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! 1 F F1 " 4567 )89* 4. * # F, F1 f : F → F1 a ∈ F, f (a) = b ∈ F1 . - ! ' k ∈ N, k ≥ 1, ϕ : U → B F ψ : U → B F1 ( )* ϕ(a) = 0, a ∈ U ; ψ(b) = 0, b ∈ U ; f (U ) ⊂ U ; ,* 3 g = ψf ϕ−1 : B → B ! g(z) = z k B. / k " f
b a, % k ! a b F F1 / bf (a) = k − 1 ! ! f a. # f : F → F1 ! - ! ' m ∈ N, m = 0, " Q ∈ F1 m F ! f 4 % * ! " Q ∈ F1 P ∈f −1(Q)(bf (P ) + 1) = m, ! P, ' Q f 4 P ! * : F ! g ≥ 1 ' ! " ."
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P ∈X bf (P ).
4 DE* 5669 1 2(g − 1) = 2n(γ − 1) + B.
>3?@ @-=.AB-13 3 S = {f (x) : x ∈ X, bf (x) > 0}. # S Y ! S 2 #! A0 2 A1 % A2 # ! % f X " X nA0 − B 2 nA1 % nA2 3 "! !
! ( A0 − A1 + A2 = 2 − 2γ, nA0 − B − nA1 + nA2 = 2 − 2g.
C DE - ! 1 ' ; π : C → C/Γ ! ! C/Γ. > Uα ⊂ C ! Γ− ! π Uα π(Uα ). 3 π : Uα → π(Uα ) (Uα , ϕ α = id) −1 (π(Uα), ϕα) ! id = ϕα πϕα : ϕα (Uα) = Uα → C, ! Uα.
= f : C → C ! ! f (z) = f (z + ω1) = f (z + ω2 ), (ω1/ω2) ∈ / R, z ∈ C, ! " " f : C/Γ → C " f = fπ. 3 f : C/Γ → C ' f : C → C ! ! ! ω1 ω2. > f ! ! f. - f − 12
5F & ,;79 56G & )GH9 ! ." ! ! f : C → C ! >3?@ @-=.AB-13 f ! " " f C/Γ 4 * # . f, f, ! #! !
§
az + b , a, b, c, d ∈ C, ad − bc = 0. cz + d √ ad − bc, ! " ad − bc = 1. # z = az+b cz+d " $ T = 1 % $& ' T ξ1, ξ2. ( c = 0 ξj = ∞, j = 1, 2. ' T ξ1, ξ2, ∞ ξ1, ξ2, ac & ) % $ T, w − ξ1 z − ξ1 =K , w − ξ2 z − ξ2 w = Tz =
K
* K T. ' b c = 0 w = T z = az+b d , ad = 1, a = d. ( ξ1 = d−a , ξ2 = ∞ T w − ξ1 = K(z − ξ1 ), K = ad . ) $ z−ξ c = 0 : W = w−ξ w−ξ Z = z−ξ , c = 0 : W = w − ξ1 Z = z − ξ1 , T W = KZ. + S Z = z−ξ z−ξ c = 0, c = 0 Z = z − ξ1 . ( $ T T ST S −1 S, S −1 S. , " % ad − bc = 1 " K + K −1 = (a + d)2 − 2. ' T = 1 " |K| = 1, K = 1; " K ∈ R+ \ {1}, " K ∈ C \ R+, |K| = 1, R+ = {K ∈ R : K > 0}. ' T - ξ. ( - & ' c = 0 ξ = a−d 2c T ∞, ξ, − dc ac , ξ, ∞ & ) % T, (w − ac )/(w − ξ) = −(ξ + dc )/(z − ξ) " " T 1/(w − ξ) = [1/(z − ξ)] ± c, +c, a + d = 2, −c, a + d = −2. ) W = 1/(w − ξ) Z = 1/(z − ξ), T W = Z ± c. . T $ ST S −1, S Z = 1/(z − ξ). ' c = 0 ∞ a = d = ±1. ' T w = z ± b. =
a c −ξ1 a c −ξ2
.
1
1
2
2
1 2
/012& ' T = 1. ( T " a + d ∈ R |a + d| > 2; T " a + d ∈ R |a + d| < 2; T " a + d = ±2; T " a + d ∈ C\R. . " T = 1 " T K ! z = 0; T 3 θ ! z = 0, K = eiθ = 1; T ! A 3 θ ! z = 0, K = Aeiθ , A = 1, A > 0; T & 1 ' T c = cT = 0. ( T (z) = (cz+d) |T (z)| z T. 4 {z ∈ 1 C : |z + dc | = |c| } = IT - T. . T −1(z) = −dz+b IT = {z ∈ C : |z − ac | = |c|1 } cz−a & /012& ' T c = cT = 0) IT IT , IT % IT & & 5& ' - L [− dc , ac ] &
2
−1
−1
−1
L DL L IT D '$ IT −1 D IT −1 D '$ a − dc qXXX ξ1 I'$ I'$ T T q qc D XX Xqa dq ξ2 qa D − dcq ξq − ξ ξ &% c c c D 2 1 D &% &% &% D D
&5
/012& 6- T
IT −1
c = cT = 0) " ! IT - L. 7 3 ! IT −2arg(a + d). + P SL(2, C) $ $ −1
w = T z = az+b cz+d , a, b, c, d ∈ C, ad − bc = 1, ! !& ( P SL(2, C) SL(2, C)/{±} ' Γ P SL(2, C)) z ∈ C, Γz = {T ∈ Γ : T z = z} z Γ 3 U (z) z " T (U (z)) ∩ U (z) = ∅ $ T ∈ Γ\Γz T (U (z)) = U (z) T ∈ Γz . 4 Ω(Γ) z ∈ C, Γ " Γ. Ω(Γ) C Λ(Γ) Γ. 4 Λ(Γ) Γz, z ∈ C. 8" Ω(Γ) " Λ(Γ) C. ' Γ P SL(2, C)) " Ω(Γ) = ∅. 9 Γ " Λ(Γ) " $ & ' Γ P SL(2, C)) " & 9 Γ P SL(2, C), : Γ, & & 3 Tn Γ " Tn → 1 n → ∞ $ C.
; Ω(Γ)/Γ Γ " " ! π : Ω(Γ) → Ω(Γ)/Γ & 4 Ω(Γ) Ωj , j = 1, 2, ... #$ - Γ. < Ω0 Γ " T Ω0 = Ω0 $ T ∈ Γ, Ω0 " Γ ! & , Γ " U & 8" Λ(Γ) ⊂ ∂U
& < Λ(Γ) = ∂U, Γ " U C\U. ( z0 z1 - Γ " 3 T ∈ Γ " T z0 = z1. 9 $ & 4 φ ⊂ Ω(Γ) - Γ, : φ Γ $ =
5 - z ∈ Ω(Γ) Γ z0 ∈ φ; 0 φ ∩ Ωj & > Γ = T : T m = 1 , -3 T z = ei z, m ∈ N, 1, T, T 2, ..., T m−1. ; - {z ∈ C : 0 ≤ argz < 2πm } 2πm % z = 0. . " " " ! & ( " Λ(Γ) = ∅ Ω(Γ) = C. ' ! π : Ω(Γ) = C → Ω(Γ)/Γ " m− C C/Γ. 6- ? :&: T1nT2m z → z + nω1 + mω2. @ $ & < & 4 Λ(Γ) = {∞} Ω(Γ) = C. ' ! π : C → C/Γ " C/Γ. , "
$ ω1 + ω2 ω1. @ 3 & ( " % 0, ω2, ω1 + ω2 ω1. ' Γ = T !
-3 T " cT = 0. ( φ Γ % $ IT IT , $ & . Λ(Γ) = {ξ1, ξ2} (ξ) Ω(Γ) = C\{ξ1, ξ2} (C\{ξ}). ' ! π : Ω(Γ) → Ω(Γ)/Γ Ω(Γ)/Γ " b1 , " A A " & & & ;& , " /012& ' Γ1 , ..., Γm φ1, ..., φm. ' " % φj φk , j = k, j, k = 1, ..., m. ( Γ = Γ1, ..., Γm , Γ1, ..., Γm, Γ1, ..., Γm, φ - $
2π m
−1
φ1 , ..., φm.
, Γ B g ≥
1 γ1, γ1, ..., γg , γg ,
-3 T1, ..., Tg -3 - -3 C" - ! 2g− D, Tj (D) ∩ D = ∅ Tj (γj ) = γj , j = 1, ..., g. ; φ B Γ D $ $ γ1, ..., γg. ' Ω(Γ)/Γ = φ/Γ $- g. # , Γ g /0D= 5E2& , Γ FGH % B (g, s, m) -3
T1, ..., Tg , U1, V1, ..., Us, Vs , W1, ..., Wm
-3
γ1 , γ1, ..., γg , γg , Λ1, ..., Λs, Δ1, Δ 1, ..., Δm, Δ m,
Λi $ Ωi, Σi, Ωi, Σi i = 1, ..., s, : - -3 C" - Δk , Δk , - ξk , k = 1, ..., m, - ! (2g + s + m)− D,
Tj (D) ∩ D = Ui(D) ∩ D = Vi (D) ∩ D = Wk (D) ∩ D = ∅,
Tj (γj ) = γj , Ui(Ωi) = Ωi , Vi(Σi) = Σi, Wk (Δk ) = Δk , j = 1, ..., g, i = 1, ..., s, k = 1, ..., m;
5 i = 1, ..., s, Ui, Vi " UiVi(z) = ViUi(z) $ z ∈ C Ui, Vi " Ui, Vi, = 0Wk ξk , k = 1, ..., m.
+ D I γ1, ..., γg, Ωi, Σi, Δk \ξk , i = 1, ..., s, k = 1, ..., m, - Γ. ; Ω(Γ)/Γ $ g + s 2m & +" $ g +s 2m FGH Γ (g, s, m), Γ = T1 , ..., Tg , U1, V1 , ..., Us, Vs, W1, ..., Wm : UiVi Ui−1Vi−1 = 1, i = 1, ..., s ,
, Γ g + m $ ! $ s $ $ 5 /0D" 1:2& ' U = {z ∈ C : |z| < 1} $ H = {z ∈ C : Imz > 0}) ' dsU = 2|dz|/(1 − |z|2 )(dsH = |dz|/Imz).
' " Γ " -3 U H), z0 ∈ U
z0 ∈ H) Γ, Γ $ z0 : Pz0 ,U (Γ) = {z ∈ U : dU (z, z0 ) ≤ dU (z, T z0), T ∈ Γ} (Pz0 ,H (Γ) = {z ∈ H : dH (z, z0 ) ≤ dH (z, T z0), T ∈ Γ}),
dU dH U H & + U H) Γ & , Γ , σ = (g, s; i1, ..., ip), , FGH Γg,s (g, s) p $ $ Γi , ..., Γi i1 , ..., ip, -3$
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f −1(Q) Q ∈ F1 F !"#$ f : F → F1 % & $ ' f1 : F1 → F f2 : F2 → F F, f : F1 → F2 f1 = f2 f F1. ( P1 ∈ F1, P ∈ F, P2 ∈ F2, P ∈ F. !"#$ F F1 f : F1 → F ) f $ * +,-,.'/01.2 $ f 3 $ ' a ∈ F F1, f a. 1 f $ . $ . P1 ∈ F1 f : F1 → F, P1 U f |U $ f 4 $ ' F, F1 f : F1 → F 4 ) $ $ P1 ∈ F1 U (P1) f |U (P ) U (P1)
F.
1
!"#$ F F1 ) f : F1 → F 4 $ . F1 f ) $ * +,-,.'/01.2 $ Σ F1,
$ ϕ : U → V ⊂ C F U1 ⊂ F1 f : U1 → U )$ . ϕ1 = ϕf : U1 → V F1 , Σ. + Σ 4 F1, )$ 1 Σ 4 F1, 3 F. 4 f ) f : U1 → U id = ϕf ϕ−1 1 V. * $ Σ F1 f : (F1, Σ ) → F )$ . id : (F1, Σ) → (F1, Σ ) ) )$ 5 Σ Σ 5 F1. . $ F F1 ) f : F1 → F 4 γ : [0, 1] → F F. . γ : [0, 1] → F1 γ, f (γ) = γ. F F1 $ f : F1 → F 4 Q ∈ F U (Q) f −1(U (Q)) = ∪j∈J Vj , Vj , j ∈ J, F1 f : Vj → U (Q) ) j ∈ J. 6 f : F1 → F γ : [0, 1] → F P0 ∈ F1 f (P0) = γ(0) γ : [0, 1] → F1 γ, γ (0) = P0 . !"#$ / 4 f : F1 → F 4 7& γ γ , P0, f (P0) = γ(0), 8& γ γ1 P1 3 P2, P1 , P1 , γ γ1 3 P2 , P2. !"#$ F F1 ) f : F1 → F 4 $ . f f $ 6 f : F1 → F
$ 6 F1 $ !"#$ ' F, F1 f : F1 → F ) n ∈ N f P ∈ F n 4 $ ! " 6 F ) )3 f : F → C % 4 &$ ! # 9 f (z) = z n + a1 z n−1 + ... + an n 4 n $ * +,-,.'/01.2 $ f : C → C, f (∞) = ∞, ) $ ∞ n, : n. 1 $ F F1 f1 : F1 → F 4 $ f1 ; 4 f2 : F2 → F P1 ∈ F1, P2 ∈ F2 < f1(P1) = f2(P2) f : F1 → F2 f (P1) = P2 . 1 ) F F. $ !"#$ F, F F f : F → F 4 $ . f F, F F. F, F1 f : F1 → F 4 $ 6 % & 5 % & ) T : F1 → F1 f (T P ) = f (P ), P ∈ F1. 9 Γ T f : F1 → F
3$ = Γ P0 , P1 ∈ F1 f (P0) = f (P1) T ∈ Γ T P0 = P1. . T P0 = P1.
!"#$ F f : F → F F. . Γ, ) T : F → F f (TP) = f (P), P ∈ F, Γ ) π1(F, O). & %>& 7# ?# !"#$ @ F F g ≥ 2, 4 F 3 π : F → F ) ) 5 U = %
{z ∈ C : |z| < 1}. = Γ π : U → F ) π1 (F, O). A 5 ) $ z0 ∈ U O ∈ F. B O
4 a1, b1, ..., ag , bg F. a1 z0 , a1 z0 3 a. . π(z0 ) = π(a), A1 ∈ Γ A1z0 = a. , b1, a2, b2, ..., ag, bg z0,
b1 , a2, b2, ..., ag , bg
B1, A2, B2, ..., Ag, Bg ∈ Γ. = π1 (F, O) = a1 , ..., bg : [a1 , b1]...[ag , bg ] = 1 ,
Γ = A1 , ..., Bg : [A1, B1]...[Ag , Bg ] = 1 ,
1 U $ 6 $C g = 2. + O F 4 (F ; a1, b1, ..., ag , bg ), $ $ π1 (F, O). .
) Γ Γ (Γ; A1, B1, ..., Ag, Bg ). ' F F π ) Γ T F ) F $ > Γ π : U → F ) U Γ ) U.
