ii 15 Abel,,,.,,,.,,, ( ) ( ) 8 24 ( ) : : ( ), ( ) 8 20 ( ) 15:30 16:10 16:30 17:00

Transcription

1 ( ), , Abel 15.,.. Jacobi, Abel,,,,. : (1),,,,,,.,, Abel. (2) Abel-Jacobi,,. (3),,,,, topics. (4),. (5) (1), Abel.,.,.,,.,..,,,,,. (C) ( ) , (B) ( ) ,, ss2007, / i

2 ii 15 Abel,,,.,,,.,,, ( ) ( ) 8 24 ( ) : : ( ), ( ) 8 20 ( ) 15:30 16:10 16:30 17:00 ( ) 17:10 18:00 ( ) Riemann,, 18:20 19:10 ( ) Riemann-Roch (1) 19:10 20:10 20:20 21:10 ( ) Riemann-Roch (2) 8 21 ( ) 7:30 8:50 9:00 9:50 ( ) Abel-Jacobi ([Tata I] ) (1) 10:10 11:00 ( ) Abel-Jacobi ([Tata I] ) (2) 11:20 12:10 ( ), ( ) Abel-Jacobi ([Iw] ) (1) 12:10 13:50 14:00 14:50 ( ), ( ) Abel-Jacobi ([Iw] ) (2) 15:10 16:00 ( ) ([Tata II] ) 16:20 17:10 ( ) genus 1 ( ) (1) 17:30 18:20 ( ) genus 1 ( ) (2) 18:30 19:30

3 iii 8 22 ( ) 7:30 8:50 9:00 9:50 ( ) sigma 10:10 11:00 ( ) Jacobi 11:20 12:10 ( ) Inversions of abelian integrals 12:20 18: ( ) 7:30 8:50 9:00 9:50 ( ) CM 2 10:10 11:00 ( ) 11:20 12:10 ( ) 12:10 13:50 14:00 14:50 ( ) Algebraic theory via schemes ([AV] ) (1) 15:10 16:00 ( ) Algebraic theory via schemes ([AV] ) (2) 16:20 17:10 ( ) (1) 17:30 18:20 ( ) (2) 18:30 19: ( ) 7:30 8:50 9:00 9:50 ( ) 10:10 11:00 ( ) Birch Swinnerton-Dyer (1) 11:20 12:10 ( ) Birch Swinnerton-Dyer (2) 12:20, ( ) [Iw] [Tata I] [Tata II] [AV] : D. Mumford : Tata lectures on theta I D. Mumford : Tata lectures on theta II D. Mumford : Abelian varieties

4 iv ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

5 v

6 vi

7 vii 1. 1 ( ) 2. Riemann-Roch 15 ( ) 3. Abel-Jacobi I 61 ( ) 4. Abel-Jacobi II 81 ( ), ( ) ( ) ( ) ( ) 8. Inversions of Abelian Integrals 191 ( ) 9. CM Abel 199 ( ) ( ) ( ) ( ) 13. Algebraic Theory of Abelian Varieties via Schemes 247 ( ) ( ) 15. Birch-Swinnerton-Dyer 291 ( )

8 viii

9 15, pp , C ( ) f = u + iv, (, u, v f ) f f x = i f x u x = v y, u y = v x.., u, v u = v = 0 (, f = 2 f x + 2 f )., 2 y2 u = 0. u, u. 1, S, A S. (i) A φ S U φ C. (ii) φ A U φ = S. (iii) U φ U ψ ψ φ 1 φ(u φ U ψ ) ψ(u φ U ψ ). (iv) S U ψ C ψ U φ U ψ φ A (iii) ψ A ( ). : Hausdorff S A R = (S, A). A φ (local coordinate), S., S, R. 1

10 2 R U, U Hausdorff, R. R. S R = (S, A) (closed), (open). (complete projective smooth algebraic curve) (cf. 2). 1., ( ).,. countable ( 2 ) ,,.,, (, ). ([1] 4.9) Hausdorff S R S. χ., n 2, n 1, n 0, χ = n 2 n 1 + n 0., χ = 2 2g g. ( ) 2. g = 0, g = 1., ( cf.1.6). ( ) ,. S 2 C { } (cf. [5]),, U 0 = C 2, U = C 2 \ {0} { }, φ 0 (z) = z, φ (z) = 1/z., P 1 P 1 (C) C L = {m 1 ω 1 + m 2 ω 2 m 1, m 2 Z} ( ).,, E C E(C). ( ) P.,.,,, ( ), (cf. [2] pp.46).

