Hecke motives associated to imaginary quadratic fields

Universit¨ at Regensburg Mathematik

p-adic Beilinson conjecture for ordinary

Hecke motives associated to imaginary quadratic fields

Kenichi Bannai and Guido Kings

Preprint Nr. 19/2010

HECKE MOTIVES ASSOCIATED TO IMAGINARY QUADRATIC FIELDS

KENICHI BANNAI AND GUIDO KINGS

1. Introduction

The purpose of this article is to give an overview of the series of papers [BK1] [BK2] concerning the p-adic Beilinson conjecture of motives associ- ated to Hecke characters of an imaginary quadratic field K, for a prime p which splits in K. The p-adic L-function for such p interpolating critical values of L-functions of Hecke characters associated to imaginary quadratic fields was first constructed by Vishik and Manin [VM], and a different con- struction usingp-adic Eisenstein series was given by Katz [Katz]. Thep-adic Beilinson conjecture, as formulated by Perrin-Riou in [PR], gives a precise conjecture concerning the non-critical values of p-adic L-functions associ- ated to general motives. The purpose of our research is to investigate the interpolation property at non-critical points of the p-adic L-function con- structed by Vishik-Manin and Katz.

For simplicity, we assume in this article that the imaginary quadratic field K has class number one and that the Hecke character ψ we consider corresponds to an elliptic curve with complex multiplication defined over Q. Letabe an integer >0. The main theorem of this article (Theorem 6.9) is a proof of thep-adic Beilinson conjecture forψ^a (see Conjecture 2.3), when the prime p ≥ 5 is an ordinary prime. The authors would like to thank the organizers Takashi Ichikawa, Masanari Kida and Takao Yamazaki for the opportunity to present our research at the RIMS “Algebraic Number Theory and Related Topics 2009” conference.

2. The p-adic Beilinson conjecture

Assume thatK is an imaginary quadratic field of class number one. Let E be an elliptic curve defined over Q. We assume in addition that E has complex multiplication by the ring of integers OK ofK. We let ψ:=ψ_E/K be the Grossencharacter ofK associated toE_K :=E⊗_QK by the theory of complex multiplication, and we denote by f the conductor ofψ.

We let M(ψ) be the motive over K with coefficients in K associated to the Grossencharacter ψ. Then we have M(ψ) = H¹(EK), where H¹(EK)

Date: June 12, 2010.

This work was supported by KAKENHI 21674001.

1

is the motive associated to EK. The Hasse-Weil L-function of M(ψ) is a function with values inK⊗_QC given by

L(M(ψ), s) = (L(ψτ, s))τ:K,→C,

where τ :K ,→ C are the embeddings of the coefficient K of M(ψ) into C and L(ψτ, s) is the HeckeL-function

L(ψτ, s) = Y

(q,f)=1

µ

1− ψτ(q) Nq^s

¶₋1

associated to the character ψτ :A^×_K −→^ψ K ,→^τ C. Here, the product is over the prime idealsq ofK which are prime tof.

For integers a > 0 and n, we let M^a = M(ψ^a) := M(ψ)^⊗Ka, which is a motive over K with coefficients in K. Then the Hasse-Weil L-function L(M^a, s) is given by the HeckeL-function

L(M^a, s) = (L(ψ_τ^a, s))_τ:K,→C

with values in K ⊗_Q C. We let M_B^a be the Betti realization of M^a, which is aK-vector space of dimension one. We fix a K-basis ω^a_B of M_B^a. The de Rham realizationM_dR^a (n) ofM^a(n) is the rank one K⊗_QK-module

M_dR^a (n) =Kω^n−a,nM

Kω^n,n−a, with Hodge filtration given by

F^mM_dR^a (n) =







M_dR^a (n) m≤ −n

Kω^n,n−a −n < m≤a−n

0 otherwise.

In what follows, we consider the case when n > a, which implies in par- ticular that our motive is non-critical. We have in this caseF⁰M_dR^a (n) = 0.

The tangent space of our motive is given by

t^a_n:=M_dR^a (n)/F⁰M_dR^a (n)∼=M_dR^a (n),

which is again a K⊗QK-module of rank one. Note that ω_tg,n^a :=ω^n−a,n+ ω^n,n−a gives a basis oft^a_n as a K⊗QK-module.

