An Elementary Proof of the

Mazur-Tate-Teitelbaum Conjecture for Elliptic Curves

Dedicated to Professor John Coates on the occasion of his sixtieth birthday

Shinichi Kobayashi^{1}

Received: November 29, 2005 Revised: February 15, 2006

Abstract. We give an elementary proof of the Mazur-Tate-Teitelbaum conjecture for elliptic curves by using Kato’s element.

2000 Mathematics Subject Classification: 11F85, 11G05, 11G07, 11G40, 11S40. Keywords and Phrases: elliptic curves, p-adic L-functions, Iwasawa theory, the Mazur-Tate-Teitelbaum conjecture, exceptional zeros, Kato’s element.

1. Introduction

The p-adic L-function L_{p}(E, s) of an elliptic curve E defined over ℚ has an extra zero at s = 1 coming from the interpolation factor at p if E has split multiplicative reduction at the prime p. The Mazur-Tate-Teitelbaum conjecture (now a theorem of Greenberg-Stevens) describes the first derivative of L_{p}(E, s) as

\frac{d}{ds}L_{p}(E, s) |_{s=1} = \frac{log_{p}(q_{E})}{ord_{p}(q_{E})} \frac{L(E, 1)}{Ω_{E}^{+}}

where q_{E} is the Tate period of E coming from the p-adic uniformization of E at p, log_{p} is the Iwasawa p-adic logarithm, Ω_{E}^{+} is the real period of E and L(E, 1) is the special value of the complex Hasse-Weil L-function at s = 1. Known proofs of this conjecture are classified into two kinds. One is, as Greenberg-Stevens [GS] did first, a proof using a global theory like Hida’s universal ordinary deformation. The other is, as Kato-Kurihara-Tsuji [KKT] or Colmez [C] did, a proof based on local theory (except using Kato’s element). Each kind of proof has its own importance but the latter type of proof makes it clear that the substantial facts behind this conjecture are of local nature. The p-adic L-function is the image of Kato’s element via a purely local morphism, the so called Coleman map or Perrin-Riou map. The extra zero phenomena discovered by Mazur-Tate-Teitelbaum is, in fact, a property of the local Coleman map. In this paper, we prove a derivative formula (Theorem 4.1) of the Coleman map for elliptic curves by purely local and elementary method and we apply this formula to Kato’s element to show the conjecture of Mazur-Tate-Teitelbaum. Of course, our proof is just a special and the simplest case of that in Kato-Kurihara-Tsuji [KKT] or Colmez [C] (they proved the formula not only for elliptic curves but for higher weight modular forms) but I believe that it is still worthwhile to write it down for the following reason. First, the important paper Kato-Kurihara-Tsuji [KKT] has not yet been published. Second, since we restrict ourselves to the case of elliptic curves, the proof is much simpler and elementary (of course, such a simple proof would be also known to specialists. In fact, Masato Kurihara informed me that Kato, Kurihara and Tsuji have two simple proofs and one is similar to ours). I hope that this paper would help those who are interested in the understanding of this interesting problem.

1Supported by JSPS Postdoctoral Fellowships for Research Abroad.

Documenta Mathematica · Extra Volume Coates (2006) 567–575

Acknowledgement: I would like to wish Professor John Coates a happy sixtieth birthday, and to thank him for his contribution to mathematics, especially to Iwasawa theory. It is my great pleasure to dedicate this article to him on this occasion. This paper was written during the author’s visit at the university of Paris 6. He would like to thank P. Colmez and L. Merel for the accommodation. He also would like to thank K. Bannai and N. Otsubo for discussion. Finally, he is grateful to the referee for his careful reading of the manuscript.

