EXPLICIT RECIPROCITY LAW FOR LUBIN-TATE GROUPS

(NOTES OF THE SEMINAR AT USC ON FEB. 13, 2001)

TAKESHI TSUJI

Let K be an imaginary quadratic field of class number 1 and let E be an elliptic curve over K with complex multiplication in the ring of integers O_{K} of K. Let ψ be the Grössencharacter of K associated to E and let p be an odd prime over which E has good reduction. In [Kato93], K. Kato defined the zeta element which is, roughly speaking, a compatible system in the Galois cohomology of abelian extensions of K with coefficients in ℤ_{p}(1) defined using the elliptic units. Then he proved that, for any positive integer k and a finite Hecke character λ, the zeta element gives exactly the value at s = 0 of the Hecke L-function L(ψ^{−k}λ, s) through the dual exponential map (see §3) for H^{1}(K ⊗_{ℚ} ℚ_{p}, F_{k}(λ)^{∗}(1)), where F_{k}(λ) denotes the padic abelian representation of Gal(\overline{K}/K) corresponding to ψ^{−k}λ (loc. cit. Chap. III, Theorem 1.2.6). A key ingredient of the proof of this theorem is the generalized explicit reciprocity law for Lubin-Tate formal groups proven in loc. cit. Chap. II, §2, which, in the above special case, describes the dual exponential maps for the Galois cohomology of abelian extensions of K with coefficients in F_{k}(1)^{∗}(1) explicitly. Here 1 denotes the trivial representation.

The generalization of the above Kato’s result on the zeta element to the Hecke character of K of the form ψ^{−k}\overline{ψ}^{j}λ (0 ≤ j < k) was proven by L. Guo [Guo99], B. Han [Han97] and K. Kimura [Kim93] in the case p splits in K. L. Guo and K. Kimura proved it by generalizing the Kato’s explicit reciprocity law. In the case p is inert, B. Han [Han97] proved a partial result: a result up to p-adic units for trivial λ. However a generalization of the Kato’s explicit reciprocity law was not known in the inert case, and it is the main theme of these notes. As a consequence of our explicit reciprocity law, we obtain the expected relation between the zeta element and L(ψ^{−k}\overline{ψ}^{j}λ, 0) (under some condition on λ in the case k − j = 2.) Our explicit reciprocity law holds for any Lubin-Tate formal group for a finite unramified extension of ℚ_{p}. However, in these notes, we will deal with only the Lubin-Tate formal group of height 2 coming from E in order to explain how the formulation of our explicit reciprocity law is inspired by the work of N. Katz [Katz78], [Katz80] on the special values of the Hecke L-functions of K.

1. Special values of Hecke L-functions of K

In this section, we will review the description of the Hecke L-functions of K as values of some algebraic or p-adic derivatives of the functions θ_{a} (0 ≠ a ∈ ℤ, (6, a) = 1) on the elliptic curve E or on the universal elliptic curve. In the case j = 0, the special values L(ψ^{−k}λ, 0) are described as values of holomorphic Eisenstein series of weight k, which is actually algebraic. However, in the case j > 0, the special values L(ψ^{−k}\overline{ψ}^{j}λ, 0) are related to real analytic Eisenstein series of weight k + j; a p-adic description of these values was obtained by N. Katz ([Katz78], [Katz80]) by giving an interpretation of the real analytic differential operator

\frac{−π}{Im(\overline{ω_{1}}ω_{2})}(\overline{ω_{1}} \frac{∂}{∂ω_{1}} + \overline{ω_{2}} \frac{∂}{∂ω_{2}} ) (1.1)

on the space of C^{∞} elliptic modular forms in terms of the Gauss-Manin connection and the Hodge decomposition of the relative de Rham cohomology of the universal elliptic curve.

First let us review the case j = 0.

Proposition 1.2 ([Kato93] Chap. III, Proposition 1.1.5). For any non-zero integer a prime to 6, there exists a unique rational function θ_{a} on E characterized by

(1) div(θ_{a}) = − ∑_{P ∈_{a}E} P + a^{2} · O.