U/Γ = F
U
A2 z0 a˜2 I z0
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b˜ 1
B1z0 π A1 z0
R
a˜1
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a1 I b1
-
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a2
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F g ≥ 2 ) U/Γ, Γ )
U, $ $ F ) 5 (U/Γ, Σ), 4 Σ 3 U, 4 (U, ϕ(z) = z). b+ 2
+ a 2
-
− R a1 ˜ ˜ Δ B1 A 1 *
I a− 2
b− 2
I z0
F = U/Γ
U
b− 1
a+ 1
b+ 1
π
a1 I b1
-
0
b2 :
a2
>$E (g = 2) B Γ, F4 γ = [a1, b1]...[ag, bg ] = 1 O ∈ F. γ z0 ∈ U, + − − + + − − γ = a+ 1 b1 a1 b1 ...ag bg ag bg = 1,
4g− U. 1 a+j , a−j aj b+j , b−j bj , j = 1, 2, ..., g. 5 1, B j : a+ → a− , B 1, ..., A g , B g ∈ Γ A j : b+ → b−, j = 1, ..., g. A j j j j 1 , B 1, ..., A g , B g : [A 1, B 1]...[A g , B g ] = 1 = Γ. π1 (F, O) ∼ = A
- (Intγ ) ∪ a+1 ∪ b+1 ∪ ... ∪ a+g ∪ b+g ) Γ. 2 >$ = Aj , Bj Aj , Bj , j = 1, ..., g, Γ. Cj = [Aj , Bj ] = −1 −1 + −1 − Aj Bj A−1 j Bj . . A1 : a1 → A1 B1 A1 a1 , a1 = a1 , A1 B1 A1 a1 = a1 , + − A1 = A1B1A−1 1 = C1 B1 . . B1 : A1 b1 = b1 → C1 b1 = b1 , 1 = C1A−1. 1 3 B 1 ∂ =
g
(C1...Cj−1aj )(C1...Cj−1Aj bj )(C1...Cj Bj aj )(C1...Cj bj ).
j=1
R
C1 A2B2 z0
˜
CAb I 1 2 2
C1A2z0
Ca ˜ I 1 2 C1z0
b1 - ?C1˜ ˜1 B
A1 B1A−1 1 z0 = C1 B1 z0
Δ A˜1
˜bgI z0
-
a ˜1
j
A
−1 ˜1 1 B1 A 1 a
-
A1 z0
A1˜b1 A1 B1z0 = A1B1 A−1 1 A1 z0
G$?
j : C1 ...Cj−1aj → C1...Cj Bj aj A j = C1...Cj Bj C −1 ...C −1, A 1 j−1
j : C1...Cj−1Aj bj → C1...Cj bj B
j = C1...Cj (C1...Cj−1Aj )−1 = C1 ...Cj A−1C −1 ...C −1, j = 1, ..., g. B 1 j j−1
% $ +4& H8#$ / F g ≥ 2 ) 5 ) Ω(Γ)/Γ,
Γ I g T1, ..., Tg , $ $ EST − (g, 0, 0).
b1
b2
b3
>$" H8# J?#$ / F g ≥ 2 ) 5 ) Ω(Γ1)/Γ1, Γ1 EST − (h, g − h, 0), 1 ≤ h < g, T1, ..., Th, U1, V1, ..., Ug−h, Vg−h.
b1
b2
'$ a3 &%
b3
>$H 6 F g = 3 4 )3 I ! EST − %8 7 :&$ - 3 U → Ω1 → Ω → F,
Ω1, Ω EST − (h, g − h, 0)
I g $ 5 Ω1 K K U K AK% & I Ω F g ≥ 2. D ∈ OAD , ) D C $ B EST − Γ1 (g, s, m) I Γ g OAD . 2 ) 4 )3 4 4
$ ' g : F1 → F ) F1 % & F. = ) %)& )3 f : F1 → C (f : F1 → C) ) %) & )3 F. ' P ∈ F g−1(P ) = {Qj : j ∈ J}, f (Qj ), j ∈ J, 5 )3 f P. ' ' F1 = C, F = C∗ g : C → C∗ ) w = g(z) = expz, id : C → C ) z = Lnw C∗, w ∈ C∗ g−1(w) Lnw = ln|w|+iargw+i2πm, m ∈ Z. + g : C → C∗ 4
V ⊂ C, 3 2πi. g : C → C∗ $ a ∈ C∗ , b ∈ C expb = a. . V0 b, U a exp : V0 → U )$ g −1 (U ) = ∪n∈Z Vn , Vn = V0 + 2πin.
' V0 A 2π, Vn g : Vn → U ) !"#$ ' k ∈ N, k ≥ 2, fk : C → C ) w = fk (z) = zk . . z = 0 ∈ C fk , w = 0 fk
z = 0 k. ( √ w = 0 k z = w,
A k− k
k
3 z = 0. 2 fk : C∗ → C∗ 4 C∗. * fk V ⊂ C∗, 3 2πk . - w = a ∈ C∗ z = b ∈ C∗ fk (b) = a(bk = a). 1 V0 b, U a fk : V0 → U )$ |w|
fk−1(U ) = V0 ∪ εV0 ∪ ... ∪ εk−1V0 ,
ε = exp 2πik k− 7$ ' V0 Vj = εj V0, j = 0, 1, ..., k − 1, fk : Vj → U )$ 6 4 )3 fk : C∗ → C∗, C∗ {0 < |z| ≤ 1}, w = a ∈ C, |a| = 1,
$ k = 2, w = f2(z) = z2, γ(t) = e2πit, t ∈ [0, 1], w = 1, γ(t) = eπit, t ∈ [0, 1], z = 1 3 z = −1 !"#$ ' Γ A {mω1 + nω2 : m, n ∈ Z}, ωω ∈/ R. . 3 π : C → C/Γ 4
$ * π V ⊂ C, Γ−5 $ * a 4 4 V0, π|V ) V0 U a + Γ C/Γ. 5 1 2
0
π −1 (U ) = ∪m,n∈Z {V0 + mω1 + nω2 },
Vm,n = V0 + mω1 + nω2 π : Vm,n → U ) !"#$ - Γ1 = T , T z = z + 2πi; 2πi )z; k Γ3 = T1, T2 : [T1, T2] = 1 , T1z = z + ω1 , T2z = z + ω2, Γ2 = T : T k = 1 , T z = (exp
7 8 !$
§
F g > 1 g ϕ F ! " ϕ1 F # $ Dif f F F
% Dif f+F
Dif f F, !& &! & '
Dif f F. (
I(F )
Dif f+F, !& " id $ $ F. %
Dif f+F (I(F ) Dif f+F ). )
Dif f+F/I(F ) "
* & g ModF. + , "
" &! & F γ F. ( F γ, " γ + γ − F \γ. γ − 2π γ + γ −. ' F " -, γ " " . + γ, " " &! & $ f : [0, 1] × S 1 → F γ = f [(1/2) × S 1], S 1 $ / τγ γ 0 τγ (P ) = P, P ∈/ Imf [0, 1] × S 1 f ) τγ (f (t, θ)) = f (t, θ + 2πt). γt = f [t × S 1], γt 2πt : 1 γ0, π γ 2π γ1 .
/ & γ F, γ 2345 6 $ ! τγ , + -, γ.
p p p p
C1 C2 C3 i
i
i
i
Z1 Z2 Z3 V1 V2 V3
Zg Zg−1 Vg p p p p
( 7
8 7 ! 3g − 1 " V1 , Z1, ..., Vg , Zg , C1, ..., Cg−1 F. # -, 9 :; 2<=5 >
ModF g > 1 $ -, V1, Z1, ..., Vg , Zg , C1, ..., Cg−1 F. . 9 2??@5 ModF
" &! &! ;
ModF g > 1. 8 " & " % # ;+ $ , $" " " $ &!
" ModF g > 1. " " " " " c d & τcτd = τdτc. " " " c d &
A? τcτdτc = τdτcτd. h & & c & & d, τd = hτch−1. 8 c d " " " " τc & d d c c. - # 2B?5
ModF $ " τV , τZ , τC R, R F. %&
ModF $ τC $
Γ + &! L1, ..., Lg ; " LL−1 = 1, Γ "
" &! L1, ..., Lg. 9 ,
Γ = L1, ..., Lg. C &
" Γ ,
" Γ D
" Γ &! ; Γ &! ; Γ. Γ = L1, ..., Lg, & α Γ , " 0 ? Li → Lj , Lj → Li Li Lj , " &! & i = j; B Lj → L−1 j ; 3 Lj → LiLj , i = j 2B15 * " Γ = L1, ..., Lg & " " " &! Γ. E ω : π1(F, O) → π1(F, O1) " " F ! γ0 O O1 ω(γ) = γ0γγ0−1, 1
1
1
1
γ & , π1 (F, O). - α1 : π1 (F, O) → π1 (F, O ) α2 : π1 (F, O1) → π1 (F, O1) " & , " !& " " ω ω α2 = ω α1ω. f F fO∗ : π1 (F, O) → π1 (F, f (O)), " $ γ → f (γ), + & " O O1 F " fO∗ fO∗ 1 , " C α
" π1 (F, O) " ! γ1 ∈ π1 (F, O) α(γ) = γ1γγ1−1 & γ ∈ π1(F, O). 2? F BB5 - ϕ ∈ I(F ), ∗ ϕO , $
" π1 (F, O).
9 ? 3 $
π1 (F, O) ∼ = Γ = A1 , B1, ..., Ag, Bg : C1...Cg = 1.
#$ , " 0 Λz : Γ → π1(F, O) A ∈ Γ z0, $ ! O,
α z0 Az0 U & αA U → U/Γ α G+ αA z0. 2? F B?5 E ω : π1 (F, O1) → π1 (F, O2) α = Λ−1 z ωΛz
" Γ, z1, z2 $ O1, O2 ϕ ∈ Dif f+F, " αϕ " "
" Γ. %& ϕ ∈ I(F ), ∗ αϕ = Λ−1 z ϕO Λz
" Γ, ϕ(O1) = O2 . % Aut+Γ IntΓ
"
" Γ 8 0
1
2
2
1
1
IntΓ Aut+Γ.
H 8 2345 :&
" π1(F ) " F *
ModF = Dif f+F/I(F ) ∼ = Aut+Γ/IntΓ, " $ ϕ → αϕ . ϕ ∈ Dif f+F, αϕ ∈ Aut+Γ αϕ $ "
" A1, B1, ..., Ag, Bg
C1...Cg $ $ " $ " (C1...Cg )−1. >
ModF g = 0 >
ModF g = 1
SL(2, Z) 2345 * &!
" ModF g > 1, $
&!
" Aut+Γ/IntΓ. 9 -$ . 2<<5 " " , &!
" Aut+Γ : i = 1, ..., g, τV : Bi → BiAi; −1 i = 1, ..., g − 1, τZ : Ai → Bi−1Bi+1Ai, Ai+1 → Ai+1Bi+1 Bi ; −1 τ Z : A g → Bg A g ; i = 1, ..., g − 1, i
i
g
τCi : Aj → Bi Aj Bi−1 , Bj → Bi Bj Bi−1, j < i, Ai → AiBi−1 .
D $
" Γ " &!
" Γ & / I, &! " " C1, ..., Cg−2, " $ & " &!
" Aut+Γ. 9 / 8 23?5 " " , " &!
" Aut+Γ : i = 1, ..., g, αi : Bi → BiAi; βi : Ai → AiBi; −1 −1 i = 1, ..., g − 1, γi : Bi → A−1 i+1 Bi Ai , Bi+1 → Bi+1Bi Ai Bi Ai+1 , −1 −1 −1 Ai+1 → A−1 i+1 Bi Ai Bi Ai+1 Bi Ai Bi Ai+1. C " αi, βi, i = 1, ..., g, γj , j = 1, ..., g − 1,
& &! & J " &!
" AutΓ, $ " δ : Aj → Bg+1−j , Bj → Ag+1−j , j = 1, ..., g, " &! & F. 9 -$ . I I 2<@5 " " &!
" g π1 (F, O) = w1, ..., w2g : [w2j−1, w2j ] = 1, j=1
W2 g
Y1 ?
I
-
Yg
) Ug i
W2g−1
Zg
U1 M W1
6 W2
Z1
W3
Y2 - `` `` U2 ``
qqq
0
qq
W4
( ?1 −1 −1 π1 (F, O) = v1 , ..., v2g : (v1v2−1v3v4−1...v2g−1v2g )(v1−1v2v3−1v4...v2g−1 v2g ) = 1,
VY4
V2 Y
q q q q q
Y
V2 g Y V2g−1
V3
V1Y
( ?? % &! v1, ..., v2g " &! w1, ..., w2g &! 0 −1 v2i−1 = (w2i−1w2i )(w2i+1w2i+3...w2g−1),
−1 )(w2i+1w2i+3...w2g−1), i = 1, ..., g. v2i = (w2i
9 &! w1, ..., w2g & &! Aut+Γ 0 i = 1, , , ., g, τUi : w2i → w2iw2i−1;
−1 −1 τZi : w2i−1 → w2i−1w2i w2i+1w2i+2w2i+1 , −1 −1 −1 w2i → w2i+1w2i+2 w2i+1 w2iw2i+1w2i+2w2i+1 , −1 −1 w2i+1 → w2i+1w2i+2 w2i+1 w2iw2i+1; −1 τYi : w2i−1 → w2i−1w2i ;
$ " wi & 2g. % [Γ, Γ]
" Γ, " $ ,
" ' " ,
" Γ. 8 [Γ, Γ] Γ. >
Γ ; C1...Cg = 1,
Γ/[Γ, Γ]
$ " 2g "
Γ/[Γ, Γ] = [A1] × [B1 ] × ... × [Ag ] × [Bg ].
D [A] = {AC : C ∈ [Γ, Γ]} $ , A
[Γ, Γ] , A Γ → Γ/[Γ, Γ]. n. " & Γ/[Γ, Γ] 8 , Aj Bim ∈ Γ j
i
n
[Aj j Bimi ] = nj [Aj ] + mi [Bi]. >
(Γ/[Γ, Γ], +)
(Z2g , +),
(R2g , +), + [Aj ] → ej ∈ Z2g [Bj ] → eg+j ∈ Z2g ,
0
" R2g . >
Γ/[Γ, Γ] " & $
" Γ. % H1(F, Z) &
F , Z. E H1(F, Z) ∼ = π1 (F, O)/[π1(F, O), π1(F, O)]. % $ ? [a1], [b1], ..., [ag], [bg ] $
[π1(F, O), π1(F, O)]
" π1(F, O). E
π1 (F, O) ∼ = Γ + 2g ∼ ∼
H1(F, Z) = (Γ/[Γ, Γ], +) = (Z , +). 6 $" Λ
" Γ Λ
" Γ/[Γ, Γ] ∼ = Z2g . * Λ $ " Λ : Z2g → Z2g . % [Λ] 0 I g . 2g " , [Λ]J t[Λ] = ±J = ± −Ig 0 D Ig g, t " ; [Λ] J ? &!