11 R, R f : R R z R U φ f(z) R V ψ, ψ(f(φ 1 (x))) x = 0. ( ). R R, R R, R = R., (local parameter)., R U C U f, U R, f(z) = t = x + iy t = f(z) U, (x(z), y(z)). U z 0 f(z 0 ) = 0 C, t z R = P 1, z 0 C, f(z) = z z 0, z 0 = f(z) = 1/z P 1 ( ) R R f z. z z = f(z) t, t z t = f(t) 0 or t = f(t) = t n (n 1). t = t n, n, f z. n > 1 z, n = 1., R U R f, f(u) R. f R z 0 w 0 R f U, f(z) = w n S n S f S S, f (covering map). : S P P U f 1 (U) f.

12 4 S f S. 2 S 1 f 1 S 1, S 2 f 2 S 2 g : S 1 S 2 g : S 1 S 2 g f 2 = f 1 g., S 1 = S 2, g. g. S 1 = S 2 = S, f 1 = f 2 = f, g, S. S f S A(S f S). S S Id S S., S. S,. S f S, G = A(S f S) S. P S, P U, G 2 g, g g(u ) g (U ) =., G.,. S S S. S, G S S (G) = G\S 1.1. S S, S f S G S P, Q G, f (P ) = f (Q ), Q = g(p ) g G. S (G) S., G 1, G 2, S (G 1 ) S (G 2 ), G 1, G 2., G H S (H), S (H 1 ) = S (H 2 ) H 1 H 2. S, S ( )., I = [t 0, t 1 ] R. γ : I X X. γ(t 0 ), γ(t 1 ).,,. γ, γ γγ. P 0 [γ], P 0.. [γ], S S f S S R, S f, R., R R.. ( ),,., R.,,,.

13 5 (1) P 1 (C) (2) C. (2-1) C = P 1 (C) \ { } (2-2) C \ {0} = P 1 (C) \ {0, } (2-3) C/L, L 2 Zω 1 + Zω 2, ω 1 /ω 2 H. (3) H. : (2-3). : (3).,,,,, ( )., 2,., 2., (3) 2n,, Ω 1., 2g., g n. (1) aa 1 (2) a 1 b 1 a 1 1 b 1 1 a 2 b 2 a 1 2 b 1 2 a h b h a 1 h b 1 h (3) a 1 a 1 a 2 a 2 a h a h., (1), (2).,, (2). ; 1.8. R α k, β k, k = 1,... g., P 0, 2 P 0, 4g (2), a k, b k α k, β k (k = 1,..., p). α k, β k P 0 S., 2, [4] S P 0 α 1, β 1,..., α g, β g., π 1 (S, P 0 )

14 6 g [α k ], [β k ] (k = 1,..., g) [α k ][β k ][α k ] 1 [β k ] 1 = 1. g,. : 1.7. ( ) g R S. g S,, g 2.., Betti (= 2g) ( or ) (= 2g or g ).,, 1.,, ( 0 ). k= R, R P 1. K(R), R. f K(R) f(z), f z., z U φ φ, f φ 1 φ(z). ( ) R. u, v., u. R ( ). R z, f R {z} ( )., U f U {z}, f R ( ) ( ). R ( ) f f.. :, (cf. [2] p.11) ( ). R f, g R ( ) {p n } f(p n ) = g(p n ), f = g.

