この
曲線の q q q の 同種写像が Pic 0 \operatorname{Pic}^0 Pic 0 を 双対同種写像の deg [ m ] = m 2 \deg [m] = m^2 deg [ m ] = m 2 と 等分 [ m ] [m] [ m ] を m m m を E [ m ] ≅ ( Z / m Z ) 2 E[m] \cong (\mathbb{Z}/m\mathbb{Z})^2 E [ m ] ≅ ( Z / m Z ) 2 である 2-同種写像と ヴェイユ対の E [ m ] ⊂ E ( K ) E[m] \subset E(K) E [ m ] ⊂ E ( K ) μ m ⊂ K \mu_m \subset K μ m ⊂ K 」を
前提 ：第2章 、14-algebraic-geometry 第8章 、04-algebra 第8章 。3.10 節では ℓ \ell ℓ 15-algebraic-number-theory 第5章 ）を
楕円曲線は 同種写像 ――を m m m [ m ] [m] [ m ] と、有限体上の ϕ q \phi_q ϕ q [ m ] [m] [ m ] の E [ m ] E[m] E [ m ] m m m 等分点の 第4章 では ∣ E ( F q ) ∣ = deg ( 1 − ϕ q ) \lvert E(\mathbb{F}_q) \rvert = \deg(1 - \phi_q) ∣ E ( F q )∣ = deg ( 1 − ϕ q ) と 第7章 では 第8章 では 2-同種写像を
本章では、曲線の deg [ m ] = m 2 \deg[m] = m^2 deg [ m ] = m 2 を [ m ] [m] [ m ] を E [ m ] ≅ ( Z / m Z ) 2 E[m] \cong (\mathbb{Z}/m\mathbb{Z})^2 E [ m ] ≅ ( Z / m Z ) 2 を
この K K K は 曲線 とは K ‾ \overline{K} K 上の K ‾ \overline{K} K 上で E E E は E E E と E ( K ‾ ) E(\overline{K}) E ( K ) も p = char K p = \operatorname{char} K p = char K p = 0 p = 0 p = 0 も The Arithmetic of Elliptic Curves を
3.1 曲線の φ : C 1 → C 2 \varphi\colon C_1 \to C_2 φ : C 1 → C 2 を φ \varphi φ が φ ∗ : K ‾ ( C 2 ) → K ‾ ( C 1 ) \varphi^{\ast}\colon \overline{K}(C_2) \to \overline{K}(C_1) φ ∗ : K ( C 2 ) → K ( C 1 ) f ↦ f ∘ φ f \mapsto f \circ \varphi f ↦ f ∘ φ は K ‾ ( C 1 ) \overline{K}(C_1) K ( C 1 ) φ ∗ K ‾ ( C 2 ) \varphi^{\ast}\overline{K}(C_2) φ ∗ K ( C 2 ) の K ‾ \overline{K} K 上超越次数 1 の
定義 3.1 （射の φ \varphi φ に deg φ = [ K ‾ ( C 1 ) : φ ∗ K ‾ ( C 2 ) ] \deg \varphi = [\overline{K}(C_1) : \varphi^{\ast}\overline{K}(C_2)] deg φ = [ K ( C 1 ) : φ ∗ K ( C 2 )] と φ \varphi φ 純非分離的 (purely inseparable) と 非分離的 と deg s φ \deg_s \varphi deg s φ deg i φ \deg_i \varphi deg i φ と deg φ = deg s φ ⋅ deg i φ \deg \varphi = \deg_s \varphi \cdot \deg_i \varphi deg φ = deg s φ ⋅ deg i φ ）。定数写像の 0 0 0 と P ∈ C 1 P \in C_1 P ∈ C 1 φ ( P ) \varphi(P) φ ( P ) での t t t に e φ ( P ) = ord P ( φ ∗ t ) e_\varphi(P) = \operatorname{ord}_P(\varphi^{\ast}t) e φ ( P ) = ord P ( φ ∗ t ) と
ここで L / M L/M L / M の M M M L L L の L s L_s L s （ [ L s : M ] [L_s : M] [ L s : M ] [ L : L s ] [L : L_s] [ L : L s ] p p p の
命題 3.2 （主張）定数でない φ : C 1 → C 2 \varphi\colon C_1 \to C_2 φ : C 1 → C 2 ψ : C 2 → C 3 \psi\colon C_2 \to C_3 ψ : C 2 → C 3 に
任意の Q ∈ C 2 Q \in C_2 Q ∈ C 2 ∑ P ∈ φ − 1 ( Q ) e φ ( P ) = deg φ \sum_{P \in \varphi^{-1}(Q)} e_\varphi(P) = \deg \varphi ∑ P ∈ φ − 1 ( Q ) e φ ( P ) = deg φ
有限個を Q ∈ C 2 Q \in C_2 Q ∈ C 2 ∣ φ − 1 ( Q ) ∣ = deg s φ \lvert \varphi^{-1}(Q) \rvert = \deg_s \varphi ∣ φ − 1 ( Q )∣ = deg s φ e ψ ∘ φ ( P ) = e φ ( P ) e ψ ( φ ( P ) ) e_{\psi \circ \varphi}(P) = e_\varphi(P) e_\psi(\varphi(P)) e ψ ∘ φ ( P ) = e φ ( P ) e ψ ( φ ( P )) deg \deg deg deg s \deg_s deg s deg i \deg_i deg i は deg φ = 1 \deg \varphi = 1 deg φ = 1 ⟺ \iff ⟺ φ \varphi φ は
(1) は 14 第8章 定理 8.5、(2) と (3) の deg \deg deg の
例 3.3 標数 ≠ 2 \neq 2 = 2 E : y 2 = f ( x ) E\colon y^2 = f(x) E : y 2 = f ( x ) f f f は x : E → P 1 x\colon E \to \mathbb{P}^1 x : E → P 1 ( x , y ) ↦ [ x : 1 ] (x, y) \mapsto [x : 1] ( x , y ) ↦ [ x : 1 ] O ↦ [ 1 : 0 ] O \mapsto [1 : 0] O ↦ [ 1 : 0 ] と K ‾ ( E ) = K ‾ ( x ) ( y ) \overline{K}(E) = \overline{K}(x)(y) K ( E ) = K ( x ) ( y ) K ‾ ( x ) \overline{K}(x) K ( x ) deg x = 2 \deg x = 2 deg x = 2 f ( x 0 ) ≠ 0 f(x_0) \neq 0 f ( x 0 ) = 0 x 0 x_0 x 0 の ( x 0 , ± y 0 ) (x_0, \pm y_0) ( x 0 , ± y 0 ) f ( x 0 ) = 0 f(x_0) = 0 f ( x 0 ) = 0 ( x 0 , 0 ) (x_0, 0) ( x 0 , 0 ) の 1 点で y y y が x − x 0 x - x_0 x − x 0 y 2 y^2 y 2 と ∞ \infty ∞ の O O O で
補題 3.4 標数 ≠ 2 \neq 2 = 2 E i : y 2 = f i ( x ) E_i\colon y^2 = f_i(x) E i : y 2 = f i ( x ) f i f_i f i は φ : E 1 → E 2 \varphi\colon E_1 \to E_2 φ : E 1 → E 2 R = u / v ∈ K ‾ ( x ) R = u/v \in \overline{K}(x) R = u / v ∈ K ( x ) u , v ∈ K ‾ [ x ] u, v \in \overline{K}[x] u , v ∈ K [ x ] は 0 0 0 S ∈ K ‾ ( x ) S \in \overline{K}(x) S ∈ K ( x ) に φ ( x , y ) = ( R ( x ) , y S ( x ) ) \varphi(x, y) = (R(x), yS(x)) φ ( x , y ) = ( R ( x ) , y S ( x )) と
deg φ = [ K ‾ ( x ) : K ‾ ( R ) ] = max ( deg u , deg v ) \deg \varphi = [\overline{K}(x) : \overline{K}(R)] = \max(\deg u, \deg v) deg φ = [ K ( x ) : K ( R )] = max ( deg u , deg v ) R ′ ≠ 0 R' \neq 0 R ′ = 0 （有理関数と φ \varphi φ は
証明. t = R t = R t = R K ‾ \overline{K} K F ( T ) = u ( T ) − t v ( T ) F(T) = u(T) - t v(T) F ( T ) = u ( T ) − t v ( T ) x x x を T T T に d = max ( deg u , deg v ) d = \max(\deg u, \deg v) d = max ( deg u , deg v ) F F F K ‾ [ t , T ] \overline{K}[t, T] K [ t , T ] の t t t に t t t を u , v u, v u , v の F F F K ‾ [ t ] [ T ] \overline{K}[t][T] K [ t ] [ T ] の 04-algebra 第6章 定理 6.20 より F F F K ‾ ( t ) [ T ] \overline{K}(t)[T] K ( t ) [ T ] で [ K ‾ ( x ) : K ‾ ( t ) ] = d [\overline{K}(x) : \overline{K}(t)] = d [ K ( x ) : K ( t )] = d φ ∗ K ‾ ( E 2 ) = K ‾ ( t , y S ) \varphi^{\ast}\overline{K}(E_2) = \overline{K}(t, yS) φ ∗ K ( E 2 ) = K ( t , y S ) ( y S ) 2 = f 2 ( t ) (yS)^2 = f_2(t) ( y S ) 2 = f 2 ( t ) y S ∉ K ‾ ( x ) yS \notin \overline{K}(x) y S ∈ / K ( x ) [ K ‾ ( t , y S ) : K ‾ ( t ) ] = 2 [\overline{K}(t, yS) : \overline{K}(t)] = 2 [ K ( t , y S ) : K ( t )] = 2 [ K ‾ ( x , y ) : K ‾ ( x ) ] = 2 [\overline{K}(x, y) : \overline{K}(x)] = 2 [ K ( x , y ) : K ( x )] = 2 と deg φ = 2 d / 2 = d \deg \varphi = 2d/2 = d deg φ = 2 d /2 = d
(2) M = K ‾ ( t , y S ) M = \overline{K}(t, yS) M = K ( t , y S ) と y = y S / S ( x ) ∈ M ( x ) y = yS/S(x) \in M(x) y = y S / S ( x ) ∈ M ( x ) M ( x ) = K ‾ ( E 1 ) M(x) = \overline{K}(E_1) M ( x ) = K ( E 1 ) F ′ ( x ) = u ′ ( x ) − t v ′ ( x ) = v ( x ) R ′ ( x ) ≠ 0 F'(x) = u'(x) - t v'(x) = v(x)R'(x) \neq 0 F ′ ( x ) = u ′ ( x ) − t v ′ ( x ) = v ( x ) R ′ ( x ) = 0 x x x F ∈ M [ T ] F \in M[T] F ∈ M [ T ] の x x x M M M 上の 0 0 0 でないから M ( x ) M(x) M ( x ) は M M M の 04-algebra 第9章 定理 9.6）なので、M ( x ) / M M(x)/M M ( x ) / M は □ \square □
3.2 フロベニウス射標数 p > 0 p > 0 p > 0 q = p r q = p^r q = p r r ≥ 1 r \geq 1 r ≥ 1 ）と E E E の a i a_i a i a i q a_i^q a i q に E ( q ) E^{(q)} E ( q ) と q q q q q q 乗写像は ( x , y ) ∈ E (x, y) \in E ( x , y ) ∈ E ( x q , y q ) ∈ E ( q ) (x^q, y^q) \in E^{(q)} ( x q , y q ) ∈ E ( q )
定義 3.5 （フロベニウス射, Frobenius morphism）ϕ q : E → E ( q ) \phi_q\colon E \to E^{(q)} ϕ q : E → E ( q ) ( x , y ) ↦ ( x q , y q ) (x, y) \mapsto (x^q, y^q) ( x , y ) ↦ ( x q , y q ) O ↦ O O \mapsto O O ↦ O q q q と E E E F q \mathbb{F}_q F q 上 E ( q ) = E E^{(q)} = E E ( q ) = E ϕ q \phi_q ϕ q E E E の
命題 3.6 ϕ q \phi_q ϕ q は deg ϕ q = q \deg \phi_q = q deg ϕ q = q ϕ q \phi_q ϕ q は
証明. K ‾ \overline{K} K は ϕ q ∗ K ‾ ( E ( q ) ) = K ‾ ( x q , y q ) \phi_q^{\ast}\overline{K}(E^{(q)}) = \overline{K}(x^q, y^q) ϕ q ∗ K ( E ( q ) ) = K ( x q , y q ) K ‾ ( E ) q : = { f q ∣ f ∈ K ‾ ( E ) } \overline{K}(E)^q := \lbrace f^q \mid f \in \overline{K}(E) \rbrace K ( E ) q := { f q ∣ f ∈ K ( E )} に f ↦ f q f \mapsto f^q f ↦ f q K ‾ ( x ) \overline{K}(x) K ( x ) K ‾ ( x q ) \overline{K}(x^q) K ( x q ) に [ K ‾ ( E ) q : K ‾ ( x q ) ] = [ K ‾ ( E ) : K ‾ ( x ) ] = 2 [\overline{K}(E)^q : \overline{K}(x^q)] = [\overline{K}(E) : \overline{K}(x)] = 2 [ K ( E ) q : K ( x q )] = [ K ( E ) : K ( x )] = 2 。補題 3.4(1) の R = x q R = x^q R = x q [ K ‾ ( x ) : K ‾ ( x q ) ] = q [\overline{K}(x) : \overline{K}(x^q)] = q [ K ( x ) : K ( x q )] = q