" H1(F, Z), Λ &! & J $ ? Λ &! & # [Λ] $
Sp(2g, Z) % det[Λ] = 1, Λ &! & det[Λ] = −1, Λ &! & F. 8
Λ
" Γ e1 , e2, ..., e2g
a
aj
b
bj
c
cj
d
dj
Λ(Aj ) = A1j1 ...Ag g B1j1 ...Bg g Cj , Λ(Bj ) = A1j1 ...Ag g B1 j1 ...Bg g Cj ,
Cj , Cj ∈ [Γ, Γ]. *
Λ ([Aj ]) =
g
ajk [Ak ] + bjk [Bk ], Λ ([Bj ]) =
k=1
g
cjk [Ak ] + djk [Bk ].
k=1
/
[Λ] =
(a)jk (b)jk (c)jk (d)jk
∈ Sp(2g, Z).
2B15 C "
" Γ &
"
" Γ/[Γ, Γ]
Aut(Γ/[Γ, Γ]) ∼ = Sp(2g, Z). / &!
" Sp(2g, Z) + 2B1 F ?445 * $ &!
" Aut+Γ/IntΓ g = 2 −1 " S = A−1 2 B1 A 1 B1 .
Λ1 Λ2 Λ3 Λ4 Λ5
A1 A1 A 1 B1 A1 A1 S −1A1S
A2 A2 A2 A2 A 2 B2 A2
B1 B1 A 1 B1 B1 B1 B1 S
B2 B2 B2 B2 A 2 B2 S −1B2
.;
τg ModF g > 1. % "
* + τg = Λ ∈ Aut+Γ : [Λ] = I2g /IntΓ. +
*
ModF, + , " " &! & F " & $ "
H1(F, Z). - g = 2
Λ ∈ Aut+ Γ : [Λ] = I4 + " Aut+ Γ
" $+ " " Λ0 Λ0 (A1) = C1A1 C1−1, Λ0(A2) = A2 , Λ0(B1) = C1 B1C1−1, Λ0(B2) = B2 ,
C1 = [A1, B1]. 6 -$ 24?5
τ2 $+ g > 2
τg $+ - g > 2 δi
" Γ, 0 < i1 < g $
IntΓ), $
* τg , δi + i Ci = j=1[Aj , Bj ]) $ 1
1
1
1
A1, B1, ..., Ai1 , Bi1 → Ci1 A1 Ci−1 , Ci1 B1Ci−1 , ..., Ci1 Ai1 Ci−1 , Ci1 Bi1 Ci−1 , 1 1 1 1
Aj , Bj → Aj , Bj j > i1, [δi ] = I2g . 1
§
1− 2− F 0− F F. 1− ω F z = x + iy U ⊂ F f (x, y), g(x, y) f (x, y)dx + g(x, y)dy z = x + iy U ⊂ F f(x, y), ∩ U = ∅ g ( x, y) ! U f( x, y)d x+ g ( x, y)d y = f (x, y)dx + g(x, y)dy.
2− Ω F
z U ⊂ F f (x, y) " f (x, y)dx ∧ dy z
U ! f(x, y), U ∩ U = ∅ dz f( x, y)d x ∧ d y = f (x, y)dx ∧ dy ⇔ f( z ) = f (z( z ))| |2 . d z x, y z, z. # ω = f dx + gdy = u(z)dz + v(z)dz, Ω = fdx ∧ dy = g (z)dz ∧ dz, dz = dx + idy, dz = dx − idy, f = u + v, g = i(u − v), dz ∧ dz = −2idx ∧ dy, f(x, y) = −2i g (z). $ k = 0 f 0− C % a naPa , na ∈ Z, Pa ∈ F ) a naf (Pa). $ k = 1 1− ω 1− % & F ' , C " I = [0, 1] → 1 dxz = x + iy, C : dy F, ω = f dx + gdy, C ω = 0 (f (x(t), y(t)) dt + g(x(t), y(t)) dt )dt. (
ω ) C
z. * ! # 1− . $ k = 2 2− Ω 2− % D F ) "
D " z, D Ω = D f (x, y)dx ∧ dy. $ C 1 % # C 1(F )) + ∂g dx + ∂f dy, dω = ( ∂x − ∂f )dx ∧ dy, )' d df = ∂f ∂x ∂y ∂y dΩ = 0;
,' ∂ ∂ ∂f
∂f = ∂f ∂z dz, ∂f = ∂z dz, ∂ω = ∂u ∧ dz + ∂u ∂v ∧ dz = ∂v ∂z dz ∧ dz, ∂ω = − ∂z dz ∧ dz, ∂Ω = 0 = ∂Ω. - d = ∂ +∂ ) ∂ 2 = ∂∂ +∂∂ = ∂ 2 = 0 d2 = 0. %.' / ω k− C 1, k = 0, 1, 2, D (k + 1)− ω= dω. ∂D
D
%. ) ' F D " F D ⊂ F ∂D 1 ω ) C , 0
" "! D. 1 ∂D ω = D dω. 1 / ω ) C F g ≥ 1, F dω = 0. $ C ∞ $ ∗ % 2"' ! " F : ∗ω = ∗(f dx + gdy) =
−gdx + f dy; ∗(udz + vdz) = −iudz + ivdz. )' ) ω ω ∈ C 1(F ) dω = 0; ,' ω ∗ω 3 4' ω F, ! f ∈ C 2(F ) df = ω; 5' ω F, ∗ω ω = ∗df f ∈ C 2(F ).
) + )' ω %' ω % '3 ,' D % ' ) D %' D; 4' % '
% ' F. / f ∈ C 2(F ) f 6 f f = ( ∂∂xf + ∂∂yf )dx ∧ dy 2
2
2
2
F. )' 7 f F f = 0 F ; ,' ) ω F, z ω = df (z) f (z) + )' 6 0 C 2(M) % ' z = x + iy
F;
,' f ∈ C 2(M) −2i∂∂f = f = d ∗ df ; 4' ) ω ∈ C 1(F ) ) ω F,
ω = df, f F F, ω = ϕ(z)dz, ϕ(z) z. 8 % 9443 4:;' + )' u F, ∂u ∂u 3 ,' ) ω = udz + vdz v = 0 u F ; 4' ) ω = udz + vdz u v !
ω1, ω2 ω = ω1 + ω2; 5') ω ω = α + i ∗ α α ) 3 <'∗(ω1 +ω2) = ∗ω1 +∗ω2; ∗(f ω) = f (∗ω); ∗∗ω = −ω; ω1 ∧∗ω2 = ω2 ∧∗ω1 = (p1p2 + q1 q2)dx ∧ dy, ωj = pj dx + qj dy, j = 1, 2; =' d = ∂x∂ dx + ∂y∂ dy, ∗d := − ∂y∂ dx + ∂x∂ dy, ∗(df ) = (∗d)f (= ∗df ); ∗dω := (∗d)ω, ∗dω = −d ∗ ω; :' ) ω ∈ C 1 ω ∗ω = −iω;
>' ω γ = ω+ω 2 " %!' 3 ?' 3 ! γ ω = γ + i ∗ γ γ. ( " !
F. $ f F
" F "0 ∂g ∂f ∂g g, 7@A ∂f ∂x = ∂y , ∂y = − ∂x F. * F " ! ) ω = udz+vdz, u, v F. $ F. A L2(D), ! ) ω D ||ω||2D =
D
ω ∧ ∗ω < ∞,
D F. . ω1, ω2 ∈ L2(D)
(ω1, ω2)D =
D
ω1 ∧ ∗ω2.
8 # (ω1, ω2)D = (ω2, ω1)D . $ ∗ (∗ω1, ∗ω2)D = (ω1, ω2)D , ∗ ||.||L (D). 8 − % "'
d ∗ 1− F. 8 E L2(F ) ! df, f ∈ C ∞(F ) B F, E ∗ = {ω ∈ L2(F ) : ∗ω ∈ E}. 1 E(E ∗) ! %' F. 8 E ⊥ (E ∗)⊥ E E ∗ L2(F ). 944; )' / α ∈ L2 (F ) ∩ C 1(F ), α ∈ (E ∗)⊥ (α ∈ E ⊥), α (α '3 ,' H(F ) = E ⊥ ∩ (E ∗)⊥, " L2(F ) = E⊕E ∗⊕H(F ), H(F ) L2(F ). $ c F ! C ∞ ! ηc F ω ∈ L2(F ), C 1, % 94C344;' 2
c
ω = (ω, ∗ηc).
( + )' F E " 3 ,' ! f F, df ∈ E ∩ H(F ), # " df = 0, f = const F. F g ≥ 2. 8 1 HDR (F ) % A F ) F. / ω 1 # [ω] HDR (F )
ω. D H1(F, Z) % F # ' ) % ' ) % ! F ) F. 1 " (c, ω) → c ω, c F ω ) F, " 1 H1(F, Z) × HDR (F ) → C.
E [c] c [ω] ω. 7 # [ω] c, 1 , ω [ω] HDR F. $ P0 ∈ F. $ P F γ P0 P, " f (P ) = γ ω. $ f (P ) γ. γ 1 P0 P. 1 c = γ 1 ∪ γ − %) ' F, c=
2g
nj N j + γ 0 ,
j=1
H1(F, Z), {N1, ..., N2g} = {a1, ..., ag, b1, ..., bg} % ' nj ∈ Z, γ0 ∼ 0 % γ0 D F ). 1 N ω = 0, j = 1, ..., 2g, " γ ω = D dω = 0, γ ω = γ ω = f (P ) 0 C 1 F df = ω. F g ≥ 1, " ! ) ω 0 j
1
0
γ ω, γ H1(F, Z), # Hom(H1(F, Z), C). 8 H1(F, Z) C " ) ω, 0 ) F. E " 1 (F ) 8 " HDR Hom(H1(F, Z), C) &
F g ≥ 1. 7 ) ηc c. $ F ! 2" 1 HDR (F ) ∼ = H(F ), [ω] ) ω ! ) ω0 F ω = ω0 + df f F. (
" c ω = c ω0 c F, ω ω0 1 F g > 0 ! F. 8 " (ω1, ω2) → F ω1 ∧ ω2 0 1 1 " HDR (F ) × HDR (F ) → C. q ∈ Z. F q−
ω F ! " z F f (z) " f (z)dzq z F. $ q = 1 * F 94434:; ' $ P ∈ F (z(P ) = 0), n ≥ 1, n ∈ N, ! ω F, F \ {P } z 1 P ; ' $ P1, P2 ∈ F (zj (Pj ) = 0, j = 1, 2) ! ω, F \ {P1, P2} z1 P1 − z1 P2. q ω P ordP ω = ordP f, ω = f (z)dzq . E f (z) = zn g(z), g(0) = 0, g z = 0, ordP ω = ord0 f = n. * F " {P ∈ F : 1 ord ω = 0} 1" resP ω := 2πi c ω = c−1 , f (z) = ∞P n n=N cn z , z(P ) = 0; c F, ! D, "! P, ω D \ {P }. n+1
1
2
944; F
ω F. 1 P ∈F resP ω = 0. $87G(G1/6H.1 8 1 F " ω " Δ1, Δ2, ..., Δk " 1 k
P ∈F
1 resP ω = 2πi j=1
∂Δj
ω = 0.
" " " 1 ! 944; P1 , ..., Pk, k > 1, k F, c1 , ..., ck j=1 cj = 0. 1 ! ω F, F \ {P1 , ..., Pk } ordP ω = −1, resP ω = cj , j = 1, ..., k. $87G(G1/6H.1 8 $ Q = Pj , j = 1, ..., k, 0 ωj , B ) Pj ) Q, F \{Pj , Q}, j = 1, ..., k. 1 ω = c1 ω1 +...+ck ωk Pj cj , j = 1, ..., k, Q (− kj=1 cj ) = 0, ω Q. 1 " 944; 6 $ @ # 944; / c F, ! F, ! % ' ω F, c ω = 0, ω + i(∗ω) F. D a b j
j
a · b :=
F
ηa ∧ ηb = (ηa, − ∗ ηb).
$ % & 9:C; a · b
H1(F, Z) × H1 (F, Z), !+ )' a · b a ,' a · b = −b · a;
b;
4' (a + b) · c = a · c + b · c; 5' (a · b) ∈ Z. ) ) H1 (F, Z) =< a1 > +...+ < bg >
2g ! a1, ..., bg, [a] @ a. $ a1, ..., ag , b1, ..., bg @ aj · bk = δjk ; aj · ak = 0, bj · bk = 0 j = k, j, k = 1, ..., g. 8 {N1, ..., N2g} = {a1, ..., ag, b1, ..., bg} % '
J = (Nj · Nk )jk =
O Ig −Ig O
,
Ig g. 6 H1(F, Z) J F. 9443:C; * F g H(F ) 2g. $87G(G1/6H.1 8 g = 0 ω P F, P0 ∈ F f (P ) = P ω, P ∈ F. 1 F f (P ) F. ! f ω = 0 F. g > 0 " Φ : ω → ( N ω, ..., N ω) H(F ) C2g . / dimC H(F ) > 2g, Φ ! ω = 0, ( " ω df, f * F g > 0. # dimC H(F ) ≤ 2g, " Φ & 8 Φ " II $ # " " j = 1, ..., 2g ω˜ j 1 Nj 0 Nk , k = j, k = 1, ..., 2g. " ω˜j = ηb , ω ˜ g+j = −ηa , j = 1, ..., g. F" # 1 A ω˜ j ! 0
1
2g
j
j
ωj "
(
Nk
ωj )jk = (δjk ) = I2g .
* 2g c1, ..., c2g 0 ω = 2g # j=1 cj ωj k = 1, ..., 2g, N ω = ck N ωk = ck . 1 J ω
F g ≥ 1 2g cω a ω, b ω, j = 1, ..., g. $ ! g c ∼ j=1(mj aj + nj bj ) F, mj , nj ∈ Z, j = 1, ..., g. $ {N1, ..., N2g} H1(F, Z) ! k
k
j
H(F )
j
{ω1 , ..., ω2g} = {ηb1 , ..., ηbg , −ηa1 , ..., −ηag } ( Nk ωj )jk = I2g .
9:C; E !
$ '(
Fωk ∧ ωj+g , j = 1, ..., g, ωk = − ωk ∧ η N j = − F ωk ∧ ωj−g , j = g + 1, ..., 2g, Nj F ωk ∧ ωj = (ωk , − ∗ ωj ) = Nk · Nj ; ( ωk ∧ ωj )kj = J,
F
F
j, k = 1, ..., 2g. [70] Γ = ((ωk , ωj ))kj J
ω1, ..., ω2g , (ωk , ωj ) = F ωk ∧ ∗ωj , 0 + )' Γ
(ωj , ωk ) = (∗ωj , ∗ωk ) = ∗ωj ∧ ∗ ∗ ωk = ωk ∧ ∗ωj = (ωk , ωj ); F
F
,' Γ ! ωk ∗ωj ! 9443:C; / θ θ˜ F g > 0,
F
g ˜ θ ∧ θ˜ = [ θ θ θ˜ − θ]. j=1
aj
bj
bj
aj
$87G(G1/6H.1 8 ) θ θ + df, f ∈ C 2(F ). 6 "
F
˜ = (θ, − ∗ θ) ˜ = (θ + df ) ∧ θ˜ = (θ + df, − ∗ θ)
F
˜ θ ∧ θ.
