16, dim V V U, V U V (c) ϕ P V, p, q ϕ = p/q, q(p ) 0, ϕ P (regular), P ϕ ϕ dom ϕ, ϕ ϕ P ϕ ϕ(p ) = p(p )/q(p ), k (= A 1 ) ϕ(p ) = 0 P ϕ (zero) 1.3, P

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1 15, pp Riemann-Roch (a) k A n = A n (k) = {(x 1,, x n ) x 1,, x n k} (affine space) A 1, A 2 k[x] = k[x 1,, X n ] n k- X = (X 1,, X n ) A n f(x) k[x], x A n, f x f(x) k[x] I, V = V (I) = {x A n f(x) = 0 ( f I)} I (affine algebraic set) k[x] I(V ) = {f k[x] f(x) = 0 ( x V )} V I(V (I)) I k[v ] = k[x]/i(v ) V (coordinate ring) f k[v ] f : V x f(x) k V (polynomial function) A n X 1,, X n (coordinate function) 1.1 (1) A n, A n (2), 1.2 A n, (Zariski ) 1 A n V, k[v ] (b) I(V ), V (affine algebraic variety) k(v ) V (function field), V (rational function) k(v ) k, k k(v ) k V (dimension), dim V f 1,, f m I(V ) P V, m n ( f i / X j (P )) i,j n dim V, P (non-singular point) (simple point) n dim V, P (singular point) V V 1, V 1 V, V 1 V I(V 1 ) I(V ) V 1 dim V 1 V V 1 V V 15

2 16, dim V V U, V U V (c) ϕ P V, p, q ϕ = p/q, q(p ) 0, ϕ P (regular), P ϕ ϕ dom ϕ, ϕ ϕ P ϕ ϕ(p ) = p(p )/q(p ), k (= A 1 ) ϕ(p ) = 0 P ϕ (zero) 1.3, P V k[v ] P, k[v ] P k[v ] k[v ] P k(v ), P P 1/ϕ, P ϕ (pole) 1.4 k[v ] P, P 2, 1.2 (a) a, b A n+1 {0} a = c b c k a b P n = A n+1 {0}/ n (n-projective space) P 1, P 2 (x 0, x 1,, x n ) A n {0} [x 0 : x 1 : : x n ], A n+1 X 0, X 1,, X n X 0 : X 1 : : X n P n k[x] = k[x 0, X 1,, X n ] f(x) f(λ X) = λ d f(x) ( λ k), f d d f, deg f I, V = V (I) = {x P n f(x) = 0 f I } I (projective algebraic set) I(V ) = (f kx f, f(x) = 0 ( x V )) V k[v ] = k[x]/i(v ) V (homogeneous coordinate ring), V (projective variety) 1.5 (1) P n, P n (2), 1.6 P n, 3 P n V, k[v ] (b) X 0 : X 1 : : X n P n (X j ) V (X j ) U j, U j = {[x 0 : ] P n x j 0} V (X j ) P n 1, U j A n V V j = V U j (U j A n ) V j, V = V j Vj, Vj = V V (X j ) (V (X j ) P n 1 ) V j X j, V j (c) n d f

3 Riemann-Roch 17 f(x 0, X 1,, X n ) = X0 d f(x 1 /X 0,, X n /X 0 ) d, f V, I(V ) I(V ) I(V ) I(V ) V V V V V (d) 1 l P n, P n 1 U = P n l A n V P n l V, U = P n l V = V U U A n, V V V V, V l V, V l V P, P l, P V V P V, V V V V l, V V 4 (1) V, (2), 1.3 (a) f(x) k(x) = k(x 0, X 1,, X n ) f(λ X) = λ d f(x) (λ k), f d, d f f d 0 f, p, q f = p/q (b) V P n, I( V ) 0 n+1 p/q q I( V ) k[x; V ] 0 p 1 /q 1, p 2 /q 2 k[x; V ] 0 p 1 q 2 p 2 q 1 I( V ), p 1 /q 1 p 2 /q 2 k( V ) = k[x; V ] 0 /, V. k( V ) k V (dimension) dim V 1.7,, (c) ϕ V P V, p, q ϕ = p/q, q(p ) 0, ϕ P, P ϕ ϕ(p ) = p(p )/q(p ) ϕ P 1.8 V V P V V P, V P P V k[v ] P ϕ dom ϕ, ϕ ϕ P ϕ, k (= A 1 ) ϕ(p ) = 0 P ϕ P V dom ϕ, 1/ϕ P (1/ϕ)(P ) = 0 P ϕ 1.9,

4 18 (d) f 1,, f m I( V ) P V, m n ( f i / X j (P )) i,j n dim V P, n dim V P 1.10 P V, P 1.4 (a) V n f 1,, f n, ϕ = (f 1,, f n ) : V P (f 1 (P ),, f n (P )) A n V A n ϕ f 1,, f n ϕ V 1, ϕ : V V 1 V V 1 ϕ : k(v 1 ) f f ϕ k(v ) ϕ V 1, ϕ, k(v 1 ) k(v ) k(v ) ϕ k(v 1 ), ϕ (degree) deg ϕ, k(v )/ϕ k(v 1 ) N k(v )/ϕ k(v 1) : k(v ) ϕ k(v 1 ), ϕ ϕ = (ϕ ) 1 N k(c)/ϕ k(c ) : k(c) k(c ) ϕ ϕ 1.11 (1) ϕ V 1, ϕ, dim V 1 dim V (2) dim V = dim V 1 = 1 ϕ ϕ, k(v )/ϕ k(v 1 ) (b) V, f 0, f 1,, f n ϕ = [f 0 : f 1 : : f n ] : V P [f 0 (P ) : f 1 (P ) : : f n (P )] P n V P n P V, g g f 0, g f 1,, g f n P P, ϕ P, P ϕ ϕ dom ϕ, ϕ ϕ V 1, ϕ : V V 1 V V 1,,,,, V ϕ, [1 : ϕ] : V P 1 ϕ [1 : ϕ] ϕ P [1 : ϕ](p ) = [1/ϕ : 1](P ) = [0 : 1], [1 : ϕ] P 1.12 (1) ϕ, [1 : ϕ] ϕ (2) ϕ, [1 : ϕ] P 1 (c) V1, V 2 ϕ : V 1 V 2, ψ : V 2 V 1, ϕ ψ, ψ ϕ, V 1 V 2 (isomorphic) V 1 V 2 ϕ, ψ (isomorphism), V 1, V 2 ϕ : V 1 V 2, ψ : V 2 V 1, ϕ ψ, ψ ϕ, ϕ, ψ (birational map), V 1

5 Riemann-Roch 19 V 2 (birational equivalent) V V A n, P n, n P 1 P 1 n, A n, P n, P 1 P , ,,, 1... P 1 X 0 : X 1, U 0 = {[x 0 : x 1 ] P 1 x 0 0} P 1 = U 0 {[0 : 1]} U 0 z = X 1 /X 0, U 0 P 1 [0 : 1] z P 1 = A 1 { } U = {[x 0 : x 1 ] P 1 x 1 0} w = X 0 /X 1 U 0 U (U 0 U = P 1 {0, }) w = 1/z, U w = 1/z P 2 X : Y : Z U = {[a : b : 1] P 2 (a, b) A 2 } x = X/Z, y = Y/Z A 2 x- {[x 0 : 0 : 1] x 0 A 1 }, y- {[0 : y 0 : 1] y 0 A 1 } l = {[a : b : 0] P 2 [a : b] P 1 } P 2 = A 2 l 2.2 (a) k 1 (affine algebraic curve), 1 (projective curve), (algebraic curve),,,, 2.6,,,, (b) 2 f(x, y) k[x, y], (f) C (affine plane curve) f C, f = 0 C C 1 f = f 1 f 2 f r, C f 1,, f r C 1,, C r C 1,, C r C f(x, Y, Z) k[x, Y, Z], (f) C (projective plane curve) f C, F = 0 C, (plane curve) C m, C, C m 1

6 20 (c) C : f(x, y) = 0 f/ x(p ) = f/ y(p ) = 0 P = (a, b) C, j 0, u, v j f (j) P (u, v) f (j) P (u, v) = (u x + v y )j f(p ) = j ( ) ji j x i y j i f(p ) ui v j i f (r) P i=0 (u, v) 0 r P (multiplicity), P r- 2, 1 r- P, f (r) P (x a, y b) = 0 (tangent cone) r, C P r- P (r 2) r ( r ), P (ordinary singular point) 2 (node) P C, P, 2 y = x, y = x C y 2 x 2 + (x, y 3 ) = , y = 0 y 2 + (x, y 3 ) = 0 y 2 x 3 + (x, y 4 ) = 0, P (cusp) 5 f(x, y) 3 C : y 2 =f(x, y) 6 F (X, Y, Z) m, C : F (X, Y, Z) = 0 (1) X X F (X, Y, Z) + Y F (X, Y, Z) + Z F (X, Y, Z) = m F (X, Y, Z) Y Z (2) P C, X F (P ) = F Y (P ) = F Z (P ) = 0 (3) P C, P X X F (P ) + Y F (P ) + Z F (P ) = 0 Y Z (4) Z- 0 P C C, C {Z 0} (d) C, P C P C l P, P C I P (C, l P ) 2 P 3 P (point of inflextion, flex) f(x, y), F (X, Y, Z), f xx f xy f x F XX F XY F XZ H f (x, y) = det f yx f yy f y H F (X, Y, Z) = det F Y X F Y Y F Y Z f x f y f F ZX F ZY F ZZ C : f(x, y) = 0, H f (x, y) = 0 C Hesse (Hessian) C : F (X, Y, Z) = 0, H F (X, Y, Z) = 0 C Hesse 2.1, Hesse (a) K k y K k(x 1,, x r ), y x 1,, x r k k, x K k (transcendental) K S k, y S, y S K S, K k(s), S k K k K S, S K k (transcendence degree) tr. deg k K K = k(s) S, K k

7 Riemann-Roch 21 (b) k n k n (rational function field), k n (algebraic function field) k K K k 1 k x K K k(x) K k(x), x 2.2 (F.K.Schmidt) 1,, k x 1 K, K k(x) K = k(x, y) K 1, x y k 0 f(x, Y ) k[x, Y ] f(x, y) = 0 C : f(x, Y ) = 0, C k(c) K 2.3, (a) C P C k[c] P, P M P k[c] P, M P P C P, k[c] P (b) P k[c] P k(c) ord P k(c) k(c) P f ord P (f) f P (order) f P P f, P f k[c] P, 1.1 (c), 1.3 (c) P k[c] P k[c] P, k[c] P, k(c),, (c) k(c) P ( 1 ) P (local parameter) P t P k(c) P Laurent, k(c) P = k((t P )) P C P P k[c] P, P 2.6 ( ), t P k(c) P C, k(c) k(t P ) , 8 (1) C 1 : y 2 = x 2 (x + 1), C 2 : y 2 = x 3, C 3 : y 2 = x 4 (x 1), (2), (3)