(2) Norm_{b}(θ_{a}) = θ_{a} for any non-zero integer b prime to a, where Norm_{b} denotes the norm map with respect to the morphism b: E → E; P ↦ b · P.

By Abel’s theorem, there exists a function satisfying (1), which is obviously unique up to K^{∗}, and it is uniquely determined by the property (2).

Choose a basis ω of Γ(E, Ω_{E}^{1}) = coLie(E) and an O_{K}-basis γ ∈ H_{1}(E(ℂ), ℤ). Then we have

ℂ/L ⟶^{≅} E(ℂ), L = O_{K} · ∫_{γ} ω (1.3)

such that ω corresponds to dz on the LHS.

Let ∂ denote the differential operators

Ω_{E}^{⊗r} ⟶ Ω_{E}^{⊗(r+1)} (r ∈ ℕ)

defined by

∂(f · η) = df ⊗ η (f ∈ O_{E}, η ∈ coLie(E)^{⊗r}).

If we identify Ω_{E}^{⊗r} with O_{E} using the basis ω^{⊗r}, then these differential operators become the operator \frac{d}{ω}, which corresponds to \frac{d}{dz} on ℂ/L via the isomorphism (1.3).

If we apply ∂ to log(θ_{a}) repeatedly, we obtain the following differential forms on E ([Wei76], [dS87] Chap. II, §3):

∂^{r}(log(θ_{a})) = (−1)^{r−1}(r−1)!(a^{2} ∑_{l∈L} \frac{1}{(z + l)^{r}} −a^{r} ∑_{l∈L} \frac{1}{(az + l)^{r}} )⊗(dz)^{⊗r} (r ≥ 3).

Let 𝔣 be the conductor of ψ. For an integer N > 0 such that 𝔣|N, if we take the values at primitive N-torsion points of the power series in the RHS, then they become the Dirichlet power series at s = 0 of the partial L-functions of ψ^{−r} for the abelian extension K(_{N}E) of K generated by N-torsion points of E (up to some powers of a period of ω). Thus we obtain a description of the special values of the partial L-functions of ψ^{−r} at s = 0 in terms of the values of ∂^{r}(log(θ_{a})) at torsion points. In fact, this also holds for r = 1 and r = 2.

In order to generalize this to ψ^{−k}\overline{ψ}^{j} (0 ≤ j < k), we need to work over the moduli spaces and our description involves some differential operators on them.

Let N be an integer≥ 4, let Y_{1}(N) be the moduli scheme over ℚ of elliptic curves with N-torsion points, let E^{univ} be the universal elliptic curve over Y_{1}(N) and let ξ_{N}: Y_{1}(N) → E^{univ} be the universal N-torsion point.

First one can easily generalize Proposition 1.2 to E^{univ}/Y_{1}(N) and obtains the function θ_{a} on E^{univ} for a non-zero integer a prime to 6. One can also define the differential operators

∂: Ω_{E^{univ}/Y_{1}(N)}^{⊗r} ⟶ Ω_{E^{univ}/Y_{1}(N)}^{⊗(r+1)} (r ∈ ℕ)

in the same way.

Set \underline{ω} := coLie(E^{univ}) and \underline{H}_{dR}^{1} := H_{dR}^{1}(E^{univ}/Y_{1}(N)). We define the real analytic differential operators

D: \underline{ω}^{⊗r} ⟶ \underline{ω}^{⊗(r+2)} (r ∈ ℕ)

to be the composite of

\underline{ω}^{⊗r} ↪ Sym^{r}\underline{H}_{dR}^{1} ⟶^{∇} Sym^{r}\underline{H}_{dR}^{1} ⊗_{O_{Y_{1}(N)}} Ω_{Y_{1}(N)}^{1} ⟵^{≅} Sym^{r}\underline{H}_{dR}^{1} ⊗_{O_{Y_{1}(N)}} \underline{ω}^{⊗2}

⟶ \underline{ω}^{⊗(r+2)}

Here the second map is the Gauss-Manin connection, the third one is the isomorphism induced by the Kodaira-Spencer isomorphism and the last one is given by the Hodge decomposition of \underline{H}_{dR}^{1}. Note that the last map is real analytic and the remaining ones are all algebraic.

