R detOK⊗Zp(H1(OK[1/S],MθZp(w+l+1))/rp(Rθ))∼= det H2( [1/S],M (w+l+1)). OK⊗Zp OK θZp Theorem 4.3. Suppose hypotheses (*). Then, there is a Z-submodule in H of rank 2 such that: θ R M (1) The map r R is an isomorphism restricted to R. θ (2) dim (Hw(MD⊗,Z(w+l)) Q)=ord L (Hw(RM ⊗,Q ),s)=2. Q B θC ⊗ s=−l S θ p (3) We have the equality r (det ( [1/D ]))= D Z[1/DK] Rθ⊗OK OK K L (Hw(M ,Q ), l)det (Hw(M ,Z(w+l)) [1/D ]) ∗S et θ p − Z[1/DK] B θ ⊗OK OK K where L (Hw(M ,Q ),s) L∗S(Hewt(Mθ,Qp),−l)=sliml S e(ts+θl)2 p →− and S is the set of places of K that divides p and the places dividing the conductor f . θ (4) We have that det ( Z )=det (RΓ( [1/S],M (w+l+1))) 1. Zp Rθ ⊗ p Zp OK θZp − If r is injective on , then r (det ( Z )) is a basis of the Z -lattice p Rθ p Zp Rθ⊗ p p det (RΓ( [1/S],M (w+l+1))) 1 Zp OK θZp − det (RΓ( [1/S],M (w+l+1) Q)[ 1]). ⊂ Qp OK θZp ⊗ − References [1] F. Bars: The local Tamagawa number conjecture for Hecke characters. Submitted. See http://mat.uab.es/∼francesc/publicac.html [2] C. Deninger: Higher regulators and Hecke L-series of imaginary quadratic fields I. Invent. Math.96,1-69(1989). [3] C. Deninger: Higher regulators and Hecke L-series of imaginary quadratic field II. Ann. of Math.(2)192,131-155(1990). [4] T. Geisser: A p-adic K-theory of Hecke characters of imaginary quadratic fields and an analogueofBeilinson’sconjectures. DukeMath.J.86,no.3,197-338(1997). [5] K. Kato: Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via BdR. InArithmeticAlgebraicGeometry(Trento, 1991), LNM1553,50-163(1993). [6] G. Kings: The Tamagawa number conjecture for CM elliptic curves. Invent. math. 143, 571-627(2001). Francesc Bars Cortina Depart. Matema`tiques, Edifici C, Universitat Autònoma de Barcelona 08193 Bellaterra. Catalonia. Spain. e-mail: [email protected]