8 22 (d), P 1 X 0 : X 1, P 1 X 0, X 1 0 0, X 0, z = X 1 /X 0 k(p 1 ) k(z) z U 0 = {X 0 0} A 1, P 1 A 1 a = [1 : a] A 1 P 1 f k(p 1 ) = k(z) a, f z = a k[p 1 ] a = {f = p/q p, q k[z], q(a) 0} k[p 1 ] a u a = z a, u a K(P 1 ) a k((u a )) = k((z a)), z = a Laurent = [0 : 1] P 1 U = {X 1 0} A 1, w = X 0 /X 1 = 1/z w z k[p 1 ] {f = p/q p, q k[z], deg p deg q}, deg q deg q f = p/q u = 1/z, k((1/z)) 2.5 Bézout (a) f, g C, D m = deg f, n = deg g, C, D C D P, C, D P A 2 P 2 A 2 x, y, C A 2, D A 2 f(x, y), g(x, y) k[[x, y]], k[[x, y]]/(f, g) k P C D, I P (C, D) (= dim k k[[x, y]]/(f, g)), 2.9 (Bézout) I P (C, D) = m n P C D (b) C D P f(p ) = 0, g(p ) = 0, f = g = 0,,,. C D 1, 2 ( ), 1, C D m n 2.10 ( ) 2 m, n, m n, m n 2.6 (a) 2,, 2,.,, 2, 2, Bézout 9,,,,,

9 Riemann-Roch 23 (b) k 2 3 f(x) k[x], C : y 2 = f(x) f, C C (elliptic curve) C C : Y 2 Z = Z 3 f(x/z) 3 [0 : 1 : 0] C 1, [0 : 1 : 0], C 3 O = [0 : 1 : 0] C 2 P, Q C Bézout, P, Q (P = Q P ) 3 R C R O R C P Q := R C O 10 (1) C : y 2 = f(x) (deg f = 3), f(x) = 0 C C (2) O C (3) P, P, O, P P (c) a 1, a 2, a 3, a 4, a 6 k, 3 C : y 2 + a 1 x y + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 k 2, y, C C 1 : y 2 = 4 x 3 + b 2 x b 4 x + b 6 b 2 = a 2 1 4a 2, b 4 = 2a 4 + a 1 a 3, b 6 = a a 6 c 4 = b b 4, c 6 = b b 2b 4 216b 6 k 3, C C 1 C 2 : y 2 = x 3 27c 4 x 54c 6 C 1 C 2 (b), C C Weierstrass C C [0 : 1 : 0], C C, C = b 2 2 b 8 8b b b 2b 6 b 6 = (c 3 4 c2 6 )/1728 b 8 = a 2 1a 6 + 4a 2 a 6 a 1 a 3 a 4 + a 2 a 2 3 a 2 4 C 0 j = j(c) = c 3 4/ = 1728 c 3 4/(c 3 4 c 2 6), C j (1) k a 1,, a 6 C C 0, k, (2) Weierstrass C, C j(c) = j(c ), 1, (b), O = [0 : 1 : 0], 1 Weierstrass C, O = [0 : 1 : 0] C 1, 1,,,, Weierstrass ( ) j-, (2). j-,

10 24 (d) k 2 4 f(x) k[x] 4 C : y 2 = f(x), f(x) = 0 C C : Y 2 Z 2 = Z 4 f(x/z), [0 : 1 : 0] 3, C, C P 0 = [0 : 1 : 0], P 0 P 0 C 0 C 4 f f(x) = a 0 x 4 + a 1 x 3 + a 2 x 2 + a 3 x + a 4 P 0 C 0 : z 2 = a 0 x 4 + a 1 x 3 z + a 2 x 2 z 2 + a 3 x z 3 + a 4 z 4 (0, 0) C 0 C 0 (0, 0) C ( ), C 0 2 C 0 C 0 : v 2 = a 0 + a 1 u + a 2 u 2 + a 3 u 3 + a 4 u 4 (= u 4 f(1/u)), f, ϕ 0 : C (x, y) (1/x, y/x 2 ) C 0, C C 0 ϕ 0, C {(0, ± a 4 )} C 0 {(0, ± a 0 )}. ϕ 0 C C 0 Ĉ Ĉ Ĉ C u = 1/x, Ĉ C u- 0 C 0 (0 ± a 0 ) 2 (0, a 0 ) P, (0, a 0 ) P P u- 0 v- a 0 P u- 0 v- a 0 k(c) = k(ĉ) = k(c 0) = k(u, v), C, u, v, u, v-, f(x) = 0 α k f(x) = a 0 (x α) 4 + a 1 (x α) 3 + a 2 (x α) 2 + a 3 (x α) f(x) = 0 a 3 0 C α : v 2 = a 3 u 3 + a 2 u 2 + a 1 u + a 0 3, ψ : C (x, y) (1/(x α), y/(x α) 2 ) C α, C C α C α C α (b) ψ C α ψ, ψ : C (x, y) [x α : y : (x α) 2 ] C α ψ C C α, C α C, C C [0 : 1 : 0], 2 (0, ± a 0 ) C α,... 4 f(x) f(x) k((1/x)) f(x) = a0 x 4 + a 1 x 3 + a 2 x 2 + a 3 x + a 4 = b 0 x 2 + b 1 x + b 2 + b 3 /x + u = y (b 0 x 2 + b 1 x), v = x y (b 0 x 3 + b 1 x 2 + b 2 x), w = x 2 y (b 0 x 4 + b 1 x 3 + b 2 x 2 + b 3 x), u b 0 w + 2 b 1 v a 4 = 0, u w v 2 b 2 w + b 3 v = 0 w 1 w, u v 2 b 0 v b 1 u v a 3 v = u 3 + b 2 u 2 + a 4 u a 4 b 2 a 0 = b 2 0 (u 0, v 0 ) = ( 2 b 0 u, 4 b 0 v), u 0, v 0 v0 2 2 b 1 u 0 v 0 2 b 0 a 3 v 0 = u b 0 b 2 u a 0 a 4 u 0 8 a 0 a 4 b 0 b 2 (u 2, v 2 ) = (u 1, v 1 b 1 u 0 b 0 a 3 ) = ( 2 b 0 u, 4 a 0 v + 2 b 0 b 1 u b 0 a 3 ), v2 2 = u3 2 + a 2 u ( 4 a 0 a 4 + a 1 a 3 ) u 2 + (a 0 a a 0 a 2 a 4 + a 2 1 a 4)

11 Riemann-Roch 25 C C : y 2 = x 3 + a 2 x 2 + ( 4 a 0 a 4 + a 1 a 3 ) x + (a 0 a a 0 a 2 a 4 + a 2 1 a 4 ) ϕ : C P (u 2 (P ), v 2 (P )) C ϕ C C, C C C C C [0 : 1 : 0], C ( 2b 0 b 2, 3b 2 0 b 3) C 2 11 (1) C Ĉ, C α, C (2) C α, Ĉ (3) C, Ĉ (e) k 2, (b), (d) f(x) k[x], C : y 2 = f(x) C C ( ) (hyperelliptic curve) ( 3.5, 4.5). C f n = deg f n 2 2 (conic), n = 3, 4 (elliptic curve), n 5 2 (a), n = 3 (b), n = 4 (d) n 5 C : y 2 = f(x) 4 n = deg f, m = [(n + 1)/2] n n = 2 m 1, n n = 2 m C C [0 : 1 : 0], C 0 : v 2 = u 2m f(1/u) C ( f ), C 0 C 0, 2 m f(x) 0 2 m, 0 2 m 1 ϕ 0 : C (x, y) (1/x, y/x m ) C 0, C C 0 ϕ 0 C C 0 Ĉ 2.12 Ĉ C,, 1, 2 n = deg f C : y 2 = f(x), f(x) = 0 α k f(x) = a 0 (x α) n + a 1 (x α) n a n 1 (x α) C : v 2 = a n 1 u n a 1 u + a 0, C, C C, C C C C C C, C C C C, deg f, C : y 2 = f(x), ι : C (x, y) (x, y) C ι, 2 ι (hyperelliptic involution) C, C, f, ι f 2, ι ι, y- 0, x- f(x) = 0

12 26 n (= deg f) ι, f n + 1, f n m = [(n + 1)/2], C : y 2 = f(x) 2m 12 (1) C : y 2 = f 1 (x) f 2 (x) 2 C : v 2 = f 1 (u) (2) C : y 2 = f(x), f(x) = 0 (f) k 2 3 C : F (X, Y, Z) = C 1, C C : F (X, Y, Z) = 0 3, H F (X, Y, Z) = 0 C Hesse F 3 H F C 9 P C, P l ( X : Y : Z ), P [0 : 1 : 0] l {Z = 0} 2.14 C Weierstrass , C, 2.12 (g) k 0 4 f(x) k[x] C : y 3 = f(x) C C : Y 3 Z = f(x/z) Z 4 C C [0 : 1 : 0] 1, C, C (h) k 0 C : y 3 = f(x) (f(x) k[x]) C, f deg f 1, 2, 3 (f), 4 (g) deg f (1) f 1, f 2, f 3 k[x], f = f 1 f2 2 f 3 3 f 1 f 2 (2) C 1 : f 2 (x) y 3 = f 1 (x), C 2 : f 1 (x) y 3 = f 2 (x) (3) C 1, C 2 C : y 3 = f(x) n 1 n 2 f 1, f 2 k[x] C 1 : f 2 (x) y 3 = f 1 (x) C 1 (f) (g), n 1 5 n s = n 1 n 2 3 C 1 C 1, (1) s > 0, [0 : 1 : 0] 1, (2) s = 0, [0 : 1 : 0] [1 : a : 0] (a a 3 = (f 1 )/(f 2 ) k ) 4, [1 : a : 0] [0 : 1 : 0] deg f 2 > 1, n 2 = 1 (3) s < 0, [0 : 1 : 0] [1 : 0 : 0] 2, [1 : 0 : 0] [0 : 1 : 0] n 2 > 1, n 2 = 1

13 Riemann-Roch 27, s = 0, s < 0,, s > 0 m = n 1 3 (f) (g) m C 1 [0 : 1 : 0] m f 2 (x) = b j x j Z s ( j! (m j)! b j X j Z n2 j ) = 0 15 s 0,, 4 r = 0, ±1 r n 1 n 2 mod 3 r = 1 f 1 (x) = x n1+1 f 1 (1/x), f 2 (x) = x n2 f 2 (1/x), r = 0, 1 f 1 (x) = x n1 f 1 (1/x), f2 (x) = x n2+1 f 2 (1/x) 2.18 C 1 : f 2 (x) y 3 = f 1 (x), C 2 : f 1 (x) y 3 = f 2 (x), C1 : f 2 (x) y 3 = f 1 (x), C2 : f 1 (x) y 3 = f 2 (x), Ĉ1 = C 1 C 2 C 1 C 2, C 1 16 C 1 (x, y) (x, ) C 2, C 1 (x, y) (1/x, ) C 1, C 1 (x, y) (1/x, ) C 2 C 1 x, y Ĉ1 = C 1 C 2 C 1 C 2, x, y : Ĉ P1 17 Ĉ1 C 1 (, Ĉ 1 C 1 x-,, x- ) (i) k 0 d > 1, n 1 n 2 f 1, f 2 k[x], C : f 2 (x) y d = f 1 (x) C C 2.19 s = n 1 n 2 d, m = n 1 d s > 0 C C [0 : 1 : 0], m f 2 (x) = b j x j Z s ( j! (m j)! b j X j Z n2 j ) = 0 18 s 0, C C,, r d/2 r n 1 n 2 mod d r = 0, ±1, C (h) 2.18 r 0, ±1,,,, r = 1 f 1 (x) = x n1+1 f 1 (1/x), f2 (x) = x n2 f 2 (1/x), r = 0, 1 f 1 (x) = x n1 f 1 (1/x), f 2 (x) = x n2+1 f 2 (1/x), 2.18 (y ) 2.20 C 1 : f 2 (x) y d = f 1 (x), C 2 : f 1 (x) y d = f 2 (x), C1 : f 2 (x) y d = f 1 (x), C2 : f 1 (x) y d = f 2 (x), Ĉ = C 1 C 2 C 1 C 2, 19 r 0, ±1 (1) C : f 2 (x) y d = f 1 (x) Ĉ 4 (2) C x, y Ĉ, x, y : Ĉ P1 (3) Ĉ C