We denote by V_∞^a(n) the R-Hodge realization of M^a(n). The Beilinson- Deligne cohomology H_D¹(K ⊗_Q R, V_∞^a(n)) is given as the cokernel of the natural inclusion

M_B^a(n)⊗QR→t^a_n⊗QR. The Beilinson regulator map gives a homomorphism

(1) r_∞:H_mot¹ (K, M^a(n))→H_D¹(K⊗_QR, V_∞^a(n)),

from the motivic cohomology H_mot¹ (K, M^a(n)) of K with coefficients in M^a(n) to H_D¹(K ⊗_Q R, V_∞^a(n)). Then r_∞ ⊗_Q R is known to be surjec- tive and is conjectured to be an isomorphism. We let c^a_n be an element ofH_mot¹ (K, M^a(n)) such that r_∞(c^a_n) generates H_D¹(K_p, V_∞^a(n)) as aK_∞:=

K⊗_QR-module. We define the complex period Ω_∞(n) of M^a(n) to be the determinant of the exact sequence

(2) 0→M_B^a(n)⊗_QR→t^a_n⊗_QR→H_D¹(K⊗_QR, V_∞^a(n))→0

for the basis r_∞(c^a_n), ω^a_tg,n, and ω^a_B. The complex period is an element in K_∞and is independent of the choice of the basis up to multiplication by an element in K^×. The value L(M^a, n) is in K⊗_QR, and the weak Beilinson conjecture for M^a(n) as proved by Deninger [De1] gives the following (see Theorem 6.2 and Corollary 6.4 for the precise statement.)

Theorem 2.1. The value

L(M^a, n) Ω_∞(n) is an element in K^×.

For any primep, the ´etale realization V_p^a(n) of our motive is aK⊗_QQp- vector space with continuous action of Gal(K/K). We fix a prime p ≥ 5 relatively prime to f such that E has goodordinary reduction at p. In this case, the ideal generated bypsplits as (p) =pp^∗inK. We fix a prime idealp ofK abovep. Then the Bloch-Kato exponential map gives an isomorphism (3) exp_p :t^a_n⊗KKp

∼=

−→H_f¹(Kp, V_p^a(n)),

and the inverse of this isomorphism is denoted by log_p. The p-adic ´etale regulator map gives a homomorphism

(4) rp :H_mot¹ (K, M^a(n))→H_f¹(Kp, V_p^a(n)),

and the maprp⊗Qp is conjectured to be an isomorphism. Assuming that the p-adic regulator map r_p is injective, we define the p-adic period Ω_p(n) of M^a(n) to be the determinant of the map log_p for the basis rp(c^a_n) and ω_tg,n^a . In other words, Ωp(n) is an element in Kp := K⊗QKp ∼=Kp

LKp^∗

satisfying

(5) log_p◦ rp(c^a_n) = Ωp(n)ω^a_tg,n.

The p-adic period Ω_p(n) is independent of the choice of basis up to multi- plication by an element inK^×.

Remark 2.2. We need to assume the injectivity of the p-adic regulatorrp

to insure that thep-adic period Ω_p(n) is non-zero. Kato has proved in [Kato]

15.15 the weak Leopoldt conjecture for any Hecke character ofK. Hence by a result of Jannsen ([Jan], Lemma 8), we may then conclude that

H²(OK[1/pfa], V_p^a(n)) = 0

for almost all n. This implies that r_p is injective for suchn.

The p-adic Beilinson conjecture as formulated by Perrin-Riou (see [Col]

Conjecture 2.7) specialized to our setting is given as follows.

Conjecture 2.3. Letabe an integer>0. Then there exists ap-adic pseudo- measure µ^a onZp with values inK_p such that the value

L_p(ψ^a⊗χⁿ_cyc) :=

Z

Z^×p

wⁿµ^a(w) in K_p for any integer n > a satisfies

L_p(ψ^a⊗χⁿ_cyc) Ω_p(n) =

µ

1− ψ(p)^a pⁿ

¶ µ

1− ψ(p^∗)^a p^a+1−n

¶Γ(n)L(ψ^a, n) Ω_∞(n) , where p is a fixed prime in K above p.

If f_a6= (1) for the conductor f_a of ψ^a, then µ^a should in fact be ap-adic measure. Note that the dependence of the pseudo-measure on the choices of the basis ω^a_tg,n and c^a_n cancel, where as the pseudo-measure depends on the choice of the basis ω_B^a. The main goal of our research is to prove that thep-adic measure constructed by Vishik-Manin and Katz gives the pseudo- measure of the above conjecture when the prime p is split in K.