In [Katz78], N. Katz showed that for a C^{∞}-modular form f ∈ Γ(Y_{1}(N)_{ℂ}, \underline{ω}^{⊗r}(C^{∞})) of weight r, if we regard it as a C^{∞} function f(ω_{1}, ω_{2}) on {(ω_{1}, ω_{2}) ∈ ℂ^{2}|Im((ω_{1})^{−1}ω_{2}) > 0} such that f((ω_{1}, ω_{2})γ) = f(ω_{1}, ω_{2}) (γ ∈ Γ_{1}(N)) and f(a^{−1}ω_{1}, a^{−1}ω_{2}) = a^{r}f(ω_{1}, ω_{2}) (a ∈ ℂ^{∗}), then the modular form D(f) of weight r + 2 corresponds to the derivative of f(ω_{1}, ω_{2}) by the differential operator (1.1). This implies ([Wei76], [dS87] Chap. II, §3) that, for 0 ≤ j < k, k − j ≥ 3, the values of

D^{j} ∘ ξ_{N}^{∗} ∘ ∂^{k−j}(log(θ_{a})) ∈ \underline{ω}^{⊗(k+j)}

at (E, α), α ∈ _{N}E(ℂ) is

(−1)^{k−j−1}(k−1)!A(L)^{−j}(a^{2} ∑_{l∈L} \frac{(\overline{z(α) + l})^{j}}{(z(α) + l)^{k}} −a^{k−j} ∑_{l∈L} \frac{(\overline{az(α) + l})^{j}}{(az(α) + l)^{k}} )⊗(dz)^{⊗(k+j)}.

Here we define A(L) to be π^{−1}Im(\overline{l_{1}}l_{2}) for a basis l_{1}, l_{2} of L such that Im(l_{2}/l_{1}) > 0. Note that we have

(\overline{ω_{1}} \frac{∂}{∂ω_{1}} + \overline{ω_{2}} \frac{∂}{∂ω_{2}} )\frac{(\overline{N^{−1}ω_{1} + nω_{1} + mω_{2}})^{j}}{(N^{−1}ω_{1} + nω_{1} + mω_{2})^{k}} = −k · \frac{(\overline{N^{−1}ω_{1} + nω_{1} + mω_{2}})^{j+1}}{(N^{−1}ω_{1} + nω_{1} + mω_{2})^{k+1}}

for integers n, m.

Thus we obtain the following real analytic description of the special values at 0 of the partial L-functions of ψ^{−k}\overline{ψ}^{j}. (For k −j = 1, 2, we need a little more delicate argument.)

Theorem 1.4. Let L be the abelian extension of K contained in ℂ and generated by N-torsion points of E. Let γ be a generator of the free O_{K}-module N^{−1}H_{1}(E(ℂ), ℤ) of rank 1, and set α := exp(γ) ∈ _{N}E(ℂ) = _{N}E(L). Let 𝔞 be a non-zero ideal of O_{K} prime to 6N and satisfying ψ(𝔞) ∈ ℤ. Set σ_{𝔞} := (𝔞, L/K) ∈ Gal(L/K). For g ∈ Gal(L/K), we denote by x_{E,gα} the point of Y_{1}(N)(ℂ) corresponding to the elliptic curve E ⊗_{K} ℂ endowed with the N-torsion point gα ∈ _{N}E(ℂ) = _{N}E(L). Then for any g ∈ Gal(L/K) and any k, j ∈ ℤ such that 0 ≤ j < k, the value of

D^{j} ∘ ξ_{N}^{∗} ∘ ∂^{k−j}(log(θ_{ψ(𝔞)})) ∈ \underline{ω}^{⊗(k+j)} (1.5) at the point x_{E,gα} is

(−1)^{k−j−1}(k − 1)!(π^{−1}A(ℂ/O_{K})N^{2})^{−j}( ∫_{γ} ω)^{−k−j} ·

\{N(𝔞)L(ψ^{−k}\overline{ψ}^{j}, g-part, 0) − ψ^{k}\overline{ψ}^{−j}(𝔞)L(ψ^{−k}\overline{ψ}^{j}, gσ_{𝔞}-part, 0)\} · ω^{⊗(k+j)}.