14 28 3 Riemann-Hurwitz 3.1 (a) C, f [1 : f] : C P [1 : f(p )] P 1 C P 1 ϕ = [f : g] : C P 1, f, ϕ = [0 : g] = [0 : 1] =, f, f ϕ = [f : g] = [1 : g/f], ϕ g/f, C P 1, k(c) { } 1 1 (b) ϕ : C C, C f k(c ) f ϕ C ϕ : k(c ) f f ϕ k(c) 3.1 (1) ϕ : C C, k(c) ϕ k(c ) (2) k- ψ : k(c ) k(c), ϕ = ψ ϕ : C C (3) k k(c) K [k(c) : K] <, C 0 ϕ : C C 0 ϕ k(c 0 ) = K (c) C, ϕ : C P n 3.2 P C ϕ 3.3 (1), (2), (3) 1 20 (1), 3.2 (2), 3.3 (d) C, C, ϕ : C C ϕ, k(c)/ϕ k(c ) ϕ deg ϕ ϕ deg ϕ = 0, deg 0 = k(c)/ϕ k(c ) ϕ (separable), k(c)/ϕ k(c ) ( ) ϕ ( ) ((purely) inseparable) k(c)/ϕ k(c ), ϕ,, deg s ϕ, deg i ϕ (e) k p (> 0) q = p r, q- Frobenius π, k π : k x x q k, π : k[x] f f π k[x], A n, P n C, I(C) π C π π : C P C π, q- Frobenius 3.4 k(c)/π k(c π ) q π q 3.5 k p > 0 ϕ : C C q = deg i ϕ, π q- Frobenius, ψ : C π C ϕ = ψ π 3.2

15 Riemann-Roch 29 (a) C, C, ϕ : C C P C ϕ(p ) C, t ϕ(p ) k(c) ϕ(p ) Q e ϕ (P ) = ord P (ϕ t ϕ(p ) ) P ϕ (ramification index) k(c) ϕ(p ) = k((t ϕ(p ) )), k( C) P ϕ k(c) k((ϕ t ϕ(p ) )) k( C) P /k((ϕ t ϕ(p ) )) 2 P ϕ (branch point), ϕ P (ramified) 1 P ϕ (unbranch point), ϕ P (unramified) ϕ, C Q C, ϕ(p ) = Q P C Q Q ϕ : C C ϕ Q, Q P k, (wild ramification), k, (tame ramification) 3.6 (1) Q C, e ϕ (P ) = deg ϕ P ϕ 1 (Q) (2) Q C, ϕ 1 (Q) = deg s ϕ (3), e ψ ϕ (P ) = e ϕ (P ) e ψ (ϕ(p )) 21,,, (b) ϕ : C C k 1 k( C)/ϕ k(c) k( C)/ϕ k(c) F (X) ϕ k(c)[x] 3.7 F (X) k((ϕ u Q )) F (X) = F 1 (X) F r (X) ϕ 1 (Q) = {P 1,, P r }, e ϕ (P j ) = deg F j P C, Q = ϕ(p ) C t P u Q k(c) k( C) k((t P )), ϕ u Q = t e P + te+1 P u Q 1 P, e = e ϕ (P ) Q + t e P, t P 3.8 f k( C), ord P (ϕ f) = e ϕ (P ) ord ϕ(p ) (f) 3.9 f C [1 : f] : C P 1 P C e f (P ) ord P (f f(p )) = e f (P ) ord P (f) = 0 P P f ord P (f) = e f (P ), f f, (c) e k ( ), (ϕ u Q ) 1/e = (t e P + te+1 P + ) 1/e = t P + t 2 P + k((t P )) k(c 1 ) P P t P, 3.10 P, t P k(c) P, t e P = ϕ u Q P ( k ) u Q ϕ u Q = t e P + k((t P )) t P k p (> 0), t P p, k((t P ))/k((ϕu Q )), k((t P ))/k((ϕ u Q )), ϕ u Q t P p 4.2

16 30, (du Q ) Q (ϕ (du Q ) Q ) P, k((t P ))/k((ϕ )), 3.3 (a),,, ( ).,,, Laurent ( ), k 0 (b) C, C C C C ( ), C C P C, f k(c) f : C P 1 P e, a = f(p ) P 1 P 1 = A 1 { } z, a A 1 u a = z a, a = u = 1/z, u a a P , P C t k( C) P, t e = f u a k(c) = k( C), t P C 3.11 C P C C f, a = f(p ), f P e P t k(c) P, P f t e = f a, P f t e = 1/f 22 C : y 2 = f(x) (f ), P a = (a, ) C (a A 1 P 1 ), x, y (c), m, n 2 k 0, m n C : y m = x n f 0 (x) (f 0 (x) k(x), f 0 (0) 0, ) P = (0, 0) C P C, P n m, (y/x n/m ) m = f 0 (x) v = y/x n/m, C P C : v m = f 0 (x) C C x-, P C x- 0 v- f 0 (0) m, P C m n 1 mod m, (y/x ) m = x f 0 (x) C P C : v m = x f 0 (x) C C x-, P C x- 0 v- 0, P C 2, C P C, 3.10 x P t k(c) P, t = x,

17 Riemann-Roch 31 t m = x v t, v = m f 0 (t) k[[t]], v = t m f 0 (t m m ) k[[t]] f 0 (t ) t, m m t = 0 v, f 0 (t) m m, f 0 (t m ) m 0 P C C, m, 1, (d) P C k[c] P,, k(c) x, y, P k[c] P, k x, y a m + b n = d (d = gcd(m, n), a, b Z) t = x a y b k(c) t m = (x a y b ) m = x am y bm = x am (x n f 0 (x)) b = x am+bn f 0 (x) b = x d f 0 (x) b t n = (x a y b ) n = x an y bn = (y m /f 0 (x)) a y bn = y am+bn f 0 (x) a = y d f 0 (x) a t k[c] P f 0 (0) 0,, d f 0 (x) k[[x]] t m/d = x d f 0 (x) b, t n/d = y ζ d f 0 (x) a ( ζ d = 1 ) x, y k[[t]] t, k(c) = k(x, y) k((t)) k((t)) t k(c) k(c) k((t)) k(c) k((t)), y d ζ, d,, C P 3.12 m, n 2 a m + b n = d (d = gcd(m, n), a, b Z) t = x a y b k(c) t P = (0, 0) C ord P (x) = m/d, ord P (y) = n/d t y d,, P d 23 m, p, q 2 gcd(m, p, q) = 1 y m = x p (1 x) q,, x, y, 24 C : y 2 = f(x) (f ), (e) Ĉ = C C 0, C 25 C : y d = f(x) (f k[x], d N) C P C 3.4 (a) ϕ : C C ϕ : C C (branched covering) (covering) ϕ (covering map), C (covering curve), C (base curve) deg ϕ (covering degree) ϕ (k( C)/ϕ k(c) ), C C deg ϕ ϕ (unbranched covering) ϕ : C C, C deg ϕ C C C deg ϕ

18 32 (b) C C (automorphism) Aut(C) ϕ : C C, C σ ϕ σ = ϕ ϕ (covering transformation) Aut( C) ϕ (covering transformation gruop) G ϕ G ϕ, deg ϕ G ϕ = deg ϕ ϕ Galois, G ϕ Galois 3.13 ϕ : C C Galois, k( C)/ϕ k(c) Galois G ϕ Gal(k( C)/ϕ k(c)) ϕ : C C Galois Q C P C e ϕ (P ), e ϕ (Q) = e ϕ (P ) Q deg ϕ 3.14 C, G Galois ϕ : C C G ϕ G 26 Galois Hilbert, Galois (,, ) 27 ϕ : C C Galois C h Galois (G f : σ G f σ h = h), C h ϕ h = h (c) C ( C P 1 ) d = gon (C) C (gonality) C d-gonal P 1, 1-gonal C 2, (hyperelliptic) 2-gonal (hyperelliptic curve). ( ), 0, 1 (elliptic curve), 2 2-gonal Riemann-Roch, 1-goanl ( ), 2-gonal 1 2-gonal C, ϕ : C P 1 2 ( ) k(c) k(p 1 ) 2 (Galois), 2 Galois G ϕ ι : C C C (hyperelliptic involution) C 2 ι ι k(c) 2 K C, 2 Galois C C K C P 1, C 2-gonal 3.5 Riemann Riemann-Hurwitz (a) k = C 1 R Riemann Riemann Riemann, Riemann Riemann C, Riemann, Riemann Ĉ = C { } T = C/Λ (Λ C ) Riemann Riemann 2,, Riemann Euler χ(r) Riemann., Riemann R (genus) g(r) (b) C R, 1, Riemann Riemann R g(r) Euler χ(r) C 2 Riemann Ĉ, 0

19 Riemann-Roch (1) Riemann Ĉ 0, Euler 2 (2) 1, Euler 0 1 (1 Betti b 1 (R)), 2 1 R, Euler Betti, 3.16 χ(r) = 2 2 g(r) (c) Riemann π : R R π, R, R Q R, π(p ) = Q P R Q Q (U, ϕ) (U R Q, ϕ : U C ), P (Ũ, ϕ), ϕ π ϕ 1 (z) = z ep e P P π (multiplicity) e P > 1 P π π, ϕ R Riemann R Riemann, Q R, Q R n π R R 3.17 (Riemann-Hurwitz) χ( R) = n π χ(r) (e P 1) 3.18 (1) π : R Ĉ 2, 4 χ(r) (2) π : R R 2, χ( R) χ(r) χ(r), χ( R) < χ(r) P R (3) 2 Ĉ, 2 28,,, Riemann-Hurwitz m, p, q 2 gcd(m, p, q) = 1 C/C : y m = x p (1 x) q Ĉ C x Ĉ x : Ĉ Ĉ (1) x C(P 1 ) = C(x), [C(Ĉ) : C(x)] (2) a Ĉ e x (P ) = m P x 1 (a) (3) x (4) (5) (6) χ(ĉ) ( Ĉ ) m, p, q χ(ĉ) = 2 ( g(ĉ) = 0) m, p, q χ(ĉ) = 0 ( g(ĉ) = 1) m, p, q 4 Riemann-Roch 4.1 (a) C C (divisor group) Div(C) Div(C) D C Z- D = n 1 P n m P m (P 1,, P m ) n 1,, n m 0, D, C D = n P P (n P Z, P n P = 0) P C

20 34 n P 0 P C D (support) supp(d) 0 0 supp(0) D deg(d) = n P D deg : Div(C) Z, Div 0 (C) P C ν P : Div(C) D n P Z, ν P P C ν P (D) 0 D (positeve (or effective) divisor) D 0 D 0 D (zero divisor), D (polar divisor) (b) C C f k(c), div(f) = P ord P (f) P f div(f) 0 f, div(f) f ord P, div : k(c) Div(C) 0, div Div 0 (C) 0,, div k (principal divisor) Div l (C). D 1, D 2, D 1 D 2 (linearly equivalent), D 1 D 2 (divisor class) D [D] Pic(C) = Div(C)/Div l (C) C (divisor class group) Picard 0 0 Pic 0 (C) 1 k k(c) div Div 0 (C) Pic 0 (C) 0, K, 1 O K K I K C K = I K /P K 0 ( ),, ( K ), 2-gonal (, ) 2, 2,,, ( ), 1, Dedekind 30, 31 C : y 2 = f(x) (f ), x, y, x a, f (x) (c) ϕ : C 1 C 2 C 2 Q ϕ (Q) = e ϕ (P ) P Div(C 1 ) ϕ(p )=Q, Div(C 2 ) ϕ : Div(D 2 ) Div(D 1 ) C 1 P ϕ (P ) = ϕ(p ) C 2 ϕ : Div(C 1 ) Div(C 2 )