The main theorem of this article (Theorem 6.9) is the proof of the above conjecture for integers n such that the corresponding p-adic regulator map rp is injective.

3. Construction of the Eisenstein class

The main difficulty in the proof of the Beilinson and p-adic Beilinson conjectures is to construct the element c^a_n ∈ H_mot¹ (K, M^a(n)) for M^a :=

M(ψ^a) and to calculate the images r_∞(c^a_n) and r_p(c^a_n) with respect to the Beilinson-Deligne and p-adic regulator maps. We will use the Eisenstein symbol as constructed by Beilinson.

We fix an integer N ≥ 3, and let M(N) be the modular curve defined over Z[1/N] parameterizing for any scheme S over Z[1/N] the pair (E, ν), whereE is an elliptic curve overS and

ν: (Z/NZ)² −→^∼= E[N]

is a full levelN-structure onE, whereE[N] is the group ofN-torsion points of E. We let pr : Ee → M be the universal elliptic curve over M with universal level N-structure eν : (Z/NZ)² ∼=E[Ne ], and consider the motivic sheafQ(1) onE. We lete

(6) H :=R¹pr_∗Q(1),

and we denote by Sym^kH the k-th symmetric product of H. Let ϕ = P

ρ∈E[N]e \{0}a_ρ[ρ] be a Q-linear sum of non-zero elements in E[Ne ]. For any integerk >0, the Eisenstein class Eis^k+2_mot(ϕ) is an element

(7) Eis^k+2_mot(ϕ)∈H_mot¹ (M,Sym^kH(1)).

Although the formalism of mixed motivic sheaves or motivic cohomology with coefficients have not yet been fully developed, one can give meaning to the above sheaves and cohomology (see [BL], [BK1] for details).

Then the classc^a_nmay be constructed from the Eisesntein class as follows.

Let K be an imaginary quadratic field of class number one, and let E be an elliptic curve defined over Q with complex multiplication by the ring of integers OK of K. We denote again by ψ the Hecke character of K corresponding to EK with conductor f. We take N ≥ 3 such that N is divisible by f. For the extension F := K(E[N]) of K generated by the coordinates of the points in E[N], we let G_{F /K} := Gal(F/K) the Galois group of F over K. We fix a level N-structure ν : (Z/NZ)² ∼= E[N] of E over F, and we denote by ν^σ the composition of ν with the action of σ ∈ G_F/K. Then for any σ ∈ G_F/K, we denote by ι^σ∗ the pull-back with respect to the F-valued point ι^σ : SpecF → M of M corresponding to (E, ν^σ). Then the image of the sum ι^∗ := P

σ∈GF /Kι^σ∗ is invariant by the action of the Galois group, hence gives a pull-back morphism

H_mot¹ (M,Sym^kH(1))−→^ι∗ H_mot¹ (K,Sym^kι^∗H(1)).

Note that on SpecK, the motivic sheafι^∗H is given by the motiveH¹(E)(1), which by definition corresponds to the motive M(ψ)(1). The structure of K-coefficients on ι^∗H gives the following decomposition.

Lemma 3.1. For integers j satisfying 0 ≤j ≤k/2, we have the decompo- sition of motives

Sym^kι^∗H = M

0≤j≤^k₂

M(ψ^k−2j) (k−j),

where we take the convention that fork= 2j, we letM(ψ⁰)(k/2)be the Tate motive Q(k/2) with coefficients in Q.

Let a > 0 be an integer and we let fa be the conductor of ψ^a. We let F_a := K(E[f_a]) be the extension of K generated by the coordinates of the points inE[fa], and we letw_F/Fa be the order of the Galois group Gal(F/Fa).

The Eisenstein classes Eis^k+2_mot(ρ) are defined for points ρ ∈ E[Ne ]\ {0} but is not defined for ρ= 0. Hence in defining c^a_n, we differentiate between the case whenf_a6= (1) andf_a= (1).

Definition 3.2. We defineϕa as follows.

(1) Iffa6= (1), then we fix a primitive fa-torsion point ρa ofE and let ϕ_a:= 1

waw_{F /Fa}[ρ_a],

where we denote again byρatheN-torsion point ofEecorresponding to ρ_a through ν and ν, ande w_a is the number of units in OK which are congruent toone modulofa.

(2) Iffa= (1), then we let ϕa:= 1

waw_F/Fa

X

ρ∈E[Ne ]\{0}

[ρ].

We define the class c^a_n as follows.