2. A structure of the group of local units in k_{∞}/ℚ_{p}.

Let k_{∞} be the (local) cyclotomic ℤ_{p}-extension of ℚ_{p} in ℚ_{p}(ζ_{p^{∞}}) := ∪_{n=0}^{∞}ℚ_{p}(ζ_{p^{n}}) with Galois group Γ and let k_{n} be its n-th layer in k_{∞} with Galois group Γ_{n}. We identify the Galois group Gal(ℚ_{p}(ζ_{p^{∞}})/ℚ_{p}) with ℤ_{p}^{×} by the cyclotomic character κ. Then Γ is identified with 1 + pℤ_{p} and the torsion subgroup Δ of Gal(ℚ_{p}(ζ_{p^{∞}})/ℚ_{p}) is regarded as μ_{p−1} ⊂ ℤ_{p}^{×}. Let U_{n}^{1} be the subgroup of O_{k_{n}}^{×} consisting of the elements which are congruent to 1 modulo the maximal ideal 𝔪_{n} of O_{k_{n}}. Following the Appendix of Rubin [R] or [Ko], for a fixed generator (ζ_{p^{n}})_{n∈ℕ} of ℤ_{p}(1), we construct a certain canonical system of local points (d_{n})_{n} ∈ \underleftarrow{lim}_{n} U_{n}^{1} and we determine the Galois module structure of U_{n}^{1} by using these points. The idea of the construction of such a system is as follows. First we consider a certain formal group F isomorphic to 𝔾̂_{m} whose formal logarithm has a certain compatible property with the trace operator of k_{∞}. Then the system of local points is essentially the image of cyclotomic units by the isomorphism F ≅ 𝔾̂_{m}. We let

ℓ(X) = log(1 + X) + ∑_{k=0}^{∞} ∑_{δ∈Δ} \frac{(X + 1)^{p^{k}δ} − 1}{p^{k}}.

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This power series is convergent in ℚ_{p}[[X]] due to the summation ∑_{δ∈Δ}. It is straightforward to see that

ℓ′(X) ∈ 1 + Xℤ_{p}[[X]], ℓ(0) = 0, (φ − p) ∘ ℓ(X) ∈ pℤ_{p}[[X]]

where φ is the Frobenius operator such that (φ ∘ ℓ)(X) = ℓ((X + 1)^{p} − 1). Hence by Honda’s theory, there is a formal group F over ℤ_{p} whose logarithm is given by ℓ, and ι(X) = exp ∘ ℓ (X) − 1 ∈ ℤ_{p}[[X]] gives an isomorphism of formal groups F ≅ 𝔾̂_{m} over ℤ_{p}. (See for example, Section 8 of [Ko].) Take an element ε of pℤ_{p} such that ℓ(ε) = p and we define

c_{n} := ι((ζ_{p^{n+1}} − 1) [+]_{F} ε).

Since this element is fixed by the group Δ, this is an element of 𝔾̂_{m}(𝔪_{n}). Then by construction, d_{n} = 1 + c_{n} ∈ U_{n}^{1} satisfies the relation

log_{p}(d_{n}) = ℓ(ε) + ℓ(ζ_{p^{n+1}} − 1) = p + ∑_{k=0}^{n} ∑_{δ∈Δ} \frac{ζ_{p^{n+1−k}}^{δ} − 1}{p^{k}}.

Proposition 2.1. i) (d_{n})_{n} is a norm compatible system and d_{0} = 1. ii) Let u be a generator of U_{0}^{1}. Then as ℤ_{p}[Γ_{n}]-module, d_{n} and u generate U_{n}^{1}, and d_{n} generates (U_{n}^{1})^{N=1} where N is the absolute norm from k_{n} to ℚ_{p}.

Proof. Since ζ_{p} − 1 is not contained in 𝔪_{n}, the group 𝔾̂_{m}(𝔪_{n}) does not contain p-power torsion points. Therefore to see i), it suffices to show the trace compatibility of (log_{p}(d_{n}))_{n}, and this is done by direct calculations. For ii), we show that (ι^{−1}(c_{n})^{σ})_{σ∈Γ_{n}} and ε generate F(𝔪_{n}) as ℤ_{p}-module by induction for n. The proof is the same as that of Proposition 8.11 of [Ko] but we rewrite it for the ease of the reader. The case n = 0 is clear. For arbitrary n, we show that ℓ(𝔪_{n}) ⊂ 𝔪_{n} + k_{n−1} and

F(𝔪_{n})/F(𝔪_{n−1}) ≅ ℓ(𝔪_{n})/ℓ(𝔪_{n−1}) ≅ 𝔪_{n}/𝔪_{n−1}.

