The local Tamagawa number conjecture on Hecke

characters

Francesc Bars ∗

Abstract

In this paper we prove the local Tamagawa number conjecture in thenon-critical case for the motives associated to Hecke characters ψθ : AK →K∗, where K is an imaginary quadratic field with cl(K) = 1, under cer-tain restrictions which originate from the Iwasawa theory of imaginaryquadratic fields.

Introduction

The local Tamagawa number conjecture for a variety X over a number field ofBloch and Kato [3], or, more precisely, for a pure Chow motive M of weight wover a number field, describes the special values of the L-function in terms ofcohomological data (see for example Kato [17] or Fontaine and Perrin-Riou [9]).

Recall that the special values of an L-function are the leading Fourier coef-ficients at integer points. Suppose we have a weight w motive M such that it’sL-function has meromorphic continuation and satisfies the expected functionalequation. Then, we say that an integer m < w

2 is non-critical if L(M, m) = 0and it is critical if L(M, m) 6= 0. We extend this definition to the integersm > w

2 + 1 by saying that m is critical for M if w − m + 1 is critical, and isnon-critical for M if w−m+1 is non-critical. The Tamagawa number conjecturecan be formulated in terms of period maps and regulator maps ([8],[17]), but inthe non-critical situation it can be formulated without the period maps (usingthe hypothetical functional equation and good compatibilities).

There are few cases proved in the non-critical situation: for the Riemannzeta function ([3, §6]), for Dirichlet motives ([4], [15]), for CM elliptic curvesdefined over the field of the endomorphism ring ([19]) or defined over Q ([3, §7],[1]).

The weak local Tamagawa number conjecture for an elliptic curve E withCM defined over the field of the endomorphism, proved by Kings [19], is re-lated to the weak local Tamagawa number conjecture for the L-function theHecke character ψ, associated to E, over the imaginary quadratic field K. Moreprecisely, Kings proves in [19] the conjecture for the motive h(ψ)(−r) withr ≥ 0 which corresponds to the special value (non-critical) for the L functionassociated to ψ at −r, where h(ψ) is the motive associated to ψ over K withK-coefficients. Then, he obtains the conjecture for the Chow motive h1(E)(−r)when one rewrites the above result with Q-coefficients. Using the functionalequation for E and good compatibilities one should obtain the conjecture forh1(E)(r + 2). We generalize the methods of Kings to Hecke characters over animaginary quadratic field K in the non-critical situation.

Consider ψ a Hecke character over an imaginary quadratic field associatedto an elliptic curve with CM defined over the endomorphism ring and let us

∗Work supported by DGI, BHA 2000-0180.

1

take the motive associated to ψaψb

with a, b ≥ 0, which has weight a + b. It isknown that the non-critical values for this motive are the integers lower thanmin(a, b) (see §3 for the precise statement). Our work is concentrated in thissituation, but note that there are results on the Tamagawa number conjecturein the critical situation (Harrison [14], Guo [12], Kimura [20], Han [13] and inbig generality by Tsuji [24]).

The aim of this paper is to prove the local Tamagawa number conjecture(under the formulation in [17]) for all the non-critical values for the L-functionof the motive associated to any Hecke character ψθ : AK → K∗ where K isan imaginary quadratic field which cl(K) = 1, (see [2, Chapter 3] for a moreconcrete result only for powers of the Grossencharacter associated to the ellipticcurve E with CM).

The basic ingredients used in the proof of this result are the specializationof the polylogarithm sheaf and the main conjecture for the Iwasawa theoryfor imaginary quadratic fields, as in [19]. The main tool in the proof is aprecise description of the image of some concrete elements in K-theory underthe Soule and Beilinson regulator maps. Using Deninger proof of the Beilinson’sconjecture for Hecke characters in [6], we only need to check the p-adic part ofthe Tamagawa number conjecture.

Let us finally give a rough sketch of the contents of this paper. In the firstsection §1 we recall the formulation of the local Tamagawa number conjecturefor pure motives using Kato’s formulation [17] for some non-critical situation.In section §2 we define the motives associated to Hecke characters coming fromelliptic curves defined over an imaginary quadratic field K. We prove a general-ization of a result of Deuring for these motives. In §3 we define the constructibleelements of Deninger [6] for these motives and we reformulate Deninger’s the-orem in the notation of the local Tamagawa number conjecture. In the nextsections we study the image of these constructible elements for our motives viathe Soule regulator map. In §4 we define a map from some Iwasawa modules tothe etale cohomology groups that appear in the conjecture. Here the Rubin’stheorem of the main conjecture of the Iwasawa theory for imaginary quadraticfields plays an important role. This section follows basically the arguments of[19, §2] with a definition of elliptic units more useful for us. In section §5 we usethe specialization of the elliptic polylogarithm to compare the image of the mapdefined in §4 with the Soule regulator map. With these ingredients we obtainthe main results of the paper, theorems 5.12 (for K-coefficients) and 5.13 (forQ-coefficients). In the appendix we study the remaining non-critical values, theones in the band of possible poles of bad Euler factors. We modify Deninger’sprojector map to define more convenient elements in K-theory and we reprovethe Beilinson’s conjecture for them. With this modification we can apply all thetechniques of §4 and §5, obtaining the main theorems of §5.

I am indebt with X.Xarles. He read different versions of this paper and told me

useful comments and suggestions resorting the writing and contents of this work and

also pointed out some inaccuracies on the first version of the paper. I thank him for

all above and moreover to give me energy to try to make me enjoy another time doing

mathematical research. It is a pleasure to thank G. Kings for many discussions about

the contents of his paper and results concerning this paper. I thank also C. Deninger

who introduce me in the L-function world. It is also a big pleasure to thank warmly

K. Rubin who clarifies me a doubt on elliptic units.

2

1 The Tamagawa number conjecture

We formulate the conjecture for the category of Chow motives over a numberfield F with coeficients in Q denoted byM(F )Q (for a more detailed explanationwe refer to [2, §1.2.3] and [17] [18]). For every M ∈ M(F )Q one expects adecomposition into direct sum of pure Chow motives in this category.

Using Kato’s notation as in [18, §2], a pure motive of weight w ∈ Z overa number field F is a finite family of 4-tuples (Xj ,mj , rj , εj) where Xj isa smooth proper scheme of pure dimension dj over F and rj are integers withw = mj−2rj , and εj is an idempotent in the ring of algebraic cycles on Xj×F Xj

with Q-coefficients modulo rational equivalence.We denote by hm(X)(r) the pure motive determined by the tuple (X,m, r,∆X),

where ∆X denotes the diagonal, regarded as the identity correspondence. Weinterpret (X,m, r, ε) as the direct sumand of hm(X)(r) corresponding to ε.For simplicity we restrict from now on our analysis to pure motives M of theform (X, m, r, ε) which we denote by Mε(r).

Fix integers m ≥ 0 and r such that m−2r = w ≤ −3 and r > inf(m, dim(X)).Let also fix a prime p of Q, p 6= 2. For every Mε(r) let associate the motiviccohomology

Hm+1M (Mε, r) := (K2r−m−1(Mε)⊗Q)(r)

which it is the r-th Adams eigenspace of the 2r−m− 1-th Quillen K-theory ofMε. We consider some realizations for the motive which are related via regulatormaps with the above motivic cohomology. Let define Hh,Q by

ε∗Hmsing(X ×Q C, (2πi)r−1Q)+,

a Betti realization, where the upper-script + means the fixed part by theGal(C/R)-action acting simultaneously on C and on Q(r − 1). The p-adicrealization for our motive corresponds to

Vp := ε∗Hmet (X ×F F,Qp(r)) = Hm

et (Mε ×F F ,Qp(r))

which is a Gal(F/F )-module, unramified outside a finite set of places of F .Consider

Hh,Z := ε∗Hmsing(X ×Q C, (2πi)r−1Z)+

which satisfy Hh,Z ⊗Z Q = Hh,Q, and

Tp := ε∗Hmet (X ×F F ,Zp(r))

a Gal(F/F )-module such that Tp⊗ZpQp = Vp. Let S be a set of finite primes ofF containing the primes lying over p and the ones where Tp is ramified. Let OF

be the ring of integers of F and write OS := OF [1/S]. Consider j : Spec(F ) →Spec(OS) the natural map and define,

Hip := Hi

et(OS , j∗Tp).

We omit j∗ if no confusion is likely. Then, there are regulator maps due toBeilinson and Soule,

rD : Hm+1M (Mε, r)⊗Q R→ Hh,Z ⊗Z R

rp : Hm+1M (Mε, r)⊗Q Qp → H1

p ⊗Zp Qp.

3

For any prime p - p in OF , we have local Euler factors which are defined by

Pp(Vp, s) := detQp(1− FrpNp−s|V Ip

p ),

where Frp is the geometric Frobenius at p and Ip is the inertia group at p. Forp | p, they are defined by

Pp(Vp, s) := detF0,p(1− ψ−1p Np−s|Dcris(Vp)),

where ψp is the arithmetic Frobenius, Dcris(Vp) = (Bcris,p ⊗ Vp)Gal(F p/Fp) andF0,p means the maximal unramified extension inside Fp. Recall that Pp(Vp, s)is conjecturally independent of the choice of the finite prime p.

The L-function of X is defined by

LS(Vp, s) :=∏

p/∈S

Pp(Vp, s)−1.

Conjecture 1.1. (cf. §4[18], conj. 2.2.7 in [17] ) Let V ∗p = Hom(Vp,Qp) be

the dual Galois module. Let p 6= 2, r, m and S be as above. Assume that

Pp(V ∗p (1), 0) 6= 0

for all p ∈ S and that LS(V ∗p (1), s) has an analytic continuation to all C, then:

1. The maps rD and rp are isomorphisms and H2p is finite.

2. dimQ(Hh,Z) = ords=0LS(V ∗p (1), s); write this number l.

3. Let η ∈ detZ(Hh,Z) be a Z-basis. There is an element ξ ∈ detQ(HM) suchthat

rD(ξ) = (lims→0

s−lLS(V ∗p (1), s))η.

This is the “Beilinson conjecture”.

4. Consider rp(ξ) ∈ detQp(H1p⊗ZpQp). Then rp(ξ) is a basis of the Zp-lattice

detZp(RΓ(OS , Tp))−1 ⊂ detQp(RΓ(OS , Vp)[−1]).

i.e.[detZp(H1

p ) : rp(ξ)Zp] = #(H2p ) = detZp(H2

p ).