Definition 3.3. For any integer a, n such that n > a > 0, we let k = 2n−a−2. Then the motive M^a(n) := M(ψ^a)(n) is a direct summand of Sym^kι^∗H(1). We define the motivic class c^a_n to be the image of Eis^k+2_mot(ϕ_a) with respect to the projection

H_mot¹ (K,Sym^kι^∗H(1))→H_mot¹ (K, M^a(n)), whereϕ_a is as in Definition 3.2.

Let p be a rational prime which does not divide f, and we take N ≥ 3 to be an integer divisible by f and prime to p. In order to prove the p- adic Beilinson conjecture, it is necessary to calculate the images of c^a_n with respect to the Beilinson-Deligne and p-adic regulator maps. The image r_∞(c^a_n) by the Beilinson-Deligne regulator map was calculated by Deninger [De1]. We will calculate the imagerp(c^a_n) by thep-adic regulator map using rigid syntomic cohomology.

Denote byM_cris^a (n) the crystalline realization ofM^a(n), which is a filtered module with a σ-linear action of Frobenius, and let H_syn¹ (K_p, M_cris^a (n)) be the syntomic cohomology of K_p with coefficients in M_cris^a (n). Then noting thatt^a_n⊗QQp=M_cris^a (n) in this case, there exists a canonical isomorphism

(8) t^a_n⊗_QQp

∼=

−→H_syn¹ (K_p, M_cris^a (n)).

If we let V_p^a(n) be the p-adic ´etale realization of M^a(n), then we have a canonical isomorphism

(9) H_syn¹ (Kp, M_cris^a (n))−→^∼= H_f¹(Kp, V_p^a(n)),

which combined with (8) gives the exponential map (3). The syntomic regulator map

r_syn:H_mot¹ (K, M^a(n))→H_syn¹ (K_p, M_cris^a (n))

defined by Besser ([Bes] §7) is compatible with the p-adic regulator r_p through the isomorphism (9) ([Bes] Proposition 9.9). Therefore, in order to calculate log_p◦r_p(c^a_n), it is sufficient to calculate the image of r_syn(c^a_n) with respect to (8). We will calculate this image using the explicit determi- nation of the syntomic Eisenstein class given in [BK1].

4. Eisenstein class and p-adic Eisenstein series

In this section, we review the explicit description of the syntomic Eisen- stein class in terms of p-adic Eisenstein series given in [BK1]. Let M :=

M(N) be the modular curve overZ[1/N] given in the previous section. We will first describe a certain real analytic Eisenstein series E_k+2,l,ϕ^∞ .

Let Γ⊂Cbe a lattice, and we denote by A the area of the fundamental domain of Γ divided by π := 3.14159· · ·. For any integer a and complex number ssatisfying Re(s)> a/2 + 1, the Eisenstein-Kronecker-Lerch series K_a^∗(z, w, s; Γ) to be the series

K_a^∗(z, w, s; Γ) :=X

γ∈Γ

∗(z+γ)^a

|z+γ|^2shγ, wi where P_∗

denotes the sum over γ ∈ Γ satisfying γ 6= −z and hz, wi :=

exp((wz−wz)/A). By [Wei] VIII §12 (see [BKT] Proposition 2.4 for the casea <0), this series forscontinues meromorphically to a function on the whole s-plane, holomorphic except for a simple pole at s= 1 when a = 0 and w∈Γ. This function satisfies the functional equation

(10) Γ(s)K_a^∗(z, w, s; Γ) =A^a+1−2sΓ(a+ 1−s)K_a^∗(w, z, a+ 1−s)hw, zi. We fix a level N-structureν : (Z/NZ)² ∼= _N¹Γ/Γ, and let ρ∈ _N¹Γ/Γ. For integers k and l, we define the real analytic Eisenstein series E_k+2,l,ρ^∞ to be the modular form on M_C := M(N) ⊗_Q C whose value at the test object (C/Γ, dz, ν) is given by

(11) E_k+2,l,ρ^∞ (C/Γ, dz, ν) :=A^−lΓ(s)K_k+l+2^∗ (0, ρ, s; Γ)¯¯

s=k+2. We let E_k+2,l,ϕ^∞ :=P

ρa_ρE_k+2,l,ρ^∞ for theQ-linear sum ϕ=P

ρa_ρ[ρ].

