A relation between p -adic L -functions and the Tamagawa number conjecture for Hecke characters

We prove that the submodule in K -theory which gives the exact value (up to Z ∗ ( p ) ) of the L -function by the Beilinson regulator map at non-critical values for Hecke characters of imaginary quadratic ﬁelds K with cl ( K ) = 1( p -local Tamagawa number conjecture) satisﬁes that the length of its coimage under the local Soul´e regulator map is the p -adic valuation of certain special values of p -adic L -functions associated to the Hecke characters. This result yields immediately, up to Jannsen’s conjecture, an upper bound for # H 2 et ( O K [1 /S ] , V p ( m )) in terms of the valuation of these p -adic L -functions, where V p denotes the p -adic realization of a Hecke motive.


Introduction
Consider a Hecke character ψ of an imaginary quadratic field K. Then there is an associated complex L-function and, for any prime p, which we impose that it is totally split in K, certain p-adic L-functions associated to the two primes above p. In this paper we find, under technical conditions, a relation between the special values of the complex L-funcions at non-critical points and some special values of the p-adic L-functions.
Deninger [6] defines a pure motive for any Hecke character ψ of an imaginary quadratic field whose L-function is equal to the L-function of the corresponding Hecke character. He also proves the Beilinson conjecture for these motives. For the p-adic L-functions Geisser proves in [8][9] a p-adic analogue of Beilinson's conjecture, relating the p-adic valuation of some special values of the p-adic L-functions with the length of a coimage module obtained from a map between Iwasawa modules. The local Tamagawa number conjecture predicts the exact value at integer points of this L-function for the Hecke character ψ (up to Z * (p) ) in terms of the corresponding motive. At non-critical points we can rewrite basically the conjecture as the Beilinson conjecture for these motives and some p-adic control. The study of the local Tamagawa number conjecture for these Hecke characters is made in [2] [3].
In this paper we compare the works [8] and [3] and we precise how in the case of Hecke characters the values for the p-adic L-functions appear in Kato's reformulation of the weak Tamagawa number conjecture. We consider the case cl(K) = 1 for simplicity but we believe that the results should generalize with similar techniques to the case p [H : K] where H is the Hilbert class field of K. The paper is basically a global-local comparison between Iwasawa modules andétale cohomology groups. Where we work with the subspace coming from theétale realization of the motivic elliptic polylogarithm [11]. A relation in the p-adic world between the elliptic polylogarithm and a special value for the p-adic L-function is studied by K. Bannai [1].
I would like to thank Xavier Xarles for his useful comments and suggestions on different versions of this paper. It is a pleasure to thank him and Enric Nart for their encouragement along my work on the Tamagawa number conjecture. Kind thanks also to the referee for his/her comments and for noticing some inaccuracies. I thank Jérôme Scherer and Xavier Xarles for proof-reading my english and for more suggestions.