Here the first isomorphism is induced by the logarithm ℓ and the last isomorphism is by (𝔪_{n} + k_{n−1})/k_{n−1} ≅ 𝔪_{n}/𝔪_{n−1}. As a set, F(𝔪_{n}) is the maximal ideal 𝔪_{n}, and we write x ∈ F(𝔪_{n}) in the form x = ∑_{δ∈Δ} ∑_{i} a_{i} ζ_{p^{n+1}}^{iδ}, a_{i} ∈ ℤ_{p}. Then for y = ∑_{δ∈Δ} ∑_{i} a_{i} ζ_{p^{n}}^{iδ} ∈ 𝔪_{n−1}, we have that x^{p} ≡ y mod pO_{k_{n}} . Therefore for k ≥ 1, we have

∑_{δ∈Δ} \frac{(x + 1)^{p^{k}δ} − 1}{p^{k}} ≡ ∑_{δ∈Δ} \frac{(x^{p} + 1)^{p^{k−1}δ} − 1}{p^{k}} ≡ ∑_{δ∈Δ} \frac{(y + 1)^{p^{k−1}δ} − 1}{p^{k}} mod 𝔪_{n}.

Hence we have ∑_{δ} \frac{(x+1)^{p^{k}δ}−1}{p^{k}} ∈ 𝔪_{n} + k_{n−1}. Since ℓ(x) is convergent, for

sufficiently large k_{0}, we have ∑_{k=k_{0}}^{∞} ∑_{δ} \frac{(x+1)^{p^{k}δ}−1}{p^{k}} ∈ 𝔪_{n}, and therefore ℓ(x) is contained in 𝔪_{n} + k_{n−1}. Since ℓ is injective on F(𝔪_{n}) (there is no torsion point in F(𝔪_{n}) ≅ 𝔾̂_{m}(𝔪_{n})) and is compatible with the Galois action, we have ℓ (𝔪_{n}) ∩ k_{n−1} = ℓ (𝔪_{n−1}). Therefore we have an injection

ℓ(𝔪_{n})/ℓ(𝔪_{n−1}) ↪ (𝔪_{n} + k_{n−1})/k_{n−1} ≅ 𝔪_{n}/𝔪_{n−1}.

By direct calculations, we have ℓ(ι^{−1}(c_{n})) ≡ ∑_{δ}(ζ_{p^{n+1}}^{δ} − 1) mod k_{n−1}. Since ∑_{δ}(ζ_{p^{n+1}}^{δ} − 1) generates 𝔪_{n}/𝔪_{n−1} as a ℤ_{p}[Γ_{n}]-module with respect to the usual addition, the above injection is in fact a bijection. Thus (ι^{−1}(c_{n})^{σ})_{σ∈Γ_{n}} generate F(𝔪_{n})/F(𝔪_{n−1}). By induction (ι^{−1}(c_{n})^{σ})_{σ∈Γ_{n}} and ε generate F(𝔪_{n}). Since 𝔾̂_{m} is isomorphic to F by ι, we have ii). □

Since Nd_{n} = d_{0} = 1, by Hilbert’s theorem 90, there exists an element x_{n} ∈ k_{n} such that d_{n} = x_{n}^{γ}/x_{n} for a fixed generator γ of Γ. We put π_{n} = ∏_{δ∈Δ}(ζ_{p^{n+1}}^{δ} − 1). Then π_{n} is a norm compatible uniformizer of k_{n}. By the previous proposition, x_{n} can be taken of the form x_{n} = π_{n}^{e_{n}}u_{n} for some integer e_{n} and u_{n} ∈ (U_{n}^{1})^{N=1}.

Proposition 2.2. In the same notation as the above, we have

p ≡ e_{n} (p − 1) log_{p} κ(γ) mod p^{n+1}.

Proof. If we put

G(X) = exp(p) · exp ∘ ℓ (X) = exp ∘ ℓ (X[+]ε) ∈ 1 + (p, X)ℤ_{p}[[X]],

then by definition

G_{σ}(ζ_{p^{m+1}} − 1) = d_{m}^{σ}

where G_{σ}(X) = G((X + 1)^{κ(σ)} − 1) for σ ∈ Γ. By Proposition 2.1 ii), u_{n} is written as a product in the form u_{n} = ∏(d_{n}^{σ})^{a}. If we put H(X) = ∏ G_{σ}(X)^{a}, then H(X) satisfies H(ζ_{p^{m+1}} − 1) = N_{k_{n}/k_{m}}u_{n} for 0 ≤ m ≤ n. We put

F (X) = ( ∏_{δ∈Δ} \frac{(X + 1)^{δκ(γ)} − 1}{(X + 1)^{δ} − 1})^{e_{n}} \frac{H((X + 1)^{κ(γ)} − 1)}{H(X)}.