Whenl= 0, thenE_k+2,0,ϕ^∞ is a holomorphic Eisenstein series of weightk+2 onM_C. From theq-expansion, we see in this case that this Eisenstein series is defined overQ, and hence defines a sectionE_k+2,0,ϕ in Γ(M_Q, ω^⊗k⊗Ω¹_M

Q) forω:= pr_∗Ω¹_e

E/M. Denote byHdR the de Rham realization of H, which is the coherentOM_Q-moduleR¹pr_∗Ω^•_e

E with Gauss-Manin connection

∇:HdR →HdR⊗Ω¹_MQ,

and let Sym^kHdR be thek-th symmetric product of HdR with the induced connection. From the natural inclusion ω^⊗k ,→ Sym^kHdR, we see that Ek+2,0,ϕ defines a section in Γ(M_Q,Sym^kHdR⊗Ω¹_MQ).

Let p be a prime number not dividing N. We denote by Hrig the fil- tered overconvergent F-isocrystal associated to H on M_Zp, which is given byHdR with an additional structure of Hodge filtration and Frobenius. Let H_syn¹ (M_Zp,Sym^kHrig(1)) be the rigid syntomic cohomology ofM_Zp with co- efficients in Sym^kHrig(1). The rigid syntomic regulator is a map

r_syn:H_mot¹ (M,Sym^kH(1))→H_syn¹ (M_Zp,Sym^kHrig(1)),

and we define the syntomic Eisenstein class Eis^k+2_syn(ϕ) to be the image by the syntomic regulator of the motivic Eisenstein class. We let M_Z^ord

p be the ordinary locus in M_Zp, and M_Q^ord

p := M_Z^ord

p ⊗_ZpQp. By [BK1] Proposition

A.16, a class inH_syn¹ (M_Z^ord_p ,Sym^kHrig(1)) is given by a pair (α, ξ) of sections α∈Γ(M_Q^ord_p , j^†Sym^kHrig(1))

ξ ∈Γ(M_Q^ord_p ,Sym^kHdR⊗_QpΩ¹_Mord) (12)

satisfying ∇(α) = (1−φ^∗)ξ. Theαfor the class (α, ξ) corresponding to the restriction to the ordinary locus of the syntomic Eisenstein class Eis^k+2_syn(ϕ) is given as follows.

We letp≥5 be a prime not dividingN, and we letMbe thep-adic mod- ular curve defined overZp parameterizing the triples (EB, η, ν) consisting of an elliptic curve E_B over ap-adic ring B, an isomorphism

(13) η:Gbm ∼=Eb_B

of formal groups over B, and a level N-structure ν. The ring of p-adic modular formsVp(Qp,Γ(N)) is defined as the global section

V_p(Qp,Γ(N)) := Γ(M,O_M)⊗_ZpQp. The q-expansion gives an injection

V_p(Qp,Γ(N)),→Qp(ζ_N)[[q]].

There exists a Frobenius actionφ^∗onVp(Qp,Γ(N)) given on theq-expansion as φ^∗ = Frob⊗σ, where Frob(q) = q^p and σ is the the absolute Frobenius acting on Qp(ζN). The Eisenstein series Ek+2,0,ϕ naturally defines an ele- ment in Vp(Qp,Γ(N)), and using the fact that the differential ∂_logq := q_dq^d preserves the space of p-adic modular forms, we let for any integer l≥0

Ek+l+2,l,ϕ :=∂_log^l _qEk+2,0,ϕ.

We let E_k+2,0,ϕ^(p) := (1−φ^∗)E_k+2,0,ϕ and E_k+l+2,l,ϕ^(p) := ∂_log^l _qE_k+2,0,ϕ^(p) for any integer l ≥ 0. Then the calculation of the q-expansion shows that we have

(14) E_k+l+2,l,ϕ^(p) = (1−p^lφ^∗)E_k+l+2,l,ϕ.

Following the method of Katz [Katz], we may construct ap-adic measure on Zp×Z^×p with values in V_p(Qp,Γ(N)) satisfying the following interpolation property.

Theorem 4.1. There exists a p-adic measureµ_ϕ onZp×Z^×p with values in V_p(Qp,Γ(N))such that

Z

Zp×Z^×p

x^k+1y^lµ_ϕ(x, y) =E_k+2,l,ϕ^(p) for integers k >0, l≥0.

Using this measure, we define E_k+2,l,ϕ^(p) forl <0 as follows.

Definition 4.2 (p-adic Eisenstein series). Let k be an integer ≥ −1. We let

E_k+2,l,ϕ^(p) :=

Z

Zp×Z^×p

x^k+1y^lµ_ϕ(x, y) ∈V_p(Qp,Γ(N)), wherel is any integer in Z.