Here A(ℂ/O_{K}) denotes the area of a fundamental parallelogram of the lattice O_{K} in ℂ.

Remark 1.6. Since the Hodge decomposition of H_{dR}^{1}(E_{ℂ}/ℂ) = H_{dR}^{1}(E/K) ⊗_{K} ℂ is defined over K (by the eigen-spaces of the action of K ≅ ℚ ⊗ End_{K}(E)), we see that the value of (1.5) at x_{E,gα} is contained in coLie(E)^{⊗(k+j)} ⊗_{K} L (Damerell’s theorem).

In order to study these values p-adically, we will replace the Hodge decomposition in the definition of the operator D by a p-adically analytic decomposition on the formal moduli space.

Let 𝔭 be an inert prime of K at which E has good reduction, let K be the completion of K at 𝔭, let E/O_{K} be the proper smooth model of the base change of E/K to K, and let E^{univ}/M be the formal moduli of E/O_{K}. Then on the PDenvelope M^{PD} of Spec(O_{K}) ↪ M, we have an isomorphism

\underline{H}_{dR}^{1} ≅ O_{M^{PD}} ⊗_{O_{K}} H_{dR}^{1}(E/O_{K})

(the Gauss-Manin connection has enough solutions on M^{PD}), and hence the Hodge decomposition of H_{dR}^{1}(E/K) (which is, in fact, defined over O_{K} because K is unramified at 𝔭 over ℚ) gives rise to a splitting of the short exact sequence:

0 ⟶ coLie(E^{univ}) ⟶ \underline{H}_{dR}^{1} ⟶ Lie(E^{univ}) ⟶ 0.

Note that the composite of

O_{M^{PD}} ⊗_{O_{K}} Lie(E) ↪ \underline{H}_{dR}^{1} → Lie(E^{univ})

is an isomorphism. By replacing the Hodge decomposition with this splitting in the definition of the operator D, we define the differential operators on M^{PD}:

D_{p}: \underline{ω}^{⊗r} ⟶ \underline{ω}^{⊗(r+2)} (r ∈ ℕ).

Let M_{N} be the kernel of the multiplication by N on E^{univ} and let ξ_{N}: M_{N} → E^{univ} ×_{M} M_{N} be the N-torsion point induced by the canonical inclusion M_{N} ↪ E^{univ}. Since M_{N} is étale over M after inverting p, the differential operators D_{p} are naturally extended to K ⊗_{O_{K}} \underline{ω}^{⊗r} → K ⊗_{O_{K}} \underline{ω}^{⊗(r+2)} on M_{N} ×_{M} M^{PD}, which we again denote by D_{p}.

For any primitive N-torsion point α ∈ E(K(_{N}E)), the morphism Spec(O_{K}) ↪ M is uniquely extended to a morphism ε_{α}: Spec(K(_{N}E) ⊗_{K} K) → M_{N} such that the inverse image of ξ_{N} is α.

Proposition 1.7. For any integers 0 ≤ j < k, the value of D^{j} ∘ ξ_{N}^{∗} ∘ ∂^{k−j}(log(θ_{a})) on Y_{1}(N) at (E, α) considered in Theorem 1.4 coincides with ε_{α}^{∗} ∘ (D_{p})^{j} ∘ ξ_{N}^{∗} ∘ ∂^{k−j}(log(θ_{a})).

By combining Theorem 1.4 with Proposition 1.7, we obtain a p-adic description of the values at s = 0 of the partial Hecke L-functions of ψ^{−k}\overline{ψ}^{j} (0 ≤ j < k).