# θ θ˜ " " " θ=
2g
2g
μj ωj , θ˜ =
j=1
μ˜j ωj ,
j=1
μj , μ˜j ∈ C, j = 1, ..., 2g. 1 F
2g
θ ∧ θ˜ =
μj μ ˜k
F
j,k=1 g
μj μ ˜k (Nj · Nk ) =
k,j=1 2g
μj μ ˜j+g (Nj · Nj+g ) +
j=1
μj μ ˜j−g (Nj · Nj−g ) =
j=g+1
g
θ
Nj
j=1
ωj ∧ ωk =
2g
Nj+g
2g
θ˜ −
Nj
j=g+1
˜ θ,
θ Nj−g
μj = N θ, μ˜j = N θ,˜ j = 1, ..., 2g. $87G(G1/6H.1 8 , . ! F Δ ∂Δ = gj=1 a+j b+j a−j b−j . 1 Δ θ = df Δ, f Δ. 1
j
F
j
θ ∧ θ˜ =
g ( j=1
a+ j
Δ
f θ˜ +
df ∧ θ˜ = b+ j
f θ˜ +
Δ
a− j
˜ = d(f θ)
f θ˜ +
b− j
∂Δ
f θ˜ =
˜ f θ).
z0 Δ, z ∈ Δ " f (z) = z + − z θ. 8 z z # aj aj ∂Δ,
0
a+ j
f θ˜ +
a− j
f θ˜ =
a+ j
˜ + f (z)θ(z)
a− j
˜ ) = f (z )θ(z
a+ j
˜ − f (z)θ(z)
a+ j
[
z
z0
a+ j
θ−
˜ = f (z )θ(z)
z
θ]θ˜ =
z0
a+ j
aj
(−
bj
˜ = [f (z) − f (z )]θ(z)
θ)θ˜ = −
aj
θ˜ ·
θ. bj
m a− j z + I bj
m b− j6
R
6
z 6 + aj
z0
G m m
b+ j
f θ˜ +
b− j
A ), # b+j b−j ,
f θ˜ = [
m
z0
bj
θ−
m z0
θ]θ˜ =
˜ θ.
θ aj
bj
1
/ θ )
F g > 0,
g 0 ≤ ||θ|| = (θ, θ) = [ θ ∗θ − θ ∗θ]. 2
aj
j=1
bj
bj
aj
8 !0 J
Γ = (γjk ) 9:C;
3)γkj =
F
ωk ∧ ∗ωj =
∗ω = , k = 1, ..., g, bk j Ng+k ∗ω j . − ak−g ∗ωj = − Nk−g ∗ωj , k = g + 1, ..., 2g;
5' F Γ " (Γ > 0). / Γ=
A B C D
Γ(tΓ = Γ) # + t A = A,t D = D, B =t C, Γ > 0 0 A > 0, D > 0. 8 2 " ∗ R %C ' !
! % '
8 ω1, ..., ω2g ∗ ! Λ 2g, ∗ωk = 2g j=1 λkj ωj , k = 1, ..., 2g, t ∗A = ΛA, A = (ω1, ..., ω2g). K ∗∗ = −id Λ2 = −I2g , −A = ∗ ∗ A = ∗(ΛA) = Λ · Λ · A. K " Λ, Γ J Γ = Λt J γlk = (ωl , ωk ) = (∗ωl , ∗ωk ) = (
2g
λlj ωj , ∗ωk ) =
j=1
2g
λlj
j=1
F
ωk ∧ ωj ,
l, k = 1, ..., 2g. 8
# 9:C; K ! ω1, ..., ω2g
" φj = ωj +i∗ωj , j = 1, ..., 2g. [70] φ1 , ..., φ2g J
" I I 1 1 (φk , φj ) = (φj , φk ) = (ωj , φk ) = 2 2
−i bj φk , j = 1, ..., g, i aj−g φk , j = g + 1, ..., 2g.
$ % 9443:C; * F
g > 0 Ω1(F ), !
F, φ1, ..., φg ,
H(F ) = Ω1(F ) ⊕ Ω1(F )
$87G(G1/6H.1 8 Ω1(F ) ∩ Ω1(F ) = {0}, ω = udz + vdz ∈ Ω1(F ), v = 0, ω = udz ∈ Ω1(F ) u = 0. " ! " ω ∈ H(F ). $ ω + i ∗ ω ∈ Ω1(F ) ω − i ∗ ω ∈ Ω1 (F ), ω + i ∗ ω ∈ Ω1(F ). 8 ω = 12 (ω + i ∗ ω) + 12 (ω − i ∗ ω). / ω = ω1 + ω2 , ωj ∈ Ω1(F ), j = 1, 2, ∗ω1 = −iω1 , ∗ω2 = −iω2 ∗ω2 = iω2 . # (ω1, ω 2) = (∗ω1, ∗ω 2) = (−iω1, iω 2) = −(ω1, ω 2 )
(ω1, ω 2 ) = 0. 8 " ω → ω R Ω (F ) Ω1 (F ), dimC Ω1(F ) = dimC Ω1(F ) = 12 dimH(F ) = g. * φ1, ..., φg C, Ω1(F ). ( φg+1, ..., φ2g " Ω1(F ). " $ % ! 9:C; * F g ≥ 1 ! ζ1, ..., ζg Ω1(F ) 1
aj ζk
= δjk , j, k = 1, ..., g.
8 #
(Ig , Ω), Ω = (πjk ), πkj =
bj
ζk ,
tΩ = Ω, JmΩ > 0. 1 Ω1(F ) a1, ..., ag , b1, ..., bg F. $ % " 9:,; F g ≥ 1, ω, ϕ Ω1(F ). / ω + ϕ = df, f ∈ C ∞(F ), ω = 0 = ϕ F. $87G(G1/6H.1 8 " ω = h(z)dz, ϕ = g(z)dz, h(z), g(z) z F. 1 ω ∧ ϕ = 0.
" ϕ = 0, i 2
F
ϕ∧ϕ=
F
|g(z)|2 dx ∧ dy > 0,
z
= x + iy. * ϕ ∧ ϕ = ϕ ∧ ω + ϕ ∧ ϕ = ϕ ∧ df. L " F ϕ∧df = 0, ϕ = 0 F. ( d(f ϕ) = df ∧ ϕ + f dϕ = df ∧ ϕ, F ϕ∧ϕ = − F d(f ϕ) = − ∂F f ϕ = 0. ! " ω = 0 " . ω = 0 = ϕ F. " a− b− ω1, ..., ωg Ω1(F ), (π1, ..., π2g), πj = t ( Nj ω1, ..., Nj ωg ), j = 1, ..., 2g. $ % # a− b− ω1 , ..., ωg Ω1(F ), R $87G(G1/6H.1 8 8 " π1, ..., π2g
R ! ! x1, ..., x2g % ' x1π1 + ... + x2g π2g = 0. 1"
x1π1 + ... + x2g π2g = 0. 8 π1 π2 ... π2g , Ω = π 1 π 2 ... π 2g
2g. ! ! " 2g Ω # x1, ..., x2g, . Ω @ 2g. 8
! ξ 1, ..., ξ g, η1, ..., ηg % ' ! 2g Ω # #
(
g
Ni j=1
j
ξ ωj +
g
η j ωj ) = 0, i = 1, ..., 2g.
j=1
g g j 1 j ω = j=1 ξ ωj , ϕ = j=1 η ωj ∈ Ω (F ), ∞ Ni (ω + ϕ) = 0, i = 1, ..., 2g. # ω + ϕ = df, f ∈ C (F ). " ) = )> ω = 0 = ϕ, # C− ω1, ..., ωg. "
M @ % ' A + )' / θ, θ˜ F, 0=
F
g ˜ θ ∧ θ˜ = [ θ θ θ˜ − θ]; aj
j=1
bj
bj
aj
= ζj , θ˜ = ζk bj ζk − bk ζj = 0, πkj = πjk , k, j = 1, ..., g, Ω = (πjk )
,' / θ = θ F, g 2 0 ≤ ||θ|| = (θ, θ) = i [ θ θ− θ θ].
θ
j=1
aj
bj
,' θ = k = 1, ..., g,
aj
bj
g
k=1 ck ζk ,
ck ∈ C,
ck
g 0 < ||θ||2 = i[ (cj (c1 π1j + ... + cg πgj ) − cj (c1 π1j + ... + cg πgj )] = j=1
2
ImΩ > 0,
ak
θ = ck ,
g
cj ck Imπjk
j,k=1 bk
θ = c1 π1k + ... + cg πgk , k = 1, ..., g.
& θ
F g > 0. / a θ b θ θ ! θ = 0 F.
$87G(G1/6H.1 8 K
2
0 ≤ ||θ|| = (θ, θ) = i
F
g θ∧θ =i [ θ θ− θ θ]. j=1
aj
bj
bj
aj
0 ||θ||2 = 0 θ = 0 F. . 6 % ' F g ≥ 0 $ ! f, ω = df # ω = 0 F f F. " " 1 ! C Ω1 (F ). A
g " ω → ( a ω, ..., a ω) Ω (F ) C . ! & a ω ω = 0 F. & g g E " C Ω1(F ) Cg . e1 = (1, 0, ..., 0), e2 = (0, 1, ..., 0), ...., eg = (0, 0, ..., 0, 1), ζ1, ..., ζg Ω1(F ). G 0 a b ! F g > 0. G G ! G " F. P, Q ∈ F, P = Q, τ = τP Q = τ (z)dz F \{P, Q}, ordP τ = −1 = ordQτ, resP τ = 1 = −resQ τ. 1 c τ (z)dz, c F, c F. 1 c ∼ c F c, c P, Q, ! n ∈ N 1
g
c τ (z)dz
− c τ (z)dz = 2πin. θ F P1, ..., Pk, k ≥ 2. " P1, ..., Pk. 0 " cj Pj ,
j = 1, ..., k,
F.* F = F \{P1, ..., Pk} ck
g k−1 + + − − j=1 aj bj aj bj + l=1 cl , a1 , ..., ag , b1 , ..., bg , c1 , ..., ck−1 F . / c F, F k−1 j=1 nj cj , nj ∈ Z. $ θ " F . 8
c
θ=
k−1
nj
j=1
cj
θ = 2πi
k−1
nj resPj θ.
j=1
/ c % ' F , c a1, ..., ag, b1, ..., bg c1 , ..., ck−1. . c θ c θ, ..., c θ. τP Q !
" a τP Q = 0, j = 1, ..., g;
N τP Q, k = 1, ..., 2g, @ τ τP Q ωP Q aj Nk 0 K @ A " @
θ = ζj , θ = τP Q, 9443:C; df = ζj Δ, Δ ζj ∧ τP Q = ∂Δ fτP Q = b τP Q. . ∂Δ f τP Q = 2πi(f (P ) − f (Q)). 1
1
k−1
j
k
j
2πi
P Q
ζj =
bj
τP Q ,
" 2πi
P Q
ζj =
bj
ωP Q −
g l=1 πjl al
Δ. G ωP Q , j = 1, ..., g.
$ θ = τP Q, θ = τRS , P, Q, R, S Δ. 0 Δ O ∈ ∂Δ P Q γ1 γ2 z Δ . * Δ θ = df, f (z) = z θ.
0
∂Δ
= 2πi[f (R) − f (S)] = 2πi f θ = 2πi[resR f θ + resS f θ]
R S
θ.
. ∂Δ
g + f θ, θ] f θ = [ θ θ − θ al
l=1
bl
bl
al
c
c O Q % +), O % −), " P. (
c
f θ = 2πi[
P
O
θ −
Q O
= 2πi θ]
P
θ.
Q
8 R
S
P
τP Q =
Q
τRS ,
" Δ = Δ \ROS, ROS O R O S Δ .
−
a O b-− 2 -2 zγ2+ a+ 1 −Q γ R1 R + + −P b1
b+ I2
a+ 2
− a− 1 I b1
A )4 G Re SR ωP Q = Re QP ωRS . * θ 0 ! n P, z(P ) = 0, B θ = zdz , n ≥ 2. * a
τP(n). 8 # ∞ (j) l z P, z(P ) = 0. θ = ζj = l=0 dl z dz P. 1 @ n
bj
j = 1, ..., g.
(n)
τP =
2πi (j) d , n − 1 n−2
$ "
θ = ζj = df, f (z) =
z
z0 ζj
Δ.
.
∂Δ
g = f θ = [ θ θ − θ θ] θ, l=1
al
bl
bl
al
bj
(j)
d f θ = 2πiresP f θ = n−2 2πi, n−1 ∂Δ
f (z) = (... +
(j) dn−2
(j)
d z n−1 1 + ...), f θ = (... + n−2 + ...)dz n−1 n − 1z
P. 0 !
6 ω1 ω2 F. 1 ϕ = ω1 − ω2 F. / d1, ..., dg a ϕ − d1ζ1 − ... − dg ζg a " " 0 F. ω1 = ω2 + d1ζ1 + ... + dg ζg . G
a
F g ≥ 1. 944; 6 " "
F g ≥ 1. $87G(G1/6H.1 8 ω c1, ..., cn P1 , ..., Pn, n ≥ 2. P0 = Pj , j = 1, ..., n. τP P , ..., τP P . 1 n n j=1 cj = 0, j=1 cj τP P
P0 cj Pj , j = 1, ..., n. / n ω − j=1 cj τP P a d1, ..., dg , ω2 = ω − g n j=1 cj τP P − k=1 dk ζk 1 944; / F g ≥ 1, 1 0
j
j
j
0
0
0
n 0
% ' F. 8 df f 944; * F !
P1, ..., Pl −2 (k) ωP = j=−n dj z j dz, k = 1, ..., l, nk ≥ 2, # 9443:C; @ A b− " F g ≥ 1. ! P1 , ..., Pn, n ≥ 2, F g ≥ 1. 1 F, F \{P1 , ..., Pn} P1 , ..., Pn, n + g − 1. n $87G(G1/6H.1 8 1 ω = j=1 cj τP P + g k=1 dk ζk , P0 = Pj , j = 1, ..., n, c1 + ... + cn = 0, dk , k = 1, ..., g, # a ω. # c1 , ..., cn−1, d1, ..., dg , # n − 1 + g. 1 k
k
j
0
§
F D = P1n ...Pkn , Pj ∈ F, nj ∈ Z, j = 1, ..., k. Div(F ) F Div(F ) ! "
# D k degD = j=1 nj . $ deg % (Div(F ), ·) (Z, +). f ∈ M ∗(F ), f F, &' %( % % (f ) = P ∈F P ord f ∈ Div(F ). & () M ∗ (F ) Div0(F ) )( & " % # DivH (F ) & () M ∗(F ). DivH (F ) & ( F. * Div(F )/DivH (F ) D D1 & " # + (D ∼ D1), D/D1 f ∈ M(F )\C, & " max{−ord f,0} (f )∞ = , (f )0 =
# P ∈F P max{ord f,0} . ,( (f ) = (f(f)) . f, g P ∈F P F (f ) = (g), (f(g)) = 1 f = cg, c ∈ C∗ = C\0, & F - f : F1 → F2 ( !# f ∗ : Div(F2) → Div(F1) . /# f ∗ ( f ∗((g)) = (f ∗(g)), g F2. 0# D1 ∼ D2 F2, f ∗(D1) ∼ f ∗(D2) F1. ω = 0 q ( % (ω) = P ∈F P ord ω . q q - q = 1 Z. ω1 ω2 q ( ( ωω ∈ M ∗(F ). $ ( (ω1) (ω2) "+ q # 1
k
P
P
0
P
∞
P
1 2
D = P ∈F P n(P ) ( n(P ) ≥ 0 P ∈ F, D ≥ 1. -+ ( D ≥ D1, DD1−1 ≥ 1. * & f ∈ M ∗ (F ) " q ω = 0) D, (f )D−1 ≥ 1((ω)D−1 ≥ 1). 1 (0)D−1 ≥ 1 & D ∈ Div(F ). 2 ( f D, f = 0, ordP f ≥ n(P ) P ∈ F. $ ( f P ∈ F, n(P ) ≥ 0. f ≥ n(P ) P, n(P ) > 0. f & ≤ −n(P ) P, n(P ) < 0. & D F L(D) = {f ∈ M(F ) : (f ) ≥ D}. r(D) & D. !# D ≤ D1 , L(D1) ⊂ L(D); /# L(1) = C r(1) = 1; 0# deg D > 0, r(D) = 0. & D ∈ Div(F ) Ωq (D), ' ω ( ω q− F (ω) ≥ D. iq (D) = dimCΩq (D) D q = 1. Ω1(D) = Ω(D) i1 (D) = i(D). & D ∈ Div(F ) degD, r(D), i(D) D, & 3 ( ω = 0 ( i(D) = r(D(ω)−1).