15 {f n } R f, f R f z 0 R. z 0 t, a 0 = f(z 0 ) f(z) a 0 = a 1 t+a 2 t 2 +, a 0 = f(z 0 )., a j 0 j f z = z 0, ν z0 (f)., 0., a 0 =, 1 f(z) = a 1t + a 2 t a j 0 j m, ν z0 (f) = m, m..,.. z = z 0, w = w 0 = ψ(φ 1 (z 0 )) d dz f(φ 1 (z)) = d dw f(ψ 1 (w)) dw dz,.,. R φ A φ(u φ ) a φ, b φ. ω : φ (a φ, b φ ), U φ U ψ φ, ψ a φ (z) = a ψ (ψ(φ 1 (z)))(ψ φ 1 ) (z) b φ (z) = a ψ (ψ(φ 1 (z)))(ψ φ 1 ) (z) 1 (1-form), ω = adz + bdz. b φ 0. ω = adz. f, a φ (z) = (f φ 1 ) (z), b φ 0 1-form. f, df. R C 1 du, du = u dz + u z dz z. f df. ω 1-form, ω φ (b φ, a φ ). Re ω = 1 2 (ω + ω), Im ω = 1 2i (ω ω) ω. ω = bdz + adz. ω = iadz + ibdz, ω. 1-form ω ω 1, ω 2 ω = ω 1 + ω 2 ( u x i u y ) dz. u du + i du = 2 u z =. adz a = {a φ } φ A., a, (, a φ ), ω = adz. ω., ω z Res(ω, z), z R

16 8. D,. f ω fω. ω 1 /ω P 1 (C) D D., U 0 = (D\{ }, φ 0 = Id) U = (D \ {0}, ζ = φ : z 1/z). ω = a dz. f = a φ0, U 0 U f(z) = a φ0 (z) = a φ (ζ(z)) dζ ( ) 1 1 dz = a φ z z, 2 f(z) = c 2 z + c 3 +,., 2. 2 z3 2 f. Abel., 1 (DFK, differential of first kind), 0 2 (DSK), 3 (DTK).,.. R p 1,..., p m p n {p n }, {p n } ( ). ([1] pp ) z 1, z 2 R 2. z 1 1, z 2 1 f K(R) z R, z n( 2), R ω z,m R, R S, S f : S S R R, f., R, R, S, S. f : S S f f # K(R ) g g K(R) g(z) = g(f(z)), f # R, R., R, R. K(R) = K(R ), R = R.

17 9.., R, R 2 z 1, z 2 z 1, z 2, z 1 1 1, z 2 1 1, Re ω z1,z 2 z 1, z 2 ω z1,z 2. ( [2] p.121): R,,, (, 0), S, φ φ(u φ ) c φ Ω : φ c φ c φ (z) = c ψ (ψ(φ 1 (z))) (ψ φ 1 ) (z) 2 z φ(u φ U ψ ), S 2 2-form. 2-form Ω c dz 2, cdzdz, c dx dy, cdx dy, i cdz dz. 2 1-form 2-form 2., 2-form 2., dz dz = (dx + idy) (dx idy) = 2idx dy. 1-form ω j = a j dz + b j dz, j = 1, 2 ω 1 ω 2 2-form (a 1 b 2 a 2 b 1 )dz dz, ω 1 ω 2., ω dω ( b dω = z a ) dz dz. z., ω = adz + bdz dω = da dz + db dz. ω, dω = , (C 1 ). α (cf. [2] p.143) α R. α R ω α. (1) α, ω α R R ω, (Re ω α ) ω = 2π ω. S α

18 10 (2) α, ω α α z 0 i 1, z 1 i 1, R \ {z 0, z 1 }, R ω,. R γ 1-form ω, ω γ ( ). ω 1 ( 2, 3 ) 1 ( 2, 3 ).. γ ( γ, aφj dz + b φj dz ). φ j γ R 1-form ω R C 1 u ω = du,., C 1-1-form ω dω = 0, (closed form) R 1-form. (1). (2) 2 γ 0, γ 1 ω = ω. γ 0 γ 1 (3) γ ω = 0. γ d(du) = 0, R., dω = 0 C 1 - u ω = du. ω ( ), ( ) f ω = df. ω, u(z) = z z 0 ω, du = ω. ( Stokes ) R C 1 1-form ω, dω = 0. :, S dω = S ω. Cauchy R ( ) γ [γ] = 0, R ω, ω = 0. ω. Cauchy. γ R