21 Riemann-Roch C f f : C P 1, f div(f) 0 = f (0) div(f) = f ( ) div(f) = f (0 ) 4.2 ϕ : C 1 C 2 D j Div(C j ), f j k(c j ), (1) deg(ϕ D 2 ) = (deg ϕ) (deg(d 2 )) (2) ϕ div(f 2 ) = div(ϕ f 2 ) (3) deg(ϕ D 1 ) = deg D 1 (4) ϕ div(f 1 ) = div(ϕ f 1 ) (5) (ψ ϕ) = ϕ ψ, (ψ ϕ) = ψ ϕ (6) ϕ ϕ = deg ϕ 4.3 ϕ : C 1 C 2, ϕ : Pic 0 (C 2 ) Pic 0 (C 1 ), ϕ : Pic 0 (C 1 ) Pic 0 (C 2 ) ϕ ϕ : Pic 0 (C 2 ) Pic 0 (C 2 ) deg(ϕ) (a) C P C k(c) P, P t k(c) P k(c) P = k((t)) f = a i t i k(c) P, df/dt = i ai t i 1 f t, P C t f df/dt, df/dt = (df/dt ) (dt /dt) (b) k(c) P (g, f), (g, f ) g df/dt = g df /dt, (g, f) P (g, f ) P 2 k(c) P (g, f), (g df)p (g df) P h k(c) P f (g df) P = (h g df) P (g df) P = (g df/dt) (dt) P, g df/dt k(c) P, k(c) P (dt) P 4.4 P, (dt) P 1 k(c) P (c) P t, t ord P (dt /dt) = 0, ord P (g df/dt) ord P ((g df) P ) = ord P (g df/dt), (g df) P P P (g df) P, P (g df) P (g df) P t (g df) P = (g df/dt) (dt) P = ( a i t i ) (dt) P t 1 (residue) Res P ((g df) P ) , t (d) f k(c) P (df) P P e = e f (P ) k ( ) 3.10, P t P k(c) P t e P = f u a (a = f(p ) P 1 ) u a P 1 a, P 1 = A 1 { } z, a A 1 u a = z a, a = u = 1/z { { a + t e (a A 1 ) e t e 1 (dt) P (a A 1 ) f = (df) t e P = df/dt (dt) P = (a = ) e t e 1 (dt) P (a = )

22 f, P e k df 0, (df) P P, P f e 1, e 1 34 k 2 C : y 2 = f(x) (f ) P C ord P (dx), ord P (dy) 4.3 (a) C k(c) 2 (g, f), (g, f ) (g, f) (g, f ), P C ((g df) P = (g df) P ) (g, f) g df, C C Ω C h k(c), h (g df) = hg df P (g df) P = 0 g df x k(c) dx 0, x k(c) (k(c)/k(x) ), P C (dx) P 4.8 dx 0 x k(c) C y dx (y k(c)), Ω C 1 k(c) (b) ϕ : C 1 C 2 ϕ : Ω C2 g df ϕ (g df) = ϕ g d(ϕ f) Ω C1 k- k(c 1 ) k(c 2 )., 4.9 ϕ : C 1 C 2, ϕ : Ω C2 Ω C1 (c) y dx C P C (y dx) P Res P ((y dx) P ) y dx P, Res P (y dx) 4.10 ( ) y dx, P 0 0 Res P (y dx) = 0 P (d) y dx C P C (y dx) P ord P ((y dx) P ) y dx P, ord P (y dx) P y dx, P y dx div(y dx) = ord P (y dx) P, C (canonical P divisor) 4.8, (C ) 2 1 ( ), k, 1 1, 2 3, Riemann-Roch ( 5.2), 4.11 (1) P C m 2, P m ω mp (2) P m, ω 2P,, ω mp

23 Riemann-Roch (1) P, Q C 1, P, Q 1, 1 ω P Q (2) P, Q C 1, ω P Q 4.13, (e) x k(c) 4.6 C, dx ( 0) ( ), k 0 ( ), ( ) k x Q 1,, Q s, e 1 (= ord Q 1 (x)),, e s x : C P1, P 1,, P r, e 1,, e r, div(dx) = (e j 1)P j (e i + 1)Q i, deg div(dx) = (e j 1) (e i + 1) j i j i, 4.15 div(dx) = (e x (P ) 1) P 2 (x), deg div(dx) = (e x (P ) 1) 2 deg x 4.4 P 1 A 1 z k(p 1 ) z P 1, 1 1 z : P 1 P 1, 4.14 (4.15), div(dz) = 2 2, P 1 ( ) dz dz 4.14, f df 2, z k(c) 1 a A 1 div(z a) = a div(dz/(z a)) = div(dz) div(z a) = 2 (a ) = a dz/(z a) a, 1 w = 1/z dz z a = 1 d(1/w) 1/w a dw dw = 1 w aw ( w 2 ) dw = ( w 1 a a 2 w a 3 w 2 ) dw, dz/(z a) 1 a t = z a dz z a = 1 d(t + a) t dt dt = t 1 dt a dz/(z a) 1, 4.16 (1) P 1 2 P 1 (2) a P 1 x k(p 1 ) a a dx k[x] (k[x] dx) (3) 2 a, b P 1 x k(p 1 ) a b 0 a b 1 dx/x Res a (dx/x) = 1, Res b (dx/x) = 1 35 z P 1 = A 1 { } f k[z] k(x) = k(p 1 ), df, d( f 1 ), dz f,,

24 L(D) (a) C D, k(c) L(D) = {f k(c) div(f) + D 0} {0} f L(D), P C ord P (f) + ν P (D) 0 ord P (f) ν P (D) f ν P (D) > 0 P, ν P (D) ν P (D) < 0 f P ν P (D) D = P, L(P ) P 1, L(2 P ) P 2, L(P Q) P 1 Q 1, L(P 2Q) P 1 Q 2 0, (L(P 2Q)) 0 L(P 2Q) = 0 L(P Q) Q P, L(P Q) L(P Q) 1 L(P ) f L(P ) k, f P, P 1, f 1 f C P L(P ) k P C, C P (1) L(D) k (2) deg(d) < 0 L(D) = {0} (3) L(0) = k div(f) = div(g) (f, g k(c)) g = c f (c k) (4) D 1 D 2 L(D 1 ) L(D 2 ) (5) D 1 D 2 = div(f) (f k(c)) L(D 2 ) g g f L(D 1 ) k (6) L(D + P )/L(D) k L(D) k k L(D) l(d) 4.18 deg(d) < 0 l(d) = 0 l(0) = 1, l(d + P ) l(d) + 1 deg(d) = 0 l(d) 1, D l(d) = 1, l(d) = C D deg D l(d) , 4.19, 4.20 (b) C ω, C K C = div(ω) D, Ω C Ω C (D) = {ω Ω C {0} div(ω) D} {0} L(K C D) f fω Ω C (D) k Ω C (D) k, l(k C D) ( ) k Ω C (0), K C, L(K C ),, 1 k(c) P C m 0 Ω C ( m P ) L(K C + m P ), P m Ω C ( m P ) m L(K C + m P ) P, Q C 1 Ω C ( P Q) L(K C + P + Q) 37 l(k C + P ) = l(k C ), l(k C + P + Q) l(k C ) + 1

25 Riemann-Roch 39 (c) k Ω C (0) (l(k C )) C (genus) g(c) k = C, ( ), 4.16 (1), (d) 4.20 deg(d) l(d) + 1, g C r(d) = g (deg(d) l(d) + 1) D, r(d) = m, deg(d) m D l(d) = deg(d) g + 1, L(D) l(d) g, g(c) Riemann-Roch, (g = g(c)) 38 D l(d) = deg D + 1,, Riemann-Roch 4.25 (Riemann-Roch) C, K C g l(d) l(k C D) = deg D g + 1 (D C ) 4.26 (1) l(d) deg D g + 1 (2) l(k C ) = g (3) deg K C = 2g 2 (4) deg D > 2g 2, l(d) = deg D g + 1 (2) g, k (1) deg D l(d) + 1 g, (4) deg D l(d) + 1 = g D g deg D l(d) + 1 Riemann-Roch g C, 4.5 (c), (d), (4) 4.24 m = 2 g(c) 1 39 Riemann-Roch, (1) l(d) > 0 deg D 0 deg D = 0 D (2) deg D = 2 g 2 l(d) g D (3) l(d) > 0 l(k C D) > 0 l(d) 1 deg D/2 (Cliford ) 4.7 Riemann-Hurwitz (a) ϕ : C 1 C 2 ϕ : Ω C2 Ω C1 P C 1, Q = ϕ(p ) C 2 Q u Q du Q, P t P f ϕ du Q = f dt P m ϕ (P ) = ord P (f) P ϕ (differntial exponent) u Q, t P m(p ) P ϕ P (ramification divisor)

26 (1) C 2 ω, ord P (ϕ ω) e ϕ (P ) ord Q (ω) + m ϕ (P ) (2) P m ϕ (P ) e ϕ (P ), m ϕ (P ) = e ϕ (P ) (1) C 2 ω ϕ ω C 1 g(c 1 ) g(c 2 ) (2) C 1 C 2 g(c 1 ) = g(c 2 ) (b) 3 Riemann Riemann-Hurwitz, Euler Riemann-Hurwitz Euler, Riemann-Hurwitz, ( 4.27), ( 4.26 (3)), Riemann, Riemann-Hurwitz 4.29 (Riemann-Hurwitz) ϕ : C 1 C 2 2 g(c 1 ) 2 = (2 g(c 2 ) 2) deg ϕ + m ϕ (P ) P C ϕ, g(c 1 ) g(c 2 ) g(c 2 ) 2 g(c 1 ) > g(c 2 ) Riemann-Hurwitz 42 C 2 ϕ : C C, ω ϕ ω = ω, ϕ = id 43 d k C : y d = f(x) ( ) 44 k 2 3 C : y 2 = x 3 + a x + b x : C P 1 y : C P 1 Riemann-Hurwitz, C 1 5 Riemann-Roch 5.1 (a) C g C D [D] P C, Φ P : C P [P P ] Pic 0 (C) P (canonical map) d, Sym g (C) C d C d /S d d Sym d (C), d Sym d (C) d d D Φ D : Sym d (C) D [D D ] Pic 0 (C) D Riemann-Roch,