The p-adic Eisenstein series satisfies the differential equation

∂_logqE_k+2,l,ϕ^(p) =E_k+3,l+1,ϕ^(p) ,

and the weight of E_k+2,l,ϕ^(p) is k+l+ 2. The syntomic Eisenstein class may be described using these p-adic Eisenstein series. The moduli problem for Mimplies that there exists a universal trivialization

η:Gbm ∼=Ebe

of the universal elliptic curve onM, which gives rise to a canonical section e

ω of ω := pr_∗Ω¹_e

E/M corresponding to the invariant differentialdlog(1 +T) on Gbm. SinceM is affine, there exists sections x and y of Ee such that the elliptic curveEe_Qp :=Ee⊗Qp is given by the Weierstrass equation

Ee_Qp :y²= 4x³−g₂x−g₃, g₂, g₃∈V_p(Qp,Γ(N))

satisfying eω=dx/y. Then the pull back of theF-isocrystalHrig toMQp is given as

Hrig =OMQpωe^∨⊕ OMQpue^∨,

with connection ∇(eu^∨) = eω^∨ ⊗dlogq, ∇(ωe^∨) = 0, Frobenius φ^∗(eω^∨) = p⁻¹ωe^∨, φ^∗(ue^∨) = eu^∨ and Hodge filtration Fil⁻¹Hrig = Hrig, Fil⁰Hrig = O_MQpue^∨, Fil¹Hrig = 0 (See [BK1] §4.3). If let ωe^m,n := eω^∨meu^∨n, then the filtered F-isocrystal Sym^kHrig(1) onM_Qp is given by the coherent module

Sym^kHrig(1) = Mk j=0

O_MQpωe^k−j,j(1)

with connection∇(ωe^k−j,j(1)) =jωe^{k−j+1,j−1}(1)⊗dlogq, Frobenius φ^∗(ωe^,k−j,j(1)) =p^j−k−1ωe^k−j,j(1),

and Hodge filtration

Fil^m(Sym^kHrig(1)) = Mk j=m+k+1

OMQpωe^k−j,j(1).

If we letαe^k+2_Eis be the section e

α^k+2_Eis (ϕ) :=

Xk j=0

(−1)^k−j

j! E_j+1,j^(p) _−k−1,ϕωe^k−j,j(1),

then we have

∇(αe^k+2_Eis (ϕ)) = (1−φ^∗)E_k+2,0,ϕ

k! eω^0,k(1)⊗dlogq.

The main result of [BK1] is the following.

Theorem 4.3 ([BK1] Theorem 5.11). For any integer k >0, the syntomic Eisenstein class

Eis^k+2_syn(ϕ)∈H_syn¹ (M_Zp,Sym^kHrig(1))

restricted to the ordinary locus H_syn¹ (M_Z^ord_p ,Sym^kHrig(1)) is represented by the pair (α, ξ) as in (12), whereξ =Ek+2,0,ϕeω^0,k(1)/k!⊗dlogq and α is a section which maps to αe^k+2_Eis (ϕ) in Γ(M_Qp,Sym^kHrig(1)).

The main ingredient in the proof of the above theorem is the character- ization of ξ by the residue, which by [BL] 2.2.3 (see also [HK] C.1.1) and the compatibility of the Beilinson-Deligne regulator map with the residue morphism shows thatξrepresents the de Rham Eisenstein class in de Rham cohomology. See [BK1] Proposition 3.6 and Proposition 4.1 for details con- cerning this point.

5. Special values of Hecke L-functions

In this section, we give in Propositions 5.2 and 5.4 the precise relation between the special values of the Hecke L-functionL(ψ^a, s) and Eisenstein- Kronecker-Lerch series. Assume that K is an imaginary quadratic field of class number one, and letE be an elliptic curve overQwith good ordinary reduction at a prime p with complex multiplication by the ring of integers OK ofK. We letψbe the Grossencharacter ofKassociated to E_K :=E⊗_Q K, and we denote by f the conductor of ψ. We fix an invariant differential ω of E defined over K. We fix once and for all a complex embedding τ :K ,→C of the base field K into C, and we let Γ be the period lattice of E :=E⊗K,τ Cwith respect toω. Then we have a complex uniformization

(15) C/Γ−→^∼= E(C)

such that the pull-back of the invariant differential ω coincides with dz.

Note that since E has complex multiplication, we have Γ = ΩOK for some complex period Ω∈C^×.