19 R g, α 1, β 1,..., α g, β g R. R ω 1, ω 2, { } g ω 2 ω 2 ω 1 = 0, β k α k β k i k=1 k=1 { g ω 1 α k α k ω 1 ω 2 β k α k ω 2 β k ω 1 γ 1-from ω γ. } = (ω 1, ω 2 ) = S ω 1 ω g 1 R α 1, β 1,..., α g, β g. (1) ω α k, k = 1,..., g A(R). ) (2) A(R) ω ( α t ω, C g. 1 α k ω, θ k = δ jk θ 1,..., θ g.. α j ( ) θ k (k = 1,..., g). T (i, j) θ i β ( j ) ( ), ω A(R), ω,..., ω = ω,..., ω T β 1 β g α 1 α g. T R {α 1, β 1,..., α g, β g }. T = t T, Im T. 2,. [3]. 2.1 k = C k (1 ) K (1) K k x, K/k(x). (2) k K. K/k(x) x (Schmidt). k, k.

20 k α, ν P (α) = 0 K ( ) P K k K x, ν P (x) 0 K 2,. x K x k ν P (x) = 0( P ). ν P (x) x P. m = ν P (x) > 0 x m, m = ν P (x) < 0 x m K P k. k = C C. 2.2 C,. K k = C. P K, u K. u P ū(p ). p P, u a mod (p) a C u P. K = {ū(p ) u K} K., ([1] 4.2) K 1, K S R K. K, R(K). K(R(K)) = K. S, K., 2.5. C K, K R K(R) K. K, R(K). 2.3,,,. R R K(R) C, R.

21 13, 1.18., ([1] pp )., K C- R = R(K).. [1],, [2],, [3], Riemann-Roch, SS2007 [4], Abel-Jacobi I, SS2007 [5],,

22 14

23 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

24 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

25 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,.

26 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 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) 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) 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

27 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 , ,,, 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.

28 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.

29 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).

30 22 (d),. P 1 X 0 : X 1, P 1 X 0, X , 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. m, n, m n, m n. 2.6 (a) 2,, 2,.,,. 2,. 2,. Bézout. 9,,,,.,.

31 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-,.

32 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)

33 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 (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 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

34 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} 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.

35 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 ) = 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) 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 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 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 ) = s 0, C C,., r d/2 r n 1 n 2 mod d. r = 0, ±1, C (h) 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 ) 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.

36 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), 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

37 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

38 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 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,

39 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 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.

40 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 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 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.

41 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 (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

42 34. n P 0 P C D (support) supp(d) 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 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 ).

43 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 = )

44 f, P e k. df 0, (df) P P, P f e 1, e 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) ( ) y dx, P 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 ) ( ), k., 1. 1, 2. 3, Riemann-Roch ( 5.2), (1) P C m 2, P m ω mp. (2) P m, ω 2P,, ω mp.

45 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, 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) = z P 1 = A 1 { }. f k[z] k(x) = k(p 1 ), df, d( f 1 ), dz f,,.

46 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) 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, (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.

47 Riemann-Roch 39 (c) k Ω C (0) (l(k C )) C (genus) g(c). k = C, ( ), (1), (d) 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) 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).

48 (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 (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,.

49 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) 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, 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 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

50 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 ( ), 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.

51 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 ).

52 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, ,, E O E, (E, O). E Weierestrass, O = [0 : 1 : 0], E

53 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 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 ).

54 46 div(dy) = div(x 0 ). dy/x 0, ( 2 ) ω E. E dx/y 0 = dy/x 0. 2 ω E = dy/x (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

55 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.

56 k 2. C Weierstrass, (Hurwitz) 2 g + 3 ϕ : C C Aut(C) S w, C. C, : f 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, (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).,

57 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,

58 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) 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 ) 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.

59 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 ( ), P, Q 1. P, Q,.

60 (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) 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 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.

61 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) 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 (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.