As our knowledge of K-theory is limited, we take a weak version of theconjecture.

Conjecture 1.2. (cf. §1.1.1 [19]) With the same hypothesis of the above con-jecture 1.1, then there is a subspace Hconstr

M in HM such that:

1. rD and rp restricted to HconstrM are isomorphisms and H2

p is finite.

2. dimQ(Hh,Z) = ords=0LS(V ∗p (1), s); write this number l.

3. There is an element ξ ∈ detQ(HconstrM ) such that

rD(ξ) = (lims→0

s−lLS(V ∗p (1), s))η.

4. The element rp(ξ) is a basis of the Zp-lattice

detZp(RΓ(OS , Tp))−1 ⊂ detQp(RΓ(OS , Vp)[−1]).

4

2 The motive associated to Hecke characters

Let K be an imaginary quadratic field and OK be its ring of integers. Denoteby dK the discriminant of K. Let E be an elliptic curve over K with CM byOK . In this section we describe some pure motives coming from a self productof the motive h1(E) and their realizations, and we prove that the L-functionsassociated to these motives correspond to Hecke characters. We obtain finallyan analog for these motives of a result of Deuring for CM elliptic curves.

Let p be an odd prime, fixed once and for all, such that E has good reductionfor all primes over p. Let S

′be the set of places that divide the conductor of

the elliptic curve f (that are the same places where E has bad reduction) andthe places that divide p.

Consider the category of Chow motives M(K) over K with morphisms in-duced by graded correspondences in Chow theory. We have then a naturalcovariant functor h from the category of smooth and projective varieties overK to M(K). Then, the motive of an elliptic curve E over K has a canonicaldecomposition h(E) = h0E⊕h1E⊕h2E with respect to the zero section, whereh0E = h(Spec(K)) and h2E = h(Spec(K))(−1). We can also consider the cat-egory MQ(K) which consist with the same object but enlarging the morphismstensoring by Q of the category M(K). For an elliptic curve we have also theabove decomposition for h(E)Q in MQ(K).

If E is an CM elliptic curve as before, the motive h1E has multiplicationby OK . Consider then the motive ⊗wh1E, for w a positive integer, whichhas multiplication by Ow := ⊗w

ZOK . Then, ⊗wh1EQ has multiplication byTw := ⊗w

QK. If we denote by Υ = Hom(K,C), observe that Tw is equal to∏θ Tθ, where θ runs through the Aut(C)-orbits of Υw = Hom(Tw,C) and Tθ

are fields. Let eθ be the idempotent corresponding to the component Tθ of Tw,and consider the motive

eθ(⊗wh1(E)Q).

Then, on the integral motive ⊗wh1E, considering the idempotent eθ as be-fore, we get that the integral motive eθ(⊗wh1(E)) which has multiplication byOθ := eθOw, a ring in Tθ.

Remark 2.1. These motives were introduced by Deninger in [6] in a more gen-eral setting. He constructs a motive for every Hecke character on AK , comingfrom the self product of h1(E) for a particular CM elliptic curve related with aweight 1 Hecke character. Under the restriction cl(K) = 1, the Hecke charac-ters on AK , are the ones which maps to K∗, with any Dirichlet character ([6,Prop.1.3.1]). Then our presentation is the general situation for Hecke charac-ters over an imaginary quadratic field with cl(K) = 1.

Let ψ : IK → K∗ be the grossencharacter associated to the elliptic curve E.Define the CM character

ψθ : IK → T ∗θby ψθ = eθ · (⊗wψ), and denote by fθ the conductor of ψθ. Observe that fθ|fsince ψθ is a sub-character of ⊗wψ.

The infinity type of this character is obtained as follows: Let fix once andfor all an embedding K → Q like in the last paragraph on [6, p.132]. We havethen a natural embedding

K → ⊗wK → Tw → Tθ

5

where the first map corresponds to the diagonal map. For any ϑ ∈ θK :=θ∩HomK(Tθ,C), (θ here means the elements of Υw which includes Tθ in C), ϑ =(λ1, . . . , λw) ∈ Υw, we set aϑ = #i|λi ∈ HomK(K,C) and bϑ = w−aϑ. Thesenumbers do not depend on the element ϑ in θK , and they determine the infinitytype for the Hecke character ψθ (cf. last paragraph 1.3 [6]). We will denotethe type of ψθ as (aθ, bθ) := (aϑ, bϑ) where ϑ is any element in θK . Moreover,in our situation, the CM field Tθ is the field generated by (λ1(K) · . . . · λw(K))which is K, where ϑ = (λ1, . . . , λw) ∈ θ, then eθ(⊗wh1(E)Q) ∈ MQ(K) hasmultiplication by K. As λi ∈ λ, λ where λ is the fixed embedding of K in C,and λi(OK) = OK , we have then that eθ(⊗h1(E)) ∈ M(K) has multiplicationby OK and we can consider Oθ = OK . We mention also that θ only contains twoϑ, because in Υw two embeddings of the same field differ by an automorphismof C, but the field Tθ is K which has only two automorphism then we have thatthe other element of θ diferent for ϑ is (λ1, . . . , λw).

Let’s denote byMθ := eθ(⊗wh1(E)),

considered as an integral Chow motive ([11, p.57]), that is and element in thecategory of M(K), and MθQ its image in MQ(K). Our objective in this sectionis to study the p-adic and Betti realizations of this motive twisted by w, and todetermine its L function in terms of the Hecke character ψθ.

The p-adic realization of the motive MθQ(w) is, by definition, Hwet(MθQ ×K

K,Qp(w)). We need to choose a lattice on it.

Lemma 2.2. The integral p-adic realization of Mθ(w), Hwet(Mθ ×K K,Zp(w)),

is isomorphic toeθ(⊗wTpE)

as free eθ(⊗wOK)-modules of rank 1 with GK-action, with the Oθ-action oneθ(⊗wTpE) given by ψθ ⊗ Zp.

Proof. Observe first that TpE is isomorphic to H1et(h

1(E)×K K,Zp(1)) by Kum-mer theory, but with GK-action given by the complex conjugation by Artinreciprocity.

The claim that eθ(⊗wTpE) is a free module of rank 1 follows because TpE isa free OK-module of rank 1 and then eθ · (TpE ⊗ . . .⊗ TpE) is a free eθ · (OK ⊗. . .⊗OK)-module of rank one.

Now, consider the natural action of GK on H1et(h1(E) ×K K,Zp(1)) =

Hom(TpE,Zp(1)). Since GK acts on the Tate module by ψ : GK → (OK⊗Zp)∗,then it acts on our GK-module via the complex conjugate. Using that

Hw((h1(E)×K K)w,Zp(w)) = H1(h1(E)×K K,Zp(1))⊗w

by [10, Prop.2.4.3 a)], and taking our idempotent, we obtain the result.

We can apply this result to the motive Mθ(w+l+1) for any integer l+1. Weget therefore a Zp-lattice in the p-adic realization of the motive Mθ,Q(w+ l+1).

We are going now to define a submodule in Hwsing(E

w×QC,Z(2πi)w+l)+ suchthat their elements are a Z-lattice in the Betti realization with Q-coefficientsfor Mθ(w + l + 1).

Observe first that H1B(E(C),Z(1)) ∼= H1

B(E ×Q C,Z(1))+ is an OK-moduleof rank 1, and hence

⊗wZH1

B(E(C),Z(1))

6

is a ⊗wOK-module of rank 1. By considering now the idempotent eθ and takingthe corresponding submodule

eθ(⊗wH1B(E(C),Z(1)))(l),

we get a Oθ := eθ(⊗wOK)-module of rank 1. This module corresponds to theZ-module Hw

B (MθC,Z(w + l)), the searched Z-lattice of Hw(MθC,Q(w + l)) forthe Betti realization of our motive in the formulation of the conjecture.

Now, we are going to study the L-function that corresponds to the p-adicrepresentation Hw

et(MθQ×K K,Qp) of MθQ. Let be S the places of K that dividefθ and the places that divide p. Define as usual

LS(Mθ, s) :=∏

l/∈S

detQp(1− FroblN l−s|(Hw

et(MθQ ×K K,Qp))Il)−1

where Frobl means the geometric Frobenius.Our goal is to compute this determinant and to relate it with the local factors

of the L-function of the Hecke character ψθ that is defined by,

LS(ψθ, s) :=∏

l/∈S

(1− ψθ(l)N ls

)−1.

Recall that the operation of the decomposition group Dp for p - p onH1

et(h1(E)×KK,Qp) is given by the operation of ψ−1|K∗p, and hence Dp operates

on Hwet(MθK ,Qp) via ψ−1

θ . On one hand, the inertia group Ip acts non-triviallyif and only if p divides the conductor fθ. On the other hand, for p - fθ, thegeometric Frobenius Frp at p acts via ψθ(p). We obtain hence the followingresult.

Lemma 2.3 (Deninger,§1.3.1[6]). Let l a finite prime of K not over p, then

detTθ⊗Qp(1− FrlN l−s|Hwet(Mθ,Qp)Il) = (1− ψθ(l)N l−s)

if l - fθ, where fθ is the conductor of the Hecke character ψθ.

We fix some restrictions for our motive Mθ(w + l + 1) once and for all. Wesuppose −w − 2l ≤ −3. Remember that, with our restriction that E is definedover K, we have #|θ| = 2, and in particular we have Tθ

∼= K and Oθ∼= OK .

The L-function for Mθ can be described by using lemma 2.3 and by takingthe norm map.

Lemma 2.4. Let l a prime of K such that l - fθ and it is prime to p. We havethen the following equality

detQp(1− FrlN l−s|(Hw(Mθ,Qp))Il) = (1− ψθ(l)N l−s)(1− ψθ(l)N l−s).

As a corollary we obtain a generalization of a result of Deuring.

Theorem 2.5. Let S be the set of the primes on K dividing fθ and primesdividing p. Then:

LS(Hw(Mθ,Qp), s) = LS(ψθ, s)LS(ψθ, s).

7

Remark 2.6. The p-adic realization Vp,w,w+l+1 = Hwet(MθQ×KK,Qp(w+l+1))

of our motive Mθ satisfies that the local Euler factors

Pp(V ∗p (1), 1) = Pp(ψθ,−l)

are different from 0 for all p ∈ S. Hence, it satisfies the hypothesis in theconjecture 1.2.