34542 67$28 - D1 ∼ D2, D1D2−1 = (f ), f 2
∈ M ∗ (F ).
L(D2) h → hf ∈ L(D1)
C ( r(D1) = r(D2). 5 Ω(D) ω0 →
ω0 ∈ L(D(ω)−1) ω + i(D) = r(D(ω)−1).
C ( - F g ≥ 1 i(1) = g.
! "9 9 # :00;( :<); - F g. 2 r(D−1) = degD − g + 1 + i(D) & D ∈ Div(F ).
D F g r(D−1) ≥ degD − g + 1. "
# $ degZ = 2g − 2.
34542 67$28 g = 0, & ( ( dz C. &( & z = ∞, dz = d 1t = − t1 dt. & degZ = deg(dz) = −2. - g > 0, & ( ( ζ1 F. 2 (ζ1) ≥ 1, 9 9 r((ζ1)−1) = deg(ζ1 ) − g + 1 + i((ζ1)). = r((ζ1)−1) = i(1) = g i((ζ1)) = r(1). -+ degZ = deg(ζ1) = 2g − 2. $ % !# r(D−1) > 0, D + . /# i(D) > 0, Z/D + 34542 67$28 !# - 0 = f ∈ L(D−1), & (f )D (f )D ≥ 1, D ∼ (f )D; /# 0 < i(D) = r( DZ ), DZ + !# - !# degD > 2g − 2, i(D) = 0; /# deg(D) = 0, r(D) ≤ 1, % r(D) = 1, D 34542 67$28 !# i(D) = r( DZ ) = 0, & deg DZ > 2
0.
/# - f ≡ 0 & D, 0 L(D). 9 > # ' f = 0 L(D), r(D) = 0; # ' f = 0 L(D), (f ) ≥ D r(D) > 0. 8 # > !# (f ) = D, & g ∈ L(D), ( fg ) ≥ 1 g = cf, ( r(D) = 1; /# f ∈ L(D) (f ) > D, deg(f ) = 0 > deg(D) = 0. - + /# D ( ' g ∈ M ∗ (F ), (g) = D. - ' degD = 0 0 ≤ r(D) ≤ 1. = g = 0, g ∈ L(D), r(D) = 1.
( r(D) = 1 degD = 0, ' g = 0, g ∈ L(D), (g) ≥ D. - & & f ∈ L(D), f = 0, f = cg ( ( (f ) = (g) ≥ D. 8 >
# (g) = D, D . # & g = 0, g ∈ L(D), (g) > D, deg(g) = 0 > degD = 0 + - 9
- 9 q≤0
1 − 2q
q>0
0
& q
1
q<0
0
q=0
1
q=1
g
g=0
g=1
g>1
q > 1 (2q − 1)(g − 1)
degD = 0, g − 1 ≤ i(D−1) ≤ g. 34542 67$28 ? ' degD = 0 % 0 ≤ r(D) ≤ 1. - 9 9 ( r(D) = degD−1 − g + 1 + i(D−1), i(D−1) = g − 1 + r(D) ≥ g − 1 i(D−1) ≤ g. &
9 -
' :<); = F g & q ∈ Z ( Ωq (1) L(Z −q ) dimΩq (1) ( 1 34542 67$28 - ω = 0 ( ( (ω) ∈ Z. 2 L(Z −q ) f → f ω q ∈ Ωq (1)
% C + 9 9 Zq r(Z ) = degZ − g + 1 + r( ) = (2q − 1)(g − 1) + r(Z q−1). Z −q
q
(1)
- g > 1. 2 > # q < 0, degZ −q = −q(2g − 2) > 0, ( r(Z −q ) = 0; # q = 0, L(1) = C r(1) = 1; # q = 1, r(Z −1) = i(1) = g; # q > 1, degZ q−1 = (q − 1)(2g − 2) > 0 "!# (
r(Z −q ) = (2q − 1)(g − 1). - g = 1. 2 "!# r(Z −q ) = r(Z q−1). ω = 0 q ( deg(ω) = (2g − 2)q = 0 & ω, ( ω. -+ ω−1 (−q)− & r(Z −q ) = r(Z q ), L(Z −q ) ∼ = Ωq (1) ∼ = Ω−q (1) ∼ = L(Z q ). $ ( r(Z q ) = r(Z q−1). = r(1) = 1, ( r(Z q ) = 1 q ∈ Z. - g = 0, degZ = −2 &' > # q ≤ 0, degZ n = n(2g − 2) = n(−2) > 0 n < 0, r(Z n) = 0 n < 0. -+ r(Z −q ) = (0 − 1)(2q − 1) + r(Z q−1) = 1 − 2q + 0, q − 1 ≤ −1 < 0; # q > 0, r(Z −q ) = (2q −1)(0−1)+r(Z q−1) = 1−2q +r(Z q−1) = 0, r(Z q−1) = r(Z −(1−q)) = 1 − 2(1 − q) = −1 + 2q, 1 − q < 1 1 − q ≤ 0. 2 ( D F g ≥ 1, degD ≥ g + 1, r(D−1) ≥ 2,
' ( " %
# & ≥ g + 1. 8 ( ' & ≥ g + 1 & P ∈ F. 9 ( r( D1 ) ≥ degD − g + 1 ≥ g + 1 − g + 1 = 2.
& P ∈ F, F g ≥ 1, r( ZP1 ) = g + m − 1 m > 1. 9 9
m
r(
1 ) = (2g − 2 + m) − g + 1 + i(ZP m ) = g + m − 1, m ZP
degZP m = 2g − 2 + m > 2g − 2 i(ZP m) = 0. = F g > 0 ' P ∈ F, ' & ( & P ∈ F i(P ) < g. 34542 67$28 P > 1 Ω(P ) ⊂ Ω(1), ( ( i(P ) = dimCΩ(P ) ≤ g. ( ( ' P ( i(P ) = g, ' & - 9 9 r(P −1) = degP − g + 1 + i(P ) = 2. 2 ( ' f & P. = % F C. & g = 0. - -+ i(P ) ≤ g − 1 < g g > 0 & P ∈ F. -
:00; = F g ≥ 1 g P1 , ..., Pg ( ' & P1, ..., Pg ( 1 &' ! 34542 67$28 & D r(D−1) ≥ 1, L(D−1) & degD + i(D) − g ≥ 0. - D = P1...Pn n P1 , ..., Pn ∈ F. 2 i(D) ≥ g − n, + n ≤ g. @ ( i(P1 ) ≤ g−1, % ' i(P1) = g−1 & P1 F. 5 Ω(P1P2 ) ⊂ Ω(P1) i(P1P2) = g − 1, g − 1 Ω(P1 ) ' & P2. = + & Ω(P1), &
Ω(P1) ' P2. - g ≥ 2 '
ϕ ∈ Ω(P1) ϕ = 0. 8% P2 ( ϕ & 2 i(P1P2) ≤ g − 2, % ' i(P1P2 ) = g − 2 P2 ∈ F. - + n ( ( g ≥ n ' n P1 , ..., Pn F ( i(P1 ...Pn) = g − n. 8 ( g 1 1 F i(P1 ...Pg ) = 0. & r( P ...P ) = 1 L( P ...P ) 2 - Ω2 F, Ωe Ω2, '
! :00;( :<); = F g dimC Ω2/Ωe = 2g. 34542 67$28 6& Ω2 a− b− Ωe. 3 Ω2 Ωe 2g (c1, ..., cg, d1, ..., dg), a− b− A C− B Ω2/Ωe C2g . -+ dim(Ω2/Ωe) ≤ 2g. g P1 , ..., Pg ∈ F i(P1...Pg ) = 0. 2 ' F, & & + ( 1 &' ! - ω1, ..., ωg ∈ Ω2 ( ωj & Pj , & - ζ1, ..., ζg 2 ω1, ..., ωg, ζ1, ..., ζg & 2g Ω2/Ωe. ( 1
g
1
g
ω = c1 ω1 + ... + cg ωg + cg+1 ζ1 + ... + c2g ζg
( ω
& P1, ..., Pg, ck = 0, k = 1, ..., g. = + + & c1 = c2 = ... = cg = 0. 5 ck , k = g + 1, ..., 2g, ( ζk , $ ( dimC(Ω2/Ωe) ≥ 2g. & ( dimC (Ω2/Ωe) = 2g. 2
$ "8 1 # :00;(:<); - F g > 0, P F.
2 ' g 1 = n1 < n2 < ... < ng < 2g
(∗)
( ' f ∈ M ∗(F ), F \ P & ni P. 34542 67$28 - P F. ? ( i(1) = g r(1) = 1. 5 r( P1 ) = 2 − g + i(P ). i(P ) = i(1) = g, ' & P. i(P ) = i(1) − 1 = g − 1, r( P1 ) = 1, '
& P, ! & P ∈ F. - % D = P n−1 D = P n. ? r(P −(n−1) ) = n − g + i(P n−1 ), r(P −n) = n + 1 − g + i(P n ).
i(P n ) = i(P n−1), r(P −n) = r(P −(n−1)) + 1 ' & n P, F \{P }. i(P n) = i(P n−1)−1, r(P −n) = r(P −(n−1)) ' ( & & n P. 2 ( i(P n) % n !( L(P −n) L(P −n) L(P −(n+1)). - + i(P n) % ( g ( i(1) = g, i(P 2g−1) = 0 " degP 2g−1 > 2g − 2). - i(P n ) !( r(P −n ) % ( ! " 1 & 6 P ). 2 2 degP n = n, r(P −n) ≥ 2 n > g "n ≥ g + 1). -+ ' ( & & P ' n. ? 9 9 r(P −g ) = g − g + 1 + i(P g ) ( r(P −g ) ≥ 2, i(P g ) > 0.
% "8 1 # :00; $' P F g ≥ 1, i(P g ) > 0. 34542 67$28 - Pn F, i(Png ) > 0, n = 1, 2, 3, ... A & P0 ∈ F. 8 z z(P0 ) = 0 U (P0). 2 ζj = fj (z)dz, j = 1, ..., g, fj , j = 1, ..., g, U (P0).
$' N ( Pn ∈ U (P0), n > N, z(Pn) = zn. ? i(Png ) > 0 ( & zn '
ϕ = c1 ζ1 + ... + cg ζg ∈ Ω(Png ),
n
2 i=1 |ci |
>0
c1 f1 (zn ) + c2 f2 (zn ) + ... + cg fg (zn) = 0
c1 f1 (zn ) + c2 f2 (zn ) + ... + cg fg (zn) = 0 ... (g−1)
c1 f1
(g−1)
(zn ) + c2 f2
(zn ) + ... + cg fg(g−1)(zn ) = 0.
2 ( 8 Wg (z) + ' zn, % zn → 0 n → ∞. = Wg (z) z U (P0). ? ( f1, ..., fg U (P0). - 2 -+ ' P F ( + & ≤ g, & F. A & 8 1 F. - g = 0, 1 ' 8 1 P0 ∈ F &' 8 1 ( ' f F & & g + 1 P0 F. ( i(P0g ) = 0 r(P0−g ) = 1. = −(g+1)
r(P0
(g+1)
) = g + 1 − g + 1 + i(P0
)≥2
' & & ≤ (g + 1) P0 F. -% + n & g + 1, n ≤ g r(P0−g ) ≥ 2. - 2 (
3 F g + (g + 1)− 5 ( 1( g + 1 ( P0 8 1 & & & P0 & ≤ g. = ( & " # F g ≥ 2 8 1 8 1 =
1 8 1 -+ F ' & 8 1 - +(
& 8 1 ( F :<)( C 0)!; &
& "D 4 E # :00; D F g ≥ 2
- 4 D F g ≥ 2 ( 84(g−1) ? ( ' F g ≥ 3 :<)( C 0)/; F 8 1 & P F, Z+ & P. - :<); > # = " P # & & ( m1 m2 f1 f2 ( m1 + m2 f1f2; # 2 8 1 g = 0, C ' f (z) = z1 &( + g = 0 . ? 8 1
F g - P F 1 < α1 < α2 < ... < αg = 2g
g P ' :<); j, 0 < j < g, αj + αg−j ≥ 2g. 34542 67$28 - ( j, 0 < j < g, αj +αg−j < 2g. 2 k, 1 ≤ k ≤ j, αk +αg−j < 2g. 2 ( 1 αj < ... < αg−j < ... < αk + αg−j < ... < αg−j + αj < αg = 2g.
αj j − 1 αk , ( αg−j αg−j + αj j − 1 (≤ 2g). 2 ( (≤ 2g) 1 ( (g − j) + (j − 1) + 1 + 1 = g + 1.
- ( ' g P F - ( α1 = 2, αj = 2j αj + αg−j = 2g j, 0 < j < g.
( α1 = 2, 2, 4, ..., 2g & g (≤ 2g), + + α1 > 2, ' j, 0 < j < g, ( αj + αg−j > 2g. 34542 67$28 g = 2, 1 %( α1 = 3 α2 = 4, ( ' j = 1, 0 < 1 < 2 α1 + α1 = 6 > 4. g = 3, {3, 4, 6}, {3, 5, 6} {4, 5, 6}, ( <( G( H 1 I - g ≥ 4. - ( αj + αg−j = 2g j, 0 < j < g. [q] 1 ≤ q( ( α1 , 2α1, ..., [
2g ]α1 α1
(1)
& (≤ 2g). 2 α1 > 2( ' 1 2g3 (< g) ' '% ≤ 2g - α ( &' "!# r, 1 ≤ r ≤ [ α2g ] < g − 1, 2g3 < g − 1 g ≥ 4, 1
&
rα1 < α < (r + 1)α1.
α1 , α2 = 2α1 , ..., αr = rα1 , αr+1 = α,
( (
αg−1 = 2g − α1 , ..., αg−r = 2g − rα1 , αg−(r+1) = 2g − α.
F ( ≥ αg−(r+1) < 2g. α1 + αg−(r+1) = α1 + 2g − α = 2g − (α − α1 ) > 2g − rα1 = αg−r .