deg ϕ q = [ K ‾ ( E ) : K ‾ ( E ) q ] = [ K ‾ ( E ) : K ‾ ( x ) ] [ K ‾ ( x ) : K ‾ ( x q ) ] [ K ‾ ( E ) q : K ‾ ( x q ) ] = 2 q 2 = q \deg \phi_q = [\overline{K}(E) : \overline{K}(E)^q] = \frac{[\overline{K}(E) : \overline{K}(x)]\,[\overline{K}(x) : \overline{K}(x^q)]}{[\overline{K}(E)^q : \overline{K}(x^q)]} = \frac{2q}{2} = q deg ϕ q = [ K ( E ) : K ( E ) q ] = [ K ( E ) q : K ( x q )] [ K ( E ) : K ( x )] [ K ( x ) : K ( x q )] = 2 2 q = q 各 f ∈ K ‾ ( E ) f \in \overline{K}(E) f ∈ K ( E ) ( T − f ) q = T q − f q ∈ K ‾ ( E ) q [ T ] (T - f)^q = T^q - f^q \in \overline{K}(E)^q[T] ( T − f ) q = T q − f q ∈ K ( E ) q [ T ] の a ↦ a q a \mapsto a^q a ↦ a q が q > 1 q > 1 q > 1 なので □ \square □
3.3 同種写像
定義 3.8 （同種写像, isogeny）楕円曲線の φ : E 1 → E 2 \varphi\colon E_1 \to E_2 φ : E 1 → E 2 φ ( O ) = O \varphi(O) = O φ ( O ) = O と と E 1 E_1 E 1 E 2 E_2 E 2 (isogenous) であると Hom ( E 1 , E 2 ) \operatorname{Hom}(E_1, E_2) Hom ( E 1 , E 2 ) End ( E ) = Hom ( E , E ) \operatorname{End}(E) = \operatorname{Hom}(E, E) End ( E ) = Hom ( E , E ) 、その Aut ( E ) \operatorname{Aut}(E) Aut ( E ) と
群演算 E × E → E E \times E \to E E × E → E は ( φ + ψ ) ( P ) = φ ( P ) + ψ ( P ) (\varphi + \psi)(P) = \varphi(P) + \psi(P) ( φ + ψ ) ( P ) = φ ( P ) + ψ ( P ) も Hom ( E 1 , E 2 ) \operatorname{Hom}(E_1, E_2) Hom ( E 1 , E 2 ) は End ( E ) \operatorname{End}(E) End ( E ) の
例 3.9 (1) [ m ] P = P + ⋯ + P [m]P = P + \cdots + P [ m ] P = P + ⋯ + P m m m [ − m ] = − [ m ] [-m] = -[m] [ − m ] = − [ m ] [ 0 ] P = O [0]P = O [ 0 ] P = O ϕ q \phi_q ϕ q Q ≠ O Q \neq O Q = O の τ Q ( P ) = P + Q \tau_Q(P) = P + Q τ Q ( P ) = P + Q は ≠ 2 \neq 2 = 2 y 2 = x 3 − x y^2 = x^3 - x y 2 = x 3 − x [ i ] ( x , y ) = ( − x , i y ) [i](x, y) = (-x, iy) [ i ] ( x , y ) = ( − x , i y ) i 2 = − 1 i^2 = -1 i 2 = − 1
定理 3.10 （同種写像は φ : E 1 → E 2 \varphi\colon E_1 \to E_2 φ : E 1 → E 2 は
証明. φ \varphi φ が φ = [ 0 ] \varphi = [0] φ = [ 0 ] である。定数でない φ ∗ : Div ( E 1 ) → Div ( E 2 ) \varphi_{\ast}\colon \operatorname{Div}(E_1) \to \operatorname{Div}(E_2) φ ∗ : Div ( E 1 ) → Div ( E 2 ) ∑ n P ( P ) ↦ ∑ n P ( φ ( P ) ) \sum n_P (P) \mapsto \sum n_P (\varphi(P)) ∑ n P ( P ) ↦ ∑ n P ( φ ( P )) は N = N K ‾ ( E 1 ) / φ ∗ K ‾ ( E 2 ) N = N_{\overline{K}(E_1)/\varphi^{\ast}\overline{K}(E_2)} N = N K ( E 1 ) / φ ∗ K ( E 2 ) に
φ ∗ ( div f ) = div ( ( φ ∗ ) − 1 N ( f ) ) ( f ∈ K ‾ ( E 1 ) × ) \varphi_{\ast}(\operatorname{div} f) = \operatorname{div}\bigl((\varphi^{\ast})^{-1} N(f)\bigr) \qquad (f \in \overline{K}(E_1)^{\times}) φ ∗ ( div f ) = div ( ( φ ∗ ) − 1 N ( f ) ) ( f ∈ K ( E 1 ) × ) が φ ∗ \varphi_{\ast} φ ∗ は Pic 0 ( E 1 ) → Pic 0 ( E 2 ) \operatorname{Pic}^0(E_1) \to \operatorname{Pic}^0(E_2) Pic 0 ( E 1 ) → Pic 0 ( E 2 ) を 第2章 の κ i : E i → Pic 0 ( E i ) \kappa_i\colon E_i \to \operatorname{Pic}^0(E_i) κ i : E i → Pic 0 ( E i ) P ↦ [ ( P ) − ( O ) ] P \mapsto [(P) - (O)] P ↦ [( P ) − ( O )] は φ ( O ) = O \varphi(O) = O φ ( O ) = O φ ∗ ( κ 1 ( P ) ) = [ ( φ ( P ) ) − ( O ) ] = κ 2 ( φ ( P ) ) \varphi_{\ast}(\kappa_1(P)) = [(\varphi(P)) - (O)] = \kappa_2(\varphi(P)) φ ∗ ( κ 1 ( P )) = [( φ ( P )) − ( O )] = κ 2 ( φ ( P )) 。すな φ = κ 2 − 1 ∘ φ ∗ ∘ κ 1 \varphi = \kappa_2^{-1} \circ \varphi_{\ast} \circ \kappa_1 φ = κ 2 − 1 ∘ φ ∗ ∘ κ 1 は □ \square □
したがって ker φ \ker \varphi ker φ は φ ≠ 0 \varphi \neq 0 φ = 0 なら φ ∘ ( ψ 1 + ψ 2 ) = φ ∘ ψ 1 + φ ∘ ψ 2 \varphi \circ (\psi_1 + \psi_2) = \varphi \circ \psi_1 + \varphi \circ \psi_2 φ ∘ ( ψ 1 + ψ 2 ) = φ ∘ ψ 1 + φ ∘ ψ 2 に End ( E ) \operatorname{End}(E) End ( E ) は φ ∘ [ m ] = [ m ] ∘ φ \varphi \circ [m] = [m] \circ \varphi φ ∘ [ m ] = [ m ] ∘ φ である。任意の α : E 1 → E 2 \alpha\colon E_1 \to E_2 α : E 1 → E 2 は τ − α ( O ) ∘ α \tau_{-\alpha(O)} \circ \alpha τ − α ( O ) ∘ α と τ α ( O ) \tau_{\alpha(O)} τ α ( O ) の
系 3.11 標数 ≠ 2 \neq 2 = 2 E i E_i E i を 0 0 0 でない φ : E 1 → E 2 \varphi\colon E_1 \to E_2 φ : E 1 → E 2 R R R 0 0 0 S S S ∈ K ‾ ( x ) \in \overline{K}(x) ∈ K ( x ) φ ( x , y ) = ( R ( x ) , y S ( x ) ) \varphi(x, y) = (R(x), yS(x)) φ ( x , y ) = ( R ( x ) , y S ( x )) と
証明. φ ( − P ) = − φ ( P ) \varphi(-P) = -\varphi(P) φ ( − P ) = − φ ( P ) x ∘ φ x \circ \varphi x ∘ φ [ − 1 ] ∗ [-1]^{\ast} [ − 1 ] ∗ y ↦ − y y \mapsto -y y ↦ − y で K ‾ ( E 1 ) \overline{K}(E_1) K ( E 1 ) K ‾ ( x ) \overline{K}(x) K ( x ) 上の y ∘ φ y \circ \varphi y ∘ φ は K ‾ ( E 1 ) = K ‾ ( x ) ⊕ y K ‾ ( x ) \overline{K}(E_1) = \overline{K}(x) \oplus y\overline{K}(x) K ( E 1 ) = K ( x ) ⊕ y K ( x ) x ∘ φ x \circ \varphi x ∘ φ ( y ∘ φ ) / y ∈ K ‾ ( x ) (y \circ \varphi)/y \in \overline{K}(x) ( y ∘ φ ) / y ∈ K ( x ) x ∘ φ x \circ \varphi x ∘ φ が c c c x x x c c c の O O O からなる y ∘ φ = 0 y \circ \varphi = 0 y ∘ φ = 0 E 2 [ 2 ] E_2[2] E 2 [ 2 ] （4 点）に φ \varphi φ は □ \square □
命題 3.12 (1) m ≠ 0 m \neq 0 m = 0 [ m ] [m] [ m ] は Hom ( E 1 , E 2 ) \operatorname{Hom}(E_1, E_2) Hom ( E 1 , E 2 ) は Z \mathbb{Z} Z End ( E ) \operatorname{End}(E) End ( E ) は
証明. (1) 標数 ≠ 2 \neq 2 = 2 で E : y 2 = f ( x ) E\colon y^2 = f(x) E : y 2 = f ( x ) f f f の e e e に T = ( e , 0 ) T = (e, 0) T = ( e , 0 ) は 第2章 命題 2.13(1)）。m m m が [ m ] T = T ≠ O [m]T = T \neq O [ m ] T = T = O ker [ 2 ] \ker[2] ker [ 2 ] は [ 2 ] [2] [ 2 ] は m = 2 k m ′ m = 2^k m' m = 2 k m ′ m ′ m' m ′ [ m ] = [ 2 ] k ∘ [ m ′ ] [m] = [2]^k \circ [m'] [ m ] = [ 2 ] k ∘ [ m ′ ] は φ ≠ 0 \varphi \neq 0 φ = 0 は ψ ∘ φ = 0 \psi \circ \varphi = 0 ψ ∘ φ = 0 ψ = 0 \psi = 0 ψ = 0 [ m ] ∘ φ = 0 [m] \circ \varphi = 0 [ m ] ∘ φ = 0 m ≠ 0 m \neq 0 m = 0 φ = 0 \varphi = 0 φ = 0 □ \square □
定理 3.13 φ : E 1 → E 2 \varphi\colon E_1 \to E_2 φ : E 1 → E 2 0 0 0 でない
すべての Q ∈ E 2 Q \in E_2 Q ∈ E 2 ∣ φ − 1 ( Q ) ∣ = deg s φ \lvert \varphi^{-1}(Q) \rvert = \deg_s \varphi ∣ φ − 1 ( Q )∣ = deg s φ P ∈ E 1 P \in E_1 P ∈ E 1 e φ ( P ) = deg i φ e_\varphi(P) = \deg_i \varphi e φ ( P ) = deg i φ ∣ ker φ ∣ = deg s φ \lvert \ker \varphi \rvert = \deg_s \varphi ∣ ker φ ∣ = deg s φ
φ \varphi φ が φ \varphi φ は ∣ ker φ ∣ = deg φ \lvert \ker \varphi \rvert = \deg \varphi ∣ ker φ ∣ = deg φ
証明. Q , Q ′ ∈ E 2 Q, Q' \in E_2 Q , Q ′ ∈ E 2 に φ ( R ) = Q ′ − Q \varphi(R) = Q' - Q φ ( R ) = Q ′ − Q と R R R を P ↦ P + R P \mapsto P + R P ↦ P + R φ − 1 ( Q ) \varphi^{-1}(Q) φ − 1 ( Q ) φ − 1 ( Q ′ ) \varphi^{-1}(Q') φ − 1 ( Q ′ ) への deg s φ \deg_s \varphi deg s φ φ ∘ τ R = τ φ ( R ) ∘ φ \varphi \circ \tau_R = \tau_{\varphi(R)} \circ \varphi φ ∘ τ R = τ φ ( R ) ∘ φ に e φ ( P + R ) = e φ ( P ) e_\varphi(P + R) = e_\varphi(P) e φ ( P + R ) = e φ ( P ) e φ e_\varphi e φ は e e e e deg s φ = deg φ e \deg_s \varphi = \deg \varphi e deg s φ = deg φ e = deg i φ e = \deg_i \varphi e = deg i φ deg i φ = 1 \deg_i \varphi = 1 deg i φ = 1 の □ \square □
3.4 双対同種写像と