2. Lifting of Coleman power series

The p-adic description of the special values of Hecke L-functions in §1 involves differential operators on the formal moduli space or on the universal elliptic curve over it. Hence such differential operators should appear in an explicit reciprocity law related to such special values, and we are naturally led to the question of lifting Coleman power series to the universal elliptic curve on the formal moduli space.

First let us recall the main theorem in the theory of Coleman power series in our special setting. We keep the notation in the previous section and set π := ψ(𝔭). First note that the completion Ê is the Lubin-Tate formal group associated to (K, π). Let H be a finite unramified extension of K, let σ be the Frobenius of H over K and let φ denote the morphism σ ⊗ [π]: O_{H} ˆ⊗_{O_{K}} Ê → O_{H} ˆ⊗_{O_{K}} Ê, where [π] denotes the multiplication by π ∈ O_{K} on Ê. φ is a lifting of the square of the absolute Frobenius. Let H_{n} be the extension of H generated by π^{n}-torsion points of Ê (or equivalently E).

Theorem 2.1 (R. Coleman [Col79]). Let η = (η_{m})_{m≥1} be a basis of the Tate module T Ê = \underleftarrow{lim}_{n}(_{π^{n}} Ê)(\overline{K}) of Ê as an O_{K}-module. Then we have an isomorphism

Γ(O_{H} ˆ⊗_{O_{K}} Ê, O_{O_{H} ˆ⊗Ê}^{∗})^{Norm_{φ}=1} ⟶^{∼} \underleftarrow{lim}_{m}O_{H_{m}}^{∗}

which sends f to ((σ^{−m}(f))(η_{m}))_{m≥1}. In the RHS, we take the projective limit with respect to the norm maps.

For a norm compatible system u = (u_{m})_{m≥1} of a unit u_{m} of H_{m}, we call the corresponding function on O_{H} ˆ⊗_{O_{K}} Ê the Coleman power series of u. Note that we have a non-canonical isomorphism Γ(O_{H} ˆ⊗_{O_{K}}E, O) ≅ O_{H}[[X]].

By the universal property, φ is naturally extended to O_{H} ˆ⊗_{O_{K}} { \atop E^{univ}} and Coleman power series can be canonically lifted to { \atop E^{univ}} as follows:

Proposition 2.2 ([Tsu00] Proposition 5.2). The specialization map

Γ(O_{H} ˆ⊗_{O_{K}} { \atop E^{univ}}, O^{∗})^{Norm_{φ}=1} ⟶^{∼} Γ(O_{H} ˆ⊗_{O_{K}}E, O^{∗})^{Norm_{φ}=1}

is an isomorphism.

3. Exponential maps and dual exponential maps

In this section, K denotes a finite extension of ℚ_{p} and K_{0} denotes the maximal unramified extension of ℚ_{p} contained in K. In [BK90], S. Bloch and K. Kato generalize the exponential maps

K ⟶^{exp} ℚ_{p} ⊗ K̂^{∗} ≅ H^{1}(K, ℚ_{p}(1)),

Lie(A_{K}) ⟶^{exp} ℚ_{p} ⊗ { \atop A(K)} ↪ H^{1}(K, ℚ_{p} ⊗ T_{p}A)

for 𝔾_{m} and an abelian scheme A over O_{K} to any crystalline representation of Gal(\overline{K}/K).

By the theory of J.-M. Fontaine ([Fon82], [Fon94a], [Fon94b]), we have the notion “crystalline” for p-adic representations of Gal(\overline{K}/K) and a fully faithful functor D_{crys} from the category of crystalline p-adic representations of Gal(\overline{K}/K) to the category MF_{K}(φ) of finite dimensional filtered φ-modules over K. Here a p-adic representation of Gal(\overline{K}/K) means a finite dimensional ℚ_{p}-vector space V endowed with a continuous and linear action of Gal(\overline{K}/K) and a finite dimensional filtered φ-module over K is a finite dimensional K-vector space D endowed with a separated and exhaustive decreasing filtration F il^{i}D (i ∈ ℤ) by K-subspaces, a K_{0}-structure D_{0} and a semi-linear automorphism φ on D_{0}.