-+ ' j, αg−r < j < 2g, ' αg−1, ..., αg−(r+1). - - # 8 g−1
αj ≥ g(g − 1),
j=1
% ( α1 = 2. 34542 67$28 ? ' 2
g−1
αj ≥ 2g(g − 1).
j=1
α1 = 2, α1 > 2( $ @ ( j ≥ 1 P F ⇔ r(P −j ) − r(P −(j−1)) = 0 ⇔ i(P j−1) − i(P j ) = 1 ⇔ ' F % j − 1 P. & P 0 = n1 − 1 < n2 − 1 < ... < ng − 1 ≤ 2g − 2, nj P F. & P
F g ≥ 2 ' ω ( & P, ordP ω = 0. ( 0 = n1 −1 P - E D ⊂ C, dimC E = n ≥ 1, z D. J ϕ1, ..., ϕn E z, ordz ϕ1 < ordz ϕ2 < ... < ordz ϕn.
(
0 ≤ μ1 = minϕ∈E {ordz ϕ}
ϕ1 ∈ E C ordz ϕ1 = μ1. 5 (n−1)− E1 = {ϕ ∈ E : ordz ϕ > μ1}. - μ2 = minϕ∈E {ordz ϕ} = ordz ϕ2 , ϕ2 ∈ E1 -
z ( μj = ordz ϕj , j = 1, ..., n. ( z, ∞ 2 ϕj (t) = k=0 akj (t−z)k U (z) ⊂ D. - ( aμ j = 1 k = j, aμ j = 0 k = j, j, k = 1, ..., n. A ordz ϕj = μj ϕj (t) = aμ j (t − z)μ + ..., aμ j = 0, ( aμ j , ϕj C aμ j = 1, j = 1, ..., n. 1
k
k
j
j
j
j
j
+ aμ 1 = 0, ϕ1 ϕ1 − aμ 1ϕ2,
aμ 1 = 0. 4 ( aμ 1 = ... = aμ 1 = 0. - ( & ) * = K K ( z, U (z), 2 & + ( + - & ( μj ≥ j − 1, 0 < 1 < ... < j − 1 μ1 < μ2 < ... < μj . 8 z E 2
2
2
3
n
τ (z) = τE (z) =
n
(μj − (j − 1)).
j=1
:<); - ϕ1, ..., ϕn & E. 2 ordz W (t) = τ (z), W (t) 8
ϕ1(t), ..., ϕn(t), ! ⎛
⎜ ⎜ W (t) = det ⎜ ⎝
ϕ1 ϕ1 ... (n−1)
ϕ1
... ϕn ... ϕn ... ... (n−1) ... ϕn
⎞
⎟ ⎟ ⎟ = det[ϕ1, ..., ϕn]. ⎠
34542 67$28 8 E n. 2 ( E, W (t) & & -+ ( z, E. $ ( & f det[f ϕ1, ..., f ϕn] = f n det[ϕ1, ..., ϕn].
( det[f ϕ1, ..., f ϕn] = ⎛
⎜ ⎜ ⎜ det ⎜ ⎜ ⎝
f ϕ1 f ϕ1 + f ϕ1 f ϕ1 + 2f ϕ1 + f ϕ1 ... (n−1) f ϕ1 + ... + f (n−1)ϕ1
... f ϕn ... f ϕn + f ϕn ... f ϕn + 2f ϕn + f ϕn ... ... (n−1) ... f ϕn + ... + f (n−1)ϕn
(1) ⎞ ⎟ ⎟ ⎟ ⎟= ⎟ ⎠
⎛ ⎜ ⎜ f det ⎜ ⎝ ⎛ ⎜ ⎜ f det ⎜ ⎝
ϕ1 f ϕ1 + f ϕ1 ... (n−1) f ϕ1 + ... + f (n−1)ϕ1
... ϕn ... f ϕn + f ϕn ... ... (n−1) ... f ϕn + ... + f (n−1)ϕn
ϕ1 f ϕ1 ... (n−1) f ϕ1 + ... + f (n−1) ϕ1
... ϕn ... f ϕn ... ... (n−1) ... f ϕn + ... + f (n−1)ϕn
⎞ ⎟ ⎟ ⎟= ⎠ ⎞ ⎟ ⎟ ⎟, ⎠
& f & $ ( ( det[f ϕ1, ..., f ϕn] =
⎛
⎜ ⎜ ⎜ 2 f det ⎜ ⎜ ⎝
... ϕn ... ϕn ... f ϕn + 2f ϕn ... ... (n−1) ... f ϕn + ... + (n − 1)f (n−2)ϕn
ϕ1 ϕ1 f ϕ1 + 2f ϕ1 ... (n−1) f ϕ1 + ... + (n − 1)f (n−2)ϕ1
⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠
- ( "!# 1 n = 1 ( τ (z) = μ1 − (1 − 1) = μ1 = ordz ϕ1 = ordz W (t).
- ( n = k, k ordz det[ϕ1 , ..., ϕk ] = (μj − (j − 1)), j=1
μj = ordz ϕj . 9 det[ϕ1, ..., ϕk+1]. ? "!# ( det[ϕ1, ..., ϕk+1] = ϕk+1 1 det[1,
ϕ2 ϕk+1 , ..., ]. ϕ1 ϕ1
ϕ ϕ 2 ϕk+1 1 det[( ϕ ) , ..., ( ϕ ϕ ϕ
2 1
ordz ϕk+1 1 det[(
2
ϕ1
) , ..., (
k+1
ϕ1
k+1 1
)]=
) ].
- &
(k + 1)μ1 +
k+1
(μj − μ1 − 1) − (j − 2) = μ1 +
j=2
( & j
k+1
(μj − (j − 1)),
j=2
μj − (j − 1) − μ1 ≥ 0.
2 ϕj , j = 1, ..., n, z, ' & 2 ( ordz det[ϕ1, ..., ϕk+1] =
k+1
(μj − j + 1).
j=1
- & & &' :00;( :<); > !# W (t) ≡ 0 ⇔ ϕ1, ..., ϕn C; /# {z ∈ D : τE (z) > 0} D; 0# = D0 ⊂ D > & z ∈ D0 ϕ1, ..., ϕn E ordz ϕj = j − 1, j = 1, .., n. - ' Ωq (F ) F g > 0 q > 0. 2 P ∈ F q− 8 1 ( % Ωq (F ) . ( ' q− F,
P 1 ( dimΩq (F ). 8 ( 1− 8 1 " # 8 1 F. g > 1, q > 0, τ (P ) P, q Ω (F ), Wq Ωq (F ) d = dq = dimΩq (F ).
$ 8 ' Wq m− ( m = (d/2)(2q − 1 + d), τ (P ) = (g − 1)d(2q − 1 + d).
P ∈F
34542 67$28 = ( Wq m− - ζ1 , ..., ζd Ωq (F ), z z = f (z) &' - ζj = ϕj (z)dz q = ϕ j ( z )d zq ,
ϕ j (f (z))f (z)q = ϕj (z). ?
det[ϕ1, ..., ϕd] = det[(ϕ 1f )(f )q , ..., (ϕ df )(f )q )] =
(
(f )m det[(ϕ 1, ..., ϕ d)f ]
m = q + (q + 1) + ... + (q + d − 1) = (d/2)(2q − 1 + d), (det[ϕ1, ..., ϕd])dz m = (det[ϕ 1, ..., ϕ d])d zm.
2 ? + ( & g > 1 ' & q− 8 1 q > 0, &
m−
% :<); = F g > 1 & ( Ω1 (F ), 1 g(g − 1)/2. A P, / 34542 67$28 ? ' τ (P ) = (g − 1)g(g + 1).
P ∈F
- 2 ≤ α1 < α2 < ... < αg = 2g g P. 2 ( nj , j = 1, ..., g, αj , j = 1, ..., g, {1, 2, ..., 2g}. & (
g 2g g g τ (P ) = (nj − j) = j− αj − j= j=1
P ∈F 2g−1
j=g+1
j−
g−1 j=1
αj ≤
j=1
j=1
j=1
3g (g − 1) − g(g − 1) = g(g − 1)/2. 2
3 ( & ! < // ( α1 = 2. 2 # "4 D # :<); - N 8 1 F g ≥ 2. 2 2g + 2 ≤ N ≤ g 3 − g.
34542 67$28 - ' ( 8 1 g(g −
8 ( 8 1 ! "+ 8 1 F ). $ ) * - ⇔ & 8 1 1, 3, ..., 2g − 1 "+ + # 8 ⇔ & 8 1 1, 2, ..., g − 1, g + 1. 8 8 1 F. ' F g ≥ 3 8 1 :<)( C 0)/; P1, P2, ..., Pn, ... F g ≥ 1 &' D0 = 1, Dj+1 = Dj Pj+1, j = 0, 1, ... $ 5 ”j”, j = 1, 2, ... : ' f F, −1 −1 Dj Dj−1 ? A ”j” + > ' 1)/2 > 0.
−1 f ∈ L(Dj−1)\L(Dj−1 )?
& "=% # :<);
& P1, P2, ..., Pn, ... F g ≥ 1 ' g nj , &' & 1 = n1 < n2 < ... < ng < 2g
(∗)
( ”j” ( j ( &' (∗). 34542 67$28 ”1” ( g > 0. -+ n1 = 1. ”j” ( −1 r(Dj−1) − r(Dj−1 ) = 1 " ( −1 −1 r(Dj ) − r(Dj−1) = 0). ? 9 9 −1 r(Dj−1) − r(Dj−1 ) = 1 + i(Dj ) − i(Dj−1).
2 ( & k > 0 : r(Dk−1 )
−
r(D0−1 )
k −1 = (r(Dj−1) − r(Dj−1 )) = j=1
k+
k
(i(Dj ) − i(Dj−1)) = k + i(Dk ) − i(D0).
j=1
- r(Dk−1) − 1 = k + i(Dk ) − g,
+ =% ( 1 &' k. -+ k > 2g − 2 ( k ( 1 &' k, k − g, degDk > 2g − 2 i(Dk ) = 0. $ ( ' g =% 1 & 2g − 1. 2 F " &# 1 & Hhol (F ) 9 ( F, " F d). - ! F g & ( 1 Ω1(1) HDR (F ) B 1 1 Ω (1) J ( dimC Hhol (F ) = g " # 1 Ω1(1) ∼ (F ). = Hhol
' :<); - F
g > 0, P1 , ..., Pk F (k ≥ 1). 1 F = F \ {P1 , ..., Pk }. 2 dimC Hhol (F ) = 2g + k − 1. 3 (
1 + Hhol(F ) F, F , % & ≤ 2g P1 1 & P2 , P3, ..., Pk. 34542 67$28 9 H1(F , Z) " F ) 2g + k − 1. D ! F , H1(F , Z), $ 1 ( dimC Hhol (F ) ≤ 2g + k − 1, H1 (F , Z) × 1 Hhol (F ) → C,
([γ], [ω]) →
ω. γ
1 A ( Hhol (F ) 1 C2g+k−1, Hhol (F ) 2g +
( ( + + B d ( ' ' -+ ( ( ! F, F , d F, F , 2g + k − 1. + k. - k = 1. n, n ≥ 2g, ' f F, F \ {P1}, &' & n P1, n(≥ 2g) P1 F. -+ & ! F, F , + & & 1 2g P1. 8 Ω(P1−2g )/d(L(P1−2g+1)). i(P1−2g ) − (r(P1−2g+1) − 1) = 3g − 1 − (g − 1) = 2g, d r( P 1 ) = 2g − 1 − g + 1 + i(P12g−1) = g " degP12g−1 = 2g − 1 > 2g − 2), 0 = r(P12g ) = −2g − g + 1 + i( P1 ) " degP12g > 0). $ ( k = 1 1 - k > 1 F = F \{P1, ..., Pk−1}, ( ( F . - =% ! < /G n ' F, ( P1 Pk , & n Pk & ≥ 2g P1, 2g + n(≥ 2g + 1) =% D2g+n = P12g Pkn. 2 F F
9 1 f = 0, f ∈ L(1/P12g Pkn) & n ≥ 1 Pk , df & n + 1 ≥ 2 Pk . 2 ( & Pk 1 & = ( τP P F , F . " F ) F . 2 1 ? :<)( H); dimCHhol (F ) ≥ 2g + k − 1. 5 ( P1 . - 1 Hhol (F ). 8 k−1
2g−1 1
2g 1
k
1
&' F : L# ζ1, ..., ζg F ; M# j = 2, 3, ..., k, k ≥ 2, τj = τP P ; C# j = 1, 2, ..., g, θj F, F &' & nj + 1 P1, (n +1) n1, ..., ng P1 F. 5 ( ng + 1 ≤ 2g θj = τP . 1 ( :<); 3 + Hhol (F ) ( % 2g + k − 1 C L#( M#( C# 34542 67$28 5 1 + + ζ + τ ( ζ ( τ ζ + τ = df, τ % (
- −n = ordP ζ. n > 0, n > 1 " # degf = degf −1(∞) = −ordP f = n−1, n−1 2 ( n ≤ 0 ζ F. 5 ζ ( F. - ) * L#(M# & Ω(1/P1...Pk ). # :<); P1 8 1 1 F, + Hhol (F ) Ω(1/P1g+1P2...Pk ). 34542 67$28 2 P1 8 1 ( ≤ g, & P1 & g + 1. ?
j
1
j
1
1
1
i(
1 1 g+1 ) = r(P P ...P ) − deg( )+g−1= 2 k 1 P1g+1P2 ...Pk P1g+1P2 ...Pk 0 + (g + 1 + k − 1) + g − 1 = 2g + k − 1
" degP1g+1P2...Pk > 0) 2g ! < /H g + 1. $ 8 & + 1 - F g ≥ 2, q ≥ 1, q ∈ N, d = dimCΩq (1), + d = g q = 1; d = (2q − 1)(g − 1) q ≥ 2. - ! < !/ ! < /0 & P ∈ F ' ω ∈ Ω1(1)\0
( P % ω. $ ( ωq ∈ Ωq (1), ' P. - ζ1, ..., ζd Ωq (1). 2 ' ( θ F CPd−1, θ(P ) % (ζ1(P ), ..., ζd(P )) 2 q− 5 1 ( ζj = ϕj (z)dzq ( θ(z) = (ϕ1(z), ..., ϕd(z)). P ∈ F CPd−1 ( Ωq (1) % & CPd−1. -+ θ ( &
:<); q− θ : F → CPd−1 ( > q = 1 F + ( g = 2 = q. 8 + & θ /! # 3 F g ≥ 2 CP3 % - F g CPd−1. 3 F ( g ! - ( θ : F → CPg−1 F + (z1, ..., zg ) CPg−1 (α1 , ..., αg ) ∈ Cg \0. D gi=1 αi zi = 0 CPg−1 θ(F ) P, P g i=1 αi ζi .