定理 3.14 （双対同種写像, dual isogeny、主張）φ : E 1 → E 2 \varphi\colon E_1 \to E_2 φ : E 1 → E 2 を m ≥ 1 m \geq 1 m ≥ 1 の φ ^ ∘ φ = [ m ] \hat{\varphi} \circ \varphi = [m] φ ^ ∘ φ = [ m ] と φ ^ : E 2 → E 1 \hat{\varphi}\colon E_2 \to E_1 φ ^ : E 2 → E 1 が 双対同種写像 ）。[ 0 ] ^ = [ 0 ] \widehat{[0]} = [0] [ 0 ] = [ 0 ] と φ , ψ : E 1 → E 2 \varphi, \psi\colon E_1 \to E_2 φ , ψ : E 1 → E 2 λ : E 2 → E 3 \lambda\colon E_2 \to E_3 λ : E 2 → E 3 に
φ ∘ φ ^ = [ deg φ ] \varphi \circ \hat{\varphi} = [\deg \varphi] φ ∘ φ ^ = [ deg φ ] E 2 E_2 E 2 λ ∘ φ ^ = φ ^ ∘ λ ^ \widehat{\lambda \circ \varphi} = \hat{\varphi} \circ \hat{\lambda} λ ∘ φ = φ ^ ∘ λ ^ φ + ψ ^ = φ ^ + ψ ^ \widehat{\varphi + \psi} = \hat{\varphi} + \hat{\psi} φ + ψ = φ ^ + ψ ^ deg φ ^ = deg φ \deg \hat{\varphi} = \deg \varphi deg φ ^ = deg φ φ ^ ^ = φ \hat{\hat{\varphi}} = \varphi φ ^ ^ = φ
参照：AEC III.6.1, III.6.2。証明の φ ^ = κ 1 − 1 ∘ φ ∗ ∘ κ 2 \hat{\varphi} = \kappa_1^{-1} \circ \varphi^{\ast} \circ \kappa_2 φ ^ = κ 1 − 1 ∘ φ ∗ ∘ κ 2 φ ∗ \varphi^{\ast} φ ∗ は κ 1 − 1 \kappa_1^{-1} κ 1 − 1 は ∑ n P ( P ) \sum n_P (P) ∑ n P ( P ) の ∑ n P P \sum n_P P ∑ n P P に φ ∗ ( ( Q ) ) = ∑ φ ( P ) = Q e φ ( P ) ( P ) \varphi^{\ast}((Q)) = \sum_{\varphi(P) = Q} e_\varphi(P) (P) φ ∗ (( Q )) = ∑ φ ( P ) = Q e φ ( P ) ( P ) と Q = φ ( P 0 ) Q = \varphi(P_0) Q = φ ( P 0 ) φ − 1 ( Q ) = P 0 + ker φ \varphi^{-1}(Q) = P_0 + \ker \varphi φ − 1 ( Q ) = P 0 + ker φ を φ ^ ( Q ) = deg i φ ( ∑ P ∈ φ − 1 ( Q ) P − ∑ R ∈ ker φ R ) = ( deg φ ) P 0 \hat{\varphi}(Q) = \deg_i \varphi \bigl(\sum_{P \in \varphi^{-1}(Q)} P - \sum_{R \in \ker \varphi} R\bigr) = (\deg \varphi)P_0 φ ^ ( Q ) = deg i φ ( ∑ P ∈ φ − 1 ( Q ) P − ∑ R ∈ k e r φ R ) = ( deg φ ) P 0 と である [ m ] [m] [ m ] φ \varphi φ で φ ∘ φ ^ ∘ φ = [ m ] ∘ φ \varphi \circ \hat{\varphi} \circ \varphi = [m] \circ \varphi φ ∘ φ ^ ∘ φ = [ m ] ∘ φ φ \varphi φ の □ \square □
系 3.15 任意の m ∈ Z m \in \mathbb{Z} m ∈ Z に [ m ] ^ = [ m ] \widehat{[m]} = [m] [ m ] = [ m ] deg [ m ] = m 2 \deg [m] = m^2 deg [ m ] = m 2
証明. [ ± 1 ] [\pm 1] [ ± 1 ] は [ ± 1 ] ∘ [ ± 1 ] = [ 1 ] [\pm 1] \circ [\pm 1] = [1] [ ± 1 ] ∘ [ ± 1 ] = [ 1 ] [ ± 1 ] ^ = [ ± 1 ] \widehat{[\pm 1]} = [\pm 1] [ ± 1 ] = [ ± 1 ] 。定理 3.14(3) と [ m ] ^ = [ m ] \widehat{[m]} = [m] [ m ] = [ m ] [ deg [ m ] ] = [ m ] ^ ∘ [ m ] = [ m 2 ] [\deg [m]] = \widehat{[m]} \circ [m] = [m^2] [ deg [ m ]] = [ m ] ∘ [ m ] = [ m 2 ] deg [ m ] = m 2 \deg [m] = m^2 deg [ m ] = m 2 □ \square □
系 3.16 （次数は φ , ψ ∈ Hom ( E 1 , E 2 ) \varphi, \psi \in \operatorname{Hom}(E_1, E_2) φ , ψ ∈ Hom ( E 1 , E 2 ) に ⟨ φ , ψ ⟩ = deg ( φ + ψ ) − deg φ − deg ψ \langle \varphi, \psi \rangle = \deg(\varphi + \psi) - \deg \varphi - \deg \psi ⟨ φ , ψ ⟩ = deg ( φ + ψ ) − deg φ − deg ψ Z \mathbb{Z} Z 双線形な ⟨ φ , φ ⟩ = 2 deg φ \langle \varphi, \varphi \rangle = 2 \deg \varphi ⟨ φ , φ ⟩ = 2 deg φ deg φ ≥ 0 \deg \varphi \geq 0 deg φ ≥ 0 φ = 0 \varphi = 0 φ = 0 の
∣ deg ( φ − ψ ) − deg φ − deg ψ ∣ ≤ 2 deg φ ⋅ deg ψ \lvert \deg(\varphi - \psi) - \deg \varphi - \deg \psi \rvert \leq 2\sqrt{\deg \varphi \cdot \deg \psi} ∣ deg ( φ − ψ ) − deg φ − deg ψ ∣ ≤ 2 deg φ ⋅ deg ψ
証明. 定理 3.14(3) と
[ ⟨ φ , ψ ⟩ ] = ( φ + ψ ) ^ ∘ ( φ + ψ ) − φ ^ ∘ φ − ψ ^ ∘ ψ = φ ^ ∘ ψ + ψ ^ ∘ φ [\langle \varphi, \psi \rangle] = \widehat{(\varphi + \psi)} \circ (\varphi + \psi) - \hat{\varphi} \circ \varphi - \hat{\psi} \circ \psi = \hat{\varphi} \circ \psi + \hat{\psi} \circ \varphi [⟨ φ , ψ ⟩] = ( φ + ψ ) ∘ ( φ + ψ ) − φ ^ ∘ φ − ψ ^ ∘ ψ = φ ^ ∘ ψ + ψ ^ ∘ φ 右辺は φ \varphi φ ψ \psi ψ の n ↦ [ n ] n \mapsto [n] n ↦ [ n ] は ⟨ ⋅ , ⋅ ⟩ \langle \cdot, \cdot \rangle ⟨ ⋅ , ⋅ ⟩ は ⟨ φ , φ ⟩ = deg ( [ 2 ] ∘ φ ) − 2 deg φ = 2 deg φ \langle \varphi, \varphi \rangle = \deg([2] \circ \varphi) - 2\deg \varphi = 2\deg \varphi ⟨ φ , φ ⟩ = deg ([ 2 ] ∘ φ ) − 2 deg φ = 2 deg φ m , n m, n m , n に
0 ≤ deg ( m φ − n ψ ) = m 2 deg φ − m n ⟨ φ , ψ ⟩ + n 2 deg ψ 0 \leq \deg(m\varphi - n\psi) = m^2 \deg \varphi - mn \langle \varphi, \psi \rangle + n^2 \deg \psi 0 ≤ deg ( m φ − n ψ ) = m 2 deg φ − mn ⟨ φ , ψ ⟩ + n 2 deg ψ deg ψ > 0 \deg \psi > 0 deg ψ > 0 と n 2 n^2 n 2 で r r r に deg φ ⋅ r 2 − ⟨ φ , ψ ⟩ r + deg ψ ≥ 0 \deg \varphi \cdot r^2 - \langle \varphi, \psi \rangle r + \deg \psi \geq 0 deg φ ⋅ r 2 − ⟨ φ , ψ ⟩ r + deg ψ ≥ 0 で、連続性から r r r でも deg φ > 0 \deg \varphi > 0 deg φ > 0 なら 0 0 0 deg φ = 0 \deg \varphi = 0 deg φ = 0 なら 1 次以下の 0 0 0 ⟨ φ , ψ ⟩ = 0 \langle \varphi, \psi \rangle = 0 ⟨ φ , ψ ⟩ = 0 。いずれに ⟨ φ , ψ ⟩ 2 ≤ 4 deg φ deg ψ \langle \varphi, \psi \rangle^2 \leq 4 \deg \varphi \deg \psi ⟨ φ , ψ ⟩ 2 ≤ 4 deg φ deg ψ deg ( φ − ψ ) − deg φ − deg ψ = − ⟨ φ , ψ ⟩ \deg(\varphi - \psi) - \deg \varphi - \deg \psi = -\langle \varphi, \psi \rangle deg ( φ − ψ ) − deg φ − deg ψ = − ⟨ φ , ψ ⟩ から □ \square □
第4章 の φ = [ 1 ] \varphi = [1] φ = [ 1 ] ψ = ϕ q \psi = \phi_q ψ = ϕ q に
3.5 等分 [ m ] [m] [ m ] を ≠ 2 , 3 \neq 2, 3 = 2 , 3 E : y 2 = x 3 + A x + B E\colon y^2 = x^3 + Ax + B E : y 2 = x 3 + A x + B と
定義 3.17 （等分 A , B , x , y A, B, x, y A , B , x , y の ψ m \psi_m ψ m m ≥ 0 m \geq 0 m ≥ 0
ψ 0 = 0 , ψ 1 = 1 , ψ 2 = 2 y , ψ 3 = 3 x 4 + 6 A x 2 + 12 B x − A 2 , ψ 4 = 4 y ( x 6 + 5 A x 4 + 20 B x 3 − 5 A 2 x 2 − 4 A B x − 8 B 2 − A 3 ) , ψ 2 m + 1 = ψ m + 2 ψ m 3 − ψ m − 1 ψ m + 1 3 ( m ≥ 2 ) , ψ 2 m = ψ m 2 y ( ψ m + 2 ψ m − 1 2 − ψ m − 2 ψ m + 1 2 ) ( m ≥ 3 ) \begin{aligned}
\psi_0 &= 0, \quad \psi_1 = 1, \quad \psi_2 = 2y, \quad \psi_3 = 3x^4 + 6Ax^2 + 12Bx - A^2, \\
\psi_4 &= 4y\,(x^6 + 5Ax^4 + 20Bx^3 - 5A^2x^2 - 4ABx - 8B^2 - A^3), \\
\psi_{2m+1} &= \psi_{m+2}\psi_m^3 - \psi_{m-1}\psi_{m+1}^3 \quad (m \geq 2), \\
\psi_{2m} &= \frac{\psi_m}{2y}\bigl(\psi_{m+2}\psi_{m-1}^2 - \psi_{m-2}\psi_{m+1}^2\bigr) \quad (m \geq 3)
\end{aligned} ψ 0 ψ 4 ψ 2 m + 1 ψ 2 m = 0 , ψ 1 = 1 , ψ 2 = 2 y , ψ 3 = 3 x 4 + 6 A x 2 + 12 B x − A 2 , = 4 y ( x 6 + 5 A x 4 + 20 B x 3 − 5 A 2 x 2 − 4 A B x − 8 B 2 − A 3 ) , = ψ m + 2 ψ m 3 − ψ m − 1 ψ m + 1 3 ( m ≥ 2 ) , = 2 y ψ m ( ψ m + 2 ψ m − 1 2 − ψ m − 2 ψ m + 1 2 ) ( m ≥ 3 ) で y 2 y^2 y 2 x 3 + A x + B x^3 + Ax + B x 3 + A x + B に ψ − 1 = − 1 \psi_{-1} = -1 ψ − 1 = − 1 と
ϕ m = x ψ m 2 − ψ m + 1 ψ m − 1 , ω m = ψ m + 2 ψ m − 1 2 − ψ m − 2 ψ m + 1 2 4 y ( m ≥ 1 ) \phi_m = x\psi_m^2 - \psi_{m+1}\psi_{m-1}, \qquad \omega_m = \frac{\psi_{m+2}\psi_{m-1}^2 - \psi_{m-2}\psi_{m+1}^2}{4y} \qquad (m \geq 1) ϕ m = x ψ m 2 − ψ m + 1 ψ m − 1 , ω m = 4 y ψ m + 2 ψ m − 1 2 − ψ m − 2 ψ m + 1 2 ( m ≥ 1 ) と ψ m \psi_m ψ m m m m 等分 と
記号は Washington, Elliptic Curves: Number Theory and Cryptography の 3.2 節に m m m ϕ m \phi_m ϕ m q q q の ϕ q \phi_q ϕ q とは m = p m = p m = p のように
補題 3.18 (1) m m m が ψ m ∈ Z [ A , B , x ] \psi_m \in \mathbb{Z}[A, B, x] ψ m ∈ Z [ A , B , x ] ψ m ∈ 2 y Z [ A , B , x ] \psi_m \in 2y\mathbb{Z}[A, B, x] ψ m ∈ 2 y Z [ A , B , x ] ψ m 2 \psi_m^2 ψ m 2 ϕ m \phi_m ϕ m x x x だけの x x x の ϕ m = x m 2 + ( 低次 ) \phi_m = x^{m^2} + (\text{低次}) ϕ m = x m 2 + ( 低次 ) ψ m 2 = m 2 x m 2 − 1 + ( 低次 ) \psi_m^2 = m^2x^{m^2 - 1} + (\text{低次}) ψ m 2 = m 2 x m 2 − 1 + ( 低次 )