To show this fact, suppose first that p|fθ. Then, the inertia group acts non-trivially on Vp,w,w+l+1, which is a one dimensional Oθ ⊗Q-module, and hence

Lp(ψθ, s) = 1

for all p|fθ, and in particular for s = −l.If p divides p, then the result follows from the fact that the abelian varieties

with CM satisfy this condition. i.e.

detQp(1− FrpNpl|Hw

et(Ew,Qp)) 6= 0,

and therefore, since the different idempotents eθ give a direct summand of thecohomology group Hw(E

w,Qp),

Lp(ψθ,−l) 6= 0.

3 The Beilinson conjecture for Hecke characters

In this section we will review briefly the study of the Beilinson conjecture forthe motive MθQ(w + l + 1) done by Deninger in [6]. First of all, recall the maintheorem in [6].

Theorem 3.1 (Deninger, 1.4.1 [6]). Let w = aθ + bθ ≥ 1. Consider aninteger l such that

−l ≤ Min(aθ, bθ) if aθ 6= bθ

−l < aθ = bθ = w/2 otherwise.

Then the L-series L(ψθ, s) has a zero of order 1 in s = −l.Moreover, there exist an element ξθ in Hw+1

M (Mθ,Q(w + l + 1)) such that,under the Deligne regulator map

rD : Hw+1M (Mθ,Q(w + l + 1)) → Hw

B (Mθ,R(w + l)),

has image

rD(ξθ) = lims→−lL(ψθ, s)

s + lηθ mod T ∗θ

in the free rank one Tθ⊗R-module Hw+1D (Mθ,R(w+l+1) = Hw

B (MθC,R(w+l)),where ηθ is a Tθ-generator of Hw

B (MθC,Q(w + l)).

Observe that the L-series L(ψθ, s) is equal to the L-function for the dualBeilinson motive of Mθ, i.e. the same motive but with action by Tθ given by itscomplex conjugation.

Let’s recall the construction of ξθ, following the results of Deninger. Wesuppose once for all that l ≥ 0. Remember that S is the set of finite primesp ∈ Spec(OK)|p|fθ or p|p.

8

Fix an algebraic differential form ω ∈ H0(E, ΩE/K). Since we have complexmultiplication, we can write the period lattice as Γ = ΩOK , where Ω ∈ C∗is the complex period. Fix an element γ in H1(E(C),Z) such that it is anOK-generator, and satisfies

Ω =∫

γ

ω.

By Poincare duality, we have that to γ it corresponds ηγ , an OK-generator forH1(E(C),Z(1)). Consider now the Oθ-generator

ηθ := (2πi)leθ(ηγ ⊗ . . .⊗ ηγ)

ofHw

B (MθC,Z(w + l)) = eθ

((⊗H1

B(E(C),Z(1)))(l)).

To construct ξθ, we will define a divisor on the torsion points of the ellipticcurve; its image by the composition of the Eisenstein map EM ([5, §8]) with theDeninger projector map KM ([6, (2.8)]) will define our ξθ.

Remember that fθ is the conductor of the Hecke character ψθ associated withMθ, and denote by f a generator of fθ (it exists since the ideal class group of Kis equal to 1). We have that

Ωf−1 ∈ f−1θ Γ

and that (Ωf−1) gives a divisor in Z[E[fθ] \ 0] defined over K(E[fθ]). Since fis the conductor of ψ and fθ|f, the divisor (Ωf−1) is defined also over K(E[f]).We will define our divisor as

βθ := NK(E[f])/K((Ωf−1)).

If aθ 6≡ bθ mod |O∗K |, we obtain that ([6, p.142,(2.11)]),

rD(KMEM(βθ)) = (−1)l−1 2l−1NK/Qfw+2lθ ψθ(f)

(2l + w)!NK/Q(fθ)l+w

Φ(f)Φ(fθ)

L′(ψθ,−l)ηθ,

where Φ(m) := |(OK/m)∗| for any ideal m of OK .This is an analog for Mθ(w + l + 1) of [19, thm.1.2.2] which corresponds to

h1(E)(1 + l + 1). Therefore, it suggests the element that we have to take toprove an analog for the p-Tamagawa number conjecture for Mθ(w + l +1), thatis, to control the coimage of the p-Soule regulator by the number of elements ofa second cohomology group.

Theorem 3.2 (Deninger, §2[6]). Suppose that aθ 6≡ bθ mod |O∗K | and thataθ, bθ, l satisfy the hypothesis of the theorem 3.1 with l ≥ 0. Define, by usingthe previous notation,

ξθ,l :=

(−1)l−1 (2l + w)!Lp(ψθ,−l)−1Φ(fθ)2l−1NK/Qflθψθ(f)Φ(f)

KM E2l+wM (βθ) ∈ Hw+1

M (Mθ,Q(w + l + 1)),

where Lp(ψθ, s) is the product of the Euler factors for the primes of K above p.Then

rD(ξθ,l) = L∗S(ψθ,−l)ηθ,

where S are the primes of K that divide fθp.

9

Definition 3.3. For aθ 6≡ bθ mod |O∗K | we define our constructible space(Hcons

M,ω,r in conjecture 1.2) by

Rθ := ξθ,lOK

Remark 3.4. In the situation aθ ≡ bθ mod|O∗K |, Deninger defines also a divisorwhose image with respect to the composition of the Eisenstein map with theprojector map satisfies the above Theorem 3.2 ([6, §5]).

As a consequence of Theorem 3.2, we have that our submodule Rθ verifiesthe Beilinson conjecture for the motive Mθ(w + l + 1).

Theorem 3.5. The OK-submodule Rθ of Hw+1M (Mθ,Q(w+ l+1)) satisfies that

detOK(rD(Rθ)) = L∗S(ψθ,−l)detOK

(HwB (Mθ,Z(w + l))

in detOK⊗R(HwB (MθC,Z(w + l))⊗ R). Here

L∗S(ψθ,−l) = lims+l→0

LS(ψθ, s)/(s + l),

and S is the set of primes dividing fθ and primes dividing p.

Proof. Observing that ηθ is a OK = Oθ-base for the free Oθ-module

HwB (Mθ ⊗K C,Z(w + l))

of rank one, the result follows.

Corollary 3.6. The submodule Rθ defined above satisfies the Beilinson con-jecture inside the p-local Tamagawa number conjecture, that is Rθ satisfies thefollowing conditions:

1. The map rD ⊗ R is a isomorphism when restricted to Rθ ⊗ R.

2. dimQ(HwB (Mθ,Z(w + l))⊗Q) = ords=−lLS(Hw(Mθ,Qp), s) = 2.

3. We have the following equality

rD(detZ(Rθ)) = L∗S(Hwet(Mθ,Qp),−l)detZ(Hw

B (Mθ,Z(w + l)))

where L∗S(Hwet(Mθ,Qp),−l) means lim

s→−lL∗S(Hw

et(Mθ,Qp), s)/(s + l)2 (this

makes sense by using theorem 3.1 and theorem 2.5).

Proof. The first and the second conditions are clear for the dimensions of thespaces involved in the Deligne regulator map, and the theorem 3.5. The thirdcondition comes from the previous theorem using the fact that, if we multiplyan Oθ = OK-module with an element L∗S(ψθ,−l) in Oθ ⊗R, the determinant ismultiplied by the norm

NOθ⊗R/R(L∗S(ψθ,−l)) = L∗S(ψθ,−l)L∗S(ψθ,−l) = L∗S(ψθ,−l)L∗S(ψθ,−l).

Using theorem 2.5, we obtain that this is equal to L∗S(Hwet(Mθ,Qp),−l).

10

4 Iwasawa theory

In this section we will define a map that relates the cohomology groups Hip,⊗w

for the p-adic lattice in the p-adic realization of Mθ(w + l + 1) defined in §2,with some Zp[[X, Y ]]-modules in Iwasawa theory.

To simplify the notation, we will denote in this section by

MθZp(m) = (eθ ⊗w H1

et(E ×K K,Zp))(m)

the p-adic lattice for the p-adic realization of Mθ(m).Suppose in the following and once and for all that p - #|O∗K | (if p > 3 this

condition is satisfied). Let Kn := K(E[pn+1]) be the field of definition of thepn+1-torsion points of E, and let K∞ := lim

←Kn its inverse limit. Denote by On

the ring of integers of these fields (respectively O∞). Then ∆ := Gal(K0/K)has order prime to p and Γ := Gal(K∞/K0) is isomorphic to Z2

p.Let G be the Galois group Gal(K∞/K); then G ∼= ∆ × Γ. Let An be the

p-part of the ideal class group of Kn, and let En be the group of global units O∗nof Kn. Let Up

n be the group of local units of Kn ⊗K Kp which are congruent to1 module the primes above p, where p is a prime of OK lying over p.

For every prime v of Kn above p, there is then an exact sequence

1 → Un,v → K∗n,v → Z× κ∗n → 1

and Upn = ⊕v|pUn,v. Here Un,v are the local units congruent to 1 modulo v and

κn is the residue class field of Kn,v.First of all, let’s recall briefly the definition of the elliptic units Cn in Kn.

For every ideal a of K prime to 6 we can define a theta function

θa : E \ ker([a]) −→ C

which has divisor N(a)(e)− ker([a]) (for the precise definition see [19, (4.2.2)]).The function θa(z) is in fact a 12-th root of the function defined in [7, II.2.4].Let fθ be a fixed ideal of OK such that O∗K → OK/fθ is injective, and supposethat fθ divides the conductor f of the elliptic curve E. Let’s denote by tfθ

agenerator for E[fθ]-torsion points as OK-module, and let a be an ideal prime to6fθ.

Definition 4.1. Let Cn be the subgroup of units generated over Z[Gal(Kn/K)]by ∏

σ∈Gal(K(fθ)/K)

θa(tσfθ+ hn),

where a runs through all ideals prime to 6pf, K(fθ) is the ray class field definedby fθ and hn is a primitive pn-torsion point (i.e. a generator of the pn-torsionpoints of E as OK-module). Define the group of elliptic units of Kn as

Cn := µ∞(Kn)Cn,

where µ∞(Kn) denotes the roots of unity in Kn.