27 Riemann-Roch (1) g = 0, Div 0 (C) = Div l (C), Pic 0 (C) = 0 (2) g 1, Φ P : C Pic 0 (C) (3) g 1, g D Φ D : Sym g (C) Pic 0 (C) (b) g = 0, 1 D l(d) = deg(d) g + 1 = deg(d) D Div 0 (C), l( D) = = 1 > 0, f L( D) div(f) + ( D) 0 div(f) D 0 div(f) = D D Div 0 (C) = Div l (C) Pic 0 (C) = 0 (c) Φ P (P ) = Φ P (Q) P, Q C [P Q] = [P P ] [Q P ] = Φ P (P ) Φ P (Q) = 0, div(f) = P Q f k(c) f Q f L(Q) 4.17 L(Q) = k, f div(f) = 0 P = Q Φ P (d) D g D Div 0 (C), l(d + D) deg(d + D) g + 1 = g g + 1, f L(D + D) D 0 = div(f) + (D + D) g, D 0 Sym g (C) Φ D (D 0 ) = [D 0 D ] = [div(f) + D + D D ] = [D], Φ D : Sym g (C) Pic 0 (C) 5.2,, (a) C g C, k, 4.11, 4.12, 4.13 k(c) x, K C = div(dx) (b) P C m 1 Ω( m P ), L(K C +m P ) y y dx Ω( m P ), L(K C + m P ) l(k C + m P ) = deg(k C + m P ) g + 1 = 2g 2 + m g + 1 = g + m 1, m 2, l(k C + m P ) > l(k C + (m 1) P ) y m L(K C + m P ) L(K C + (m 1)P ) y m dx P m (c) P, Q C 1 Ω C ( P Q), L(K C + P + Q) y y dx Ω C ( P Q), L(K C + P + Q) l(k C + P + Q) = deg(k C + P + Q) g + 1 = 2g g + 1 = g + 1 > g = l(k C ), y P Q L(K C + P + Q) L(K C ) y P Q dx P, Q 1,, P 1 Q 1, y P Q dx P, Q 1, 1 (d) ω C ω div(ω) P C, m = ord P (ω) t P 4.12 ω jp T j Laurent b 2,, b m k, ω (b m ω mp + + b 2 ω 2P ) Laurent t m t 2 ω P 1, ω = ω ( ) P 1,, P r 1 P j ω c j c c r = 0 P j, P r 1, 1

28 42 ω j, ω = ω (c 1 ω c r 1 ω r 1 ) ω P 1,, P r 1 0, P 1,, P r 1 P r c c r = 0, P r ω (a) C 0 D l(d) = deg D + 1 P C l(p ) = 2 > l(0) = 1 x L(P ) k x x P, deg(x) = 1 x : C P 1 C P 1 (b) x 2 2 P, L(2 P ) l(2 P ) = 3, L(2 P ) 1, x, x 2 L(m P ) m x m 0 L(m P ) = k[x] a A 1 P 1, x(p a ) = a P a C x(p ) =, P a a P 1 a P a x : C P 1, C = {P a a P 1 } x(p 0 ) = 0 P 0 x div(x) = P 0 P 1/x L(P 0 ) (x a)(p a ) = x(p a ) a = 0 x a L(P ), div(x a) = P a P 1/(x a) L(P a ) a A 1 P 1 t a = x a P a 1 P a a = t a = 1/x P a A 1 u a = t a = x a, a = u a = 1 a, b P 1, div(u b /u a ) = P b P a f k(c) div(f) = P b1 + + P br P a1 P ar f 0 = u b1 u br /u a1 u ar div(f) f f 0 k(c) = k(x) (c) C, k(c) = k(x) k(x), Aut(C) Aut(k(x)/k) PSL 2 (k) k = C σ : x 1/x, τ : x (x 1)/x PSL 2 (C) 3 S 3 C(C) = C(x) S 3 K = C(x) S3 K C 0, C 0 K 6 C C 0 C 0 0 u = x + τ x + τ 2 x k(x) u τ u C(x) τ [C(x) : C(u)] = 3 C(x) τ = C(u) C(x) = C(u)(x), x C(u) 3 X 3 u X 2 + (u 3) X + 1 = 0 Shanks 3, 3 u + σ u, u σ u S 3 - K = C(x) S3 u + σ u = 3 ( ), w = u σ u C [C(x) : C(w)] = 6 K = C(w) C(x) = C(w)(x), x C(w) X 6 3 X 5 + (w 3) X 4 (2 w 11) X 3 + (w 3) X 2 3 X + 1 = 0 S 3 - u C(w) U 2 3 U + w = 0, C(u) x u ( ), 6 45 σ : x 2/x, τ : x 2 (x 1)/x PSL 2 (C) 4 D 4 D 4 - C(x)/C(x) D4 46 P 1 4 {0, 1, 1, } PSL 2 (C) G, C(P 1 )/C(P 1 ) G G 2 H C(P 1 )/C(P 1 ) H

29 Riemann-Roch (a) C 1, P C l(0) = 1, l(n P ) = n (n 1), L(0) = L(P ) = k L(2 P ) k l(2 P ) = 2, x L(2 P ) k L(2 P ) 1, x x 2 P l(3 P ) = 3, y L(3 P ) L(2 P ) L(3 P ) 1, x, y y 3 P x 2 4 P, L(4 P ) 1, x, y, x 2 0, 2 P, 3 P, 4 P k 1, x, y, x 2 4 k L(4 P ). L(5 P ) 1, x, y, x 2, x y, L(6 P ) 1, x, y, x 2, xy, x 3, m 0 L(m P ) = k[x] + k[x] y (b) y 2 6 P y 2 L(6 P ) 6 k L(6 P ) 1, x, x 2, x 3, y, xy, y 2 y 2 +a 1 x y+a 3 y = a 0 x 3 +a 2 x 2 +a 4 x+a 6 a 0, a 1,, a 6 k 6 P y 2 x 3, x 3 a 0 x, y, (0 ) x, y a 0, x 3 1 (c) C x, y, 3 E : Y 2 + a 1 X Y + a 3 Y = X 3 + a 2 X 2 + a 4 X + a 6, Weierstrass, E Weierstrass 5.2 (1) C x, y E (2) ϕ : C P (x(p ), y(p )) E, C Ē (E ), Y 2 + a 1 X Y + a 3 Y = X 3 + a 2 X 2 + a 4 X + a 6 C ( E ) Weierstrass E X, Y C E ϕ X = x, ϕ Y = y, k(c) x, y, 5.3 C 1 (1) P C, 2 P x, 3 P y (2) L(m P ) 1, x,, x [m/2], y, xy,, x [(m 3)/2] y (3) x, y y 2 + a 1 x y + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 a 1,, a 6 k (4) C k x, y k(c) = k(x, y) (a) Φ : C P [P P ] Pic 0 (C), 5.1 (2), (3) Φ, Pic 0 (C) C P, Q C Φ(P ) + Φ(Q) = Φ(R) R C P Q := R C, C P D = P +Q P deg(d) = 1 l(d) = 1 f L(D) div(f)+d 1, div(f) + D = R (R C) R P D P P + Q 2P = P P + Q P, Φ(R) = Φ(P ) + Φ(Q) L(P + Q P )

30 44 (b) L(P + Q P ) P, Q 1 L(P + Q) l(p + Q) = 2, L(P + Q), P P, Q L(P + Q), P Q = R R x L(2 P ) k 2 Galois x : C P 1 Galois G x ι P C P = ι(p ) P C (P P ) x x(p ) P x(p ) = x(p ) P x x(p ) P x (P P ), div(x x(p )) = P +P 2 P P x, P 2 x x(p ) P 2 div(x x(p )) = 2 P 2 P div(x x(p )) = P +P 2 P [P P ] = [P P ] Φ(P ) = Φ(P ) P x 2 Φ(P ) = 0. Φ(P ) + Φ(Q) = Φ(R), Φ(R ) = Φ(R) Φ(P ) + Φ(Q) + Φ(R ) = 0 P + Q + R 3 P 0 f L(3 P ) div(f) = P + Q + R 3 P. L(3 P ) 1, x, y f x, y 1 C Ē C x, y X, Y, x, y 1 f = 0 f = 0, f P, Q, R f = 0 P, Q C l P Q P = Q l P C l C P, Q, R Φ(P ) + Φ(Q) = Φ(R ), P Q = R. R x(r ) = x(r), x x(r) = 0 C (c) x(p ) x(q) L(P +Q P ) P, Q a+b x+c y = 0, div(f) > P + Q 3 P a + b x + c y div( (x x(p )) (x x(q)) ) > P P Q, h = (a + b x + c y)/(x x(p ))(x x(q)) L(P + Q P ) P, P t = x x(p ) h t a + b x(p ) + c y(p ) h = t x(p ) x(q) 1 + P, P t 2 = x x(p ) y = y(p ) + b 1 t + t, h = c b 1 x(p ) x(q) t 1 + t 1, h P x(p ) = x(q) P, Q, L(P + Q P ), C (d), 1,, 1, 1, 1 0 2,, E O E, (E, O) E Weierestrass, O = [0 : 1 : 0], E 5.6 1

31 Riemann-Roch 45 (a) Riemann-Roch,, 1 k 5.2, Weierstrass ( 1 ),,, E : y 2 + a 1 x y + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 x : E P 1 2 Galois, Galois G x ι ι(p ) = P x ι = x x(p ) = x(p ) y(p ), y(p ) y 2 y 2 +a 1 x(p ) y+a 3 y = x(p ) 3 + 2, y(p ) + y(p ) = a 1 x(p ) a 3 y(p ) = y(p ) a 1 x(p ) a 3, ι : E P = (x, y) P = (x, y a 1 x a 3 ) E P x (P = P ), y(p ) = y(p ) y 2 y 2 +a 1 x(p ) y+a 3 y = x(p ) 3 + k 2, P x- (a 1 x + a 3 ) 2 4(x 3 + a 2 x 2 + a 4 x + a 6 ) = 0, y- y = (a 1 x(p ) + a 3 )/2 k 2, x- a 1 x + a 3 = 0 k 2 y, (y ) 2 = (x 3 ) x, 3 x 3 (P 1, P 2, P 3 ), P = [0 : 1 : 0] 49 Riemann-Hurwitz, 2 x : E P 1, k 2 4, k 2 2 (b) k 2 x P, 2 x : C P 1, P 1, P 2, P , 4.3, E dx, (dx) = P 1 + P 2 + P 3 3 P, 1 deg div(dx) = 0 = 2 g(e) 2, y 0 = 2 y + a 1 x + a 3 L(3 P ) y 0 E y, E y y 0 x, div(y 0 ) = P 1 + P 2 + P 3 3 P ω E = dx/y 0 ω E E Néron div(ω E ) = div(dx) div(y 0 ) = 0, E ω E E ω div(ω) = 0, ω/ω E E 1 k f k(e) div(f ω E ) = div(f) P, P f f ω E P m 0 L(m P ) = k[x] + k[x] y, P (k[x] + k[x] y) ω E P E P l(0) = l(p ) = 1 < l(2 P ) = 2 < l(3 P ) = 3, P x P, y P, P 2 3 P, P (k[x P ] + k[x P ] y P ) ω E P, Q C (P Q) 1, L(P + Q) ω E L(P + Q) 5.5 (c), P, Q 1 L(P + Q) ω E l(p + Q) = 2, P, Q 1 k 2 50 P P x P = 1/(x x(p )) L(2 P ), y P = (y 0 y 0 (P ))/(x x(p )) 2 L(3 P ) (c) k 3 ( 2 ) 3 y dy x 0 = 3x 2 + 2a 2 x + a 4 a 1 y (E x )