証明. (1) 漸化式に ψ 0 , … , ψ 4 \psi_0, \dots, \psi_4 ψ 0 , … , ψ 4 は ψ 2 m + 1 \psi_{2m+1} ψ 2 m + 1 の m m m が ψ m + 2 ψ m 3 \psi_{m+2}\psi_m^3 ψ m + 2 ψ m 3 ψ m − 1 ψ m + 1 3 \psi_{m-1}\psi_{m+1}^3 ψ m − 1 ψ m + 1 3 ）は ( 2 y ) 4 (2y)^4 ( 2 y ) 4 の y y y を y 2 y^2 y 2 x 3 + A x + B x^3 + Ax + B x 3 + A x + B に Z [ A , B , x ] \mathbb{Z}[A, B, x] Z [ A , B , x ] に ψ 2 m \psi_{2m} ψ 2 m の m m m が ψ m / ( 2 y ) ∈ Z [ A , B , x ] \psi_m/(2y) \in \mathbb{Z}[A, B, x] ψ m / ( 2 y ) ∈ Z [ A , B , x ] で ψ m ± 2 \psi_{m \pm 2} ψ m ± 2 を 2 y 2y 2 y の m m m が ψ m ∓ 1 2 \psi_{m \mp 1}^2 ψ m ∓ 1 2 を 4 y 2 4y^2 4 y 2 の ψ 2 m ∈ 2 y Z [ A , B , x ] \psi_{2m} \in 2y\mathbb{Z}[A, B, x] ψ 2 m ∈ 2 y Z [ A , B , x ] m ≤ 16 m \leq 16 m ≤ 16 では x x x に y y y に 3 / 2 3/2 3/2 を y 2 = x 3 + ⋯ y^2 = x^3 + \cdots y 2 = x 3 + ⋯ の ψ m \psi_m ψ m の m x ( m 2 − 1 ) / 2 m x^{(m^2 - 1)/2} m x ( m 2 − 1 ) /2 m m m m y x ( m 2 − 4 ) / 2 m y x^{(m^2 - 4)/2} m y x ( m 2 − 4 ) /2 m m m ( m 2 − 1 ) / 2 (m^2 - 1)/2 ( m 2 − 1 ) /2 」を n n n に ( n 2 − 1 ) / 2 (n^2 - 1)/2 ( n 2 − 1 ) /2 に ( m + 2 ) 2 − 1 2 + 3 ⋅ m 2 − 1 2 = ( 2 m + 1 ) 2 − 1 2 \frac{(m+2)^2 - 1}{2} + 3 \cdot \frac{m^2 - 1}{2} = \frac{(2m + 1)^2 - 1}{2} 2 ( m + 2 ) 2 − 1 + 3 ⋅ 2 m 2 − 1 = 2 ( 2 m + 1 ) 2 − 1 ）、係数は
( m + 2 ) m 3 − ( m − 1 ) ( m + 1 ) 3 = 2 m + 1 , m 2 ( ( m + 2 ) ( m − 1 ) 2 − ( m − 2 ) ( m + 1 ) 2 ) = 2 m (m + 2)m^3 - (m - 1)(m + 1)^3 = 2m + 1, \qquad \frac{m}{2}\bigl((m + 2)(m - 1)^2 - (m - 2)(m + 1)^2\bigr) = 2m ( m + 2 ) m 3 − ( m − 1 ) ( m + 1 ) 3 = 2 m + 1 , 2 m ( ( m + 2 ) ( m − 1 ) 2 − ( m − 2 ) ( m + 1 ) 2 ) = 2 m で ψ m 2 \psi_m^2 ψ m 2 の m 2 x m 2 − 1 m^2 x^{m^2 - 1} m 2 x m 2 − 1 ϕ m \phi_m ϕ m x m 2 x^{m^2} x m 2 の m 2 − ( m + 1 ) ( m − 1 ) = 1 m^2 - (m + 1)(m - 1) = 1 m 2 − ( m + 1 ) ( m − 1 ) = 1 □ \square □
定理 3.19 （[ m ] [m] [ m ] の m ≥ 1 m \geq 1 m ≥ 1 P = ( x , y ) ∈ E P = (x, y) \in E P = ( x , y ) ∈ E に
[ m ] P = ( ϕ m ( x ) ψ m ( x ) 2 , ω m ( x , y ) ψ m ( x , y ) 3 ) [m]P = \left( \frac{\phi_m(x)}{\psi_m(x)^2},\ \frac{\omega_m(x, y)}{\psi_m(x, y)^3} \right) [ m ] P = ( ψ m ( x ) 2 ϕ m ( x ) , ψ m ( x , y ) 3 ω m ( x , y ) )
参照：Washington 3.2 節（証明は ℘ \wp ℘ σ \sigma σ 関数を
例 3.20 m = 2 m = 2 m = 2 λ = ( 3 x 2 + A ) / ( 2 y ) \lambda = (3x^2 + A)/(2y) λ = ( 3 x 2 + A ) / ( 2 y ) と x ( 2 P ) = λ 2 − 2 x x(2P) = \lambda^2 - 2x x ( 2 P ) = λ 2 − 2 x の ( 3 x 2 + A ) 2 − 8 x ( x 3 + A x + B ) (3x^2 + A)^2 - 8x(x^3 + Ax + B) ( 3 x 2 + A ) 2 − 8 x ( x 3 + A x + B ) ϕ 2 = 4 x ( x 3 + A x + B ) − ψ 3 \phi_2 = 4x(x^3 + Ax + B) - \psi_3 ϕ 2 = 4 x ( x 3 + A x + B ) − ψ 3 が x 4 − 2 A x 2 − 8 B x + A 2 x^4 - 2Ax^2 - 8Bx + A^2 x 4 − 2 A x 2 − 8 B x + A 2 ψ 2 2 = 4 y 2 \psi_2^2 = 4y^2 ψ 2 2 = 4 y 2 A , B A, B A , B を m = 2 m = 2 m = 2 の [ m ] P [m]P [ m ] P の P P P を [ m + 1 ] P [m + 1]P [ m + 1 ] P の 2 ≤ m ≤ 7 2 \leq m \leq 7 2 ≤ m ≤ 7 で m ≤ 8 m \leq 8 m ≤ 8 では p p p F p \mathbb{F}_p F p 上の m ≤ 40 m \leq 40 m ≤ 40 [ m ] P [m]P [ m ] P と
系 3.21 (1) ϕ m \phi_m ϕ m ψ m 2 \psi_m^2 ψ m 2 x x x の p ∤ m p \nmid m p ∤ m [ m ] [m] [ m ] は ∣ E [ m ] ∣ = m 2 \lvert E[m] \rvert = m^2 ∣ E [ m ]∣ = m 2 ψ m 2 = m 2 ∏ P ∈ E [ m ] ∖ { O } ( x − x ( P ) ) \psi_m^2 = m^2 \prod_{P \in E[m] \setminus \lbrace O \rbrace} (x - x(P)) ψ m 2 = m 2 ∏ P ∈ E [ m ] ∖ { O } ( x − x ( P ))
証明. (1) 定理 3.19 より x ∘ [ m ] = ϕ m / ψ m 2 x \circ [m] = \phi_m/\psi_m^2 x ∘ [ m ] = ϕ m / ψ m 2 なので、系 3.11 と deg [ m ] \deg [m] deg [ m ] ϕ m / ψ m 2 \phi_m/\psi_m^2 ϕ m / ψ m 2 を deg ϕ m = m 2 > deg ψ m 2 \deg \phi_m = m^2 > \deg \psi_m^2 deg ϕ m = m 2 > deg ψ m 2 δ > 0 \delta > 0 δ > 0 の deg [ m ] = m 2 − δ \deg[m] = m^2 - \delta deg [ m ] = m 2 − δ と
(2) ( ϕ m / ψ m 2 ) ′ (\phi_m/\psi_m^2)' ( ϕ m / ψ m 2 ) ′ の ϕ m ′ ψ m 2 − ϕ m ( ψ m 2 ) ′ \phi_m'\psi_m^2 - \phi_m(\psi_m^2)' ϕ m ′ ψ m 2 − ϕ m ( ψ m 2 ) ′ x 2 m 2 − 2 x^{2m^2 - 2} x 2 m 2 − 2 の m 2 ⋅ m 2 − ( m 2 − 1 ) m 2 = m 2 ≠ 0 m^2 \cdot m^2 - (m^2 - 1)m^2 = m^2 \neq 0 m 2 ⋅ m 2 − ( m 2 − 1 ) m 2 = m 2 = 0 [ m ] [m] [ m ] は ∣ E [ m ] ∣ = m 2 \lvert E[m] \rvert = m^2 ∣ E [ m ]∣ = m 2 x ∘ [ m ] x \circ [m] x ∘ [ m ] の [ m ] − 1 ( O ) = E [ m ] [m]^{-1}(O) = E[m] [ m ] − 1 ( O ) = E [ m ] の P ∈ E [ m ] ∖ { O } P \in E[m] \setminus \lbrace O \rbrace P ∈ E [ m ] ∖ { O } ϕ m ( x ( P ) ) ≠ 0 \phi_m(x(P)) \neq 0 ϕ m ( x ( P )) = 0 ord P ψ m ( x ) 2 = 2 \operatorname{ord}_P \psi_m(x)^2 = 2 ord P ψ m ( x ) 2 = 2 ord P ( x − x ( P ) ) \operatorname{ord}_P(x - x(P)) ord P ( x − x ( P )) 2 P ≠ O 2P \neq O 2 P = O 2 P = O 2P = O 2 P = O x ( P ) x(P) x ( P ) ψ m 2 \psi_m^2 ψ m 2 の ± P \pm P ± P が ψ m 2 \psi_m^2 ψ m 2 の x ∘ [ m ] x \circ [m] x ∘ [ m ] が E [ m ] E[m] E [ m ] に m 2 − 1 m^2 - 1 m 2 − 1 と m 2 m^2 m 2 を □ \square □
2 ≤ m ≤ 5 2 \leq m \leq 5 2 ≤ m ≤ 5 ϕ m \phi_m ϕ m ψ m 2 \psi_m^2 ψ m 2 の Δ m 2 ( m 2 − 1 ) / 6 \Delta^{m^2(m^2 - 1)/6} Δ m 2 ( m 2 − 1 ) /6 Δ = − 16 ( 4 A 3 + 27 B 2 ) \Delta = -16(4A^3 + 27B^2) Δ = − 16 ( 4 A 3 + 27 B 2 ) ）に deg [ m ] = m 2 \deg [m] = m^2 deg [ m ] = m 2 を
3.6 等分点の
定理 3.23 p ∤ m p \nmid m p ∤ m E [ m ] ≅ ( Z / m Z ) 2 E[m] \cong (\mathbb{Z}/m\mathbb{Z})^2 E [ m ] ≅ ( Z / m Z ) 2
証明. 系 3.21(2)（標数 2, 3 では m m m の d d d に ∣ E [ d ] ∣ = d 2 \lvert E[d] \rvert = d^2 ∣ E [ d ]∣ = d 2 E [ m ] E[m] E [ m ] m m m で E [ m ] ≅ Z / n 1 Z ⊕ ⋯ ⊕ Z / n k Z E[m] \cong \mathbb{Z}/n_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/n_k\mathbb{Z} E [ m ] ≅ Z / n 1 Z ⊕ ⋯ ⊕ Z / n k Z 1 < n 1 ∣ ⋯ ∣ n k ∣ m 1 < n_1 \mid \cdots \mid n_k \mid m 1 < n 1 ∣ ⋯ ∣ n k ∣ m 04-algebra 第7章 定理 7.16）と m > 1 m > 1 m > 1 と ℓ ∣ n 1 \ell \mid n_1 ℓ ∣ n 1 を E [ ℓ ] ≅ ( Z / ℓ Z ) k E[\ell] \cong (\mathbb{Z}/\ell\mathbb{Z})^k E [ ℓ ] ≅ ( Z / ℓ Z ) k の ℓ 2 \ell^2 ℓ 2 k = 2 k = 2 k = 2 n 1 n 2 = m 2 n_1n_2 = m^2 n 1 n 2 = m 2 n i ∣ m n_i \mid m n i ∣ m n 1 = n 2 = m n_1 = n_2 = m n 1 = n 2 = m □ \square □
K = C K = \mathbb{C} K = C E ( C ) ≅ C / Λ E(\mathbb{C}) \cong \mathbb{C}/\Lambda E ( C ) ≅ C /Λ 19-riemann-surfaces 第2章 ）で、E [ m ] = 1 m Λ / Λ E[m] = \frac{1}{m}\Lambda/\Lambda E [ m ] = m 1 Λ/Λ である。定理 3.23 は、これが
定理 3.24 （主張）標数 p > 0 p > 0 p > 0 と e ≥ 1 e \geq 1 e ≥ 1 E [ p e ] = { O } E[p^e] = \lbrace O \rbrace E [ p e ] = { O } と e ≥ 1 e \geq 1 e ≥ 1 E [ p e ] ≅ Z / p e Z E[p^e] \cong \mathbb{Z}/p^e\mathbb{Z} E [ p e ] ≅ Z / p e Z と
参照：AEC III.6.4。証明の ∣ E [ p e ] ∣ = deg s [ p e ] = ( deg s [ p ] ) e \lvert E[p^e] \rvert = \deg_s [p^e] = (\deg_s [p])^e ∣ E [ p e ]∣ = deg s [ p e ] = ( deg s [ p ] ) e [ p ] = ϕ ^ p ∘ ϕ p [p] = \hat{\phi}_p \circ \phi_p [ p ] = ϕ ^ p ∘ ϕ p ϕ p \phi_p ϕ p は deg s [ p ] = deg s ϕ ^ p \deg_s [p] = \deg_s \hat{\phi}_p deg s [ p ] = deg s ϕ ^ p deg ϕ ^ p = p \deg \hat{\phi}_p = p deg ϕ ^ p = p の 1 1 1 p p p p p p ∣ E [ p e ] ∣ = p e \lvert E[p^e] \rvert = p^e ∣ E [ p e ]∣ = p e p p p E [ p ] E[p] E [ p ] が p p p E [ p e ] E[p^e] E [ p e ] も □ \square □
前者の (ordinary) と 第4章 定義 4.16・定理 4.17）。