Denote by En and Cn the closures of En ∩ Upn respectively Cn ∩ Up

n in Upn.

Define

A∞ := lim←An, E∞ := lim

←En, C∞ := lim

←Cn, Up

∞ := lim←Up

n

11

where the limits are taken with respect to the norm maps.Denote by Mp

∞ the maximal abelian p-extension of K∞ which is unramifiedoutside of the primes above p, and write X p

∞ := Gal(Mp∞/K∞). Define the

Iwasawa algebraZp[[G]] := lim

←Zp[Gal(Kn/K)]

which has a natural action of Zp[∆].For any irreducible Zp-representation χ of ∆, consider

eχ :=1

#∆

∑

τ∈∆

Tr(χ(τ))τ−1 ∈ Zp[∆]

and for every Zp[∆]-module Z denote by Zχ := eχZ.We define

Λχ := Zp[[G]]χ = Rχ[[Γ]]

where Rχ is the ring of integers in the unramified extension of Zp of degreedim(χ).

We denote by Λ := Op[[Γ]] where Op = OK ⊗ Zp. The modules Aχ∞ and

Eχ

∞/Cχ

∞ are torsion Λχ-modules. The classical theorem for the main conjecturein Iwasawa theory, using the determinant notation instead of the characteristicideal (see [18, Proposition 6.1]), states:

Theorem 4.2 (Rubin, theorem 4.1 in [21]). Let p - #O∗K .i) Suppose that p splits in K. Then

detΛχ(Aχ∞) = detΛχ(Eχ

∞/Cχ

∞).

ii) Suppose that p remains prime or ramifies in K and that χ is nontrivial onthe decomposition group of p in ∆. Then

detΛχ(Aχ∞) = detΛχ(Eχ

∞/Cχ

∞).

Remark 4.3. Rubin uses for the above theorem 4.2 in [21], another definitionfor the elliptic units as the ones defined above, but the result remains true withour definition of elliptic units. Let me briefly explain the reasons for this claim.The original definition [21] defines Cn(F ) with F an abelian extension of K by

NFK(g)/F θa(τ)σ−1|τ, g, a as follows, and σ ∈ Gal(F/K),

where g runs any ideal of OK such that O∗K injects into OK/g, τ is an exact gdivision point and a be an ideal prime to 6g. Denote by I the augmentation idealfor Gal(K∞/K) and by S the annihilator in Zp[[X, Y ]] of µp∞(K∞), the groupof p-power roots of unity in K∞. Then the elliptic units in [21] are isomorphicas Zp[[X,Y ]]-modules to IS (cf. [21, thm 7.7]) . We observe, taking invariantsby χ, if χ is not trivial, that I becomes invertible; this implies that we canchange the Rubin’s definition of elliptic units for another one without multiplyour theta function by σ − 1. Our elliptic units are isomorphic to S, then usingthe observation by the ideal augmentation we can modify them multiplying byσ − 1. After this modification, our elliptic units are included in the ones ofRubin in [21], and the Λχ-determinant remains equal with this modification. Iffθ = f the conductor of the elliptic curve, is known by the specialist theorem

12

4.2 is true with the definition 4.1. If fθ divides f, we can relate our definition ofelliptic units (4.1) for the fix ideal fθ with the definition of elliptic units (4.1)with f as a fix ideal. We obtain that the one defined by f is exactly multiplyby

∏l(σ

−1l − 1) the one with fθ, where σl is the Frobenius at l, where l runs

in the prime ideals of K which divide f but do not divide fθ. Then we relateboth definitions multiplying with an element of the augmentation ideal. Thenwe have an inclusion:

C∞,f ⊂ C∞,fθ⊂ C∞,

because we modify our definition by the augmentation ideal and the element thatmultiply belongs to this ideal. Now, we take χ-invariant and Λχ-determinants.We know that the first determinant coincides with the third one, because thetheorem 4.1 is satisfied in the case fθ = f for our definition of elliptic units, thenas consequence the determinant of the middle coincides also with the third one,obtaining then our claim.

Let X∞ be the Galois group for the maximal abelian p-extension Mp∞ of K∞

over K∞ which is unramified outside of the primes above p. Define also

U∞ := Up∞ × Up∗

∞

if p = pp∗ is split, andU∞ := Up

∞if p is inert or ramified. Similarly, let Yn be the p-adic completion of (Kn⊗Qp)∗

and Y∞ := lim←Yn. We have an inclusion U∞ ⊂ Y∞. Using Class field theory

we have an exact sequence

0 → E∞/C∞ → U∞/C∞ → X∞ → A∞ → 0, (1)

where C∞ is diagonally embedded into Up∞ × Up∗

∞ if p is split.We can consider moreover the representations χ : ∆ → O∗K . For any topo-

logical group B, we denote by B the completion by the ideal p. Denote thenby

Bχ := b ∈ B ⊗OK |σb = χ(σ)b ∀σ ∈ ∆.Using this notation, we have also an idempotent

eχ :=1

#∆

∑

τ∈∆

Tr(χ(τ))τ−1 ∈ OK ⊗ Zp[∆].

Using the sequence (1) one gets easily [19, cor. 2.1.5]:

Corollary 4.4. Under the same hypothesis of theorem 4.2, and if Xχ∞ denotes

eχX∞, we have thatdetΛχ(Xχ

∞) = detΛχ(Uχ∞/Cχ

∞).

To control the coimage of the inclusion of U∞ ⊂ Y∞ we will use the followingresult, [19, lemma 2.1.6].

Lemma 4.5. Let p be a prime such that p - NK/Qf. If p splits in K, theinclusion U∞ → Y∞ is an isomorphism and if p is inert or ramified in K, thereis an exact sequence

0 → U∞ → Y∞ → Zp[∆/∆p] → 0

where ∆p is the decomposition group of p in ∆.

13

Recall that OS = OK [1/S]. Here and in the following S denotes the set ofprimes of K which divide fθ plus the primes of K above p, and S′ denotes theset of primes of K above p and the primes which divide the conductor f of theelliptic curve E. Denote by Sp the set of finite primes over p, and by On,Sp

the ring of integers of Kn where all the primes above p are inverted. We definethen O∞,Sp := lim

→On,Sp . Similarly we define On,S and O∞,S , where the primes

above S are inverted in Kn or K∞ respectively.After introducing all the above notations, we are going to define a map in

the spirit of Soule:

ep : C∞ ⊗Zp (eθ ⊗w TpE)(l) → H1(OS , (eθ ⊗w TpE)(l + 1)).

All makes sense because MθZp(w + l) is unramified outside S and it is consideras OS-sheaf. Denote in the following by Op := OK⊗Zp and by Oθ,p := Oθ⊗Zp.

Using the definition of MθZp(w)(l + 1), we have that

H1(OS , (eθ ⊗w TpE)(l + 1)) = lim←

H1(OS , (eθ ⊗w E[pr+1])(l + 1)).

Define ep in the following way. Consider (θr)r a norm compatible system ofelliptic units and an element (tr)r of lim

←(eθ(⊗wE[pr+1]))(l), then we define

ep((θr ⊗ tr)r) := (NormKr/K(θr ⊗ tr))r.

It is well defined because θr ⊗ tr is an element in

O∗r,S/(Or,S)pr+1 ⊗ (eθ(⊗wE[pr+1]))(l) ⊂ H1(Or,S , (eθ(⊗wE[pr+1]))(l + 1))

Definition 4.6. The Soule elliptic elements are the elements in the image ofthe map

ep : (C∞ ⊗MθZp(w + l))G → H1(OS , MθZp(w + l + 1))

where G = Gal(K(E[p∞])/K).

We need to find a finite field extension field of K such that MθZp(w + m)is unramified in all places outside p. We know that the elliptic curve has goodreduction over K0 at all the places not dividing p. And hence, using Serre-Tatetheorem, the Tate module of the elliptic curve is unramified in those places. Inparticular, our tensor product of Tate modules twisted by m, corresponding toMθZp(w + m), is unramified in all the places not dividing p in K0. Therefore,K0 verifies the conditions.

For a GK-module M , we define

H1(K∞ ⊗Qp,M) := lim←

H1(Kn ⊗Qp,M) = lim←⊕v|pH1(Kn,v,M).

Suppose that M is a Op := OK⊗Zp-module. We will use the following notation:If M has and action of absolute Galois group GK of K,

M ′ := HomOp(M,Qp/Zp(1)⊗Zp Op)

with the natural action of GK ; and if M has an action of a group G

M∗ := HomOp(M,Qp/Zp ⊗Zp Op)

with the natural action of G.We follow closely the argument in [19, §2], but now the p-adic integer real-

ization is MθZp(w) instead of the Tate module of the elliptic curve E.

14

Proposition 4.7. There are isomorphisms of Op[[G]]-modules

X∞ ⊗ZpMθZp

(w + l) ∼= H1(O∞,Sp,MθZp

(w + l + 1)′)∗

Y∞ ⊗Zp MθZp(w + l) ∼= H1(K∞ ⊗Qp,MθZp(w + l + 1)′)∗.

Proof. See [19, Proof prop. 2.2.3] or [2, Proof prop. 3.4.7].

Proposition 4.8. The groups

H2(K∞ ⊗Qp,MθZp(w + l + 1)′) and H2(O∞,Sp

,MθZp(w + l + 1)′)

are zero.

Proof. See [19, Proof prop. 2.2.4] or [2, Proof prop. 3.4.8].

Let’s now recall [19, lemma 2.2.6].

Lemma 4.9. Let M be a perfect complex of Λ = Op[[Γ]]-modules. Then hereare canonical isomorphism

M∗ ⊗LΛ Op∼= RΓ(Γ, M)∗

where the right hand side is the (continuous) group cohomology of Γ and M∗ =Hom(M,Qp/Zp ⊗Op).

Using this lemma, we obtain the following result.

Corollary 4.10. There are exact triangles

(Y∞ ⊗MθZp(w + l))⊗LΛ Op → RΓ(K0 ⊗Qp,MθZp(w + l + 1)′

)∗ [−1]

→ RΓ(Γ,H0(K∞ ⊗Qp,MθZp(w + l + 1)′)

)∗[−1]

and

(X∞ ⊗MθZp(w + l))⊗LΛ Op → RΓ(O0,Sp ⊗Qp,MθZp(w + l + 1)′)∗[−1]

→ RΓ(Γ, H0(O∞,Sp ,MθZp(w + l + 1)′)

)∗[−1].