The crystalline conjecture, which is now a theorem, asserts that for any proper smooth scheme over O_{K}, the p-adic representation H_{ét}^{m}(X\overline{K}, ℚ_{p}) is crystalline and the corresponding filtered φ-module is canonically isomorphic to the de Rham cohomology H_{dR}^{m}(X_{K}/K) endowed with the Hodge filtration and the K_{0}-structure with φ given by the crystalline cohomology of the special fiber of X.

The essential image of the functor D_{crys} is stable under extensions and hence D_{crys} defines a natural injective homomorphism:

Ext_{MF_{K}(φ)}^{1}(K, D) ↪ H^{1}(K, V)

for any crystalline representation V of Gal(\overline{K}/K) and D = D_{crys}(V). On the other hand, we have a natural map:

D/F il^{0}D → Ext_{MF_{K}(φ)}^{1}(K, D)

sending the class of a ∈ D to K ⊕ D whose filtration is twisted by a:

F il^{i}(K ⊕ D) = \begin{cases}F il^{i}D & (i > 0), \\ F il^{i}D ⊕ K · (1, a) & (i ≤ 0). \end{cases}

The K_{0}-structure and the Frobenius are the direct sums. We define the exponential map:

exp: D/F il^{0}D → H^{1}(K, V)

to be the composite of these two maps. Using Tate duality, we obtain the dual exponential map:

exp^{∗}: H^{1}(K, V) → F il^{0}D.

Note that in the case V = ℚ_{p} ⊗ T_{p}A for an abelian scheme A over O_{K}, D is canonically isomorphic to the dual of H_{dR}^{1}(A_{K}/K) and D/F il^{0}D is isomorphic to Lie(A_{K}).

4. Explicit reciprocity law

Before giving our generalized explicit reciprocity law, we shall review how the classical explicit reciprocity law for ℚ_{p}(μ_{p^{n}}) (n ≥ 1) by Artin-Hasse and K. Iwasawa can be reformulated in terms of the dual exponential maps for the trivial p-adic representation ℚ_{p}.

By the definition of the dual exponential map, we have

u ∪ c = Trace_{ℚ_{p}(μ_{p^{n}})/ℚ_{p}}(log(u) exp^{∗}(c))

for u ∈ ℚ_{p} ⊗ ℤ_{p}[μ_{p^{n}}]^{∗} ⊂ H^{1}(ℚ_{p}(μ_{p^{n}}), ℚ_{p}(1)) and c ∈ H^{1}(ℚ_{p}(μ_{p^{n}}), ℚ_{p}). Hence the classical explicit reciprocity law says that the image of f ∈ (ℤ_{p}[[X − 1]]^{∗})^{Norm_{φ}=1} under the composite of

(ℤ_{p}[[X − 1]]^{∗})^{Norm_{φ}=1} ⟶^{∼}_{Coleman} \underleftarrow{lim}_{m}ℤ_{p}[μ_{p^{m}}]^{∗} → \underleftarrow{lim}_{m}H^{1}(ℚ_{p}(μ_{p^{m}}), ℤ_{p}(1))

⟶^{∼} \underleftarrow{lim}_{m}H^{1}(ℚ_{p}(μ_{p^{m}}), ℤ/p^{m}ℤ(1)) ⟶^{∼}_{t^{⊗(−1)}} \underleftarrow{lim}_{m}H^{1}(ℚ_{p}(μ_{p^{m}}), ℤ_{p}/p^{m}ℤ)

⟵^{∼} \underleftarrow{lim}_{m}H^{1}(ℚ_{p}(μ_{p^{m}}), ℤ_{p}) ⟶^{proj} H^{1}(ℚ_{p}(μ_{p^{n}}), ℤ_{p}) ⟶^{exp^{∗}} ℚ_{p}(μ_{p^{n}})

is

p^{−n}(X \frac{d}{dX} ) log(f)|_{X=ζ_{n}}

for n ≥ 1. Here we choose a generator t = (ζ_{m})_{m≥1} of ℤ_{p}(1) and the first map is defined by taking the values at X = ζ_{m} (m ≥ 1).

