2 R U, U Hausdorff, R. R. S R = (S, A) (closed), (open). (complete projective smooth algebraic curve) (cf. 2). 1., ( ).,. countable ( 2 ) ,,.,,

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1 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

2 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).

3 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.

4 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.,,,.

5 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 )

6 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.

7 {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

8 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.

9 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

10 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

11 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.

12 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.

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

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

ii 15 Abel,,,.,,,.,,, ( ) ( ) 8 24 ( ) : : ( ), ( ) 8 20 ( ) 15:30 16:10 16:30 17:00 ( ), 2007 8 20 24 5, Abel 15.,.. Jacobi, Abel,,,,. : (1),,,,,,.,, Abel. (2) Abel-Jacobi,,. (3),,,,, topics. (4),. (5) (1), Abel.,.,.,,.,..,,,,,. (C) ( ) 16540002, (B) ( ) 16340012.,,.. 2008 1 20 ss2007,