! - F
g CPg−1. 2 F ( F " & # 2g − 2 " && 2g − 2 % #
§
{a1, ..., ag, b1, ..., bg} = {a, b} F g ≥ 1 ζ1 , ..., ζg 1 Ω (1), a ζj = δjk , k, j = 1, ..., g. Ω = (πjk ), (πjk = b ζj ) b L(F ) Z, ! 2g (Ig , Ω). " e(1) , ..., e(g), π (1) , ..., π (g). # L(F ) !$ g j=1 mj e(j) + gj=1 nj π(j), mj , nj ∈ Z,
Im + Ωn, m = t (m1 , ..., mg ), n = t (n1 , ..., ng ) ∈ Zg . %
J(F ) = Cg /L(F ) $
& $ F. g
$$ $ g ' & ϕ : F → J(F ) $ $ ( k
k
t
ϕ(P ) =
P P0
t
ζ= (
P
P0
ζ1 , ...,
P
P0
ζg ) ∈ Cg ,
P0 ( $ F, $ $ $
ϕ ( F J(F ),
) F. *+,-,#.'/0#1 γ1 γ2
$!2 P0 3 P, γ1γ2−1 (a, b) t(m, n) $ m, n ∈ Zg . 4
γ1
t
( ζ) −
γ2
(t ζ) = Im + Ωn ∈ L(F ).
. z $
U (P ), z(P ) = 0, ϕ = (ϕ1, ..., ϕg), $ ζj = ηj dz, ϕj (z) = PP ζj + 0z ηj (z)dz dϕdz = ηj (z). # ϕ P, (( ! P. 5 4 ! $ ϕ $ ) *$ ! n ∈ Z, n ≥ 1, F n = F × ... × F Fn
n−
F. & $ ϕ : Fn → J(F ), $ $ D = P1 ...Pn, j
0
ϕ(D) =
n j=1
t
ϕ(Pj ) = (
n j=1
Pj P0
ζ1 , ...,
n j=1
Pj P0
ζg )modL(F ) ∈ J(F ).
# ϕ(P0) = 0,
ϕ(Fn+1) ⊃ ϕ(Fn) ⊃ ... ⊃ ϕ(F1) = ϕ(F ).
ϕ P0 F. 5 Div0(F ) 6 7 8 ϕ : Div(F ) → J(F ),
( ϕ(D) = rj=1 ϕ(Pj ) − sj=1 ϕ(Qj ) $ D = QP ...P ...Q , P0 F. 65 , $8 9::;<7= D ∈ Div(F ). # D F g ≥ 1,
degD = 0 ϕ(D) = 0. *+,-,#.'/0#1 D 2 f ∈ M ∗ (F ) (f ) = D. # degD = 0, ! $ f *$ D = 1 f - 1
r
1
s
D = P1α1 ...Pkαk /Qβ1 1 ...Qβr r , k ≥ 1, r ≥ 1, $ Pj = Ql $ j, l, Pj = Pl Qj = Ql $ j = l, kj=1 αj = rj=1 βj ≥ 1. > $ 2
Pj , Qj
- dff (( ! resP dff = ordP f. ! k r df −( αj τPj P0 − βj τQj P0 ), f j=1 j=1
P0 = Pj , P0 = Ql $ j, l, P0 (( *$ cj j = 1, 2, ..., g,
k
r
g
df = αj τPj P0 − βj τQj P0 + cj ζj . f j=1 j=1 j=1
! a dff = cl , $
$ $ (( τP Q,
l
bl
Pj Qj g k r df = 2πi( αj ζl − βj ζl ) + cj πjl , f P P 0 0 j=1 j=1 j=1
l = 1, df /f = d(logf ), ...,df g. 0 df al f = 2πiml , bl f = 2πinl, l = 1, ..., g, ml , nl ∈ Z. 4 l$ $ ϕ(D) k j=1
αj
Pj P0
ζl −
r
βj
Qj
1 ζl = 2πi
P0
j=1
g
g
df 1 − cj πjl = nl − mj πjl . f 2πi j=1 j=1
bl
0 ϕ(D) = 0modL(F ). - D 7 4 ϕ(D) P0 $ ϕ. *$ $ D, $ ( ! f ! (f ) = D. 1 Q0 Pj Ql , P0 2 aj bj , j = 1, ..., g. f (P ) = exp(
k
αj
P Q0
j=1
τPj P0 −
r
βj
P
τQj P0 +
Q0
j=1
g
cj
P Q0
j=1
ζj ) = exp
P
τ,
Q0
c1, ..., cg # f ( ( F (f ) = D,
2 $ 1 al
τ=
bl
τ=
k
αj
al
j=1 k
αj
2πi(
j=1
bl
αj
βj
τPj P0 −
r
P0
ζl −
r j=1
τQj P0 + cl = cl ,
βj
j=1 Pj
al
j=1
j=1 k
τPj P0 −
r
βj
bl
τQj P0 +
Qj P0
g
cj πjl =
j=1
ζl ) +
g j=2
cj πjl .
*$ f F a τ, b τ 2πi. ? ϕ(D) = 0
nl + gj=1 mj πjl , nj , mj ∈ Z, γj γj ,
$!2 P0 3 Pj 3 Qj
1$ cj = −2πimj , j = 1, ..., g, f F. # 9::;<7= *$ F g = 1 ϕ : F → J(F ) ( ( *+,-,#.'/0#1 @ P, Q ∈ F, P = Q. . ϕ(P ) = ϕ(Q) J(F ), ϕ(P/Q) = 0 J(F )
l = 1, ..., g.
l
l
, $ P/Q 2 ( $ ( $ f 3 (f ) = P/Q. A F ( ! 4 ϕ @
(
B (
$
F ! J(F )
C C 0 *$ F g ≥ 2 ϕ $$ $ @ ( F
ϕ(F ) J(F ). *+,-,#.'/0#1 D @ $ 2 E & $ 4 $ ) F. 4 ϕ(F )
g ≥ 2,
g
J(F ). 0 D g F, D = P1...Pg , Pj ∈ F, j = 1, ..., g. E E
r(D−1) = 1+i(D) ≥ 1. * D, degD = g, $
i(D) > 0 r(D−1) > 1. 9<7= D ∈ Div(F ), D ≥ 1, degD = g, 2 D g, D Fg , D + D $2 g F. E ϕ : Fg → J(F ). 4 C C D0 = P1...Pg ∈ Fg , Pj = Pk , j = k, j, k = 1, ..., g, i(D0) = 0. Uj
! Pj
tj , tj (Pj ) = 0. 1 4
ζk = ηkj (tj )dtj , k = 1, ..., g, ϕ (0, ..., 0) Cg Cg (
(z1, ..., zg ) → K + (ϕ1(z1 , ..., zg ), ..., ϕg(z1 , ..., zg )), z ϕl (z1 , ..., zg ) = gj=1 0 j ηlj (tj )dtj , K = ϕ(D0). ! ∂ϕl (0, ..., 0) = ηlk (0) = ζl (Pk ). ∂zk
-
# $ $ ϕ D0 = P1...Pg ⎞ ζ1 (P1 ) ζ1 (P2 ) ... ζ1 (Pg ) ⎝ ... ... ... ... ⎠ . ζg (P1 ) ζg (P2 ) ... ζg (Pg ) ⎛
D D0 $ $ 4 $ g. 0
( ϕ : U1 × ... × Ug → Cg ( (0, ..., 0) U (K). c = t (c1, ..., cg ) ∈ Cg $ # $
N K + c/N ∈ U (K) 2 ! Q1, ..., Qg
ϕ(Q1...Qg ) = K + c/N, N (ϕ(Q1...Qg ) − K) = c. # c ϕ, D g 3 ϕ(D) = N (ϕ(Q1...Qg ) − K).
E (Q(P...Q...P) ) P . # E E 1
1
N g N g g 0
(P1...Pg )N (Q1...Qg )N P0g r( ) = 1 + i( ) ≥ 1. (Q1...Qg )N P0g (P1 ...Pg )N
F 2 f ∈ L( (Q(P...Q...P) ) P ) 2 D g
1
1
N g N g g 0
D(P1 ...Pg )N (f ) = . (Q1...Qg )N P0g
, $
ϕ(D) + N ϕ(P1...Pg ) = N ϕ(Q1...Qg ).
# 2 $ & !2$ ! 6+ & 8 9::;<7= + $ J(F )
g. " # . i(D0) > 0, D0 = P1 ...Pg & (
U (D0) Fg . $ 9::;<7= B (J(F ), +) ( ( Div0(F )/DivH (F ) 7 F g ≥ 1. # J(F ) ( P ic0 (F ) = Div0(F )/DivH (F ) P ic(F ) = Div(F )/DivH (F ) ∼ = DivH (F ) × Z. ( ( deg : D → degD, D ∈ Div(F ).
g ≥ 2,
9<7= F $ $ ! q ≥ 1 (
q (( Ωq (1).
$ $$ $
*+,-,#.'/0#1 d = dimCΩq (1). D
1 < d < ∞. ω1 ω2 4 Ωq (1) 2 (( 1 ω1 ! ( q (( G (( ! (ω1), @ Ωq (1) ! ) 4 ω2 $ C C Ωq (1). - f = ωω $ $ ( $ ( $ F q(2g − 2). α
$$!2
$ $ $ f. #
α 2 $ $ f
4 ( $ f α q(2g − 2) F. " $$!$ $ (( ω1 − αω2, $ $ 6ω1 − αω28 q(2g − 2). " ( q (( $ $$ $ Ωq (1), ! (( ω1 2 6 α) (( $
( q (( $ Ωq (1). ω q− (( $ P1, ..., Pq(2g−2) ω1, ..., ωd Ωq (1). 1 Dj Pj , j = 1, ..., q(2g−2), Dj
zj , zj (Pj ) = 0, j = 1, ..., q(2g − 2). 1
ω = φj (zj )dzjq ωk = φkj (zj )dzjq , k = 1, ..., d. 5 $ $ ω φj (0) = 0 φj ! γj = ∂Dj $ j. 0 |φj (zj )|
mj > 0 γj $ ! j. # mkj |φkj (zj )| γj . 1 ε
ε dk=1 mkj < mj , j = 1, ..., q(2g − 2). # $ ! η1, ..., ηd, |ηk | < ε, (( θ = ω+ dk=1 ηk ωk
Dj , j = 1, ..., q(2g − 2). " E γj
1 2
|
d k=1
ηk φkj (zj )| ≤
d k=1
|ηk |mkj < ε
d k=1
mkj < mj ,
j = 1, ..., q(2g −2). " q(2g −2), $$!$ $ $ θ, $$ $ & θ = ω + dk=1 ηk ωk
$! ω q (( $ #
§
! "
#
$ F " % & D, degD = 2 r(D−1 ) ≥ 2. ! % F f
' ( ) * % ' +( f F # 2g + 2. ,# F g ≤ 2 # " -.$/0/1,234. 5 D 6 F. " * * r(D−1 ) = 2 − g + 1 + i(D) ≥ 2
g ≤ 1. - g = 2. 4# ω = 0 # " F. " deg(ω) = 2g − 2 = 2 (ω) = P Q. . i(P Q) = 1, % r((P Q)−1) = 2 − g + 1 + i(P Q) = 2. 5 789: 5 F " g ≥ 2. " z F 6 #
# # z. $ "% z ' ( 4;
F. g ≥ 2 % 2g + 2 4;
789% < +9+: = " F g ≥ 2 # 4;
# q− 4;
F #" q > 1. ! "% # 4;
# 4;
1, 3, ..., 2g − 1. 5 & # ω " < P 2g−2. . ωq " q− P q(2g−2). q(2g − 2) > (2q − 1)(g − 1) − 1, P # q− 4;
F.
5 z 6 " F g ≥ 1 P1 , ..., P2g+2 F. > " #& % z(Pj ) = ∞, j = 1, ..., 2g + 2. 5 2g+2 & w = j=1 (z − z(Pj )) F. 789: ? w g+1 F P1 ...P2g+2/Qg+1 1 Q2 , " Q1 Q2 z. 3 & z w
" F % 2
2
2g+2
SF = {(z, w) ∈ C : w =
(z − z(Pj ))}.
j=1
5 # " SF , # g SF # F. 5 "
g = 2 : +( > C (e1, e2 ), (e3, e4 ) (e5, e6 ), -
b1
a1
R
e1
e2
b2
a2
-
R
e3
e4
e5
e6
P uc.14
0 # % & @ @ % ; @ @ % ej = z(Pj ), j = 1, 2, ..., 6.
6( 3 @ @A
b-1 a-2
a-1 P1
P2
b-2
P3
P4
P5
P6
P uc.15
-
z wdz , j = 0, ..., g − 1, # # # " " F g ≥ 1. -.$/0/1,234. B % z(Pj ) = 0, j = 1, ..., 2g + 2, 3 Q4 (z) = Q Q1 Q2 . C % C = % # " F. ) (dz) = P1Q...P21Q2g+2
% 2 2 j
z j dz ) = Q1g−j−1Q2g−j−1Qj3 Qj4 ≥ 1 ( w
0 ≤ j ≤ g − 1. 3 j 0 % z wdz j ≥ g # g F
Q1 , Q2. 5 j = g z wdz # # " F
Q1, Q2, j Q1 = Q2. 5 j ≥ g + 1 z wdz # # " F
Q1 Q2 j − g + 1(≥ 2). 5 % 7DDE+DE+9DEF8:% " " " F g ≥ 2, w2 = (z − e1 )(z − e2 )...(z − e2g+1),
" ej = ek , j = k, ej = ∞, j, k = 1, ..., 2g + 1. 1 ? f1 = z−e ' i, i = 1, ..., 2g + 1) # i " (z = ei , w = 0), # - % z = a = ∞, a = ei , 1 i = 1, ..., 2g + 1, t = z − a f1 = t+a−e i √
t = 0; z = ei % t = z − ei f1 = t12 2 " t = 0; z = ∞, t = √1z f1 = 1−tt 2ei
t = 0. . % (z = ei , w = 0), i = 1, ..., 2g + 1, 4;
F, "
f1 2 ≤ g. f2 = z z = ∞ " % t = √1z f2 = z = t12 t = 0. 5 (z = ∞, w = ∞) 4;
4 " # 2g+2 4;
F. 0 % 1 ei f1 = z−e i (z = ei , w = 0) # " " F. 5 " F, # 6 P0 P0 # " 5 z(P0 ) = a = ej #" j = 1, ..., 2g + 1, a = ∞, " t = z − a w z = a : w = f (z) = b0 + b1(z − a) + b2(z − a)2 + ..., b0 = f (a) = 0.
(a)+f (a)(z−a) " f (z)+f(z−a) dz " 2 f (z) P0 - %
1 1 2 )dz = [2b + 2b (z − a) + b (z − a) + ...]( 0 1 2 (z − a)2 b0 + b1(z − a) + ... 1 1 b1 2 [2b + 2b (z − a) + b (z − a) + ...][ − (z − a) + ...]dz = 0 1 2 (z − a)2 b0 b20 2 1 2b1 2b0b1 ( [ + − 2 ] + ...)dz = 2 (z − a) z − a b0 b0 2 + c0 + c1 (z − a) + ...)d(z − a). ( (z − a)2 5 z = b = a
A G( b = ej , j = 1, ..., 2g + 1, t = z − b, % # dz # t = 0; √ #( b = e1 , t = z − e1, t2 = (z − e1 ), dz = 2tdt dz √ 2tdt2 , % t = 0; " f (z) t (t +e −e ) j=1
1
j
b = ej , j = 2, 3, ..., 2g + 1; ( b = ∞, t = √1z , z = t12 , dz = − t23 dt
1 b0 + b1 (z − a) ]dz = [1 + (z − a)2 f
t4 t2 b0 + b1 − at2 b1 2g−1 −2dt (1 + ) 3 t (1 − t2 a)2 t 2 j (1 − ej t )
t = 0. √ dz 5 z(P0 ) = e1 , " t = z − e1 (z−e 1 )f P0 6 - % t2 = z − e1 , dz = 2tdt dz 2tdt = = (z − e1 )f 3 t j=1 (z − ej ) 2 ( t2
1
2 j=1 (t
+ e1 − ej )
)dt =
2 ( t2
1 j=1 (e1
− ej )
+ c2 t2 + ...)dt.