By abuse of notation, we will denote by ψ and ψ the complex Hecke characters ψ_τ and ψ_τ associated toψ, whereτ is the fixed embedding given above. Let−dK denote the discriminant ofK, so thatK=Q(√

−dK). The Hecke characterψis of the form ψ((u)) =ε(u)ufor any u∈ OK prime tof, whereε: (OK/f)^×→K^× is a primitive character on (OK/f)^×.

Let χ : (OK/f_χ)^× → K^× be a primitive character of conductor f_χ, and letf_χ be a generator offχ. Then for anyuinOK, we define theGauss sum

G(χ, u) by

G(χ, u) := X

v∈OK/fχ

χ(v) exp

³

2πiTr_K/Q

³

uv/f_χp

−d_K

´´

(see [Lan] Chapter 22 §1), where we extend χ to a function on OK/fχ by taking χ(u) := 0 for anyu ∈ OK not prime tofχ. We let G(χ) :=G(χ,1).

Then the standard fact concerning Gauss sums are as follows (see for exam- ple [Lan] Chapter 22 §1.)

Lemma 5.1. Let the notations be as above.

(1) We have |G(χ)|²=N(fχ).

(2) For anyu∈ OK, we have G(χ, u) =χ(u)G(χ).

As in§3, we leta >0 be an integer andfa be the conductor ofψ^a. Then the finite part ε^a of ψ^a is a primitive character ε^a : (OK/f_a)^× → C^× of conductor fa. We fix a generator fa offa and we denote by byG(ε^a, u) the corresponding Gauss sum for anyu inOK.

We let the notations be as in §3. In particular, we let wa be the number of units in OK which are congruent to one mod fa, and we let w_F/Fa be the order of the Galois group Gal(F/F_a). We again let N ≥ 3 be a ratio- nal integer divisible by f and prime to p, and F := K(E[N]). We fix an isomorphism ν: (Z/NZ)² ∼= _N¹Γ/Γ.

We first consider the case whenfa6= (1). We letρa:= Ω/fabe a primitive f_a-torsion point, which corresponds through the uniformization (15) to a point ρ_a6= 0∈E(K). We then have the following.

Proposition 5.2. Suppose fa6= (1). Then we have (−1)^a

waw_F/Fa

X

σ∈Gal(F/K)

E_n,a^∞_−n,ρσ

a(C/Γ, dz, ν) = G(ε^a)Ω^a

A^a−n|Ω|²ⁿΓ(s)L(ψ^a, s)|s=n. Proof. Letw0 be the number of units in OK. By definition, we have

L(ψ^a, s) = 1 w₀

X

u∈OK

ψ^a(u) N(u)^s = 1

w₀ X

u∈OK

ε^a(u)u^a N(u)^s

Then Lemma 5.1 (2) gives the equalityε^a(u) =G(ε^a, u)/G(ε^a). If we expand the definition of the Gauss sum, we see that

L(ψ^a, s) = 1 w0

X

u∈OK

v∈OK/fa

ε^a(v)u^a N(u)^s exp

µ

2πiTr_K/Q

µ uv fa

√−d_K

¶¶

.

Noting thatOK is preserved by complex conjugation, we see that the above is equal to

(−1)^a w₀

X

v∈OK/fa

X

u∈OK

ε^a(v)u^a

|u|^2s exp µ 2π

√d_K µuv

f_a −uv f_a

¶¶

.

For any σ ∈ Gal(F/K), we have ρ^σ_a = ρ^σ_a⁰, where σ⁰ is the class of σ in Gal(F_a/K). If σ⁰_v := (v, F_a/F) is the element in Gal(F_a/K) corresponding tov∈(OK/fa)^× through theinverse of the Artin map, then by the theory of complex multiplication, we have ρ^σa^0v =ψ(v)ρ_a.Hence

X

σ∈Gal(F/K)

K_a^∗(0, ρ^σ_a, s; Γ) =w_F/Fa X

σ⁰∈Gal(Fa/K)

K_a^∗(0, ρ^σ_a⁰, s; Γ)

=w_F/Faw_a w0

X

v∈(OK/fa)^×

X

γ∈Γ

γ^a

|γ|^2shγ, ψ(v)ρ_ai

=w_F/Faw_a w0

X

v∈(OK/fa)^×

X

γ∈Γ

ε^a(v)γ^a

|γ|^2s hγ, vρai.