例 3.25 補題 3.18(1) より ψ m \psi_m ψ m は m o d p \bmod p mod p で p p p の y 2 = x 3 + x + 1 y^2 = x^3 + x + 1 y 2 = x 3 + x + 1 ψ 5 ≡ 2 ( x 2 + 2 x + 4 ) 5 ( m o d 5 ) \psi_5 \equiv 2(x^2 + 2x + 4)^5 \pmod 5 ψ 5 ≡ 2 ( x 2 + 2 x + 4 ) 5 ( mod 5 ) y 2 = x 3 + 1 y^2 = x^3 + 1 y 2 = x 3 + 1 ψ 5 ≡ 4 ( m o d 5 ) \psi_5 \equiv 4 \pmod 5 ψ 5 ≡ 4 ( mod 5 ) ψ 7 ≡ 3 ( x 3 + 3 ) 7 ( m o d 7 ) \psi_7 \equiv 3(x^3 + 3)^7 \pmod 7 ψ 7 ≡ 3 ( x 3 + 3 ) 7 ( mod 7 ) y 2 = x 3 − x y^2 = x^3 - x y 2 = x 3 − x ψ 7 ≡ 6 ( m o d 7 ) \psi_7 \equiv 6 \pmod 7 ψ 7 ≡ 6 ( mod 7 ) である。系 3.21 の x ∘ [ p ] x \circ [p] x ∘ [ p ] の 2 deg i [ p ] 2\deg_i[p] 2 deg i [ p ] E [ p ] ∖ { O } E[p] \setminus \lbrace O \rbrace E [ p ] ∖ { O } の x x x 座標は ψ p \psi_p ψ p の deg i [ p ] \deg_i [p] deg i [ p ] である。たとえば x 2 + 2 x + 4 x^2 + 2x + 4 x 2 + 2 x + 4 の 2 根が 4 点を ∣ E [ 5 ] ∣ = 5 \lvert E[5] \rvert = 5 ∣ E [ 5 ]∣ = 5 と deg i [ 5 ] = 5 \deg_i [5] = 5 deg i [ 5 ] = 5 を ψ p \psi_p ψ p が E [ p ] = { O } E[p] = \lbrace O \rbrace E [ p ] = { O } （超
3.7 2-同種写像と 位数 2 の ( 0 , 0 ) (0, 0) ( 0 , 0 ) を 第8章 の
命題 3.26 （2-同種写像）標数 ≠ 2 \neq 2 = 2 a , b ∈ K a, b \in K a , b ∈ K b ( a 2 − 4 b ) ≠ 0 b(a^2 - 4b) \neq 0 b ( a 2 − 4 b ) = 0 E : y 2 = x ( x 2 + a x + b ) E\colon y^2 = x(x^2 + ax + b) E : y 2 = x ( x 2 + a x + b ) E ′ : Y 2 = X ( X 2 − 2 a X + a 2 − 4 b ) E'\colon Y^2 = X(X^2 - 2aX + a^2 - 4b) E ′ : Y 2 = X ( X 2 − 2 a X + a 2 − 4 b ) と
φ ( x , y ) = ( y 2 x 2 , y ( b − x 2 ) x 2 ) , φ ^ ( X , Y ) = ( Y 2 4 X 2 , Y ( a 2 − 4 b − X 2 ) 8 X 2 ) \varphi(x, y) = \left( \frac{y^2}{x^2},\ \frac{y(b - x^2)}{x^2} \right), \qquad \hat{\varphi}(X, Y) = \left( \frac{Y^2}{4X^2},\ \frac{Y(a^2 - 4b - X^2)}{8X^2} \right) φ ( x , y ) = ( x 2 y 2 , x 2 y ( b − x 2 ) ) , φ ^ ( X , Y ) = ( 4 X 2 Y 2 , 8 X 2 Y ( a 2 − 4 b − X 2 ) ) は φ : E → E ′ \varphi\colon E \to E' φ : E → E ′ φ ^ : E ′ → E \hat{\varphi}\colon E' \to E φ ^ : E ′ → E ker φ \ker \varphi ker φ ker φ ^ \ker \hat{\varphi} ker φ ^ は { O , ( 0 , 0 ) } \lbrace O, (0, 0) \rbrace { O , ( 0 , 0 )} φ ^ ∘ φ = [ 2 ] E \hat{\varphi} \circ \varphi = [2]_E φ ^ ∘ φ = [ 2 ] E φ ∘ φ ^ = [ 2 ] E ′ \varphi \circ \hat{\varphi} = [2]_{E'} φ ∘ φ ^ = [ 2 ] E ′ が
証明. 2 つの 3 次式の b 2 ( a 2 − 4 b ) b^2(a^2 - 4b) b 2 ( a 2 − 4 b ) 16 b ( a 2 − 4 b ) 2 16b(a^2 - 4b)^2 16 b ( a 2 − 4 b ) 2 0 0 0 φ \varphi φ E ′ E' E ′ に X = y 2 / x 2 = ( x 2 + a x + b ) / x X = y^2/x^2 = (x^2 + ax + b)/x X = y 2 / x 2 = ( x 2 + a x + b ) / x X − a = ( x 2 + b ) / x X - a = (x^2 + b)/x X − a = ( x 2 + b ) / x
X 2 − 2 a X + a 2 − 4 b = ( X − a ) 2 − 4 b = ( x 2 − b ) 2 x 2 , X ( X 2 − 2 a X + a 2 − 4 b ) = y 2 ( x 2 − b ) 2 x 4 = Y 2 X^2 - 2aX + a^2 - 4b = (X - a)^2 - 4b = \frac{(x^2 - b)^2}{x^2}, \qquad X(X^2 - 2aX + a^2 - 4b) = \frac{y^2(x^2 - b)^2}{x^4} = Y^2 X 2 − 2 a X + a 2 − 4 b = ( X − a ) 2 − 4 b = x 2 ( x 2 − b ) 2 , X ( X 2 − 2 a X + a 2 − 4 b ) = x 4 y 2 ( x 2 − b ) 2 = Y 2 E ′ E' E ′ ( a , b ) (a, b) ( a , b ) ( − 2 a , a 2 − 4 b ) (-2a, a^2 - 4b) ( − 2 a , a 2 − 4 b ) に E ′ E' E ′ に y 2 = x ( x 2 + 4 a x + 16 b ) y^2 = x(x^2 + 4ax + 16b) y 2 = x ( x 2 + 4 a x + 16 b ) に ( x , y ) ↦ ( x / 4 , y / 8 ) (x, y) \mapsto (x/4, y/8) ( x , y ) ↦ ( x /4 , y /8 ) E E E に φ ^ \hat{\varphi} φ ^ 次数と x ∘ φ = ( x 2 + a x + b ) / x x \circ \varphi = (x^2 + ax + b)/x x ∘ φ = ( x 2 + a x + b ) / x は 1 − b / x 2 ≠ 0 1 - b/x^2 \neq 0 1 − b / x 2 = 0 だから、補題 3.4 より x ∘ φ x \circ \varphi x ∘ φ O O O ( 0 , 0 ) (0, 0) ( 0 , 0 ) で φ ^ \hat{\varphi} φ ^ も
φ ^ ∘ φ = [ 2 ] \hat{\varphi} \circ \varphi = [2] φ ^ ∘ φ = [ 2 ] φ ^ ( φ ( P ) ) \hat{\varphi}(\varphi(P)) φ ^ ( φ ( P )) x x x Y 2 4 X 2 = y 2 ( b − x 2 ) 2 / x 4 4 y 4 / x 4 = ( x 2 − b ) 2 4 y 2 \dfrac{Y^2}{4X^2} = \dfrac{y^2(b - x^2)^2/x^4}{4y^4/x^4} = \dfrac{(x^2 - b)^2}{4y^2} 4 X 2 Y 2 = 4 y 4 / x 4 y 2 ( b − x 2 ) 2 / x 4 = 4 y 2 ( x 2 − b ) 2 λ = ( 3 x 2 + 2 a x + b ) / ( 2 y ) \lambda = (3x^2 + 2ax + b)/(2y) λ = ( 3 x 2 + 2 a x + b ) / ( 2 y ) と x ( 2 P ) = λ 2 − a − 2 x x(2P) = \lambda^2 - a - 2x x ( 2 P ) = λ 2 − a − 2 x の ( 3 x 2 + 2 a x + b ) 2 − 4 ( a + 2 x ) ( x 3 + a x 2 + b x ) = ( x 2 − b ) 2 (3x^2 + 2ax + b)^2 - 4(a + 2x)(x^3 + ax^2 + bx) = (x^2 - b)^2 ( 3 x 2 + 2 a x + b ) 2 − 4 ( a + 2 x ) ( x 3 + a x 2 + b x ) = ( x 2 − b ) 2 で P P P φ ^ ( φ ( P ) ) = ± 2 P \hat{\varphi}(\varphi(P)) = \pm 2P φ ^ ( φ ( P )) = ± 2 P y y y 座標の ρ = y ( φ ^ φ ( P ) ) / y ( 2 P ) ∈ K ‾ ( x ) \rho = y(\hat{\varphi}\varphi(P))/y(2P) \in \overline{K}(x) ρ = y ( φ ^ φ ( P )) / y ( 2 P ) ∈ K ( x ) は ± 1 \pm 1 ± 1 を ± 1 \pm 1 ± 1 f = x 3 + a x 2 + b x f = x^3 + ax^2 + bx f = x 3 + a x 2 + b x と y y y で y y y x → ∞ x \to \infty x → ∞ で
y ( φ ^ φ ( P ) ) y = ( b − x 2 ) ( ( a 2 − 4 b ) x 4 − f 2 ) 8 x 2 f 2 → 1 8 , y ( 2 P ) y = 3 x 2 + 2 a x + b 2 f ( x − ( x 2 − b ) 2 4 f ) − 1 → 9 8 − 1 = 1 8 \frac{y(\hat{\varphi}\varphi(P))}{y} = \frac{(b - x^2)\bigl((a^2 - 4b)x^4 - f^2\bigr)}{8x^2f^2} \to \frac{1}{8}, \qquad \frac{y(2P)}{y} = \frac{3x^2 + 2ax + b}{2f}\left(x - \frac{(x^2 - b)^2}{4f}\right) - 1 \to \frac{9}{8} - 1 = \frac{1}{8} y y ( φ ^ φ ( P )) = 8 x 2 f 2 ( b − x 2 ) ( ( a 2 − 4 b ) x 4 − f 2 ) → 8 1 , y y ( 2 P ) = 2 f 3 x 2 + 2 a x + b ( x − 4 f ( x 2 − b ) 2 ) − 1 → 8 9 − 1 = 8 1 なので ρ = 1 \rho = 1 ρ = 1 φ ∘ φ ^ ∘ φ = [ 2 ] ∘ φ \varphi \circ \hat{\varphi} \circ \varphi = [2] \circ \varphi φ ∘ φ ^ ∘ φ = [ 2 ] ∘ φ φ \varphi φ の φ ∘ φ ^ = [ 2 ] \varphi \circ \hat{\varphi} = [2] φ ∘ φ ^ = [ 2 ] □ \square □
y y y 座標を φ ^ \hat{\varphi} φ ^ φ \varphi φ の
例 3.27 E : y 2 = x 3 − 2 x E\colon y^2 = x^3 - 2x E : y 2 = x 3 − 2 x a = 0 a = 0 a = 0 b = − 2 b = -2 b = − 2 E ′ : Y 2 = X 3 + 8 X E'\colon Y^2 = X^3 + 8X E ′ : Y 2 = X 3 + 8 X P = ( 2 , 2 ) P = (2, 2) P = ( 2 , 2 ) に φ ( P ) = ( 1 , − 3 ) \varphi(P) = (1, -3) φ ( P ) = ( 1 , − 3 ) 9 = 1 + 8 9 = 1 + 8 9 = 1 + 8 φ ^ ( 1 , − 3 ) = ( 9 / 4 , − 21 / 8 ) \hat{\varphi}(1, -3) = (9/4, -21/8) φ ^ ( 1 , − 3 ) = ( 9/4 , − 21/8 ) λ = 10 / 4 \lambda = 10/4 λ = 10/4 x ( 2 P ) = 25 / 4 − 4 = 9 / 4 x(2P) = 25/4 - 4 = 9/4 x ( 2 P ) = 25/4 − 4 = 9/4 y ( 2 P ) = 5 2 ( 2 − 9 4 ) − 2 = − 21 8 y(2P) = \frac{5}{2}(2 - \frac{9}{4}) - 2 = -\frac{21}{8} y ( 2 P ) = 2 5 ( 2 − 4 9 ) − 2 = − 8 21 φ ^ ( φ ( P ) ) = 2 P \hat{\varphi}(\varphi(P)) = 2P φ ^ ( φ ( P )) = 2 P が
定理 3.28 （ヴェルの ≠ 2 , 3 \neq 2, 3 = 2 , 3 E : y 2 = x 3 + A x + B E\colon y^2 = x^3 + Ax + B E : y 2 = x 3 + A x + B G ⊂ E G \subset E G ⊂ E を G 2 = { Q ∈ G ∣ Q ≠ O , 2 Q = O } G_2 = \lbrace Q \in G \mid Q \neq O,\ 2Q = O \rbrace G 2 = { Q ∈ G ∣ Q = O , 2 Q = O } G ∖ ( G 2 ∪ { O } ) = R ⊔ ( − R ) G \setminus (G_2 \cup \lbrace O \rbrace) = R \sqcup (-R) G ∖ ( G 2 ∪ { O }) = R ⊔ ( − R ) と R R R を S = G 2 ∪ R S = G_2 \cup R S = G 2 ∪ R と Q = ( x Q , y Q ) ∈ S Q = (x_Q, y_Q) \in S Q = ( x Q , y Q ) ∈ S に t Q = 3 x Q 2 + A t_Q = 3x_Q^2 + A t Q = 3 x Q 2 + A Q ∈ G 2 Q \in G_2 Q ∈ G 2 t Q = 2 ( 3 x Q 2 + A ) t_Q = 2(3x_Q^2 + A) t Q = 2 ( 3 x Q 2 + A ) Q ∈ R Q \in R Q ∈ R u Q = 4 y Q 2 u_Q = 4y_Q^2 u Q = 4 y Q 2 t = ∑ Q ∈ S t Q t = \sum_{Q \in S} t_Q t = ∑ Q ∈ S t Q w = ∑ Q ∈ S ( u Q + x Q t Q ) w = \sum_{Q \in S} (u_Q + x_Qt_Q) w = ∑ Q ∈ S ( u Q + x Q t Q ) と E ′ : Y 2 = X 3 + ( A − 5 t ) X + ( B − 7 w ) E'\colon Y^2 = X^3 + (A - 5t)X + (B - 7w) E ′ : Y 2 = X 3 + ( A − 5 t ) X + ( B − 7 w ) は
X = x + ∑ Q ∈ S ( t Q x − x Q + u Q ( x − x Q ) 2 ) , Y = y d X d x X = x + \sum_{Q \in S} \left( \frac{t_Q}{x - x_Q} + \frac{u_Q}{(x - x_Q)^2} \right), \qquad Y = y\,\frac{dX}{dx} X = x + Q ∈ S ∑ ( x − x Q t Q + ( x − x Q ) 2 u Q ) , Y = y d x d X は G G G の E → E ′ E \to E' E → E ′ ∣ G ∣ \lvert G \rvert ∣ G ∣ ）を