The generalized explicit reciprocity law of Bloch-Kato in [BK90] §2 and K. Kato in [Kato93] Chap. II, §2 asserts that, if we replace t^{⊗(−1)} by t^{⊗(−r)} (r ≥ 2) and ℤ_{p} by ℤ_{p}(1 − r), then we obtain

\frac{1}{(r − 1)!}p^{−nr}((X \frac{d}{dX} )^{r} log(f))|_{X=ζ_{n}}.

The classical explicit reciprocity law of Artin-Hasse and K. Iwasawa is generalized by A. Wiles [Wil78] to a finite extension K of ℚ_{p} and the abelian extensions K generated by torsion points of the Lubin-Tate formal group associated to K and a uniformizer π of K. In [Kato93], K. Kato also showed that the above generalized explicit reciprocity law still holds if we replace ℚ_{p}(μ_{p^{n}}) by the extension of K generated by π^{n}-torsion points of the Lubin-Tate group, t by a basis η of the Tate module T of the Lubin-Tate group as an O_{K}-module and the derivation X \frac{d}{dX} by ∂ defined in the same way as in §1. If we apply this to K and E, then by §1, we see that the translations of θ_{a} by torsion points of order prime to p are related to the values at s = 0 of the partial Hecke L-functions of ψ^{−r} (r ≥ 1) through the dual exponential maps for T^{⊗(−r)}(1).

Now we will give the precise statement of our generalization of the above Kato’s explicit reciprocity law. We keep the notation of §1 and §2. Let T be the Tate module of Ê, let \overline{T} be the same ℤ_{p}-representation of Gal(\overline{K}/K) as T with the action of O_{K} twisted by the unique non-trivial automorphism of K/ℚ_{p}, and set T_{k,j} := T^{⊗(−k)} ⊗_{O_{K}} \overline{T}^{⊗j} for integers 0 ≤ j < k. Choose and fix a basis η = (η_{n})_{n≥1} of T = \underleftarrow{lim}_{n}(_{π^{n}} Ê)(\overline{K}) and let \overline{η} be the same element as η regarded as an element of \overline{T}.

Theorem 4.1 ([Tsu00] Theorem 6.3). For any integers 0 ≤ j < k, the image of an element f of Γ(O_{H} ˆ⊗_{O_{K}} { \atop E^{univ}}, O^{∗})^{Norm_{φ}=1} under the composite of the maps

Γ(O_{H} ˆ⊗_{O_{K}} { \atop E^{univ}}, O^{∗})^{Norm_{φ}=1} ⟶^{∼} Γ(O_{H} ˆ⊗_{O_{K}} Ê, O^{∗})^{Norm_{φ}=1} ⟶^{∼}_{Coleman} \underleftarrow{lim}_{m}O_{H_{m}}^{∗}

⟶ \underleftarrow{lim}_{m}H^{1}(H_{m}, ℤ_{p}(1)) ⟶^{η^{⊗(−k)}⊗\overline{η}^{⊗j}} \underleftarrow{lim}_{m}H^{1}(H_{m}, T_{k,j}(1))

⟶^{proj} H^{1}(H_{n}, T_{k,j}(1)) ⟶^{exp^{∗}} (coLie(E))^{⊗(k+j)} ⊗_{K} H_{n}

is

(−1)^{j} \frac{1}{(k − 1)!}π^{−nk}\overline{π}^{nj} · ε_{η_{n}}^{∗} ∘ D_{p}^{j} ∘ ξ_{p^{n}}^{∗} ∘ ∂^{k−j}(log(σ^{−n}(f))).