Then we have

G(X) ≡ F (X) mod \frac{(X + 1)^{p^{n+1}} − 1}{X}

since they are equal if we substitute X = ζ_{p^{m+1}} −1 for 0 ≤ m ≤ n. Substituting X = 0 in this congruence and taking the p-adic logarithm, we have that p ≡ e_{n}(p − 1) log_{p} κ(γ) mod p^{n+1}. □

3. The Coleman map for the Tate curve.

We construct the Coleman map for the Tate curve following the Appendix of [R] or Section 8 of [Ko]. See also [Ku]. In this section we assume that E is the Tate curve

E_{q} : y^{2} + xy = x^{3} + a_{4}(q)x + a_{6}(q)

where q = q_{E} ∈ ℚ_{p}^{×} satisfying |q|_{p} < 1 and

s_{k}(q) = ∑_{n≥1} \frac{n^{k}q^{n}}{1 − q^{n}} , a_{4}(q) = −s_{3}(q), a_{6}(q) = −\frac{5s_{3}(q) + 7s_{5}(q)}{12}.

Then we have the uniformization

ϕ : ℂ_{p}^{×}/q^{ℤ} ≅ E_{q}(ℂ_{p}), u ↦ (X(u, q), Y (u, q))

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where

X(u, q) = ∑_{n∈ℤ} \frac{q^{n}u}{(1 − q^{n}u)^{2}} − 2s_{1}(q),

Y (u, q) = ∑_{n∈ℤ} \frac{(q^{n}u)^{2}}{(1 − q^{n}u)^{3}} + s_{1}(q).

(Of course, we put ϕ(q^{ℤ}) = O.) This isomorphism induces the isomorphism of the formal groups ϕ̂: 𝔾̂_{m} ≅ Ê. It is straightforward to see that the pull back by ϕ̂ of the invariant differential ω_{E} = \frac{dx}{2y+x} on Ê with the parameter t = −x/y

is the invariant differential ω𝔾̂_{m} = \frac{dX}{1+X} on 𝔾̂_{m} with the parameter X = u − 1. Hence ϕ̂ is given by the power series t = expÊ ∘ log(1 + X) − 1 ∈ ℤ_{p}[[X]]. From now we identify 𝔾̂_{m} with Ê by ϕ̂. In particular, we regard c_{n} ∈ 𝔾̂_{m}(𝔪_{n}) in the previous section as an element of Ê(𝔪_{n}). Let T = T_{p}E be the p-adic Tate module of E and V = T ⊗ ℚ_{p}. The cup product induces a non-degenerate pairing of Galois cohomology groups

( , )_{E,n} : H^{1}(k_{n}, T) × H^{1}(k_{n}, T^{∗}(1)) → H^{2}(k_{n}, ℤ_{p}(1)) ≅ ℤ_{p}.

If there is no fear of confusion, we write ( , )_{E,n} simply as ( , )_{E}. By the Kummer map, we regard Ê(𝔪_{n}) as a subgroup of H^{1}(k_{n}, T). Then we define a morphism Col_{n} : H^{1}(k_{n}, T^{∗}(1)) → ℤ_{p}[Γ_{n}] by

z ⟼ ∑_{σ∈Γ_{n}} (c_{n}^{σ}, z)_{E,n} σ.

This morphism is compatible with the natural Galois action and since the sequence (c_{n})_{n} is norm compatible, Col_{n} is also compatible for n with respect to the corestrictions and the natural projections. We define the Coleman map

Col : \underleftarrow{lim}_{n} H^{1}(k_{n}, T^{∗}(1)) ⟶ Λ = ℤ_{p}[[Γ]]

as the projective limit of Col_{n} over all n. We recall the dual exponential map. For every n let tan(E/k_{n}) denote the tangent space of E/k_{n} at the origin, and consider the Lie group exponential map

exp_{E,n} : tan(E/k_{n}) → E(k_{n}) ⊗ ℚ_{p}.