If in the definition of ep only take the norm maps to K0, we get a map ofcomplexes

ep : C∞ ⊗MθZp(w + l) → H1(O0,Sp ,MθZp(w + l + 1))

Recall that, for weight reasons, we have that H0(O0,Sp , MθZp(w + l + 1)) = 0.So, we obtain a map of complexes

ep : (C∞ ⊗MθZp(w + l))⊗LΛ Op → RΓ(O0,Sp , MθZp(w + l + 1))[1]

Lemma 4.11. -The following diagram is commutative

15

(C∞ ⊗MθZp(w + l))⊗LΛ Opep−→ RΓ(O0,Sp ,MθZp(w + l + 1))[1]

↓ ↓(Y∞ ⊗MθZp(w + l))⊗LΛ Op −→ RΓ(K0 ⊗Qp,MθZp(w + l + 1)′)∗[−1]

α ↓ ↓(X∞ ⊗MθZp

(w + l))⊗LΛ Op −→ RΓ(O0,Sp,MθZp

(w + l + 1)′)∗[−1]

where the map α is induced by the natural map Y∞/C∞ → X∞.

Proof. See [19, Proof lemma 2.2.8] or [2, Proof lemma 3.4.11].

We consider in the following the representation χ of the group ∆ given bythe action of ∆ in

HomOp(MθZp

(w + l),Op).

We impose that the above representation is irreducible. If (aθ, bθ) denotes theinfinity type for ψθ, then ψθ is irreductible if p− 1 - aθ − bθ and p splits in K orif p + 1 - aθ − bθ and p is inert in K.

Definition 4.12. A character χ of the group ∆ is called a good character if itis irreducible and it satisfies the Iwasawa main conjecture (that is, if χ satisfiesthat detΛχ(Aχ

∞) ∼= detΛχ(Eχ

∞/Cχ

∞)), and that

Uχ∞ ∼= Yχ

∞.

These conditions are always satisfied when the prime p splits in K. If p isinert or ramified, χ is good if χ is non trivial on the decomposition group of pin ∆ (by theorem 4.2(b)) and Zp[∆/∆p]χ = 0 by lemma 4.5. We impose oncefor all that χ is not the cyclotomic character and that χ is a good character.

Lemma 4.13. Observe that our character χ is equal to (ψθκl)−1 where κ is the

cyclotomic character. Suppose that p splits in K and suppose that p−1 - aθ+l+1or p− 1 - bθ + l + 1 or p− 1 - aθ − bθ. Then χ is not the cyclotomic character.

Proof. Since p is split in K we have that p = pp∗, where p is not equal to p∗. Let∆p be the Galois group Gal(K(E[p])/K); it is a subgroup of the decompositiongroup since p is totally ramified in ∆p. Observe that ψθ ⊗ Zp = ψΩ1 ⊕ ψΩ2 ,since p is split. It is known ψΩ1

|∆p = κbθ (see for example [10, §2.5]), so we getthat our character is different for κ as long as #∆p = p− 1 - bθ + l + 1, since κis a generator for the character group of ∆p.

Using the same kind of argument for p∗ applied also to the character ψΩ1

we obtain the same result but with aθ instead of bθ. Thus, we will obtain thecyclotomic character only in the case that p−1 | aθ + l+1 and p−1 | bθ + l+1.

We can also use the same argument for ψΩ2 , obtaining the same simultaneousarithmetic conditions, i.e. p− 1|l + bθ + 1 and p− 1|aθ + l + 1. We refer to [11,p.220,pp.223-234] for more details on the above characters.

The goal of the rest of the section is to show that the map ep induces anisomorphism of Op-modules

detOp((Cχ

∞ ⊗MθZp(w + l))⊗LΛ Op) ∼= detOp(RΓ(OS ,MθZp(w + l + 1)))−1.

Proposition 4.14. With the same hypothesis as above, we have that

detOp(RΓ(G,H0(K∞ ⊗Qp, MθZp(w + l + 1)′))) ∼= Op

detOp(RΓ(G, H0(O∞,Sp ,MθZp(w + l + 1)′))) ∼= Op

16

Proof. It follows from [10, prop. 2.4.6], that the action of G on

MθZp(w + l) ∼= (e(OK ⊗ · · ·OK)⊗Qp) ∼= Oθ,p

is via the characterψθ : G → O∗θ,p.

We have hence a surjection of Op-modules ρ : Op[[Γ]] → MθZp(w + l). Thusker(ρ) is an ideal of height 2 because Γ ∼= Z2

p. We know that detR is determinedby the ideals of height 1 for the ring R (cf. [17, 2.1.4]). This implies that

detOp(MθZp

(w + l)⊗LOp[[G]] Op) ∼= Op. (2)

In fact, since ∆ is finite and G ∼= Γ×∆, we have the equality

MθZp(w + l)⊗LOp[[G]] Op

∼= (MθZp(w + l))∆ ⊗LOp[[Γ]] Op.

Since we know that ker(ρ) has height 2, we have detOp[[Γ]]((MθZp(w + l))∆) ∼=Op[[Γ]] and so detOp((MθZp(w + l))∆ ⊗LOp[[Γ]] Op) ∼= Op. This checks (2). Weconclude by using lemma 4.9.

Using the isomorphismYχ∞ ∼= Uχ

∞

coming from the property that χ is a good character, and applying the derivedfunctor RΓ(∆, ) to the sequences in corollary 4.10 it proves:

Corollary 4.15.

detOp((Uχ∞ ⊗Op MθZp(w + l))⊗LΛ Op) ∼=

detOp

(H0(∆, RΓ(K0 ⊗Qp,MθZp(w + l + 1)′)∗[−1])

)

and thatdetOp

((Xχ∞ ⊗Op MθZp(w + l))⊗LΛ Op

) ∼=detOp

(H0(∆, RΓ(O0,Sp ⊗Qp,MθZp(w + l + 1)′)∗[−1])

)

So, if we apply the functor RΓ(∆, ) to the triangle

RΓ(O0,Sp ,MθZp(w + l + 1)) → RΓ(K0 ⊗Qp,MθZp(w + l + 1)′)∗[−2]

→ RΓ(O0,Sp ,MθZp(w + l + 1)′)∗[−2],

we obtain the following result.

Corollary 4.16. There is an isomorphism of determinants

detOp(H0(∆, RΓ(O0,Sp ,MθZp(w + l + 1))))−1 ∼=

detOp((Uχ∞ ⊗Op MθZp(w + l))⊗LΛ Op)detOp((Xχ

∞ ⊗Op MθZp(w + l))⊗LΛ Op)−1

We need now to find the relation between O0,Sp and O0,S .

17

Lemma 4.17. The restriction map induces an equality of determinants

detOp(H0(∆, RΓ(O0,Sp

,MθZp(w + l + 1))) ∼=

detOp(H0(∆, RΓ(O0,S ,MθZp(w + l + 1)))) ∼= detOp(RΓ(OS , MθZp(w + l + 1))).

Proof. Consider the exact triangle

RΓ(O0,Sp,MθZp

(w + l)) → RΓ(O0,S ,MθZp(w + l))

→ ⊕v∈S\SpRΓk(v)(Ov,MθZp

(w + l))[1]

where Ov is the local ring at v. Since TpE is unramified at the places of K0 inS \ S0, the same is true for eθ(⊗TpE)(l). By purity we have that

RΓk(v)(Ov,MθZp(w + l)) ∼= RΓ(k(v),MθZp(w + l)).

But we have that

H0(∆,⊕v∈S\SpRΓ(k(v), MθZp(w + l)) = 0.

To show this result, observe that

H1(k(v), MθZp(w + l)) ∼= MθZp(w + l)Gal(k(v)/k(v))

and H0 = 0 since we are under the hypothesis that −w − 2l ≤ −3. Now, let vo

be a prime dividing v in K0 and let ∆v0 be the stabilizer of vo. Since Iv0 ⊂ ∆v0

acts non trivially in the coinvariants

MθZp(w + l)Gal(k(v)/k(v))

because v0|fθ, we obtain the result.

As a consequence we have the main result of this section.

Theorem 4.18. Suppose that p is an odd prime, prime to NK/Qfθ and to#|O∗K |. Let χ the ∆-representation on HomOp(Mθ(w + l),Op) be a good repre-sentation. Then, the map ep induces an isomorphism of Op-modules

detOp((Cχ

∞ ⊗Op MθZp(w + l))⊗LΛ Op) ∼= detOp(RΓ(OS ,MθZp(w + l + 1)))−1.

Proof. See [19, Proof thm.2.2.12] or [2, Proof thm.3.4.19].

5 The comparison between the map rp and ep inthe constructible K-elements

Let’s start recalling the result of Kings on the specialization of the elliptic poly-logarithm sheaf, which is a new key in the proof of the Tamagawa numberconjecture.

Let E be an elliptic curve over a base scheme T , and denote by π thestructural morphism, i.e. π : E → T which is proper and smooth. ConsiderU = E \ e, where e is the zero section of E. There exist on U a lisse pro-sheaf(i.e. a projective limits of lisse sheaves), which is called the elliptic polylogarithm

18

sheaf and denoted by PolQp ([19, §3.2]). Let us consider t a N -torsion point inE diferent of e. Then are defined too some projections prt and σ ( [19, §3.5.1],[19, §3.5.3] respectively), which define the p-adic k-Eisenstein class associatedto a torsion point t by

(σkprtt∗PolQp

),

for any integer k. This element is defined inside the cohomology group

H1(T, SymkHQp),

where HQp := HomT (R1π∗Qp,Qp). The definition of the p-adic Eisensteinclasses is extended by linearity to any divisor formed by N -torsion points ([19,Def. 3.5.9]). The main part of the result of Kings is the explicit computationof these Eisenstein classes. He compares this classes with the elements comingfrom a connecting map of the cohomology of some tori which appears when weconsider torsors in the elliptic curve. Let us be a little more precise. ConsiderHn := ker[pn] as a scheme over T . Let us consider the map multiplicationby pn, pn : En → E, where En is the elliptic curve E but which has pn-torsors. Consider the characteristic group I[Hn] := ker(pn,∗Z → Z) which isthe characteristic group to a torus THn . In this situation we have a connectingmap δ as follows ([19, (10),§4.1]):

δ : H0(Hn, THn) → H1(Hn, THn [pr]). (3)

Using this connecting morphism, we can express the Eisenstein classes explicitly.