The Tamagawa number conjecture for Hecke characters
Let E be a fixed elliptic curve with CM, which is defined over an imaginary quadratic field K, with CM by O K , the ring of integers of K. Note that this implies cl(K) = 1. Associated to this elliptic curve there is a Hecke character ϕ of the imaginary quadratic field K with conductor f, an ideal of O K that coincides with the conductor of the elliptic curve E. Let us introduce the motives that we will use and the result on the weak local Tamagawa number conjecture for them. Consider the category of Chow motives M Q (K) over K with morphisms induced by graded correspondences in Chow theory tensored with Q. Then, the motive of the elliptic curve E has a canonical decomposition for w a strictly positive integer, which has multiplication by T w := ⊗ w Q K. Observe that T w has a decomposition θ T θ as a product of fields T θ , where θ runs through the Aut(C)-orbits of Υ w = Hom(T w , C), where Υ = Hom(K, C). This decomposition defines some idempotents e θ and gives also a decomposition of the motive and its realizations. Let us fix once and for all an immersion λ : K → C as in [6, p.135].
The L-function associated to the motive e θ (⊗ w h 1 (E) Q ) corresponds to the L-function associated to ψ θ = e θ (⊗ w ϕ) : A * K → K * ([6, §1.3.1]) a CM-character (for equivalent definitions of Hecke characters over an imaginary quadratic field, see [8, §2.2]), which, with the fixed embedding λ, corresponds to ϕ a ϕ b , where a, b ≥ 0 are integers such that w = a + b. The pair (a, b) is the infinite type for ψ θ . We note that there are different θ with the same infinite type. Every θ gives two elements of Υ w , one given by the infinite type ϑ ∈ θ ∩ Hom K (T w , C) and the other coming from the other embedding.
We introduce the notation which is considered as an integral Chow motive (we can consider e θ as an idem- Since we want to consider the values at positive integers of the L-function associated to the above motive in the non-critical strip band, we take integers l ≥ 0 and we will study the Tamagawa number conjecture for the motive This conjecture predicts the value of the L-function for M θ at w + l + 1. Via the functional equation proved by Hecke for the Hecke characters, the study of these values is equivalent to the study of the value of the L-function at −l since M θ has weight w. We will formulate the Tamagawa number conjecture and our result on Hecke characters using this functional equation instead of the original formulation made by Bloch and Kato [4]. In this reformulation, the Tamagawa number conjecture predicts the cardinal of the coimage for the Beilinson regulator map as well as for the Soulé regulator map at every prime number p, of a lattice in a K-theory group associated to the motive. Let us introduce all these objects. The K-theory group corresponding to our motive M θ (w + l + 1) is where the K-groups are the Quillen K-groups and the superscript indicates the Adam's filtration on them. Let S be a finite set of primes of K, which contains the primes above p, and such that the p-adic realization of the motive, We have a Beilinson regulator map, where the cohomology group on the right is the Betti realization for our motive, which coincides with e θ (⊗ w Q H 1 (E(C), Q(1))(l). We have inside this Qvector space a Z-lattice given by We have also, for every prime number p, the Soulé regulator map: where F r means the geometric Frobenius and I p is the inertia group at p [3, After all these preliminaries we can formulate the local weak Tamagawa number conjecture at p for odd primes for our motives: Conjecture 2.1. Let us fix a prime number p = 2 and let S be the finite primes Finally, suppose that this representation is irreducible, that it is not the cyclotomic character, and that it satisfies the hypothesis of [12, theorem 4 (w+l+1)) and the injectivity on the Soulé regulator map.
Let us write down explicitly the construction of H constr M in theorem 2.3, and briefly indicate a sketch of the proof. The elements in K-theory for these motives were first constructed by Deninger in [5]. Denote by Γ = ΩO K the lattice of the elliptic curve, and let f ∈ O K be a generator for f θ . Then we have that We obtain next a divisor defined over K by taking the norm: Deninger constructs a map from the divisors to H M as a composition of an Eisenstein symbol map with a projection map (see [6, §2]), Then, we define R θ by where Φ(m) := |(O K /m) * | and P p is the product of the Euler factor of ψ θ at primes over p (see [3, remark 2.6] for the non-vanishing of these Euler factors).
We take H constr M equal to R θ ⊗ Q. For the proof of the theorem 2.3 we need to study the image of H constr M with respect to the Beilinson and Soulé regulator maps. In the case of the Beilinson regulator map this study is basically made by Deninger in [6]. We compare the Soulé regulator map with a map from an Iwasawa module to a global Galois cohomology group, where T p E means the Tate module of the elliptic curve E at p, C ∞ are the elliptic units and G is the Galois group Gal(K ∞ /K) (since K ∞ = ∪ n∈N K(E[p n ])). Let us introduce here the definition of this elliptic units for our latter interest.

Definition 2.4. Let C n be the subgroup of units generated over Z[Gal(K(E[p n ]))] by
where a runs through all ideals prime to 6pf, t f is a primitive f-torsion point and h n is a primitive p n -torsion point, and θ a is the classical theta function (see [7,II]). Define the group of elliptic units of K n := K(E[p n ]) as If U p n is the group of local units at K n congruent to 1 for all place v of K n over p (called local principal units), denote by C n the closure of C n ∩ U p n in U p n . Considering the limit with respect to the norm maps we obtain: Remark 2.5. The definition above coincides with the one in [2] but it differs from the one in [3]. Theorem 2.3 however remains valid with this definition as well, using a similar argument as in remark 4.3 [3].
The main Iwasawa conjecture proved by Rubin [12] and the specialization of the elliptic polylogarithm sheaf proved by Kings [11] allows to compare the image of the map e p with the image of H constr M under the Soulé regulator map. This concludes the proof of the local weak Tamagawa number conjecture.