The cotangent space cotan(E/k_{n}) is generated by the invariant differential ω_{E} over k_{n}, and we let ω_{E}^{∗} be the corresponding dual basis of tan(E/k_{n}). Then there is a dual exponential map

exp_{E,n}^{∗} : H^{1}(k_{n}, V^{∗}(1)) ⟶ cotan(E/k_{n}) = k_{n} ω_{E},

which has a property

(x, z)_{E,n} = Tr_{k_{n}/ℚ_{p}} logÊ(x) exp_{ω_{E},n}^{∗}(z)

for every x ∈ Ê(𝔪_{n}) and z ∈ H^{1}(k_{n}, V^{∗}(1)). Here exp_{ω_{E},n}^{∗} = ω_{E}^{∗} ∘ exp_{E,n}^{∗}. If there is no fear of confusion, we write exp_{ω_{E},n}^{∗}(z) as exp_{ω_{E}}^{∗}(z). Then using the identification ϕ̂: 𝔾̂_{m} ≅ Ê, the morphism Col_{n} is described in terms of the dual exponential map as follows.

Col_{n}(z) = ∑_{σ∈Γ_{n}} (c_{n}^{σ}, z)_{E,n} σ

= ∑_{σ∈Γ_{n}}( Tr_{k_{n}/ℚ_{p}} log_{p}(d_{n}^{σ}) exp_{ω_{E}}^{∗}(z) ) σ

= ( ∑_{σ∈Γ_{n}} log_{p}(d_{n}^{σ}) σ) ( ∑_{σ∈Γ_{n}} exp_{ω_{E}}^{∗}(z^{σ}) σ^{−1}) .

Let G_{n} be the Galois group Gal(ℚ_{p}(ζ_{p^{n}})/ℚ_{p}) and let χ be a finite character of G_{n+1} of conductor p^{n+1} which is trivial on Δ. Then we have

∑_{σ∈Γ_{n}} log_{p}(d_{n}^{σ}) χ(σ) = \begin{cases}τ(χ) & if χ is non-trivial, \\ 0 & otherwise \end{cases}

where τ(χ) is the Gauss sum ∑_{σ∈G_{n+1}} χ(σ) ζ_{p^{n+1}}^{σ}. Hence for χ ≠ 1, we have

χ ∘ Col(z) = τ(χ) ∑_{σ∈Γ_{n}} exp_{ω_{E}}^{∗}(z^{σ}) χ(σ)^{−1}.

Kato showed that there exists an element z^{Kato} ∈ \underleftarrow{lim}_{n} H^{1}(k_{n}, T^{∗}(1)) such that

∑_{σ∈Γ_{n}} exp_{ω_{E}}^{∗}((z^{Kato})^{σ}) χ(σ)^{−1} = e_{p}(\overline{χ})\frac{L(E, \overline{χ}, 1)}{Ω_{E}^{+}}

where e_{p}(χ) is the value at s = 1 of the p-Euler factor of L(E, χ, s), that is, e_{p}(χ) = 1 if χ is non-trivial and e_{p}(χ) = (1 − \frac{1}{p}) if χ is trivial. (See [Ka], Theorem 12.5.) Hence we have

χ ∘ Col(z^{Kato}) = τ(χ)\frac{L(E, \overline{χ}, 1)}{Ω_{E}^{+}}

if χ is non-trivial. The p-adic L-function L_{p}(E, s) is written of the form

L_{p}(E, s) = L_{p,γ}(E, κ(γ)^{s−1} − 1)

for some power series L_{p,γ}(E, X) ∈ ℤ_{p}[[X]]. If we identify Λ = ℤ_{p}[[Γ]] with ℤ_{p}[[X]] by sending γ ↦ 1 + X, then it satisfies an interpolation formula

χ ∘ L_{p,γ}(E, X) = τ(χ)\frac{L(E, \overline{χ}, 1)}{Ω_{E}^{+}}.

Since an element of Λ has only finitely many zeros, we conclude that

Col(z^{Kato})(X) = L_{p,γ}(E, X).

Here we denote Col(z^{Kato}) by Col(z^{Kato})(X) to emphasis that we regard Col(z^{kato}) as a power series in ℤ_{p}[[X]]. Note that we have 1 ∘ Col(z) = 0 for the trivial character 1, or Col(z)(0) = 0, namely, any Coleman power series Col(z)(X) for the Tate curve has a trivial zero at X = 0.