32 46 div(dy) = div(x 0 ) dy/x 0, ( 2 ) ω E E dx/y 0 = dy/x 0 2 ω E = dy/x 0 51 (1) k 2, 3 dx/y 0 = dy/x 0 (2) ω E, E (d), E 5.4 E Weierstrass 1 (1) dx/(2y + a 1 x + a 3 ), dy/(3 x a 2 x + a 4 a 1 y) E, (2) ω E div(ω E ) = 0 (3) E k ω E 1 k (4) x P, y P P E, P 2, 3 (P = P, x P = x, y P = y ). P (k[x P ]+k[x P ] y P ) ω E (5) P, Q C (P Q) 1 L(P + Q) ω E P, Q 1 2 k 5.7 Weierstrass (a) C g, P C P, P n L(n P ), P n L(n P ) P n, L(n P ) L((n 1) P ), l(n P ) > l((n 1) P ) g = 0 P g = 1, n 2, P n 5.5 (Weierstrass) C g 2 (g 2), P C (1) P (2) g n 1,, n g, n, P, P n (b) l((g + 1) P ) g + 1 g + 1 = 2 l((g + 1) P ) 2, f L((g + 1) P ) f, f P, f P (c) P n, l(n P ) > l((n 1) P ), l(n P ) = l((n 1) P ) l(n P ) = n g + 1 (n > 2g 2), n 2g l(n P ) > l((n 1) P ) P n ( 2g) l((2g 1) P ) = 2g 1 g + 1 = g, 1 = l(0) l(p ) l(2 P ) l((2g 2) P ) l((2g 1) P ) = g l(n P ) l((n 1) P ) 0 1, 2g 1 g 1 1 g 0 l(n P ) = l((n 1) P ) n n 1,, n g, 5.5 (2) (d) l(n P ) = l((n 1) P ) n P (gap value), {n 1,, n g } (n 1 < < n g ) P (gap sequence) P

33 Riemann-Roch 47, {1,, g} {1,, g} P C Weierstrass g = 0 C, g = 1 {1} 0 1 Weierstrass C C Weierstrass, Hurwitz, Weierstrass Φ D : Sym g (C) Pic 0 (C), 1 1, Sym g (C) ( ) (E ) Sym g (C) C C, Weierstrass C E Weierstrass C E Bézout C E, Weierstrass (e) k 0 C ω 1,, ω g, Wronski (Wronskian) W (ω 1,, ω g ) P C t, ω i = f i dz m = g(g + 1)/2, f (j) i f i t j W (ω 1,, ω g ) P = det(f (j) i ) i,j (dt) m P m ( m ) W (ω 1,, ω g ) = (W (ω 1,, ω g ) P ), Wronski m ( m ) Wronski W (ω 1,, ω g ) W C, Weierstrass 5.7 (1) Weierstrass W C (2) deg(w C ) = g(g 2 1), P C ν P (W C ) = n n g m (3) W C (W C 0), Weierstrass Weierstrass, g(g 2 1) g(g 2 1), {1, 2,, g 1, g + 1} Weierstrass, 2 g, g N {n 1,, n g } (1 n 1 < < n g < 2 g), n n g g 2 Weierstrass P ν P (W C ) m = g(g + 1)/2, ν P (W C ) g(g 1)/2, Weierstrass 2g (Hurwitz) Weierstrass 2g + 2, g(g 2 1) Weierstrass g(g 2 1), 2g + 2 g = 2 2g + 2 = 6, g(g 2 1) = 6, 2 Riemann-Roch, ( 5.12) , (a) C g ( 2), Weierstrass w C Aut(C), C Weierstrass

34 k 2 C Weierstrass, 5.12 (Hurwitz) 2 g + 3 ϕ : C C 5.13 Aut(C) S w, C C, : f 5.12 P C h L((g + 1) P ) k h 0 = h ϕ h, h 0 = 0 f(p ) = P (b) C C k(c), Aut(C) k(c) G Aut(C) d k(c) G- k(c) G k(c)/k, [k(c) : k(c) G ] = G < ϕ (C G ) = k(c) G C G ϕ : C C G C G G C (quotient curve) ϕ : C C G Galois G Galois ϕ P 1,, P s C G ϕ Galois, P j ϕ (e j ), e j ϕ d. Riemann-Hurwitz 2 g 2 d (2 g(c G ) 2 + (1 e 1 1 ) + + (1 e 1 s )) k 0,, g ( 2) g(c G ), e 1,, e s, 5.15 (Hurwitz) k 0 g 2, Aut(C) 84 (g 1) 54 k 0, 84 (g 1), 48 (g 1), 40 (g 1), ( ),, C 55 k 0 C 2, 48 (, 84 ) 5.9 (a) g ( 2) C P 1 2 ( 2 ) x : C P 1 2 x C D D 2 l(d) 2 L(D) 1, x, l(d) = 2 1, x m g l(m D) = deg(m D) g + 1 = 2m g + 1, l(g D) = g + 1 x m L(m D) L((m 1) D) (m 1), l(0) = 1 < l(d) = 2 < l(2 D) < l(3 D) < < l((g 1) D) < l(g D) = g m g l(m D) = m + 1 L(m D) m x. l((g + 1) D) = g + 3 = l(g D) + 2, L((g + 1) D) g + 1 x y L((g + 1) D),

35 Riemann-Roch L(m D) = { k + k x + + k x m (1 m g) k + k x + + k x m + k y + k xy k x m g 1 y (m g + 1) y 2 2(g + 1) D y 2 L(2(g + 1) D) g + 1 a 1 k[x] 2g + 2 a 2 k[x] y 2 + a 1 (x) y + a 2 (x) = C, C 0 : y 2 + a 1 (x) y + a 2 (x) = 0 56 k 2 (1) C 0 : y 2 + a 1 (x) y + a 2 (x) = 0 (2) g C x, y C 0 (b) a 1, a 2 k[x] deg a 1 g + 1, deg a 2 2g + 2, C : y 2 + a 1 (x) y + a 2 (x) = 0 C : v 2 + (u g+1 a 1 (1/u)) v + u 2g+2 a 2 (1/u) = 0, C ϕ : C (x, y) (1/x, y/x g+1 ) C, ϕ C C Ĉ = C C C C x-, C u- 0 a 1 (x) x g+1 c 1, a 2 (x) x 2g+2 c 2 C u- 0 v- v 2 + c 1 v + c 2 = 0 k, C c 2 1 4c 2 = 0 1, c 2 1 4c Galois x : C P 1 Galois ι : C P = (x, y) P = (x, y a 1 (x)) C ι u- 0 P = (0, v 0 ) C, ι P = (0, v 0 c 1 ) C (c), k 2 C, f(x) k[x] y 2 = f(x) C, f 1, f 2 d Q 1,, Q d x- x(q 1 ),, x(q d ) f(x) = 0, α 1,, α d Q j t 2 j = x α j (t j k(c) P ) Q j y 0 f(x) t j, f(x) = b 1 t 2 j + b 2 4 j + (b 1 0) y 2 = f(x) y = b 1 t j + ( ) ±, t j ±, y ± ord Qj (x α j ) = 2, ord Qj (y) = 1 (x ) P C, a = x(p ) A 1 x- a P P x- P, P P a, P a x P a P a t Pa = x a P a y- y(p a ) 0 f(x) t Pa, f(x) = b 0 + b 1 t Pa + (b 0 0) (y(p )) 2 = f(x(p ) = f(a) = b 0, y = y(p a ) + ( ) ord Pa (x a) = 1, ord Pa (y) = 0 d P, u, v- (0, 0) x(p ) = (1/u)(P ) = 1/u(P ) P x, (y/x m )(P ) = u(p ) = 0, y P ord P (y) x m P m ord P (x), P 2 x, P t 2 = 1/x (t k(c) P ) y 2 = f(x) = a 0 t 2d + (a 0 0), y = a 0 t d + ( ) ord P (x) = 2, ord P (y) = d d 2, u, v- (0, ± a 0 ) (0, a 0 ) P, (0, a 0 ) P u = y/xm, P a0, P a 0, P 2 x,

36 50 P t = 1/x (t k(c) P ) y 2 = f(x) = a 0 t d + (a 0 0), y = a 0 t d/2 + ( ) ord P (x) = 1, ord P (y) = d/ (1) d = deg f, div(x) = P 0 + P 0 2 P, div(y) = Q Q d d P (2) d = deg f, div(x) = P 0 + P 0 (P + P ), div(y) = Q Q d (d/2) (P + P ) (d) d = deg f ( d = 2g+1), P P,, k[x, y] y 2 = f(x), k[x]+k[x] y y L((2g +1) P ) n 2g L(n P ) k[x] x L(2 P ) L(0) = L(P ) = k, L(2 P ) = L(3 P ) = k + k x,, L((2g 2) P ) = k + k x + + k x g 1, L(g P ) = k + k x + + k x g n P (n 2g + 1 = d), n x n/2, n x (n d)/2 y. P {1, 3, 5,, 2g 1} d = deg f ( d = 2g +2), P L(n P ) l(n P ), L(n(P + P )) ( L(n P )) P P,, x, y k[x, y] y = f(x), k[x] + k[x] y y L((g + 1)(P + P )), L(n (P + P )) k[x] (n g) k[x] ι, k[x] P P. P k[x] P L(n P ) = k (n g) g, P {1, 2, 3,, g} C d P, d P x P C {1, 3, 5,, 2g 1} {1, 2,, g} C Weierstrass 2g , RAut(C) = Aut(C)/ ι C (reduced automorphism group) 5.20 RAut(C),,, S 2g+2 (e) k 2 C g ( 2), x : C P 1 2 (a) L(m D) (m 1, D x ), C Weierstrass, x 2 Galois, 1 2 Riemann-Hurwitz, 2g + 2 a 1,, a 2g+2 P 1 x 5.21 ) C C 0 : y 2 = (x a 1 ) (x a 2 ) (x a 2g+2 ) ( a j = 5.10 (a) k 2, C : y 2 = f(x) g ( 2) f k[x], d = 2g + 1, 2g + 2 dx 2 Galois x : C P 1 Q 1,, Q d, d P x, d P 2 d 2 P, P 1

37 Riemann-Roch div(dx) = { Q Q d 3 P Q Q d 2 P 2 P (d ) (d ) d (P ), Q d+1 = P d d + 1 = 2g + 2, d d = 2g + 2 d Q 1,, Q 2g+2 x d P = P, dx d div(dx) = Q Q 2g+2 2 P 2 P (b) y div(y) = Q Q 2g+2 (g + 1) (P + P ), div( dx y ) = (g 1) (P + P ) div(x) = P 0 + P 0 P P, 0 n g 1 div(x n dx y ) = n(p 0 + P 0) + (g 1 n) (P + P ) 5.23 C, dx y, x dx y,, xg 1 dx y g k- (c) P (P = P ), P k[x] + k[x] y ω P, ω/ dx y P P (k[x] + k[x] y) dx y P (P P ), P ω ord P (ω) = n (n 1), f ω = ω/ dx y n P + (g 1) (P + P ) n = 1, f ω g (P + P ) L(g (P + P )) k[x], f ω P P P 1, n = 2, f ω (g + 1) P + (g 1) P, f ω L((g + 1) (P + P )) = k + k x + + k x g+1 + k y P P u = 1/x, v = y/x g+1 (0, a 0 ) P (0, a 0 ) P P y a 0 x g+1 P t = 1/x y Laurent y = a 0 t g 1 + b 1 t g + b 2 t g+1 + f P = y + a 0 t g 1 b 1 t g, h L(2 P + (g 1) (P + P )) f dx P y P 2 n 2 l(n P + (g 1) (P + P )) = g + n 1 l((g 1) (P + P )) = l(p + (g 1) (P + P )) = g, n ( 2), P n P n g + n 1 k- (d) 2 P, Q C l(p + Q + (g 1) (P + P )) = 2g g + 1 = g + 1 l((g 1)(P + P )) = g, f P Q L(P + Q + (g 1) (P + P )) L((g 1) (P + P )) ω P Q = f dx P Q y P, Q 1 P, Q 1 g + 1 k- p = x(p ), q = x(q) P, Q, p q, p, q, ω P Q f P Q, h P Q = f P Q (x p)(x q), div(h P Q ) P Q + (g + 1)(P + P ) h P Q L((g + 1) (P + P )) = k + k x + + k x g+1 + k y P, Q P, Q h P Q = a + b x + c y = 0. f P Q = h P Q /(x p)(x q) f P Q L(P + Q (g 1)(P + P )), ω P Q = f dx P Q y P, Q 1 57 ( ), P, Q 1 P, Q,