Theorem 5.1 (Kings, theorem 4.2.9 in [19]). Let p be a prime number,and let E be an elliptic curve over a base scheme T where p is invertible.

Let β be any divisor in E of the form

β :=∑

t∈E[N ](T )\ent(t)

and consider [a] : E → E any isogeny relatively prime to Np.Then, for any m > 0, the p-adic Eisenstein class

Na(a⊗mNa− 1)(β∗PolQp)m ∈ H1(T, SymmHQp(1))

is given by

± 1m!

(δ∑

t∈E[N ](T )\ent

∑

[pn]tn=t

θa(−tn)tn⊗m)n

where tn is the projection of tn to E[pn] and δ is the Sym-extension of the bound-ary map H0(Hn, THn) → H1(Hn, THn [pr]) where Hn := ker[pn] is consider as ascheme over T and THn is the torus with character group I[Hn] := ker(pnZ→Z) (3).

The following result relates the image of EmM(β) by the Soule regulator with

the polylogarithmic sheaf.

Theorem 5.2. Let β be as in the previous theorem. Then we have that

rp(EmM(β)) = −N2m(β∗PolQp)m

in H1(T, SymmHQp(1)).

19

Proof. The same proof of [19, Theorem 1.2.5] with m instead of 2k + 1 works.See also [2, proof Theorem 3.5.2].

We are going to apply these results above to the divisor βθ = NK(f)/K((t)),where t := Ωf−1 is a fθ-torsion point. Take N = NK/Qfθ, m = w + 2l, T = OS

and HQp = TpE ⊗ Qp, using the notations of the previous theorem 5.1. Leta ⊂ OK be an ideal prime to 6pf, and consider the isogeny given by ψ(a).Finally, θa means the classical theta function.

To simplify the notation, define for any tr ∈ E[pr]

γ(tr)m :=< tr,√

dK tr >⊗m

where <, > means the Weil pairing. Our objective is the computation of

KM Ew+2lM (βθ)

Remember that we are under the restriction aθ 6≡ bθ mod|O∗K |.Considering the diagram [6, (2.8)]

H2l+w+1M (Sym2l+wh1E,Q(w + 2l + 1))

(∆CM )l×id)∗−→ H2l+w+1M (El+w,Q(2l + w + 1))

KM ↓ ↓ pr∗Hw+1M (MθQ,Q(w + l + 1))

eθ←− Hw+1M (h1(E)⊗w,Q(l + w + 1)),

with pr the projection in the last w components and ∆CM : E → E × E givenby e 7→ (e,

√dKe). We obtain a map in Galois cohomology given by

H1(OS , Sym2l+wHQp(1)) →H1(OS , (eθSymwHQp)(l + 1)) = H1(OS ,Hw(Mθ ×K K,Qp(w + l + 1)))

such that

KM(ψ(a)⊗2l+wSym2l+wHQp(1)) = eθ(⊗wψ(a))NalSymwHQp(l + 1).

Theorem 5.3. Let p be a prime number such that p - 6N fθ. Let θ be theidempotent satisfying #|θ| = 2 and such that the infinity type (aθ, bθ) satisfiesaθ 6≡ bθ mod |O∗K |. For a prN fθ-torsion point tr, denote by tr its projection toE[pr]. Then, if t = Ωf−1, we have the following equality

Na(ψθ(a)Nal+1 − 1

)rp(ξθ,l) =

(−1)lLp(ψθ,−l)−1NTθ/Qf3l+2wθ Φ(fθ)

2l−1ψθ(f)Φ(f)

(δNK(f)/K

∑prtr=t

θa(−tr)⊗ eθ(⊗w tr)⊗ γ(tr)l

)

r

Proof. Using theorem 3.5 and the theorems above, we have that

rp(ξθ,l) =(−1)l−1(2l + w)!Lp(ψθ,−l)−1Φ(fθ)

2l−1NTθ/Qflθψθ(f)Φ(f)KM(E2l+w

M (β))

=(−1)l(2l + w)!Lp(ψθ,−l)−1NTθ/Qf3l+2w

θ Φ(fθ)2l−1ψθ(f)Φ(f)

KM(β∗PolQp)w+2l.

Using the same notation as above, we have

KM(tr⊗2l+w) = eθ(⊗w tr)⊗ γ(tr)l

Finally, applying Kings’ theorem 5.1, we obtain the desired identity.

20

We want to rewrite the previous formula in terms of the norm map of theextension K(fθ)K(E[pn])/K. For technical reasons, we will work with f insteadof fθ since then we can use that K(E[pnf]) = K(pnf), the ray class field, becausef is the conductor of E and divides the ideal fpn ([7, Prop.1.6]).

Fix a prime p of K where E has good reduction, and take π = ψ(p). Denoteby

Hpr,t := tr ∈ E[prf]|πrtr = t.

We write tr = (tr, π−rt) ∈ E[prf] = E[pr]⊕ E[f]. Define a filtration of Hpr,s as

F ir,t := tr ∈ Hp

r,s|πr−itr = 0.Theorem 5.4. Let p be as above and tr = (tr, π−rt) ∈ F 0

r,s \ F 1s,t. Suppose

O∗ → (O/fθ)∗ is injective. Denote the Euler factor of ψθ at p evaluated at −lby Lp(ψθ,−l). Then

Lp(ψθ,−l)−1

NK(f)/K

∑

sr∈Hpr,t

θa(−sr)⊗ eθ(⊗wsr)⊗ γ(sr)l

r

=

(NK(prf)/K

(θa(−tr)⊗ eθ(⊗w tr)⊗ γ(tr)l

))r

in H1(OS , eθ(TpE(1))(l)⊗Qp) for all a relatively prime to pf.

Proof. The identification HomOp(TpE,Op) ∼= TpE(−1) is via the conjugatelinear Op-action on the right side. Hence ψ(p)tr = tr−1. We have the equality

(ψθ(p)/Np−l)iNK(prf)/K(pr−if)(θa(−tr)⊗ eθ(⊗w tr)⊗ γ(tr)l) =

NK(prf)/K(pr−if)(θa(−tr)⊗ eθ(⊗wψ(p)itr)⊗ γ(ψ(p)

itr)l) =

(NK(prf)/K(pr−if)(θa(−tr)))⊗ eθ(⊗w ˜tr−i)⊗ γ( ˜tr−1)l) =

θa(−( ˜tr−i, πi−rt))⊗ eθ(⊗w ˜tr−i)⊗ γ( ˜tr−i)l,

where the last equality uses the distribution relation for θa ([7, II 2.5]).The Galois group of K(pr−if)/K(f) acts simply transitively on F i

r,t \ F i+1r,t .

We get hence that

(ψθ(p)/Np−l)iNK(prf)/K(f)(θa(−tr)⊗ eθ(⊗w tr)⊗ γ(tr)l) =∑

tr−i∈F ir,t\F i+1

r,t

θa(−( ˜tr−i, πi−rt))⊗ eθ(⊗w ˜tr−i)⊗ γ( ˜tr−i)l.

We know by [7, Prop. II.2.4.ii)] that we have the equality θa(−( ˜tr−i, πi−rt)) =

θa(−( ˜tr−i, π−rt))σi

p with σp is the Frobenius at p in the Galois group of K(f)/K.This and the fact that NK(f)/K is the sum over all Galois translates, which acttrivially on ˜tr−i, gives that

(ψθ(p)/Np−l)iNK(prf)/K(θa(−tr)⊗ eθ(⊗w tr)⊗ γ(tr)l) =

NK(f)/K

∑

tr−i∈F ir,t\F i+1

r,t

θa(−( ˜tr−i, π−rt))⊗ eθ(⊗w ˜tr−i)⊗ γ( ˜tr−i)l

,

Adding this equalities with respect to i and increasing r if necessary we get theresult.

21

Lemma 5.5. Suppose that θ has infinity type (w, 0) or (0, w) and (#|O∗K |, w) =1. Then

O∗K → (OK/fθ)∗

is injective.

Proof. Let u be and element in O∗K , u 6= 1 and consider the idele defined byx∞ = 1 and xp = u for all finite places p of K. Then ψw(x) = ψw(u−1x) =uw 6= 1 if (w, |O∗K |) = 1. So, by definition of the conductor of ψθ, we obtainthat u 6≡ 1(mod fθ), hence the result desired for the type (w, 0). For the type(0, w) the proof is the same but with ψ instead of ψ.

Corollary 5.6. With the same hypothesis of theorem 5.3 but with p - 6N f andif O∗K → (OK/fθ)∗ is injective, we have the equalities

Na(ψθ(a)Nal+1 − 1)rp(ξθ,l) =

± N f3l+2wθ Φ(fθ)

2l−1ψθ(f)Φ(f)δ(NK(E[pr ])K(f)/Kθa(−tr)⊗ eθ(⊗w tr)⊗ γ(tr)l

)r

=

± N f3l+2wθ Φ(fθ)

2l−1ψθ(f)Φ(f)|Gal(K(f)/K(fθ))|·

δ(NK(E[pr])K(fθ)/Kθa(−tr)⊗ eθ(⊗w tr)⊗ γ(tr)l

)r

where tr is a primitive prfθ-torsion point with prtr = t and a is relative primeto pf.

Proof. If p is inert or prime the first equality is deduced from the previoustheorem. If p split, it decomposes in a p and a p∗ part. Putting together theprevious result with p and with p∗, we have the first equality.

For the second equality, we have that

NK(E[pr ])K(f)/K(E[pr ])K(fθ)θa(−tr) =∏

σ∈Gal(K(f)K(E[pr])K(pr)/K(fθ)K(E[pr])K(pr))

θa(−tr)σ

since K = K(1) and hence K(f) is disjoint with K(pr) over K, and we alsoknow that K(f) = K(E[f]) is disjoint with K(E[pr]) over K. Moreover, sinceθa(−tr) ∈ K(f)K(pr) = K(fpr) and (f, p) = 1, we have that the norm is equalto ∏

τ∈Gal(K(fpr)/K(fθpr))

θa(−tr)τ .

But θa(−tr) ∈ K(fθpr) because −tr is a point of fθpr-torsion. Hence we get the

second equality.