4. The first derivative of the Coleman map.

We compute the first derivative of the Coleman map Col(z)(X). By Tate’s uniformization, there is an exact sequence of local Galois representations

(1)  0 → T_{1} → T → T_{2} → 0

where T_{1} = T_{p}Ê ≅ ℤ_{p}(1) and T_{2} ≅ ℤ_{p}. The cup product induces a nondegenerate paring

H^{1}(k_{n}, T_{1}) × H^{1}(k_{n}, T_{1}^{∗}(1)) → H^{2}(k_{n}, ℤ_{p}(1)) ≅ ℤ_{p}.

With the identification by ϕ̂: T_{1} ≅ ℤ_{p}(1), this is in fact the cup product pairing of 𝔾_{m}

( , )_{𝔾_{m},n} : H^{1}(k_{n}, ℤ_{p}(1)) × H^{1}(k_{n}, ℤ_{p}) → H^{2}(k_{n}, ℤ_{p}(1)) ≅ ℤ_{p}.

If there is no fear of confusion, we write ( , )_{𝔾_{m},n} simply as ( , )_{𝔾_{m}}. Since c_{n} ∈ Ê(k_{n}) ⊂ H^{1}(k_{n}, T_{1}), we have

(c_{n}^{σ}, z)_{E,n} = (d_{n}^{σ}, π(z))_{𝔾_{m},n}

for z ∈ H^{1}(k_{n}, T^{∗}(1)) where π is the morphism induced by the projection T^{∗}(1) → T_{1}^{∗}(1). Tate’s uniformization ϕ also induces a commutative diagram

H^{1}(k_{n}, V^{∗}(1)) ⟶^{exp_{E}^{∗}} k_{n} ω_{E} ⟶^{ω_{E}^{∗}} k_{n}
π↓  ↓
H^{1}(k_{n}, V_{1}^{∗}(1)) ⟶^{exp_{𝔾m}^{∗}} k_{n} ω_{𝔾_{m}} ⟶^{ω_{𝔾m}^{∗}} k_{n}

where ω_{𝔾_{m}} is the invariant differential of 𝔾_{m} which is \frac{dX}{1+X} on 𝔾̂_{m}, and ω_{𝔾_{m}}^{∗} is the dual basis for ω_{𝔾_{m}}. We also put exp_{ω_{𝔾m}}^{∗} = ω_{𝔾_{m}}^{∗} ∘ exp_{𝔾_{m}}^{∗}. Now we compute the derivative. With the same notation as the previous section, we have

Col_{n}(z) = ∑_{σ∈Γ_{n}} (c_{n}^{σ}, z)_{E,n} σ = ∑_{σ∈Γ_{n}} (d_{n}^{σ}, π(z))_{𝔾_{m},n} σ

= ∑_{σ∈Γ_{n}} ((x_{n}^{γ}/x_{n})^{σ}, π(z))_{𝔾_{m},n} σ

= (γ^{−1} − 1) ∑_{σ∈Γ_{n}} (x_{n}^{σ}, π(z))_{𝔾_{m},n} σ.

Therefore by the identification ℤ_{p}[X]/((X + 1)^{p^{n}} − 1) ≅ ℤ_{p}[Γ_{n}], X ↦ γ − 1, we have

\frac{Col(z)(X)}{X} ≡ −\frac{1}{γ} ∑_{σ∈Γ_{n}} (x_{n}^{σ}, π(z))_{𝔾_{m},n} σ mod \frac{(X + 1)^{p^{n}} − 1}{X}.

Hence

Col(z)′(0) ≡ − (Nx_{n}, π(z))_{𝔾_{m},0} mod p^{n}.

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Since Nx_{n} = p^{e_{n}}N(u_{n}) = p^{e_{n}} and by Proposition 2.2, we have

(Nx_{n}, π(z))_{𝔾_{m}} = e_{n} (p, π(z))_{𝔾_{m}} ≡ \frac{p}{(p − 1) log_{p} κ(γ)} (p, π(z))_{𝔾_{m}} mod p^{n}.

Taking limit for n, we have that

(2)  Col(z)′(0) = −\frac{p}{(p − 1) log_{p} κ(γ)} (p, π(z))_{𝔾_{m}}.