38 (a) C g, k 2, C P Weierstrass (P = P ) C, 2g + 1 f(x) k[x], C : y 2 = f(x) g P Φ = Φ gp : Sym g (C) Pic 0 (C) 5.1 (3) 5.24 Φ (b) P C P P u P = x x(p ), u P = 1 D = n P P, D = n P P u D = u np P D d, 5.25 div(u D ) = D + D 2d P D 1, D 2 Sym g (C) L(D 1 D 2 ) h h u D1 Φ(D 1 ) = Φ(D 2 ) l(d 1 D 2 ) > 0 L(2g P D 1 D 2 ), l(2g P D 1 D 2 ) = l(d 1 D 2 ) L(2g P D 1 D 2) L(2g P ) = k + k x + + k x g k[x], 5.26 D 1, D 2 Sym g (C) Φ(D 1 ) = Φ(D 2 ), 2g D 1 + D 2 g D Sym g (C) D 1 +D 2 = D + D 5.27 g = 2, Φ = Φ 2P Φ 1 (0) = {P + P P C} P 1 (c) Φ : Sym g (C) Pic 0 (C), Pic 0 (C) Sym g (C) D 1, D 2 Sym g (C) D = 3 g P D 1 D 2, D g l(d) deg D g+1 = g g+1 = 1 > 0, h L(D) D 3 = div(h)+d, D 3 g D 3 Sym g (C) l(d) = 1 h D 3 l(d) > 1 D 3 Sym g (C) h, h, D 3, Φ(D 1 ) + Φ(D 2 ) + Φ(D 3) = [D 1 g P ] + [D 2 g P ] + [D 3 g P ] = [D 1 + D 2 + D 3 3 g P ] = [div(h)] = 0, Φ(D 3 ) = Φ(D 1 ) Φ(D 2 ) = Φ(D 1) + Φ(D 2 ) 5.15, D 3 P + P 2 P D 4, Φ(D 4 ) = Φ(D 3 ), D 4 Sym g (C) h D 1, D 2 Sym g (C) D 1 D 2 = D 4, Sym g (C) (d), L(D) (D = 3 g P D 1 D 2 ) h L(D) L(3 g P ), D 1 + D 2 h L(3 g P ) L(3 g P ) h 1 (x) + h 2 (x) y (h 1, h 2 k[x], deg h 1 3g/2, deg h 2 (g 1)/2) D 1 + D 2, h 1 (x) + h 2 (x) y 2g + 1, D 1 + D 2 x- 2 g (P P 2g ) P 1,, P 2g h 1 (x) + h 2 (x) y = 0, 2g (2g + 1), h = h 1 (x) + h 2 (x) y, h L(D) (e), 2, 2, (ambig class) 2. D Sym g (C), Φ(D) = [D g P ] = [D g P ] = Φ(D ) = Φ(D), Φ(D) (Φ(D) = Φ(D)) 2 Φ(D) = 0 Pic 0 (C) 2 [P P ] = [P P ] = [P P ], P Weierstrass 2 [P P ] = 0 P Weierstrass Q 1,, Q 2g+1, [Q 1 P ],, [Q 2g+1 P ] 2

39 Riemann-Roch 53 div(y) = Q Q 2g+1 (2g + 1) P [Q 1 P ] + + [Q 2g+1 P ] = 0 [Q 1 P ],, [Q 2g+1 P ] m (1 m 2g), [Q P ] + + [Q P ] = [(Q + + Q ) m P ] L(m P ) L(2g P ) = k + k x + + k x g L(m P ) Q j (Q + + Q ) m P [Q 1 P ],, [Q 2g P ] 2 2g (2,, 2) 5.28 g ( 2) C, Pic 0 (C) 2, 2 2g 2- Q 1,, Q 2g+2 C Weierstrass, [Q i Q 2g+2 ] (1 i 2g) (a) C 2, K C ( ) P C l(0) = l(p ) = 1, m 3 l(m P ) = m 1 l(2 P ) 1 2 l(2 P ) = 2 {1, 3}, P Weierstrass L(2 P ) x x 2, C l(2 P ) = 1 {1, 2}, P Weierstrass l(p + P ) = 2 P C L(P + P ) 2, C , (b) C 2, 2 k ω 1, ω 2 P ω 1 h = ω 2 /ω 1, div(ω 1 ), div(ω 2 ) h L(div(ω 1 )) k = L(0) L(div(ω 1 )), L(div(ω 1 )) k + k h P h, a = h(p ) k ω P = (h a) ω 1, ω P P L(div(ω 1 ) P ) L(div(ω 1 )) 5.30 P C, P ω P 5.31 (1) ι : C P P = div(ω P ) P C C (2) P C Weierstrass, ι- (P = P ) (3) Φ KC : Sym 2 (C) Pic 0 (C), Sym 2 (C) { } Pic 0 (C) {0} (4) {P + P P C} (5) P + P x, ι Galois x : C P 1 Galois (6) C, ι (7) C ι , y 2 + a 1 (x) y + f 2 (x) = 0 (a 1, a 2 k[x], deg a 1 3, deg a 2 6) 58 C 2 P C, L(m (P + P )) (m = 1,, 5) L(2 (P + P ) + P ), (c) C 2 K C Φ KC : Sym 2 (C) Pic 0 (C) P +Q Sym 2 (C) P +Q K C = (P Q )+(Q+Q K C ) P Q Φ KC (P + Q) = [P Q ] [P Q ] K C

40 Ψ : C C (P, Q) [P Q] Pic 0 (C), Ψ 1 (0) = {(P, P ) C C} C 59, C 2 C RAut(C) P 1 (1 ), Weierstrass 6 k 0 2 C, ,15 48, 24 Weierstrass 6, 6 S 6 24, 2 x : C P 1, C Weierstrass P 1,, P 6 3 x(p 1 ) = 0, x(p 2 ) = 1, x(p 3 ) = x(p 4 ) = a, x(p 5 ) = b, x(p 6 ) = c, {0, 1,, a, b, c}. τ : z 1 1 z 3, 0, 1, b = 1 1 a, c = a a 1, τ(a) = b, τ(b) = c, τ(c) = a, τ {0, 1,, a, 1 1 a, a a 1} 3 C a : y 2 = x (x 1) (x a) (x 1 1 a ) (x a a 1) τ Z/3Z, 2 σ : z a z z a a + 1 {0, 1,, a, 1 1 a, a a 1} 2 σ C a 2 στ = τ 2 σ, σ, τ S 3 a = 2, 2 σ 2 : z z 2 2z 1 {0, 1,, 2, 1, 1/2} 2 σ 2 τ, (σ 2 τ) 5 σ = σ (σ 2 τ), σ, τ, σ 2 12 D 12 C 2 : y 2 = x (x 2 1) (x 2) (x 1/2) D 12 a = 1 4 σ 4 : z 1 z 1 1 {0, 1,, 1, (1 + 1)/2, 1 + 1}, σ, τ, σ 4 4 S 4 C 1 : y2 = x (x 1) (x 1) (x ) (x 1 1) S 4 S 4 4! = 24, 2 4, D 8, S {1}, Z/2Z, Z/5Z, D 8, S 3, D 12, S 4 2 k 0,

41 Riemann-Roch X 0 (23) (a) SL 2 (Z) H H = H P 1 (Q) SL 2 (Z) Γ 0 (23) = { ( a b) SL2 (Z) c 0 mod 23} c d Y 0 (23) = Γ 0 (23)\H Riemann X 0 (23) = Γ 0 (23)\H Riemann, Y 0 (23) Y 0 (23) C, X 0 (23) τ H X 0 (23) (τ) Γ 0 (23)\P 1 (Q) X 0 (23) (cusp) X 0 (23) (i ) (0) 2 P 1 (Q) i, H P 1 (Q) i X 0 (23) Γ 0 (23) (modular curve) X 0 (23) 2 Γ 0 (23) 2 (cusp form), X 0 (23) Dedekind η- η- H, η(τ) = q 1/24 n=1 η(τ + 1) = e 2πi/24 η(τ), (1 q n ) (q = e 2πiτ ) η( 1/τ) = i τ η(τ) 5.36 (1) f 23 (τ) = η(τ) 2 η(23τ) 2 Γ 0 (23) 2 (2) Hecke T 2 f 23 T 2 (τ), 2πi f 23 (τ) dτ, 2πi f 23 T 2 (τ) τ, Γ 0 (23) 2 C (b) X 0 (23) 2πi f 23 (τ) dτ, Y 0 (23) X 0 (23) (i ) q = e 2πiτ 2πi f 23 (τ) dτ, 2πi f 23 (τ) dτ dq = q 2q2 q 3 + 2q 4 + q 5 + 2q 6 2q 7 2q 9 2q 10 + q q q 15 +, (i ) 1 Riemann-Roch 2, (0) 2πi f 23 (τ) dτ 1 X 0 (23) div(2πi f 23 (τ) dτ) = (i ) + (0). (i ) + (0), (i ) (0), 2πi f 23 T 2 (τ) dτ, 2πi f 23 T 2 (τ) dτ dq = 1 q + q2 2q 4 3q 5 + q 7 + 2q 8 + 4q 9 2q q q q 13 + (i ) (0) 2πi f 23 (τ) dτ 2πi f 23 T 2 (τ) dτ K = (i ) + (0) x = 2πi f 23 T 2 (τ) dτ/2πi f 23 (τ) dτ, x L(K), (i ) q x, x = 2πi f 23 T 2 (τ) dτ = f 23 T 2 (τ) = q 2πi f 23 (τ) dτ f 23 (τ) q + 7q q q q q q 7 +, x (i ) 1 X 0 (23) y L(3 K), X 0 (23) y 2 = (x 6 ) X 0 (23) dx y, x dx y div( dx y ) = K, y dx y = 2πi f 23(τ) dτ x x dx y = 2πi f 23 T 2 (τ) dτ, y q y = dx 2πi f 23 (τ) dτ = q f 23 (τ) dx dq = q 3 2q 2 q q + 228q q q 4 + x, y y 2 = (x 6 ) y 2 (x 6 ) = 0 q- 0 x x, y y 2 = x 6 2x 5 23x 4 50x 3 58x 2 32x 11 x x 1, 5.37 Y 0 (23) y 2 = x 6 8x 5 + 2x 4 + 2x 3 11x x 7, X 0 (23) X 0 (23) ( ) A 0 (23) C