See §1 for the definition of ∂, D_{p}, ξ_{p^{n}} and ε_{η_{n}}. Here \overline{π} denotes the image of π ∈ K under the unique non-trivial automorphism of K/ℚ_{p}.

If we apply this theorem to the translation of θ_{a} by torsion points of E of order prime to p, then, by §1, the images are related to the special values at s = 0 of the partial L-functions of ψ^{−k}\overline{ψ}^{j}. By the similar argument as in [Kato93] Chap. III, we see that the Kato’s zeta element gives (−1)^{k−j−1} times the special value at 0 of the Hecke L-function of ψ^{−k}\overline{ψ}^{j}λ (for integers 0 < j ≤ k and a finite Hecke character λ) through the dual exponential map for H^{1}(K, F_{k,j}(λ)^{∗}(1)) under some condition on λ in the case k − j = 2 ([Tsu00] Theorem 12.8). Here F_{k,j}(λ) denotes the p-adic representation of Gal(\overline{K}/K) such that, for any prime ideal 𝔭 of O_{K} prime to the conductor of ψ^{−k}\overline{ψ}^{j}λ, F_{k,j}(λ) is unramified at 𝔭 and the action of the geometric Frobenius at 𝔭 is given by the multiplication by (ψ^{−k}\overline{ψ}^{j}λ)(𝔭).

The above explicit reciprocity law can be generalized to any Lubin-Tate formal group for a finite unramified extension of ℚ_{p} using the formal moduli space of the Lubin-Tate group. See [Tsu00] for details.

References

[BK90] S. Bloch and K. Kato. L-functions and Tamagawa number of motives. In The Grothendieck Festschrift, vol. I, pages 333–400. Birkhäuser, 1990.

[Col79] R. Coleman. Division values in local fields. Inv. math., 53:91–116, 1979.

[dS87] E. de Shalit. Iwasawa theory of elliptic curves with complex multiplication. Perspectives in Mathematics, Vol. 3. Academic Press, 1987.

[Fon82] J.-M. Fontaine. Sur certains types de représentations p-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate. Ann. of Math., 115:529– 577, 1982.

[Fon94a] J.-M. Fontaine. Le corps des périodes p-adiques. In Périodes p-adiques, Séminaire de Bures, 1988, Astérisque 223, pages 59–111, 1994.

[Fon94b] J.-M. Fontaine. Représentations p-adiques semi-stables. In Périodes p-adiques, Séminaire de Bures, 1988, Astérisque 223, pages 113–183, 1994.

[Guo99] L. Guo. Hecke characters and formal group characters. In Topics in number theory (University Park, PA, 1997), Math. Appl. 467, pages 181–192. Kluwer Acad. Publ., 1999.

[Han97] B. Han. On Bloch-Kato conjecture of Tamagawa numbers for Hecke characters of imaginary quadratic number field. PhD thesis, University of Chicago, 1997.

[Katz78] N. Katz. p-adic L-functions for CM fields. Inv. math., 49:199–297, 1978.

[Katz80] N. Katz. p-adic L-functions, Serre-Tate local moduli, and ratios of solutions of differential equations. In Proceedings of the ICM Helsinki, 1978, Vol. I, pages 365–371. Academia Scientiarum Fennica, 1980.

[Kato93] K. Kato. Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via B_{dR}. Part I. In Arithmetic algebraic geometry, Trento, 1991, Lecture Notes in Math. 1553, pages 50–163. Springer, 1993.

[Kim93] K. Kimura. Special values of Hecke L-functions of imaginary quadratic fields and explicit reciprocity law (in japanese). Master’s thesis, University of Tokyo, 1993.

[Tsu00] T. Tsuji. Explicit reciprocity law and formal moduli for Lubin-Tate formal groups, 2000, preprint.

[Wei76] A. Weil. Elliptic functions according to Eisenstein and Kronecker. Springer, 1976.

[Wil78] A. Wiles. Higher explicit reciprocity laws. Ann. Math., 107:235–254, 1978.

Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan

   E-mail address: t-tsuji@ms.u-tokyo.ac.jp