Next we compute (p, π(z))_{𝔾_{m}}. We consider the exact sequence

H^{1}(ℚ_{p}, T^{∗}(1)) ⟶^{π} H^{1}(ℚ_{p}, T_{1}^{∗}(1)) ⟶^{δ_{2}} H^{2}(ℚ_{p}, T_{2}^{∗}(1))

induced by (1), and a diagram

H^{1}(ℚ_{p}, T_{1}) × H^{1}(ℚ_{p}, T_{1}^{∗}(1)) ⟶^{( ,)_{𝔾m}} H^{2}(ℚ_{p}, ℤ_{p}(1)) = ℤ_{p}
δ_{1}↑  ^{δ_{2}}↓  ↓
H^{0}(ℚ_{p}, T_{2}) × H^{2}(ℚ_{p}, T_{2}^{∗}(1)) ⟶^{( ,)_{𝔾m}} H^{2}(ℚ_{p}, ℤ_{p}(1)) = ℤ_{p}.

It is straightforward to see that the connecting morphism δ_{1} is given by

H^{0}(ℚ_{p}, T_{2}) = ℤ_{p} → ℚ_{p}^{×} ⊗ ℤ_{p} = H^{1}(ℚ_{p}, T_{1}), 1 ↦ q_{E} ⊗ 1.

Hence for w ∈ H^{1}(ℚ_{p}, T_{1}^{∗}(1)), we have

(q_{E} ⊗ 1, w)_{𝔾_{m}} = (δ_{1}(1), w)_{𝔾_{m}} = (1, δ_{2}(w))_{𝔾_{m}}.

In particular, if w comes from H^{1}(ℚ_{p}, T^{∗}(1)), namely, it is of the form π(z), then

(3)  (q_{E} ⊗ 1, w)_{𝔾_{m}} = (q_{E} ⊗ 1, π(z))_{𝔾_{m}} = (1, δ_{2} ∘ π(z))_{𝔾_{m}} = 0.

On the other hand, if we put q_{E} = p^{ord_{p}(q_{E})} ρ u_{q} where ρ ∈ μ_{p−1} and u_{q} ∈ 1 + pℤ_{p}, we have

(4)  (q_{E} ⊗ 1, w)_{𝔾_{m}} = ord_{p}(q_{E}) (p, w)_{𝔾_{m}} + (u_{q}, w)_{𝔾_{m}}

(5) = ord_{p}(q_{E}) (p, w)_{𝔾_{m}} + log_{p}(u_{q}) exp_{ω_{𝔾m}}^{∗} (w).

Hence by (3) and (5) we have

(6) (p, π(z))_{𝔾_{m}} = −\frac{log_{p}(u_{q})}{ord_{p}(q_{E})} exp_{ω_{𝔾m}}^{∗} (π(z)) = −\frac{log_{p}(q_{E})}{ord_{p}(q_{E})} exp_{ω_{E}}^{∗}(z).

Combining (2) and (6), we obtain

Theorem 4.1. For z ∈ \underleftarrow{lim}_{n} H^{1}(k_{n}, T^{∗}(1)), the first derivative of the Coleman map Col(z) is given by

\frac{d}{dX} Col(z)(X) |_{X=0} = \frac{p}{(p − 1) log_{p} κ(γ)} \frac{log_{p}(q_{E})}{ord_{p}(q_{E})} exp_{ω_{E}}^{∗}(z).

Now if E/ℚ has split multiplicative reduction at p, then we may assume that E is locally the Tate curve for some q_{E} ∈ ℚ_{p}^{×}. We apply the above formula to Kato’s element z = z^{Kato}. Since exp_{ω_{E}}^{∗}(z^{Kato}) = (1 − \frac{1}{p})\frac{L(E,1)}{Ω_{E}^{+}}, we have Corollary 4.2. Let L_{p,γ}(E, X) be the power series in ℤ_{p}[[X]] such that L_{p}(E, s) = L_{p,γ}(E, κ(γ)^{s−1} − 1). Then

\frac{d}{dX} L_{p,γ}(E, X) |_{X=0} = \frac{1}{log_{p} κ(γ)} \frac{log_{p}(q_{E})}{ord_{p}(q_{E})} \frac{L(E, 1)}{Ω_{E}^{+}},

or

\frac{d}{ds}L_{p}(E, s) |_{s=1} = \frac{log_{p}(q_{E})}{ord_{p}(q_{E})} \frac{L(E, 1)}{Ω_{E}^{+}}.

References

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Shinichi Kobayashi

Graduate School of Mathematics Nagoya University Furo-cho Chikusa-ku Nagoya 464-8602 Japan shinichi@math.nagoya-u.ac.jp 576

Documenta Mathematica · Extra Volume Coates (2006)