Our assertion follows from the fact that Γ = ΩOK, A = |Ω|²√

d_K/2π and the definition (11) of the Eisenstein-Kronecker-Lerch series. ¤ The right hand side of Proposition 5.2 may be used to express the Hecke L-function on the other side of the functional equation as follows.

Lemma 5.3. We have

(16) 1

w_aw_F/Fa

X

σ∈Gal(F/K)

E_n,a^∞_−n,ρσ

a(C/Γ, dz, ν)

= A¹⁻ⁿN(fa)^a+1−nΩ^a

f^a_a|Ω|^2(a+1−n) Γ(s)L(ψ^a, s)¯¯¯

s=a+1−n. Proof. We have by definition

X

v∈OK/fa

K_a^∗(ψ(v)ρa,0, a+ 1−s; Γ) = X

v∈OK/fa

X

γ∈Γ

∗ (ψ(v)ρa+γ)^a

|ψ(v)ρa+γ|^2(a+1−s)

= N(fa)^a+1−sΩ^a

f^a_a|Ω|^2(a+1−s) L(ψ^a, a+ 1−s) for Re(s)< a/2, hence for anys∈Cby analytic continuation. Our assertion follows from the functional equation

Γ(s)K_a^∗(0, ψ(v)ρ_a, s; Γ) =A^a+1−2sΓ(a+ 1−s)K_a^∗(ψ(v)ρ_a,0, a+ 1−s; Γ) and the definition (11) of the Eisenstein-Kronecker-Lerch series. ¤

The case when fa= (1) is given as follows.

Proposition 5.4. Suppose fa= (1). Then we have 1

wa

X

ρ∈E[N]\{0}

E_n,a^∞_−n,ρ(C/Γ, dz, ν)

=

µN^a+2 N²ⁿ −1

¶ Ω^a

A^a−n|Ω|²ⁿΓ(s)L(ψ^a, s)|s=n.

Proof. By definition, we have X

ρ∈N¹Γ/Γ

K_a^∗(0, ρ, s; Γ) =X

γ∈Γ

X

ρ∈N¹Γ/Γ

γ^a

|γ|^2shγ, ρi= N^a+2 N^2s

X

γ∈Γ

γ^a

|γ|^2s, where the last equality follows from the equality

X

ρ∈_N¹Γ/Γ

hγ, ρi= (

N² γ ∈NΓ 0 otherwise

and the fact that complex conjugation acts bijectively on Γ. Our assertion follows from the definition (11) of the Eisenstein-Kronecker-Lerch series. ¤

Similarly to Lemma 5.3, we have the following.

Lemma 5.5. We have (17) 1

wa

X

ρ∈E[N]\{0}

E_n,a−n,ρ^∞ (C/Γ, dz, ν))

=

µN^a+2 N²ⁿ −1

¶ A¹⁻ⁿΩ^a

|Ω|^2(a+1−n) Γ(s)L(ψ^a, s)¯¯¯

s=a+1−n.

6. The Main Result

In this section, we give an outline of the proof of our main theorem. We will mainly deal with the case when fa 6= (1), as the case for fa = (1) is essentialy the same except for the factor (N^a+2/N²ⁿ−1). We first calculate the p-adic and complex periods Ωp(n) and Ω(n). From the definition of c^a_n and from the compatibility of the syntomic regulator with respect to pull- back morphisms, the restriction of the syntomic Eisenstein class through the decomposition of Lemma 3.1 gives the image by the syntomic regulator of the element c^a_n inH_mot¹ (K, M^a(n)).

Let the notations be as in the previous section. We denote byω^∗ the class inH_dR¹ (E/C) corresponding todz/A, which is in fact a class in H_dR¹ (E/K).

Let k= 2n−a−2. Thenω^k−j+1,j+1 :=ω^∨k−jω^∗∨j(1) for 0≤j ≤k form a basis of Sym^kι^∗HdR(1). The relation between the basisωe^m,n andω^m,n is given byωe^m,n = Ωⁿ_p^−mω^m,n. In what follows, let ϕa be as in Definition 3.2.

By Theorem 4.3, the pull-back of the syntomic Eisenstein class Eis^k+2_syn(ϕa) toH_syn¹ (K_p,Sym^kι^∗H(1)) is expressed by the element

ι^∗αe^k+2_Eis (ϕa) = Xk j=0

(−1)^k−j

j! Ω^2j_p ^−kE_j+1,j^(p) _−k−1,ϕ

a(E, ω, ν)ω^k−j+1,j+1.