42 56 x, y 61 x = f 23 T 2 (τ)/f 23 (τ), y = dx/2πi f 23 (τ) dτ q-, x f(x) y 2 f(x) q- y 2 f(x) = 0 62 Hecke T 2, 2 2 C (f 23 (τ), f 23 T 2 (τ) ) (f 23 T 2 ) T 2 (τ) = f 23 (τ) f 23 T 2 (τ), T 2 63 [P P ] Pic0 (X 0 (23)) 2, 3, 4, X 0 (22) (a), Γ 0 (22) X 0 (22) = Γ 0 (22)\H Riemann X 0 (22) 2 f 11 (τ) = η(τ) 2 η(11τ) 2 = q 2q 2 q 3 + 2q 4 + q 5 + 2q 6 2q 7 + Γ 0 (11) 2, Γ 0 (22) 2 f 22 (τ) = f 11 (2τ) = η(2τ) 2 η(22τ) 2 = q 2 2q 4 q 6 + 2q 8 + q q 12 2q 14 + Γ 0 (22) 2, f 11 (τ) 5.38 (1) X 0 (22) (i ), (0), (1/2), (1/11) 4 (2) X 0 (22) 2πi f 11 (τ) dτ 2πi f 22 (τ) dτ (0) + (1/11) (i ) + (1/2) x = f 11 (τ)/f 22 (τ), y = dx/2πi f 22 (τ) dτ, x L(K), y L(3 K) (K = (i ) + (1/2)), y 2 = (x 6 ) 5.39 X 0 (22) y 2 = x x x x x x , A 0 (22) C x, y X 0 (22) (i ), (1/2) 2, (0), (1/11) (0, 8), (0, 8) (i ) y = x 3 6x 2 10x x 1 88x x 3 + 1/x Laurent 64 (1) τ H (i ), (1/11), (0), (1/2) X 0 (22), q= exp(2πi τ), q 11 = exp(2πi (τ/(1 11τ)) /2), q 0 = exp(2πi ( 1/τ) /22), q 2 = exp(2πi (τ/(1 2τ)) /11) (2) x, y, (b) C = X 0 (22), y 2 = x 6 +12x 5 +56x x x x+64 x, y ι : (x, y) (x, y) (y/8) 2 = (x/2) 6 + 6(x/2) (x/2) /2(x/2) (x/2) 2 + 6(x/2) + 1, C Weierstrass x- x 4/x C σ, σ σ : (x, y) (4/x, 8y/x 3 ), σ : (x, y) (4/x, 8y/x 3 ) σ 2 = id, σ = σι = ισ σ ι 2 C Aut(C) (2, 2)- ι, σ Aut(C) σ, σι, ι C σ, C σ,

43 Riemann-Roch 57 C ι ι 2 x : C P 1, C ι P 1 C σ, C σ f : C C σ, g : C C σ 2 Galois, Galois σ, σι. C σ, C σ 2, C σ 0 (C σ P 1 ), σ X 0 (22) 2, C σ P 1 C σ 1 C σ 1 f : C C σ, g : C C σ, f : Pic 0 (C σ ) Pic 0 (C) f : Pic 0 (C) Pic 0 (C σ ) g : Pic 0 (C σ ) Pic0 (C) g : Pic 0 (C) Pic 0 (C σ ) f, g, f f Pic 0 (C σ ), g g Pic 0 (C σ ) 2- f, g 2 1 2, 2 P 1 C f, G f = σ σ(p 1 ) = P 1 σ(p 1 ) = σ(ι(p 1)) = σι(p 1 ) = ι σ(p 1 ) = ι(σ(p 1 )) = ι(p 1 ) = P 1, P 1 g P 2, P 2 C P 1, P 1 = ( 2, ±4 2), P 2, P 2 = (2, ±44 2) 5.40 (1) P C f(σι(p )) = f(p ), g(σ(p )) = g(p ) (2) ker f = f [P 2 P 2], ker g = g [P 1 P 1] 2 (3) f g, g f 5.41 ϕ = (f, g ) : Pic 0 (C) [D] (f [D], g [D]) Pic 2 (C σ ) Pic 0 (C σ) ψ = f + g : Pic 0 (C σ ) Pic 0 (C σ ) ([D 1], [D 2 ]) f [D 1 ] + g [D 2 ] Pic 0 (C), ψ ϕ Pic 0 (C σ ) Pic 0 (C σ), ϕ ψ Pic 0 (C) 2- (c) C σ C(C σ ) A 0 (22) = C(x, y) σ - σ(x) = 4/x, σ(y) = 8 y/x 3, w = x + 4/x, z = y (x 2)/x 2 σ- A 0 (22) A 0 (22) σ = C(C σ ) C(w, z), [A 0 (22) : C(C σ )] = [A 0 (22) : C(w, z)] = 2 C(C σ ) = C(w, z) C σ z 2 = (w 4) (w w w + 52), 2.6 (d) 2, C σ v 2 1 = u u u u 1 = 484/(w 4) 2 = 484 x/(x 2) 2, v 1 = 484 z 1 /(w 4) 2 = 484 y/(x 2) 3,,, Weierstrass u = 121 x/(x 2) , v = (121 y/(x 2) 3 1)/2, f = (u, v) : X 0 (22) P (u(p ), v(p )) P 2 Weierstrass v 2 + v = u 3 u 2 10 u 20, C σ, X 0 (11), C σ X 0 (11) u, v i X 0 (22) q = e 2πiτ, u = q q q q q q q 7 + v = q 5687 q q q q q q 7 C σ ω = du/(2v + 1) ω = ( q + q q 4 q q 7 4 q q 9 q 11 2 q 12 4 q 13 + q q q 17 + ) dq q q Γ 0 (11) 2 f 11 (τ) = η(τ) 2 η(11τ) 2 q-, ω = 2πi (f 11 (τ) + 2 f 22 (τ)) dτ

44 58, X 0 (11), f 11 C σ X 0 (11), Weierstrass, C(u, v) ( A 0 (22)) A 0 (11) (Γ 0 (11) : Γ 0 (22)) = 3, A 0 (11) A 0 (22) 3, u, v A 0 (11) u, v level 22 11, f f 22, Γ 0 (22) 2 Hecke Atkin-Lehner W 2, W 11, f 11 W 2 = 2 f 22, f 22 W 2 = 1/2 f 11, f 11 W 11 = f 11, f 22 W 11 = f 22, (f f 22 ) W 2 = f f 22, (f f 22 ) W 11 = (f f 22 ) C σ, f : X 0 (22) C σ P 2, (modular parametrization) C σ, Γ 0 (22) 2 Hecke f f 22 level 22 C σ, Atkin-Lehner W 2 level 11 f 11, level 22 f f 22 ( ) ( ) W 2 =, W = f W (τ) = 2 (22τ + 2) 2 f(w 2τ), f W 11 (τ) = 11 (22τ + 11) 2 f(w 11 τ) Atkin-Lehner η-, f 11 W 2, f 11 W 11, f 22 W 2, f 22 W (1) f = (u, v) : X 0 (22) C σ, C σ (2) (i ), (1/2), (0), (1/11) X 0 (22) f (3) C σ f((i )) C σ 5 (4) Pic 0 (X 0 (22)), [(i ) (1/2)] 0 67 (1) w, z q-, z 1/w Laurent (2) s = (z + w 2 + 4w)/2, t = z (w + 2)/2 + w 3 /2 + 3w 3 w 16 s, t q- (3) s, t, C σ X 0 (11) (4) s, t C σ, f 11, f C σ = X 0(22)/ σι, (1) u = x/(x + 2) 2, v = (y/(x + 2) 3 1)/2 C(X 0 (22)) σι (2) u, v X 0 (22) (i ) q Laurent (3) C(C σ) = A 0 (22) σι = C(u, v), u, v (4) C σ, q (5) C σ f 11, f 22 (d) f : X 0 (22) C σ, (i ) X 0 (22) C σ. C σ (i ), (i ) (Weierstrass ) ( ), X 0 (22) C σ, (i ) X 0 (22) q C σ L( ), L( ) X 0 (22), L σ ( ) C σ f = (u, v) : X 0 (22) C σ P 2, (i ) X 0 (22) C σ (Weierstrass ) u, v A 0 (22) (i ) X 0 (22) f Galois, u, v Galois u 0, v 0 C(C σ ) f u 0 = u, f v 0 = v u 0, v 0 C σ Weierstrass, σ C σ u 0 L σ (2 σ ), v 0 L σ (3 σ ) σ (i ) X 0 (22) σ, (1/11) = σ(i ) σ

45 Riemann-Roch 59 D σ = f σ = (i ) + (1/11) Div(X 0 (22)) D σ σ, σ L(m D σ ) (m Z) L(m D σ ) ± = {h L(m D σ ) σ h = ±h}, L(m D σ ) = L(m D σ ) + L(m D σ ) m h 0 L σ (m σ ), div(f h 0 ) = f div(h 0 ) f ( m σ ) = m D σ, f h 0 σ, L σ (m σ ) h 0 f h 0 L(m D σ ) + f : C(C σ ) A 0 (22), L σ (m σ ) C σ 1 l σ (m ) = m dim L(m D σ ) + = m (m N) l(m D σ ) = 2 m 1 dim L(m D σ ) = m 1 dim L(m D σ ) + = m, u L(2 D σ ) + C, v L(3 D σ ) + L(2 D σ ) +, f = (u, v) : X 0 (22) P (u(p ), v(p )) C σ P 2 2 Weierstrass C σ, (i ) X 0 (22) ( σ ) (i ) f, σ C σ q 0 f q 0 = q, f A 0 (22) (i ) C(C σ ) σ, q σ C σ (e) K = (i ) + (1/2), K 0 = (0) + (1/11), D σ = (i ) + (1/11) Div(X 0 (22)) (a), x = f 11 (τ)/f 22 (τ), y = dx/2πi f 22 (τ) dτ A 0 (22), x L(K), y L(3 K), A 0 (22) = C(x, y) X 0 (22) y 2 = x x x x x x , (i ), (1/2) (i ) q, y 1/x Laurent (y = x 3 6 x 2 10 x x 1 88 x x 3 + ) η-, (1/2) X 0 (22) y 1/x, (i ) (1/2), 1/x, y = x x x x x x 3 v 1 = y (x x x) v 1 K(3 K) ord (1/2) (v 0 ) 0, v 1 L(3 (i )) l(3 (i )) = = 2, L(3 (i )) = C + C v 1 σ (i ) = (1/11), σ v 1 L(3 (1/11)), L(3 (1/11)) = C+C σ v 1 v = (v 1 +σ v 1 )/2, v L(3 D σ ) + L(2 D σ ) + x σ v 1 (1/2) x σ v 1 = 8 x x x x x x 5 + u 1 = x σ v 1 +8 x, L((i )+2 (1/11)) = C+C u 1 L(2 (i )+(1/11)) = C+ Cσ u 1 L(2 D σ ) = C+C u 1 +C σ u 1 u = (u 1 +σ u 1 )/8, u L(2 D σ ) + C u L(2 D σ ) + C, v L(3 D σ ) + L(2 D σ ) +, f = (u, v) : X 0 (22) P (u(p ), v(p )) C σ P 2 f (i ) X 0 (22) C σ 2, f C σ P 2 C σ : v v = u 3 19 u u 284 (i ) X 0 (22) q, u, v ω = du/(2v + 17) u = 1/q q + 2 q 2 + q q 4 3 q 5 6 q q q 9 + v = 1/q 3 + 1/q 7 2 q + 2 q 2 4 q q q 6 7 q q q 9 ω = ( q + q q 4 q q 7 4 q q 9 q 11 2 q 12 4 q 13 + q q q 17 + ) dq q ω = 2πi (f 11 (τ) + 2 f 22 (τ)) dτ C σ u, v, q-

46 60 69 C σ P 2, C σ X 0 (11) : y 2 + y = x 3 x 2 10 x 20 C σ u, v, (1/11) X 0 (22) q (1) (0) X 0 (22) C σ 2 f 0 = (u 0, v 0 ) : X 0 (22) C σ (2) u 0, v 0 (0) X 0 (22) q 0 (3) (i ) X 0 (22) C σ 2 g = (u, v) : X 0 (22) C σ (4) u, v, ω (i ) X 0 (22) q = exp(2πi τ) (5) (0) X 0 (22), (3), (2)