参照：Washington（第2版）の E ′ E' E ′ の X X X Y Y Y の ellisogeny の G G G の
例 3.29 (1) y 2 = x 3 − x y^2 = x^3 - x y 2 = x 3 − x G = { O , ( 0 , 0 ) } G = \lbrace O, (0, 0) \rbrace G = { O , ( 0 , 0 )} t = − 1 t = -1 t = − 1 w = 0 w = 0 w = 0 E ′ : Y 2 = X 3 + 4 X E'\colon Y^2 = X^3 + 4X E ′ : Y 2 = X 3 + 4 X X = x − 1 / x X = x - 1/x X = x − 1/ x a = 0 a = 0 a = 0 b = − 1 b = -1 b = − 1 φ \varphi φ Y Y Y の E : y 2 = x 3 + 1 E\colon y^2 = x^3 + 1 E : y 2 = x 3 + 1 ψ 3 = 3 x ( x 3 + 4 ) \psi_3 = 3x(x^3 + 4) ψ 3 = 3 x ( x 3 + 4 ) の ( 0 , ± 1 ) (0, \pm 1) ( 0 , ± 1 ) は G = { O , ( 0 , ± 1 ) } G = \lbrace O, (0, \pm 1) \rbrace G = { O , ( 0 , ± 1 )} R = { ( 0 , 1 ) } R = \lbrace (0, 1) \rbrace R = {( 0 , 1 )} t = 0 t = 0 t = 0 w = 4 w = 4 w = 4
E ′ : Y 2 = X 3 − 27 , ( x , y ) ↦ ( x 3 + 4 x 2 , y ( x 3 − 8 ) x 3 ) E'\colon Y^2 = X^3 - 27, \qquad (x, y) \mapsto \left( \frac{x^3 + 4}{x^2},\ \frac{y(x^3 - 8)}{x^3} \right) E ′ : Y 2 = X 3 − 27 , ( x , y ) ↦ ( x 2 x 3 + 4 , x 3 y ( x 3 − 8 ) ) （u = x 3 u = x^3 u = x 3 と ( u + 4 ) 3 − 27 u 2 = ( u + 1 ) ( u − 8 ) 2 (u + 4)^3 - 27u^2 = (u + 1)(u - 8)^2 ( u + 4 ) 3 − 27 u 2 = ( u + 1 ) ( u − 8 ) 2 なので E ′ E' E ′ に E E E E ′ E' E ′ は Cremona の = [ 0 , − 1 , 1 , − 10 , − 20 ] = [0, -1, 1, -10, -20] = [ 0 , − 1 , 1 , − 10 , − 20 ] を ( 5 , 5 ) (5, 5) ( 5 , 5 ) の = [ 0 , − 1 , 1 , − 7820 , − 263580 ] = [0, -1, 1, -7820, -263580] = [ 0 , − 1 , 1 , − 7820 , − 263580 ] に
3.8 ヴェイユ対m ≥ 2 m \geq 2 m ≥ 2 p ∤ m p \nmid m p ∤ m と T ∈ E [ m ] T \in E[m] T ∈ E [ m ] に m ( T ) − m ( O ) m(T) - m(O) m ( T ) − m ( O ) は m T = O mT = O m T = O 第2章 の div ( f ) = m ( T ) − m ( O ) \operatorname{div}(f) = m(T) - m(O) div ( f ) = m ( T ) − m ( O ) と f f f が m T ′ = T mT' = T m T ′ = T と T ′ T' T ′ を [ m ] [m] [ m ] は [ m ] ∗ ( T ) − [ m ] ∗ ( O ) = ∑ R ∈ E [ m ] ( ( T ′ + R ) − ( R ) ) [m]^{\ast}(T) - [m]^{\ast}(O) = \sum_{R \in E[m]} \bigl((T' + R) - (R)\bigr) [ m ] ∗ ( T ) − [ m ] ∗ ( O ) = ∑ R ∈ E [ m ] ( ( T ′ + R ) − ( R ) ) で、これも m 2 T ′ = O m^2T' = O m 2 T ′ = O div ( g ) \operatorname{div}(g) div ( g ) と f ∘ [ m ] f \circ [m] f ∘ [ m ] g m g^m g m は f ∘ [ m ] = g m f \circ [m] = g^m f ∘ [ m ] = g m と S ∈ E [ m ] S \in E[m] S ∈ E [ m ] g ( X + S ) m = f ( [ m ] X + [ m ] S ) = f ( [ m ] X ) = g ( X ) m g(X + S)^m = f([m]X + [m]S) = f([m]X) = g(X)^m g ( X + S ) m = f ([ m ] X + [ m ] S ) = f ([ m ] X ) = g ( X ) m g ( X + S ) / g ( X ) g(X + S)/g(X) g ( X + S ) / g ( X ) μ m \mu_m μ m に X X X の
定義 3.30 （ヴェイユ対, Weil pairing）e m ( S , T ) = g ( X + S ) / g ( X ) ∈ μ m e_m(S, T) = g(X + S)/g(X) \in \mu_m e m ( S , T ) = g ( X + S ) / g ( X ) ∈ μ m と e m : E [ m ] × E [ m ] → μ m e_m\colon E[m] \times E[m] \to \mu_m e m : E [ m ] × E [ m ] → μ m m m m 次の と
定理 3.31 （ヴェイユ対の
双線形：e m ( S 1 + S 2 , T ) = e m ( S 1 , T ) e m ( S 2 , T ) e_m(S_1 + S_2, T) = e_m(S_1, T) e_m(S_2, T) e m ( S 1 + S 2 , T ) = e m ( S 1 , T ) e m ( S 2 , T ) e m ( S , T 1 + T 2 ) = e m ( S , T 1 ) e m ( S , T 2 ) e_m(S, T_1 + T_2) = e_m(S, T_1) e_m(S, T_2) e m ( S , T 1 + T 2 ) = e m ( S , T 1 ) e m ( S , T 2 )
交代的：e m ( T , T ) = 1 e_m(T, T) = 1 e m ( T , T ) = 1 e m ( S , T ) = e m ( T , S ) − 1 e_m(S, T) = e_m(T, S)^{-1} e m ( S , T ) = e m ( T , S ) − 1
非退化：すべての S ∈ E [ m ] S \in E[m] S ∈ E [ m ] e m ( S , T ) = 1 e_m(S, T) = 1 e m ( S , T ) = 1 T = O T = O T = O
ガロア同変：E E E K K K 上 σ ∈ Gal ( K ‾ / K ) \sigma \in \operatorname{Gal}(\overline{K}/K) σ ∈ Gal ( K / K ) に e m ( S , T ) σ = e m ( S σ , T σ ) e_m(S, T)^{\sigma} = e_m(S^{\sigma}, T^{\sigma}) e m ( S , T ) σ = e m ( S σ , T σ ) P σ P^{\sigma} P σ は σ \sigma σ を
両立性：S ∈ E [ m m ′ ] S \in E[mm'] S ∈ E [ m m ′ ] T ∈ E [ m ] T \in E[m] T ∈ E [ m ] e m m ′ ( S , T ) = e m ( m ′ S , T ) e_{mm'}(S, T) = e_m(m'S, T) e m m ′ ( S , T ) = e m ( m ′ S , T )
参照：AEC III.8.1。文献に e m e_m e m の ellweilpairing(E, S, T, m) は e m ( T , S ) = e m ( S , T ) − 1 e_m(T, S) = e_m(S, T)^{-1} e m ( T , S ) = e m ( S , T ) − 1 を g g g を 第5章 例 5.17）。
系 3.32 (1) e m ( S , T ) e_m(S, T) e m ( S , T ) が 1 の m m m 乗根に S , T ∈ E [ m ] S, T \in E[m] S , T ∈ E [ m ] が E E E K K K 上 E [ m ] ⊂ E ( K ) E[m] \subset E(K) E [ m ] ⊂ E ( K ) μ m ⊂ K \mu_m \subset K μ m ⊂ K
証明. (1) 位数 m m m の T T T を S ↦ e m ( S , T ) S \mapsto e_m(S, T) S ↦ e m ( S , T ) の μ m \mu_m μ m の μ d \mu_d μ d d ∣ m d \mid m d ∣ m S S S e m ( S , d T ) = e m ( S , T ) d = 1 e_m(S, dT) = e_m(S, T)^d = 1 e m ( S , d T ) = e m ( S , T ) d = 1 d T = O dT = O d T = O d = m d = m d = m
(2) (1) の S , T S, T S , T ζ = e m ( S , T ) \zeta = e_m(S, T) ζ = e m ( S , T ) を S , T ∈ E ( K ) S, T \in E(K) S , T ∈ E ( K ) σ ∈ Gal ( K ‾ / K ) \sigma \in \operatorname{Gal}(\overline{K}/K) σ ∈ Gal ( K / K ) に σ ( ζ ) = e m ( S σ , T σ ) = ζ \sigma(\zeta) = e_m(S^{\sigma}, T^{\sigma}) = \zeta σ ( ζ ) = e m ( S σ , T σ ) = ζ ζ ∉ K \zeta \notin K ζ ∈ / K K K K は ζ \zeta ζ の ζ ′ \zeta' ζ ′ を σ ( ζ ) = ζ ′ \sigma(\zeta) = \zeta' σ ( ζ ) = ζ ′ と σ \sigma σ が ζ ∈ K \zeta \in K ζ ∈ K μ m = ⟨ ζ ⟩ ⊂ K \mu_m = \langle \zeta \rangle \subset K μ m = ⟨ ζ ⟩ ⊂ K □ \square □
例 3.33 (1) Q \mathbb{Q} Q の 1 の ± 1 \pm 1 ± 1 Q \mathbb{Q} Q 上の E [ m ] ⊂ E ( Q ) E[m] \subset E(\mathbb{Q}) E [ m ] ⊂ E ( Q ) と m ≤ 2 m \leq 2 m ≤ 2 に y 2 = x 3 − x y^2 = x^3 - x y 2 = x 3 − x E [ 2 ] = { O , ( 0 , 0 ) , ( ± 1 , 0 ) } ⊂ E ( Q ) E[2] = \lbrace O, (0, 0), (\pm 1, 0) \rbrace \subset E(\mathbb{Q}) E [ 2 ] = { O , ( 0 , 0 ) , ( ± 1 , 0 )} ⊂ E ( Q ) y 2 = x 3 + 1 y^2 = x^3 + 1 y 2 = x 3 + 1 の 3 等分点の Q \mathbb{Q} Q に Q ( 2 3 , ζ 3 ) \mathbb{Q}(\sqrt[3]{2}, \zeta_3) Q ( 3 2 , ζ 3 ) μ 3 \mu_3 μ 3 を F q \mathbb{F}_q F q p ∤ m p \nmid m p ∤ m の E [ m ] ⊂ E ( F q ) E[m] \subset E(\mathbb{F}_q) E [ m ] ⊂ E ( F q ) m ∣ q − 1 m \mid q - 1 m ∣ q − 1 」と μ m \mu_m μ m は m m m F q × \mathbb{F}_q^{\times} F q × は q − 1 q - 1 q − 1 の の
3.9 自己準同型環
定理 3.34 （自己準同型環の End ( E ) \operatorname{End}(E) End ( E ) Z \mathbb{Z} Z 、(ii) 虚二次体の
参照：AEC III.9 節。整環 (order) とは、Q \mathbb{Q} Q 上有限次元の Z \mathbb{Z} Z Q \mathbb{Q} Q 上張る 定符号四元数環 とは、α 2 , β 2 \alpha^2, \beta^2 α 2 , β 2 が β α = − α β \beta\alpha = -\alpha\beta β α = − α β と Q + Q α + Q β + Q α β \mathbb{Q} + \mathbb{Q}\alpha + \mathbb{Q}\beta + \mathbb{Q}\alpha\beta Q + Q α + Q β + Q α β の End ( E ) \operatorname{End}(E) End ( E ) が Z \mathbb{Z} Z 加群である φ ↦ φ ^ \varphi \mapsto \hat{\varphi} φ ↦ φ ^ が φ ^ φ = [ deg φ ] \hat{\varphi}\varphi = [\deg \varphi] φ ^ φ = [ deg φ ] が E E E (complex multiplication, CM) を p > 0 p > 0 p > 0 で (iii) が E E E が 第4章 定理 4.17）、その End ( E ) \operatorname{End}(E) End ( E ) p p p だけで p p p ∞ \infty ∞ だけで （包含に