Relation with Geisser's p-adic analogue of Beilinson's conjectures
We want to relate theorem 2.3 with the results of Geisser (see [8] or [9]). We impose once and for all that p splits in K, with p = pp * , p = p * . Observe that M θ ⊗ Z p has multiplication by O θ ⊗ Z p and, as p splits, it decomposes in two idempotents. Denote by e Ω 1 and e Ω 2 the two idempotents which give the following decomposition in the category of Chow motives with coefficients in Z p . We have also a direct sum decomposition of the map (Soul) p = (Soul) Ω1 ⊕ (Soul) Ω2 , comes easily from the localization exact sequence.
Observe that ψ θ ⊗ Z p = ψ Ω 1 ⊕ ψ Ω 2 and they satisfy ψ Ω 1 = ψ Ω 2 . If (a θ , b θ ) denotes the infinity type of ψ θ , and we suppose that Ω 1 comes from p and Ω 2 from p * by using the identification O θ = O K , then the infinity types for ψ Ω 1 and ψ Ω2 are (a θ , b θ ) and (b θ , a θ ) respectively. Here the infinity types come from morphisms to C p , (see [9] for more details). Let us denote by ι : Theorem 3.2 (Geisser, theorem 9.1 [9]). Suppose that p > 3w + 2l where (a i , b i ) is the infinity type for ψ Ωi , κ is the cyclotomic character for G and G is the (ψ Ω i κ l ) −1 -component of the two variable p-adic L-function (see [9, p. 227] for an explicit definition or see below).
We define now the Geisser elliptic units; and prove that they coincide with the elliptic units introduced in 2.4. The Geisser elliptic units are a modification of the elliptic units one can find in the book of de Shalit [7]. We follow [7,III §1]. Consider the ideal of K given by g := fp * n . We define C n,m as the group generated by the primitive Robert units of conductor gp m . Then C n,m is generated by θ a (t f + h n,m ) where h n,m is a point of p * n p m -torsion, with (a, 6gp) = 1. If µ ∞ denotes the roots of unity in K(gp m ), consider the group C n,m := C n,m µ ∞ (K(gp m )). Define next C gp m = C fp * n p m as the closure of C n,m ∩ U p gp m in U p gp m where U p gp m are the local principal units at K(gp m ) for the places over p. Define also C f(p * ) n := lim ←− m C fp * n p m , and where these limits are defined with respect to the norm maps.
The Geisser elliptic units over K are finally defined by taking the norm map from K(f) to K of C(f). We will denote these elliptic units by C Rob .

Lemma 3.3. The Geisser elliptic units coincide with the elliptic units 2.4.
Proof. First of all, notice that where the projective limit (with respect to the norm maps) on the left is taken over K(fp m (p * ) n ), and on the right over K(E[p m p * n ]). This equality comes from the fact that From the above we obtain the second equality holds because the projectives limits coming from the bicomplex U n,m := N K(f)/K U fp * m p n coincide, and as K(f)/K is unramified at p we obtain the last equality. We compare now the definitions of elliptic units. For similar reasons we have, Let us observe that N K(f)/K C n,n coincides with C n possibly up to roots of unity.
Thus as K(f)/K in unramified at primes above p, C n and N K(f)/K C f(p * ) n p n could differ only up to roots of unity. Since we take intersection with the principal units at places over p, where the norm map of K(f)/K is surjective, both definitions must coincide.
We recall now some facts on p-adic L-functions in our situation. Let us denote by D the ring of integers of the maximal unramified extension of K p . All characters of finite groups of order prime to p have values in D.
Observe that the Galois group Γ := Gal(K ∞ /K(E[p])) is isomorphic to a product Γ 1 ×Γ 2 of two copies of Z p , Γ 1 being the Galois group Gal(K(E[p ∞ ])/K(E[p])) and Γ 2 the analogue for p * . Let γ i be a generator of Γ i , and let κ i be the character of Γ i giving the action on the torsion points of the elliptic curve. Denote by u i the image of γ i in Z p .
We recall the connections between measures and power series. We have the isomorphism mapping a measure µ in Λ(Γ, D) to the power series β).
Let µ be a measure in Λ(G, D) for G = Gal(K(E[p ∞ ])/K), and let χ be a character of Gal(K(E[p])/K). We denote the power series associated to the χ-component of µ by G(χ −1 , T 1 , T 2 ). Then we obtain that We denote G(χκ a 1 κ b 2 , T 1 , T 2 ) by G(χ, T 1 , T 2 ) for a character χκ a 1 κ b 2 of G (in particular for the characters ψ Ω i κ l , see [9, §4 and §9]) By the interpolation theorem [7, theorem 4.14], G(χ, u a Proposition 3.4. With the hypotheses of theorems 2.3 and 3.2, the length of the coimage of ι • r p (R θ ) in H 1 (K p , M θZ p (w + l + 1)) is equal to the p-adic valuation of Proof. It is easy to check any element in the module (C ∞ ⊗ M Ωi (w + l)) G is an elliptic unit. This is because M Ω i is a free Z p -module of rank 1 endowed with a Galois action ( [8, proposition 2.4.6]) and the coinvariants come from the representation Hom Z p (M Ω i (w + l), Z p ). Therefore The comparison map between the image of e p and of r p , see [3, corollary 5.9 ], yields e p ((C ∞ ⊗ M θZ p (w + l)) G ) = r p (R θ ⊗ Z p ).
Hence, we obtain the following equality The lenght of the coimage of e Ω i (C ∞ ) is known factor by factor: it is equal to G(ψ Ω i κ l , u −a i −1