Now we want to show that the elements

(NK(E[pr])K(fθ)/Kθa(tr)⊗ eθ(⊗w tr)⊗ γ(tr)l)r

generate (Cχ

∞ ⊗ MθZp(w + l))Γ, where a is prime to 6pf and where χ is therepresentation of ∆ on HomOp(MθZp(w + l),Op), that we suppose it is a goodrepresentation. Here we use MθZp with the same notation as in section 4.

We suppose from now on that the natural map O∗K → (OK/fθ)∗ is injective.

22

Proposition 5.7. Consider p - 6N fθ and a an ideal in Op, which is prime to6pf and such that ψθ(a)Nal+1 6≡ 1(mod p). Then the Op[[Γ]]-module

Cχ

∞ ⊗Op(eθ(⊗wTpE)(l))

is generated by (θa(tr)⊗eθ(⊗w tr)⊗γ(tr)l)r, where tr is a primitive prfθ-divisionpoint.

Remark 5.8. Before we begin with the proof, let’s remark that the existenceof an ideal a satisfying the conditions of the proposition 5.7 is equivalent to thecondition that the ∆-representation χ is not the cyclotomic character (whichis one of the hypothesis that we imposed before for χ). To show this fact, justnotice that the Op-action on Op[[Γ]] is given by complex conjugation because theOp-action on eθ(⊗wTpE) is given by complex conjugation on Op.

Proof. Let b be another ideal prime to 6pf. Take σa = [a,Kn/K] and σb =[b,Kn/K]. Then, by the properties of the theta function, we have that

(σa − ψθ(a)Nal+1)(θb(tn)⊗ eθ(⊗w tn)⊗ γ(tn)l) =

ψθ(a)Nal(θb(tn)σa−Na ⊗ eθ(⊗w tn)⊗ γ(tn)l) =

ψθ(a)Nal(θa(tn)σb−Nb ⊗ eθ(⊗w tn)⊗ γ(tn)l).

Now, it is enough show that (σa − ψθ(a)Nal+1) is invertible in Op[[Γ]] = Λ,because Cχ

∞ is a torsion free Λ-module since U∞ is torsion free ([22, Prop.11.4]),and we have an inclusion of C∞ in U∞.

But the element σa corresponds to 1 on Op/p and thus σa − ψθ(a)Nal+1 isinvertible in Λ because 1 6≡ ψθ(a)Nal+1 mod p.

It remains prove that eθ(tr)⊗γ(tr) generates MθZp(w+ l). This follows sinceMθZp(w) is one dimensional and hence to prove that it generates Zp(l) one canuse the same proof as in [19, p.623].

Corollary 5.9. Assume that p > NK/Qf. Then the image of Rθ by rp inH1(OS , eθ(⊗wTpE)(l + 1))⊗Qp coincides with

ep((Cχ

∞ ⊗ eθ(⊗wTpE)(l))Γ).

Proof. As|Gal(K(f)/K(fθ))|N f3l+2w

θ Φ(fθ)/2l−1ψθ(f)Φ(f)

is prime to p, it follows from the definition of ep and Corollary 5.6.

Let’s note the following lemma.

Lemma 5.10. The canonical map

(C∞⊗MθZp(w + l))⊗LOp[[G]]Op → (C∞⊗MθZp(w + l))G ∼= (Cχ

∞⊗MθZp(w + l))Γ

is an isomorphism and moreover (Cχ

∞ ⊗MθZp(w + l))Γ ∼= Op.

Proof. We observe that the proof of proposition 5.7 shows that Cχ

∞ ∼= Λχ =Op[[Γ]] is a free Λχ-module of rank 1. This implies, as in [19, lemma 5.2.3],that (Cχ

∞ ⊗MθZp(w + l))Γ ∼= Op. The claim follows since the previous moduleis induced and hence the higher Tor-terms vanish.

23

As a consequence, we get the part 4 of conjecture 1.2 for the constructiblesubspace Rθ.

Corollary 5.11. The map

Rθ ⊗ Zp → RΓ(OS ,MθZp(w + l + 1)⊗Qp)[1]

induced by rp, gives an isomorphism

detOpRθ

∼= detOpRΓ(OS ,MθZp

(w + l + 1))−1

This proves, taking NormK/Q, the conjecture 1.2 under the hypothesis thatthe Soule regulator is not zero for the elements in K-theory constructed for themotive Mθ(w + l + 1). Let us first write the same result but over OK .

Theorem 5.12. Let p be a prime different from 2 and 3 (hence, in particular,p - #|O∗K |), and p > NK/Qf.Consider l a non-negative integer. Suppose thatψθ has infinity type (aθ, bθ) with aθ, bθ non-negative integers, such that aθ 6≡bθ mod|O∗K | and w = aθ + bθ ≥ 1 verifies −w − 2l ≤ −3. Suppose that O∗K →(OK/fθ)∗ is injective.

Suppose moreover that the representation χ of ∆ in HomOp(Hw(Mθ ×K

K,Zp(w + l)),Op) is a good representation (see the definition in 4.12) which isnot the cyclotomic character.

If we denote by MθZp(w + m) = Hw(Mθ ×K K,Zp(w + m)), then, there isan OK-submodule Rθ ⊂ Hw+1

M (Mθ,Q(w + l + 1)) of rank 1 such that:

1. detOK(rD(Rθ)) ∼= L∗S(ψθ,−l)detOK

(HwB (MθC,Z(w + l)))

in detOK⊗R(HwB (MθC,Z(w + l))⊗ R).

2. The map rp induces an isomorphism

detOK⊗Zp(Rθ) ∼= detOK⊗Zp(RΓ(OK [1/S],MθZp(w + l + 1))−1.

Here

L∗S(ψθ,−l) = lims→−l

LS(ψθ, s)s + l

,

and S is the set of primes of K dividing p and the ones dividing fθ.Moreover, if rp is injective on Rθ, the second part can be written as

detOK⊗Zp(H1(OK [1/S], MθZp(w + l + 1))/rp(Rθ)) ∼=detOK⊗ZpH2(OK [1/S],MθZp(w + l + 1)).

Proof. It is a direct consequence of the theorem 3.5 and the above corollary5.11.

As a consequence we obtain part of the local Tamagawa conjecture 1.2 forthe motive Mθ(w + l + 1).

Theorem 5.13. Let p be a prime different from 2 and 3 (hence, in particular,p - #|O∗K |), and p > NK/Qf. Consider l a non-negative integer. Supposethat ψθ has infinity type (aθ, bθ) with aθ, bθ non-negative integers, such thataθ 6≡ bθ mod|O∗K | and w = aθ + bθ ≥ 1 verifies −w − 2l ≤ −3. SupposeO∗K → (OK/fθ)∗ is injective.

Moreover, suppose that χ, the representation of ∆ in HomOp(MθZp(w +l),Op), is a good representation which is not the cyclotomic character.

Then, there is a submodule Rθ in Hw+1M (Mθ,Q(w + l + 1)) such that:

24

1. The map rD ⊗ R is an isomorphism restricted to Rθ ⊗ R.

2. dimQ(HwB (MθC,Z(w + l))⊗Q) = ords=−lLS(Hw(Mθ,Qp), s) = 2.

3. We have the equality

rD(detZ(Rθ)) = L∗S(Hwet(Mθ,Qp),−l)detZ(Hw

B (Mθ,Z(w + l)))

where

L∗S(Hwet(Mθ,Qp),−l) = lim

s→−l

LS(Hwet(Mθ,Qp), s)(s + l)2

and S is the set of places of K that divides p and the places dividing theconductor fθ.

4. We have that

detZp(Rθ) = detZp

(RΓ(OK [1/S],MθZp(w + l + 1)))−1.

If rp is injective on Rθ, then rp(detZ(Rθ)) is a basis of the Zp-lattice

detZp(RΓ(OK [1/S], MθZp(w + l + 1)))−1

⊂ detQp(RΓ(OK [1/S],MθZp(w + l + 1)⊗Q)[−1]).

Remark 5.14. Theorem 5.13 is part of the local Tamagawa number conjecture1.2 for Hecke characters. To prove the general conjecture it remains to provethe finiteness of the second cohomology group and the bijectivity of the p-Souleregulator map.

1. The finiteness of H2p := H2(O[1/S],MθZp(w+l+1)) follows from a general

conjecture of Jannsen [16].

Following the argument of Wingberg in [25], is easy to prove that if p is aregular prime for the field K(E[p]), then this Galois cohomology group isfinite. See [2, Appendix B] or an incoming preprint of the author.

2. Moreover, on the bijectivity of the Soule regulator we make the followingremarks.

(a) Deninger proves the Beilinson conjecture for the above Hecke char-acters. He computes the dimension of the space Hw

B (Mθ,Q(w + l))which it is mapped the Beilinson regulator map. He constructs ele-ments on the K-theory group HM, generating a subspace which hasthe same dimension of Hw

B (Mθ,Q(w + l)). These subspace of con-structed elements are denoted in the paper by Hconstr

M . It remains toprove on the Beilinson conjecture on Hecke characters of imaginaryquadratic fields that the K-theory group HM coincides with Hconstr

M .On the Soule regulator map, the image of Hconstr

M tensor by Q shouldmap (if the Soule regulator map is an isomorphism) to a free groupH1(OK [1/S],MθZp(w + l + 1))⊗Q of the same rank as Hconstr

M . Wenote that the group H1(O[1/S],MθZp(w + l +1))⊗Q has exactly thisrank when H2

p is finite, and bigger if not, [16, cor. 1].

25

(b) We observe that, if ep is injective, then rp is injective on Rθ, becausethe image of ep is an Oθ⊗Zp-module of rank 1 and Rθ is a Oθ⊗Zp-module of rank 1. Let us suppose that H2

p is finite. Then, similararguments as in [19, §5.2.2] gives the injectivity for ep like in thesituation of a Hecke character of type (1, 0) which is made in [19].

As a conclusion, for regular primes p, we obtain in full generality the conjecture1.2 for Hecke characters of imaginary quadratic fields.