例 3.35 (1) 標数 ≠ 2 \neq 2 = 2 E : y 2 = x 3 − x E\colon y^2 = x^3 - x E : y 2 = x 3 − x [ i ] [i] [ i ] [ i ] ∘ [ i ] = [ − 1 ] [i] \circ [i] = [-1] [ i ] ∘ [ i ] = [ − 1 ] を [ i ] ^ = [ i ] − 1 = − [ i ] \widehat{[i]} = [i]^{-1} = -[i] [ i ] = [ i ] − 1 = − [ i ] と a , b ∈ Z a, b \in \mathbb{Z} a , b ∈ Z に [ deg ( a + b [ i ] ) ] = ( a − b [ i ] ) ( a + b [ i ] ) = a 2 + b 2 [\deg(a + b[i])] = (a - b[i])(a + b[i]) = a^2 + b^2 [ deg ( a + b [ i ])] = ( a − b [ i ]) ( a + b [ i ]) = a 2 + b 2 Z [ i ] ⊂ End ( E ) \mathbb{Z}[i] \subset \operatorname{End}(E) Z [ i ] ⊂ End ( E ) End ( E ) \operatorname{End}(E) End ( E ) Q ( i ) \mathbb{Q}(i) Q ( i ) の Z [ i ] \mathbb{Z}[i] Z [ i ] に End ( E ) = Z [ i ] \operatorname{End}(E) = \mathbb{Z}[i] End ( E ) = Z [ i ]
(2) 標数 ≠ 2 , 3 \neq 2, 3 = 2 , 3 E : y 2 = x 3 + 1 E\colon y^2 = x^3 + 1 E : y 2 = x 3 + 1 ρ ( x , y ) = ( ζ 3 x , y ) \rho(x, y) = (\zeta_3 x, y) ρ ( x , y ) = ( ζ 3 x , y ) ζ 3 \zeta_3 ζ 3 は 1 の x y ≠ 0 xy \neq 0 x y = 0 の P P P P , ρ ( P ) , ρ 2 ( P ) P, \rho(P), \rho^2(P) P , ρ ( P ) , ρ 2 ( P ) は Y = y Y = y Y = y E E E の 3 交点なので O O O 。よって 1 + ρ + ρ 2 1 + \rho + \rho^2 1 + ρ + ρ 2 [ 0 ] [0] [ 0 ] ρ 2 + ρ + 1 = 0 \rho^2 + \rho + 1 = 0 ρ 2 + ρ + 1 = 0 。(1) と deg ( a + b ρ ) = a 2 − a b + b 2 \deg(a + b\rho) = a^2 - ab + b^2 deg ( a + b ρ ) = a 2 − ab + b 2 Z [ ζ 3 ] ⊂ End ( E ) \mathbb{Z}[\zeta_3] \subset \operatorname{End}(E) Z [ ζ 3 ] ⊂ End ( E ) 1 − ρ 1 - \rho 1 − ρ は ρ ( P ) = P \rho(P) = P ρ ( P ) = P と { O , ( 0 , ± 1 ) } \lbrace O, (0, \pm 1) \rbrace { O , ( 0 , ± 1 )} で、例 3.29(2) と Y 2 = X 3 − 27 Y^2 = X^3 - 27 Y 2 = X 3 − 27 E E E − 3 -3 − 3 に Q ( − 3 ) \mathbb{Q}(\sqrt{-3}) Q ( − 3 ) E E E に
(3) E E E F q \mathbb{F}_q F q 上 ϕ q ∈ End ( E ) \phi_q \in \operatorname{End}(E) ϕ q ∈ End ( E ) 第4章 の ϕ q 2 − a q ϕ q + q = 0 \phi_q^2 - a_q\phi_q + q = 0 ϕ q 2 − a q ϕ q + q = 0 を p ≡ 3 ( m o d 4 ) p \equiv 3 \pmod 4 p ≡ 3 ( mod 4 ) y 2 = x 3 − x y^2 = x^3 - x y 2 = x 3 − x End ( E ) \operatorname{End}(E) End ( E ) は
3.10 テイト加群と ℓ ≠ p \ell \neq p ℓ = p を [ ℓ ] : E [ ℓ n + 1 ] → E [ ℓ n ] [\ell]\colon E[\ell^{n+1}] \to E[\ell^n] [ ℓ ] : E [ ℓ n + 1 ] → E [ ℓ n ] に T ℓ ( E ) = l i m ← n E [ ℓ n ] T_\ell(E) = \varprojlim_n E[\ell^n] T ℓ ( E ) = lim n E [ ℓ n ] ℓ \ell ℓ (Tate module) と T ℓ ( E ) ≅ Z ℓ 2 T_\ell(E) \cong \mathbb{Z}_\ell^2 T ℓ ( E ) ≅ Z ℓ 2 Z ℓ \mathbb{Z}_\ell Z ℓ ℓ \ell ℓ C \mathbb{C} C T ℓ ( E ) ≅ Λ ⊗ Z ℓ T_\ell(E) \cong \Lambda \otimes \mathbb{Z}_\ell T ℓ ( E ) ≅ Λ ⊗ Z ℓ と「格子の E E E K K K 上 G K = Gal ( K ‾ / K ) G_K = \operatorname{Gal}(\overline{K}/K) G K = Gal ( K / K ) が E [ ℓ n ] E[\ell^n] E [ ℓ n ] に ℓ \ell ℓ
ρ E , ℓ : G K → Aut ( T ℓ ( E ) ) ≅ GL 2 ( Z ℓ ) , ρ ˉ E , ℓ : G K → Aut ( E [ ℓ ] ) ≅ GL 2 ( F ℓ ) \rho_{E, \ell}\colon G_K \to \operatorname{Aut}(T_\ell(E)) \cong \operatorname{GL}_2(\mathbb{Z}_\ell), \qquad \bar{\rho}_{E, \ell}\colon G_K \to \operatorname{Aut}(E[\ell]) \cong \operatorname{GL}_2(\mathbb{F}_\ell) ρ E , ℓ : G K → Aut ( T ℓ ( E )) ≅ GL 2 ( Z ℓ ) , ρ ˉ E , ℓ : G K → Aut ( E [ ℓ ]) ≅ GL 2 ( F ℓ ) を m o d ℓ \bmod \ell mod ℓ 還元）。ヴェイユ対の σ ( ζ ℓ ) = ζ ℓ det ρ ˉ E , ℓ ( σ ) \sigma(\zeta_\ell) = \zeta_\ell^{\det \bar{\rho}_{E,\ell}(\sigma)} σ ( ζ ℓ ) = ζ ℓ d e t ρ ˉ E , ℓ ( σ ) ℓ \ell ℓ 進版も y 2 = x 3 + 1 y^2 = x^3 + 1 y 2 = x 3 + 1 Q ( E [ 3 ] ) = Q ( 2 3 , ζ 3 ) \mathbb{Q}(E[3]) = \mathbb{Q}(\sqrt[3]{2}, \zeta_3) Q ( E [ 3 ]) = Q ( 3 2 , ζ 3 ) ρ ˉ E , 3 \bar{\rho}_{E,3} ρ ˉ E , 3 の S 3 \mathfrak{S}_3 S 3 に ζ 3 \zeta_3 ζ 3 を
有限体上では ϕ q \phi_q ϕ q T ℓ ( E ) T_\ell(E) T ℓ ( E ) に X 2 − a q X + q X^2 - a_qX + q X 2 − a q X + q に の ρ E , ℓ \rho_{E,\ell} ρ E , ℓ の の の
まとめ
曲線の ϕ q \phi_q ϕ q は q q q の 同種写像は Pic 0 \operatorname{Pic}^0 Pic 0 を 0 0 0 でない ∣ ker φ ∣ = deg s φ \lvert \ker \varphi \rvert = \deg_s \varphi ∣ ker φ ∣ = deg s φ ∣ ker φ ∣ = deg φ \lvert \ker \varphi \rvert = \deg \varphi ∣ ker φ ∣ = deg φ 双対同種写像（主張）から deg [ m ] = m 2 \deg [m] = m^2 deg [ m ] = m 2 と「次数は ∣ deg ( φ − ψ ) − deg φ − deg ψ ∣ ≤ 2 deg φ deg ψ \lvert \deg(\varphi - \psi) - \deg \varphi - \deg \psi \rvert \leq 2\sqrt{\deg \varphi \deg \psi} ∣ deg ( φ − ψ ) − deg φ − deg ψ ∣ ≤ 2 deg φ deg ψ を
等分 [ m ] P = ( ϕ m / ψ m 2 , ω m / ψ m 3 ) [m]P = (\phi_m/\psi_m^2, \omega_m/\psi_m^3) [ m ] P = ( ϕ m / ψ m 2 , ω m / ψ m 3 ) m ≤ 8 m \leq 8 m ≤ 8 で ϕ m \phi_m ϕ m ψ m 2 \psi_m^2 ψ m 2 は p ∤ m p \nmid m p ∤ m [ m ] [m] [ m ] は p ∤ m p \nmid m p ∤ m E [ m ] ≅ ( Z / m Z ) 2 E[m] \cong (\mathbb{Z}/m\mathbb{Z})^2 E [ m ] ≅ ( Z / m Z ) 2 p p p E [ p e ] E[p^e] E [ p e ] 0 0 0 （超 Z / p e Z \mathbb{Z}/p^e\mathbb{Z} Z / p e Z y 2 = x ( x 2 + a x + b ) y^2 = x(x^2 + ax + b) y 2 = x ( x 2 + a x + b ) の 2-同種写像と [ 2 ] [2] [ 2 ] 。有限部分群を ヴェイユ対は E [ m ] ⊂ E ( K ) E[m] \subset E(K) E [ m ] ⊂ E ( K ) μ m ⊂ K \mu_m \subset K μ m ⊂ K End ( E ) \operatorname{End}(E) End ( E ) Z \mathbb{Z} Z 、虚二次体の y 2 = x 3 − x y^2 = x^3 - x y 2 = x 3 − x Z [ i ] \mathbb{Z}[i] Z [ i ] y 2 = x 3 + 1 y^2 = x^3 + 1 y 2 = x 3 + 1 Z [ ζ 3 ] \mathbb{Z}[\zeta_3] Z [ ζ 3 ] を ρ E , ℓ \rho_{E,\ell} ρ E , ℓ が
演習問題
問題 3.1 ★ 系 3.16 を φ , ψ : E 1 → E 2 \varphi, \psi\colon E_1 \to E_2 φ , ψ : E 1 → E 2 に deg ( φ + ψ ) + deg ( φ − ψ ) = 2 deg φ + 2 deg ψ \deg(\varphi + \psi) + \deg(\varphi - \psi) = 2\deg \varphi + 2\deg \psi deg ( φ + ψ ) + deg ( φ − ψ ) = 2 deg φ + 2 deg ψ を
解答
双線形性より deg ( φ ± ψ ) = deg φ + deg ψ + ⟨ φ , ± ψ ⟩ = deg φ + deg ψ ± ⟨ φ , ψ ⟩ \deg(\varphi \pm \psi) = \deg \varphi + \deg \psi + \langle \varphi, \pm\psi \rangle = \deg \varphi + \deg \psi \pm \langle \varphi, \psi \rangle deg ( φ ± ψ ) = deg φ + deg ψ + ⟨ φ , ± ψ ⟩ = deg φ + deg ψ ± ⟨ φ , ψ ⟩ で、2 式を y 2 = x 3 − x y^2 = x^3 - x y 2 = x 3 − x [ 1 ] [1] [ 1 ] [ i ] [i] [ i ] deg ( 1 + [ i ] ) + deg ( 1 − [ i ] ) = 4 \deg(1 + [i]) + \deg(1 - [i]) = 4 deg ( 1 + [ i ]) + deg ( 1 − [ i ]) = 4 で、例 3.35(1) より
問題 3.2 ★ E : y 2 = x 3 − x E\colon y^2 = x^3 - x E : y 2 = x 3 − x Q \mathbb{Q} Q ψ 3 \psi_3 ψ 3 を E ( Q ) E(\mathbb{Q}) E ( Q ) に
解答
A = − 1 A = -1 A = − 1 B = 0 B = 0 B = 0 と ψ 3 = 3 x 4 − 6 x 2 − 1 \psi_3 = 3x^4 - 6x^2 - 1 ψ 3 = 3 x 4 − 6 x 2 − 1 E [ 3 ] ∖ { O } E[3] \setminus \lbrace O \rbrace E [ 3 ] ∖ { O } の x x x 座標は ψ 3 \psi_3 ψ 3 の ± 1 , ± 1 / 3 \pm 1, \pm 1/3 ± 1 , ± 1/3 での − 4 -4 − 4 − 44 / 27 -44/27 − 44/27 で、どれも x x x 座標は x 2 = 1 ± 2 / 3 x^2 = 1 \pm 2/\sqrt{3} x 2 = 1 ± 2/ 3
問題 3.3 ★ E : y 2 = x 3 + 1 E\colon y^2 = x^3 + 1 E : y 2 = x 3 + 1 Q \mathbb{Q} Q E [ 2 ] E[2] E [ 2 ] E [ 3 ] E[3] E [ 3 ] の Q \mathbb{Q} Q に Q ( E [ 2 ] ) \mathbb{Q}(E[2]) Q ( E [ 2 ]) Q ( E [ 3 ] ) \mathbb{Q}(E[3]) Q ( E [ 3 ]) Q \mathbb{Q} Q 上の
解答
x 3 + 1 = ( x + 1 ) ( x 2 − x + 1 ) x^3 + 1 = (x + 1)(x^2 - x + 1) x 3 + 1 = ( x + 1 ) ( x 2 − x + 1 ) E [ 2 ] = { O , ( − 1 , 0 ) , ( ( 1 ± − 3 ) / 2 , 0 ) } E[2] = \lbrace O, (-1, 0), ((1 \pm \sqrt{-3})/2, 0) \rbrace E [ 2 ] = { O , ( − 1 , 0 ) , (( 1 ± − 3 ) /2 , 0 )} Q ( E [ 2 ] ) = Q ( − 3 ) \mathbb{Q}(E[2]) = \mathbb{Q}(\sqrt{-3}) Q ( E [ 2 ]) = Q ( − 3 ) で ψ 3 = 3 x ( x 3 + 4 ) \psi_3 = 3x(x^3 + 4) ψ 3 = 3 x ( x 3 + 4 ) x = 0 x = 0 x = 0 ( 0 , ± 1 ) (0, \pm 1) ( 0 , ± 1 ) x = − 4 3 ζ 3 k x = -\sqrt[3]{4}\zeta_3^k x = − 3 4 ζ 3 k k = 0 , 1 , 2 k = 0, 1, 2 k = 0 , 1 , 2 y 2 = − 3 y^2 = -3 y 2 = − 3 y = ± − 3 y = \pm\sqrt{-3} y = ± − 3 O O O で 9 点（定理 3.23 と 2 3 = ( 4 3 ) 2 / 2 \sqrt[3]{2} = (\sqrt[3]{4})^2/2 3 2 = ( 3 4 ) 2 /2 − 3 = 2 ζ 3 + 1 \sqrt{-3} = 2\zeta_3 + 1 − 3 = 2 ζ 3 + 1 Q ( E [ 3 ] ) = Q ( 2 3 , ζ 3 ) \mathbb{Q}(E[3]) = \mathbb{Q}(\sqrt[3]{2}, \zeta_3) Q ( E [ 3 ]) = Q ( 3 2 , ζ 3 ) x 3 − 2 x^3 - 2 x 3 − 2 の μ 3 ⊂ Q ( E [ 3 ] ) \mu_3 \subset \mathbb{Q}(E[3]) μ 3 ⊂ Q ( E [ 3 ]) ζ 3 \zeta_3 ζ 3 を
