E(K) open

prob-

lem S b : E(K)/mE(K)×

E[m]K^{∗}/K^{∗}^{m} m = 2 E/K

m ≥ 2 E[m] ⊆

E(k)

k K K E/K

E(K)/mE(K) E(K)

E m E(K)

κ : E(K)×Gal(\overline{K}/K)E[m],

P ∈ E(K)

Q ∈

E Q ∈

E(\overline{K})

E [m]Q =

P κ(P, σ) :=

Q^{σ}− Q Q′ [m]Q = [m]Q′ =

P Q^{′σ}−

Q′ = Q^{σ}− Q [m](Q′− Q) =

0 Q−

Q′ ∈ E[m] ⊆ E(K) Q′− Q (\overline{K}/K) (Q′− Q)^{σ} = Q′− Q Q^{′σ}− Q′ = Q^{σ}−

Q κ κ κ mE(K)

(\overline{K}/L)

L K [m]^{−1}E(K)

L = K([m]^{−1}E(K)) κ E(K)/mE(K)×Gal(L/K)E[m].

    Proof: κ P ∈ E(K) κ Q ∈

E(\overline{K}) [m]Q =

P Q ∈

E(K)

P ∈ mE(K)

κ(P, σ) =

0 Q σ P ∈ mE(K) κ σ ∈ (K) σ L/K

P ∈ E(K)

Q [m]Q =

P κ(P, σ) =

0 Q^{σ} =

Q [m]^{−1}E(K)

L σ σ ∈

Gal(\overline{K}/L) σ ∈

Gal(\overline{K}/L)

[m]^{−1}E(K) κ