A The remaining non-critical values

In the main body of the paper, we restricted the study of the Tamagawa numberconjecture to the motives Mθ(w + l + 1) with l ≥ 0, basically because of theformulation in §1 of the Tamagawa number conjecture (more precisely for thecondition r = w + l+1 > inf(m,X) = w). Deninger proves also the Beilinson’sconjecture for Mθ(w+ l+1) without the condition l ≥ 0. Then, the formulationfor the above motives is under this restriction in order to avoid the poles in thebad Euler factors. But, in our case, there are no poles in the bad Euler factors,that correspond to primes in S (see remark 2.6). Therefore we can ask whichis the image of some K-theory elements by the regulators maps for values withl < 0 and −w − 2l ≤ −3.

The restriction l ≥ 0 for Mθ(w+l+1) with −w−2l ≤ −3 does not include allthe non-critical values associated to the Hecke character ψθ (we restrict to thesituation aθ 6≡ bθ(mod|O∗K |) for simplicity). It remains to study the conjecturefor the motives Mθ(w + l + 1) which are related with the non-critical valuesassociated to ψθ for the integers l such that

0 < −l ≤ min(aθ, bθ)

(see theorem 3.1 for the exact non-critical values up to functional equation).In this appendix we construct elements in K-theory for Mθ(w + l + 1) with

0 < −l ≤ min(aθ, bθ) and computate the Beilinson regulator and the Souleregulator in them, obtaining the same result as in the main theorems in §5.

Deninger [6, pp.142-144]constructs also elements in K-theory for the motiveMθ(w + l + 1) with l < 0 and proves the Beilinson conjecture. In this situationhe constructs a projector map KM, without using complex multiplication. Theproblem of this construction is that in the image by this projector map of thespecialization of the elliptic polylogarithm, the Weil pairing appearing in §5corresponds to γ(tr) =< tr, tr > and the arguments in proposition 5.7 does notgeneralize. We modify the Deninger’s projector map by K′M, and with use of theK-theory elements K′M(EM(βθ)), we prove Beilinson’s conjecture. Hence, thearguments in §4 an §5 apply straightforward in order to obtain a determinantcomparison for the image of the Soule regulator map of these elements.

We follow the same notation introduced in the paper, and let us followthe arguments through §3 to §5 with l < 0 but we remain the other hypothesisappearing. Let us fix w ≥ 1 and l < 0 such that −w−2l ≤ −3 and let us considerthe motive Mθ(w+l+1). With the fixed embedding we have ϑ = (λ1, . . . , λw) ∈θ and set I1 = i|λi ∈ HomK(K,C) and I2 = i|λi /∈ HomK(K,C) and wehave now that 0 < |l| ≤ #I1 = aθ and 0 < |l| ≤ #I2 = bθ. Denote by∆ = id1× id2 : E → E×E the diagonal map and by ∆CM = id1,CM × id2,CM :

26

E → E × E given by e 7→ (e, (√

dK)e) where we understand√

dK ∈ End(E).Let us choose exactly |l| elements in the sets I1 and I2 denote it in increasingorder i1, . . . , i|l| ∈ I1 and j1, . . . , j|l| ∈ I2. Let us define the projector mappr : Ew+l → Ew+2l by the projection of the first w + 2l-components of Ew+l

and let us define (id×∆|l|) : Ew+l → Ew (which it depends of the choice in thesets I1 and I2) by (e1, . . . , ew+2l, ew+2l+1, . . . , ew+l) 7→ (eα1 , . . . , eαw) where eαs

is defined as follows:

• if αs appears in one component of the set of tuples L := (i1, j1), . . . , (i|l|, j|l|)then

eαs:=

id1(ew+2l+m) if αs = imid2(ew+2l+m) if αs = jm

,

• in the other case, then it is defined by eαs := eny with 1 ≤ ny ≤ w + 2lsuch that αs = ny +

∑1 where the sum runs the naturals that appear in

some component of the elements of L and which are lower than αs.

We define also the map (id × ∆|l|CM ) similarly as the map (id × ∆|l|) but

interchanging idi by idi,CM .We define the projector map K′M by

H2l+w+1M (Sym2l+wh1E,Q(w + 2l + 1))

pr∗−→ H2l+w+1M (E2l+w+|l|,Q(2l + w + 1))

K′M ↓ ↓ (id×∆|l|CM )∗

Hw+1M (MθQ,Q(w + l + 1))

eθ←− Hw+1M (h1(E)⊗w,Q(l + w + 1))

(Deninger defines KM with a similar diagram as above but with the map(id×∆|l|)∗ instead of the map (id×∆|l|

CM )∗). The next result is a modificationof Deninger’s result [6, pp.143-145].

Theorem A.1. Suppose that aθ 6≡ bθ mod |O∗K | and that aθ, bθ, l satisfy thehypothesis of the theorem 3.1 with l < 0. Define, up to ± by using the §3notation,

ξθ,l := ± (√

dK)2l(2l + w)!Lp(ψθ,−l)−1Φ(fθ)2−1NK/Qflθψθ(f)Φ(f)

K′M E2l+wM (βθ)

which belongs to Hw+1M (Mθ,Q(w + l + 1)). Then

rD(ξθ,l) = L∗S(ψθ,−l)ηθ,

where S are the primes of K that divide fθ and p.

Proof. We will follow closely Deninger’s papers [5] and [6], we follow also for thisproof his notation where his n is our w + 2l. We need to modify the calculationin [6, (2.13)Lemma] for K′M instead of KM, up to a sign because our order ofthe factors of the map (id×∆|l|

CM ), one obtains (up to sign)

1(2πi)w

∫

Ew

K′D(ξ) ∧ dz(ε) =

Bε

√dK

|l|(

nn + |l| − |ε|

)−1

A(Γ)n+|l|cn+|l|−|ε|

27

(we use the calculation at the top of [5, p.63]). Then we obtain, following theargument [6, p.143-144] and our conditions of §3 with l < 0,

rD(K′MEM(βθ)) = tθL∗(ψθ, l)ηθ

with tθ given by (up to sign)

2−1NK/Qflθψθ(f)Φ(f)(√

dK)2l(2l + w)!Φ(fθ).

By the remark 2.6 we obtain the result.

Following §3 we define for l < 0 the constructible space by

Rθ := ξθ,lOK ,

with ξθ,l given by the theorem A.1. Let observe that with this notation we canfollow straightforward all the results and proofs of §3 and §4. In §5 we needto compute K′M Ew+2l

M (βθ), and for this we only need to observe from thedefinition of K′M, our projector has these two properties,

K′M(ψ(a)⊗2l+wSym2l+wHQp(1)) = eθ(⊗wψ(a))NalSymwHQp(l + 1),

and KM(tr⊗2l+w) = eθ(⊗w tr) ⊗ γ(tr)l with γ(tr) =< tr,

√dK tr >. After this

observation all the results of §5 and the proof follows straightforward up to apower of 2 and dK . The reader could make these modifications which followfrom our definition of Rθ. Therefore we obtain the local Tamagawa numberconjecture with K-coefficients,

Theorem A.2. Let p be a prime different from 2 and 3 (hence, in particular,p - #|O∗K |), and p > NK/Qf. Suppose that ψθ has infinity type (aθ, bθ) withaθ, bθ non-negative integers, such that aθ 6≡ bθ mod|O∗K | and w = aθ + bθ ≥ 1verifies −w − 2l ≤ −3 with l < 0. Suppose that O∗K → (OK/fθ)∗ is injective.

Suppose moreover that the representation χ of ∆ in HomOp(Hw(Mθ ×K

K,Zp(w + l)),Op) is a good representation (see the definition in 4.12) which isnot the cyclotomic character.

If we denote by MθZp(w + m) = Hw(Mθ ×K K,Zp(w + m)), then, there isan OK-submodule Rθ ⊂ Hw+1

M (Mθ,Q(w + l + 1)) of rank 1 such that:

1. detOK(rD(Rθ)) ∼= L∗S(ψθ,−l)detOK

(HwB (MθC,Z(w + l)))

in detOK⊗R(HwB (MθC,Z(w + l))⊗ R).

2. The map rp induces an isomorphism

detOK⊗Zp(Rθ) ∼= detOK⊗Zp(RΓ(OK [1/S],MθZp(w + l + 1))−1.

Here

L∗S(ψθ,−l) = lims→−l

LS(ψθ, s)s + l

,

and S is the set of primes of K dividing p and the ones dividing fθ.Moreover, if rp is injective on Rθ, the second part can be written as

detOK⊗Zp(H1(OK [1/S], MθZp(w + l + 1))/rp(Rθ)) ∼=detOK⊗ZpH2(OK [1/S],MθZp(w + l + 1)).

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And the local Tamagawa number conjecture with Q-coefficients,

Theorem A.3. Let p be a prime different from 2 and 3 (hence, in particular,p - #|O∗K |), and p > NK/Qf. Suppose that ψθ has infinity type (aθ, bθ) withaθ, bθ non-negative integers, such that aθ 6≡ bθ mod|O∗K | and w = aθ + bθ ≥ 1verifies −w − 2l ≤ −3 with l < 0. Suppose O∗K → (OK/fθ)∗ is injective.

Moreover, suppose that χ, the representation of ∆ in HomOp(MθZp

(w +l),Op), is a good representation which is not the cyclotomic character.

Then, there is a submodule Rθ in Hw+1M (Mθ,Q(w + l + 1)) such that:

1. The map rD ⊗ R is an isomorphism restricted to Rθ ⊗ R.

2. dimQ(HwB (MθC,Z(w + l))⊗Q) = ords=−lLS(Hw(Mθ,Qp), s) = 2.

3. We have the equality

rD(detZ(Rθ)) = L∗S(Hwet(Mθ,Qp),−l)detZ(Hw

B (Mθ,Z(w + l)))

where

L∗S(Hwet(Mθ,Qp),−l) = lim

s→−l

LS(Hwet(Mθ,Qp), s)(s + l)2

and S is the set of places of K that divides p and the places dividing theconductor fθ.

4. We have that

detZp(Rθ) = detZp(RΓ(OK [1/S],MθZp(w + l + 1)))−1.

If rp is injective on Rθ, then rp(detZ(Rθ)) is a basis of the Zp-lattice

detZp(RΓ(OK [1/S], MθZp(w + l + 1)))−1

⊂ detQp(RΓ(OK [1/S],MθZp(w + l + 1)⊗Q)[−1]).

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Francesc Bars CortinaDepart. Matematiques,Edifici C,Universitat Autonoma de Barcelona08193 Bellaterra, Barcelona,Spain.e-mail: francesc@mat.uab.es

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