問題 3.4 ★ ≠ 2 \neq 2 = 2 E : y 2 = x 3 − x E\colon y^2 = x^3 - x E : y 2 = x 3 − x と x ( ( 1 + [ i ] ) P ) = − i 2 ⋅ x 2 − 1 x x((1 + [i])P) = -\frac{i}{2} \cdot \frac{x^2 - 1}{x} x (( 1 + [ i ]) P ) = − 2 i ⋅ x x 2 − 1 を deg ( 1 + [ i ] ) = 2 \deg(1 + [i]) = 2 deg ( 1 + [ i ]) = 2 ker ( 1 + [ i ] ) = { O , ( 0 , 0 ) } \ker(1 + [i]) = \lbrace O, (0, 0) \rbrace ker ( 1 + [ i ]) = { O , ( 0 , 0 )} を φ : E → E ′ : Y 2 = X 3 + 4 X \varphi\colon E \to E'\colon Y^2 = X^3 + 4X φ : E → E ′ : Y 2 = X 3 + 4 X と E ′ → E E' \to E E ′ → E ( X , Y ) ↦ ( u − 2 X , u − 3 Y ) (X, Y) \mapsto (u^{-2}X, u^{-3}Y) ( X , Y ) ↦ ( u − 2 X , u − 3 Y ) u 2 = 2 i u^2 = 2i u 2 = 2 i ）の 1 + [ i ] 1 + [i] 1 + [ i ] x x x 座標が
解答
(1) x ≠ 0 x \neq 0 x = 0 P P P [ i ] P = ( − x , i y ) [i]P = (-x, iy) [ i ] P = ( − x , i y ) x x x 座標は λ = ( i y − y ) / ( − 2 x ) = ( 1 − i ) y / ( 2 x ) \lambda = (iy - y)/(-2x) = (1 - i)y/(2x) λ = ( i y − y ) / ( − 2 x ) = ( 1 − i ) y / ( 2 x ) x ( P + [ i ] P ) = λ 2 − x + x = ( 1 − i ) 2 ( x 3 − x ) / ( 4 x 2 ) = − i ( x 2 − 1 ) / ( 2 x ) x(P + [i]P) = \lambda^2 - x + x = (1 - i)^2(x^3 - x)/(4x^2) = -i(x^2 - 1)/(2x) x ( P + [ i ] P ) = λ 2 − x + x = ( 1 − i ) 2 ( x 3 − x ) / ( 4 x 2 ) = − i ( x 2 − 1 ) / ( 2 x ) 。分子と − i 2 ( 1 + x − 2 ) ≠ 0 -\frac{i}{2}(1 + x^{-2}) \neq 0 − 2 i ( 1 + x − 2 ) = 0 だから、補題 3.4 より x = 0 x = 0 x = 0 の ( 0 , 0 ) (0, 0) ( 0 , 0 ) O O O x ( φ ( P ) ) = y 2 / x 2 = ( x 2 − 1 ) / x x(\varphi(P)) = y^2/x^2 = (x^2 - 1)/x x ( φ ( P )) = y 2 / x 2 = ( x 2 − 1 ) / x u − 2 = 1 / ( 2 i ) = − i / 2 u^{-2} = 1/(2i) = -i/2 u − 2 = 1/ ( 2 i ) = − i /2 なので ( u − 3 Y ) 2 = u − 6 ( X 3 + 4 X ) (u^{-3}Y)^2 = u^{-6}(X^3 + 4X) ( u − 3 Y ) 2 = u − 6 ( X 3 + 4 X ) ( u − 2 X ) 3 − u − 2 X (u^{-2}X)^3 - u^{-2}X ( u − 2 X ) 3 − u − 2 X に u 4 = − 4 u^4 = -4 u 4 = − 4 u 4 = ( 2 i ) 2 = − 4 u^4 = (2i)^2 = -4 u 4 = ( 2 i ) 2 = − 4 1 + [ i ] 1 + [i] 1 + [ i ] ± \pm ± を φ \varphi φ と
問題 3.5 ★ R \mathbb{R} R 上の E E E E [ m ] ⊂ E ( R ) E[m] \subset E(\mathbb{R}) E [ m ] ⊂ E ( R ) と m ≥ 2 m \geq 2 m ≥ 2 m = 2 m = 2 m = 2 に m = 2 m = 2 m = 2 の
解答
系 3.32(2) より μ m ⊂ R \mu_m \subset \mathbb{R} μ m ⊂ R R \mathbb{R} R に ± 1 \pm 1 ± 1 m ≤ 2 m \leq 2 m ≤ 2 y 2 = x 3 − x y^2 = x^3 - x y 2 = x 3 − x E [ 2 ] = { O , ( 0 , 0 ) , ( ± 1 , 0 ) } ⊂ E ( R ) E[2] = \lbrace O, (0, 0), (\pm 1, 0) \rbrace \subset E(\mathbb{R}) E [ 2 ] = { O , ( 0 , 0 ) , ( ± 1 , 0 )} ⊂ E ( R )
問題 3.6 ★ p ≥ 5 p \geq 5 p ≥ 5 と ψ p \psi_p ψ p x ( p 2 − 1 ) / 2 x^{(p^2 - 1)/2} x ( p 2 − 1 ) /2 の p p p 0 0 0 に ∣ E [ p ] ∣ ≤ p \lvert E[p] \rvert \leq p ∣ E [ p ]∣ ≤ p を
解答
P ∈ E [ p ] ∖ { O } P \in E[p] \setminus \lbrace O \rbrace P ∈ E [ p ] ∖ { O } x ∘ [ p ] = ϕ p / ψ p 2 x \circ [p] = \phi_p/\psi_p^2 x ∘ [ p ] = ϕ p / ψ p 2 ϕ p \phi_p ϕ p は ψ p ( x ( P ) ) = 0 \psi_p(x(P)) = 0 ψ p ( x ( P )) = 0 p p p は E [ p ] E[p] E [ p ] に E [ p ] ∖ { O } E[p] \setminus \lbrace O \rbrace E [ p ] ∖ { O } の x x x ( ∣ E [ p ] ∣ − 1 ) / 2 (\lvert E[p] \rvert - 1)/2 (∣ E [ p ]∣ − 1 ) /2 個の ψ p \psi_p ψ p x x x の ψ p ≠ 0 \psi_p \neq 0 ψ p = 0 である（定理 3.19 の ψ p 2 ⋅ ( x ∘ [ p ] ) = ϕ p \psi_p^2 \cdot (x \circ [p]) = \phi_p ψ p 2 ⋅ ( x ∘ [ p ]) = ϕ p とも 1 1 1 の 0 0 0 でない）。補題 3.18 の ψ p \psi_p ψ p の ( p 2 − 1 ) / 2 (p^2 - 1)/2 ( p 2 − 1 ) /2 以下で、その p p p 0 0 0 だから ( p 2 − 1 ) / 2 (p^2 - 1)/2 ( p 2 − 1 ) /2 ∣ E [ p ] ∣ < p 2 \lvert E[p] \rvert < p^2 ∣ E [ p ]∣ < p 2 E [ p ] E[p] E [ p ] F p \mathbb{F}_p F p 上の ∣ E [ p ] ∣ = deg s [ p ] ≤ p 2 \lvert E[p] \rvert = \deg_s[p] \leq p^2 ∣ E [ p ]∣ = deg s [ p ] ≤ p 2 ∣ E [ p ] ∣ ≤ p \lvert E[p] \rvert \leq p ∣ E [ p ]∣ ≤ p
問題 3.7 ★ E E E F q \mathbb{F}_q F q 上の E ( F q n ) = ker ( 1 − ϕ q n ) E(\mathbb{F}_{q^n}) = \ker(1 - \phi_q^n) E ( F q n ) = ker ( 1 − ϕ q n ) を ϕ q \phi_q ϕ q E ( F ‾ q ) E(\overline{\mathbb{F}}_q) E ( F q ) の q q q である。これが
解答
(1) ϕ q n = ϕ q n \phi_q^n = \phi_{q^n} ϕ q n = ϕ q n a ∈ F ‾ q a \in \overline{\mathbb{F}}_q a ∈ F q F q n \mathbb{F}_{q^n} F q n に a q n = a a^{q^n} = a a q n = a と ϕ q n ( P ) = P ⟺ P ∈ E ( F q n ) \phi_q^n(P) = P \iff P \in E(\mathbb{F}_{q^n}) ϕ q n ( P ) = P ⟺ P ∈ E ( F q n ) ϕ q \phi_q ϕ q は deg s ϕ q = 1 \deg_s \phi_q = 1 deg s ϕ q = 1 deg i ϕ q = q \deg_i \phi_q = q deg i ϕ q = q で、定理 3.13(1) より q q q である。「核の = = = 次数」は
問題 3.8 ★ p ≡ 3 ( m o d 4 ) p \equiv 3 \pmod 4 p ≡ 3 ( mod 4 ) を E : y 2 = x 3 − x E\colon y^2 = x^3 - x E : y 2 = x 3 − x F ‾ p \overline{\mathbb{F}}_p F p 上で ϕ p ∘ [ i ] = − [ i ] ∘ ϕ p \phi_p \circ [i] = -[i] \circ \phi_p ϕ p ∘ [ i ] = − [ i ] ∘ ϕ p を End ( E ) \operatorname{End}(E) End ( E ) が
解答
i p = i 3 = − i i^p = i^3 = -i i p = i 3 = − i ϕ p ( [ i ] ( x , y ) ) = ( − x p , − i y p ) \phi_p([i](x, y)) = (-x^p, -iy^p) ϕ p ([ i ] ( x , y )) = ( − x p , − i y p ) [ i ] ( ϕ p ( x , y ) ) = ( − x p , i y p ) [i](\phi_p(x, y)) = (-x^p, iy^p) [ i ] ( ϕ p ( x , y )) = ( − x p , i y p ) ϕ p ∘ [ i ] = − [ i ] ∘ ϕ p \phi_p \circ [i] = -[i] \circ \phi_p ϕ p ∘ [ i ] = − [ i ] ∘ ϕ p 2 [ i ] ∘ ϕ p = 0 2[i] \circ \phi_p = 0 2 [ i ] ∘ ϕ p = 0 End ( E ) \operatorname{End}(E) End ( E ) は [ i ] , ϕ p ≠ 0 [i], \phi_p \neq 0 [ i ] , ϕ p = 0 なので 第4章 定理 4.18(2)）。
問題 3.9 ★ E : y 2 = x 3 − x E\colon y^2 = x^3 - x E : y 2 = x 3 − x Q = ( 1 , 0 ) Q = (1, 0) Q = ( 1 , 0 ) を j j j 不変量を E E E と
解答
Q ∈ G 2 Q \in G_2 Q ∈ G 2 t Q = 3 − 1 = 2 t_Q = 3 - 1 = 2 t Q = 3 − 1 = 2 u Q = 0 u_Q = 0 u Q = 0 t = 2 t = 2 t = 2 w = x Q t Q = 2 w = x_Qt_Q = 2 w = x Q t Q = 2 E ′ : Y 2 = X 3 − 11 X − 14 E'\colon Y^2 = X^3 - 11X - 14 E ′ : Y 2 = X 3 − 11 X − 14 X = x + 2 / ( x − 1 ) X = x + 2/(x - 1) X = x + 2/ ( x − 1 ) 4 ( − 11 ) 3 + 27 ( − 14 ) 2 = − 5324 + 5292 = − 32 4(-11)^3 + 27(-14)^2 = -5324 + 5292 = -32 4 ( − 11 ) 3 + 27 ( − 14 ) 2 = − 5324 + 5292 = − 32 j ( E ′ ) = 1728 ⋅ ( − 5324 ) / ( − 32 ) = 287496 = 66 3 ≠ 1728 = j ( E ) j(E') = 1728 \cdot (-5324)/(-32) = 287496 = 66^3 \neq 1728 = j(E) j ( E ′ ) = 1728 ⋅ ( − 5324 ) / ( − 32 ) = 287496 = 6 6 3 = 1728 = j ( E ) 第1章 定理 1.15(1) より（K ‾ \overline{K} K E ′ E' E ′ は Cremona の
問題 3.10 ★ φ : E 1 → E 2 \varphi\colon E_1 \to E_2 φ : E 1 → E 2 を ℓ \ell ℓ であると ker φ ^ = φ ( E 1 [ ℓ ] ) \ker \hat{\varphi} = \varphi(E_1[\ell]) ker φ ^ = φ ( E 1 [ ℓ ]) を
解答
R ∈ ker φ R \in \ker \varphi R ∈ ker φ ℓ R = φ ^ ( φ ( R ) ) = O \ell R = \hat{\varphi}(\varphi(R)) = O ℓ R = φ ^ ( φ ( R )) = O ker φ ⊂ E 1 [ ℓ ] \ker \varphi \subset E_1[\ell] ker φ ⊂ E 1 [ ℓ ] φ \varphi φ E 1 [ ℓ ] E_1[\ell] E 1 [ ℓ ] ℓ 2 \ell^2 ℓ 2 ）に ker φ \ker \varphi ker φ ℓ \ell ℓ 、定理 3.13(2)）なので、像の ℓ \ell ℓ φ ^ ( φ ( P ) ) = ℓ P = O \hat{\varphi}(\varphi(P)) = \ell P = O φ ^ ( φ ( P )) = ℓ P = O より ker φ ^ \ker \hat{\varphi} ker φ ^ に ∣ ker φ ^ ∣ = deg s φ ^ ≤ deg φ ^ = ℓ \lvert \ker \hat{\varphi} \rvert = \deg_s \hat{\varphi} \leq \deg \hat{\varphi} = \ell ∣ ker φ ^ ∣ = deg s φ ^ ≤ deg φ ^ = ℓ （定理 3.14(4)）なので