5 z = b = e1
A
( b = e2, e3 , ..., e2g+1, t = z − b, f (b) = 0 # dz #
t = 0; √ #( b = eν , ν = 1, t = z − eν , t2 = z − eν , 2tdt = dz dz 2tdt = = (z − e1 )f 2 (t + eν − e1 )( j=ν (z − ej ))t (t2 + eν − e1 )(
t = 0; ( b = ∞, t =
√1 z
2dt
2 j=ν (t
+ eν − ej ))
% dz = − t23 dt
− t23 dt dz = = (z − e1 )f 1 1 ( t2 − e1 ) j ( t2 − ej ) −
1 2dt 2g t (1 − t2 e1 ) 2 j (1 − t ej )
t = 0. g = % z(P0 ) = ∞, t = √1z z fdz
" P0 P0 . - % 1 (− t23 dt) z g dz 2 t−2 dt t2g = − 2 (1 + c2 t2 + ...)dt. = = −2 f t 1 2 j (1 − t ej ) j ( t2 − ej )
4 z = b = ∞ " % & 5 " % &
+ P0 + P1 . 0 P0 % " z(P0 ) = ∞. 5 P1 # % z(P1 ) = a = eν , ν = 1, 2, ..., 2g + 1, a = ∞. 4 P1 t = z − a f (z) = 0 dz b0 + b1(z − a) + b2 (z − a)2 + ... " 12 f (z)+b z−a f (z) " - % 1 1 f (a) 1 (1 + )dz = (1 + c1 (z − a) + ...)d(z − a) 2z − a f (z) z−a
P1 . 0 P0 t =
√1 z
;
1 − t23 dt f (a) )= (1 + 1 2 t12 − a j ( t2 − ej ) −
dt f (a)t2g+1 dt 1 ) = − (1 + (1 + c1 t + ...). t 1 − t2 a t 2 j (1 − t ej )
4 % P0 P1 # 1 P1 % z(P1 ) = eν , " dz F. - % z = eν , # 12 z−e ν √ t = z − eν 1 dz 2tdt dt = 2 = , 2 z − eν 2t t z = ∞, t = √1z , z = t12
1 dz 1 2dt dt(1 + c1 t + ...) ( . =− ) = − 2 z − eν 2t 1 − t2 eν t
5 eν " # " n ≥ 3. = % d dz dz ( )= deν z − eν (z − eν )2
P1 (z(P1 ) = eν ) " P0 # - % z = eν , t2 = z − eν , ;
dt 2tdt t4 = 2 t3 , z = ∞, t =
√1 , z z
=
1 t2 ,
;
− t23 t4 dt − t23 dt = (1 − eν t2)2 ( t12 − eν )2
t = 0. - #" g ≥ 3 & " 4 "
% 7DD:% " F % "# {(z, f ) : f 4 = z 4 − 1}. 5 *
F % 1, i, −1, −i, % g = 3, 2g −2 = 4(0−2)+4(4−1). = "
dz zdz f dz , , . f3 f3 f3
- "
√ z = ±1, ±i z = ∞. 4 z = 1% t = 4 z − 1, % t4 = z − 1, 4t3dt = dz f 4 = t4 (t4 + 2)(t4 + 1 + i)(t4 + 1 − i). . dz 4dt = 3, f3 [(t4 + 2)(t4 + 1 + i)(t4 + 1 − i)] 4 zdz 4(t4 + 1)dt = 3, f3 [(t4 + 2)(t4 + 1 + i)(t4 + 1 − i)] 4 4tdt f dz = 1, f3 [(t4 + 2)(t4 + 1 + i)(t4 + 1 − i)] 2 # t = 0. / "
z = ±i, −1. .
z = ∞. 4 t = 1z , 4 " dz = − dtt2 , f 4 = 1−t t4 . ) dz tdt zdz dt f dz dt = − , = − , = − 3 3 1 f3 (1 − t4 ) 4 f 3 (1 − t4 ) 4 f 3 (1 − t4 ) 2
t = 0. ! C, & " ; 1, z, f. 5 # # Ω1(1). 0 % ; " " f, ; " " z. 5
; " ' ( M(F ) F. 4 " ; dz zdz z g−1 dz " # ; # % f , f , ..., f z, M(F ) g ≥ 2. 789: = "
F g ≥ 2 # " # (2g − 1)− (3g − 3)− "
! "% " z k+j dz 2 , 0 ≤ k + j ≤ 2(g − 1), w2
(1)
(2g − 1)− +( 2g − 1 = 3g − 3 ⇐⇒ g = 2; 6( #
Ω2 (1) " F g ≥ 3, '+( # & j 2
z wdz , j = 0, ..., g − 3, g − 2 " F. - % z j dz z j dz 2 )=( )(dz) = ( w w P1 ...P2g+2 = Q1g−j−1Q2g−j−1Qj3Qj4 Q21Q22
Q1g−j−3Q2g−j−3Qj3Qj4 P1 ...P2g+2 ≥ 1
g − j − 3 ≥ 0, 0 ≤ j ≤ g − 3. = " F g ≥ 2 q ≥ 3 q− # "
# (q(g − 1) + 1)−
(2q − 1)(g − 1)− Ωq (1). - " # # " q− z j1 ...z jq q dz , 0 ≤ jk ≤ g − 1, k = 1, ..., q. (2) wq " 0 ≤ j1 +...+jq ≤ q(g−1) # # (q(g−1)+1)−
* (2q − 1)(g − 1) = q(g − 1) + 1 %
q = 2, g = 2. # % q ≥ 3
" # # g ≥ 2. .# C # ϕ : F −→ J(F ) H " # > "% e ∈ J(F ) n(n ∈ N), ne = 0 ' J(F )), me = 0 0 < m < n. !789: 5 F " g ≥ 2 4# # P0 ϕ 4;
F " ϕ(P ) 6% P 4 ;
F. -.$/0/1,234. 5 P (= P0 ) & 4;
F. 3 & f F, (f )∞ = P 2 . 2 P0 # f, (f − f (P0)) = PP02 . . P02 ∼ P 2 /# 2ϕ(P ) = ϕ(P 2) = ϕ(P02 ) = 0 J(F ), ϕ(P ) = 0 H ϕ. 5 "789: 5 F g ≥ 2. " F " % & J F, J 2 = id, 2g + 2 1 g ≥ 2, " 4;
1 F " g ≥ 2, 2
w =
2g+2
(z − ej ),
j=1
" e1 , ..., e2g+2 z− % " J, # (z, w) → (z, −w), z −1 (ej ), j = 1, ..., 2g + 2, 0 z −1 (∞) F. 5 " F g ≥ 2,
w2 = z(z − 1)
2g−1
(z − λk ),
k=1
" λ1 , ..., λ2g−1 C\{0, 1}, P1 = z −1 (0), P2 = z −1 (1), Pj+2 = z −1 (λj ), j = 1, ..., 2g − 1, P2g+2 = z −1 (∞). 5
# $789% DI9: "
" " $ "% "
§
D = P ∈F P α(P ) , α(P ) ∈ Z, α(P ) = 0 F
g ≥ 1. α(P ) P D. |D| D D1 D. !"# $%!&' ( ) * D + F g ≥ 1 |D| ) + ) P L(D−1) * L(D−1)). (,-./.0123405,% D1 ∈ |D|, * D1 ≥ 1 D1 = D(f ) f = 0 6 6
F, D1 ≥ 1 # f ∈ L(D−1). ,) # ) f ∈ L(D−1) D1 = D(f ) ≥ 1
D1 D. ( 6
f g L(D−1) # f = cg, c ∈ C∗. % 2 DV |D| # + 7 P V P L(D−1), * V + + L(D−1). 8 (degD = d, dimCV = r + 1).
9
DV + # )7 DV . 1 P + ) DV , (f )D ≥ P P f ∈ V, # V ⊂ L( D ). 5 # P D # P ) |D|, ) D1 ∈ |D| P, ) f ∈ L(D−1) ) 7 P, L( D1 ) = L( DP ). % &&' + F g ≥ 1 : ; 1 degD < 0, |D| = ∅; < 1 D = 1, |1| = 1, ) = > |D| ) # ) * P ∈ F dimCL( DP ) = dimCL( D1 ) − 1; ? ,) A ) @ # |D|,
# |D| ) % A+ A |D|. 0 * L( DA ) = L( D1 ). ( +
% 0 A ≥ 1, DA ≤ D
A L( D ) ⊂ L( D1 ). ,) # f ∈ L( D1 ), D(f ) ≥ 1 D(f ) ∈ |D|. /+ (f )D = AD, D ≥ 1, # (fA)D = D ≥ 1 f ∈ L( DA ). | DA | |D|. B+ # ) |D| ) 6 = C ( ) * D, degD ≥ 2g, |D| ) % ( # dimL( DP ) = dimL( D1 ) − 1 ) P ∈ F. 0 degD deg DP @ 2g − 1, i(D) = 1 P D i( D P ) = 0, # dimL( D ) = degD + 1 − g dimL( D ) = deg( P ) + 1 − g = D - |Z| ) % (+ # 6
P ∈ F. A # L( PZ ) = L( Z1 ), # r( PZ ) = r( Z1 ) − 1 = g − 1. E+ r( P1 ) = 1 6 F +F 1 = r( P1 ) = r( PZ ) + degP + 1 − g
dimL( PZ ) = g − 1; ! D = P1...Pn, n ≥ 1. 0 * 7 * 6 66 ω F # (ω) # 7 D. ( * D L( D1 ) + @ s(≥ 1), D s − 1 G ) G % 9 # 6
!"'% D +
F g ≥ 1. 0 * r( D1 ) ≥ s, ) * * * D, degD ≤ s − 1, 7 D # DD ∼ D. - * # # D ) ) * * Fs−1 (s − 1)+ F ). 5 ) B ) ϕ : Fn → J(F ) + J(F ). 1 F +
g ≥ 1, ) ) * * * * ) {a, b} H1(F, Z), # * * * ) ζ1, ..., ζg Ω1(1), Cg /L(F ). 5 # + 7 ) ) {a, b}, 6 * *% 4 # ) # 6
# ) {a, b} F, # ) @ L(F ), ) * * + ) B ) # %% * *
6 % +
(F, {a, b}) (J(F ), L(F )) *# * ) ) P0 )+
B ) % ,) Wn = ϕ(Fn). B # W0 = {0}, Wn ⊂ Wn+1 Wg = J(F ). Wnr = ϕ({D ∈ Fn : r(D−1) ≥ r + 1}) = ϕ(Fnr ),
%% J(F ), 7 ) ϕ D # degD = n, r(D−1) ≥ r + 1. E .) # W1 = ϕ(F ) 6 F ϕ
J(F ). , K = (KF ) = ϕ(Z) + J(F ). g−1 ! K = W2g−2 .
(,-./.0123405,% D + # degD = 2g−2, D ≥ 1. 0 * r(D−1) ≥ g ⇔ i(D) ≥ 1 ⇔ D + % + % # @ J(F ) % !"# $% ;C"' : ; D ∈ Fn. 0 * ) ) ϕ : Fn → Wn ⊂ J(F ) D * n + 1 − r(D−1)(= n − ν); < 4 ϕ−1(ϕ(D)) ) ϕ + D * ) Fn + ν = r(D−1) − 1, + ) * * CP ν ;
> , ) ϕ : Fn → Wn 6 Fn \ Fn1 Wn \ Wn1; ? n ≤ g G )7* G D ∈ Fn i(D) = g − n; C 1 1 ≤ n ≤ g − 1, u ∈ Wn ) Wn,
u ∈ Wn1; 1 D F r Wn+1 , 1 ≤ n ≤ g − 1, + 2n − g ≤ r ≤ n − 1, r = n − 1 *# F + * + % " ??# % ;D<'% H {D ∈ Fg : r(D−1) = 1} Fg F g ≥ 2. (,-./.0123405,% F +F D, degD = g, r( D1 ) = g − g + 1 + i(D) ≥ 1. F @ Fg , %% {D ∈ Fg : r( D1 ) ≥ 1 + 1} = Fg1.
E 7 # dimWg1 ≤ g − 2. @ Fg \ Fg1, ) * 6 6 ϕ Wg \ Wg1 Wg . % * ) S J(F ) # ) +
6 6 J(F ), + S, ) S. # !"# % ;CC'% n ≤ g Wn + + * ) J(F ) n, Wnr + + * ) J(F ), # 2r ≤ n. $ !"'% ( ) * a ∈ J(F ) −Wg−1 − a = Wg−1 − a − K
(,-./.0123405,% ( ) * * D, degD = g − 1, D g − 1 # DD ∼ Z. , ϕ(D) + ϕ(D ) = K Wg−1 = K − Wg−1. % % H% A !"'% A * + F g ≥ 3 ) * q ≥ 2 q− ) 66 * + Ωq (1). 4 #
% & 47 ω1 , ω2 ∈ Ω1(1),
)7 % (,-./.0123405,% 5) ) ω1 ∈ Ω1(1),
P1 , ..., Pr ) * % 1 # ) * ω2 ∈ Ω1(1) ) ) P1 , ..., Pr . F Ω(P1) ∪ ... ∪ Ω(Pr ). - )+ (g − 1)− # # )I (g − 1)− ) Ω1(1). % 0 ) # 7 ω2 ∈ Ω1 (1), P1 , ..., Pr. 2 % ' ( 66 ω1 ω2 ;%;"%J )
7 % (,-./.0123405,% ( 66 ω2 0 P1, ..., Pr. + 6 f = ωω F (2g − 2)− C # * z ∈ C f −1(z) # f −1(0) ∼ f −1(∞) ∼ f −1(z). A f # 66 # ) @ % ( # (ω1) = f −1(0), (ω2) = f −1(∞) # )
a 1 2
b (a = b) # f −1(a) f −1(b)
)7 % 0 + Af +B ) + ) w = Cf , f = a
+D f = b w = 0 w = ∞ % 66 + Aω1 + Bω2 Cω1 + Dω2 @ )7 % 2 % !"# $% ;C&'% 47 ω1, ω2 ∈ Ω1(1),
7 )7 P0,
# F * % (,-./.0123405, ;%;"%!% *# F * + % 0 * F 7 D, degD = g − 2, # i(D) = 2 ) Q ∈ F, i(DQ) = 1. , * % ) * D ∈ Fg−2 7 Q ∈ F # i(DQ) ≥ 2. 0 ) # ) * v ∈ Wg−2 7 x ∈ W1 # 1 1 1 1 (−W1) dimWg−1 v + x ∈ Wg−1 , v ∈ Wg−1 − x ⊂ Wg−1 ≥ g − 3. 1 A )7
# dimWg−1 ≤ g − 3; @ * % % ω1, ω2 Ω(D), # ;%;"%J# ) * 6 66 ω3, )7 ω1ω2.
D # ωω + 6 6 F, g i( (ωD ) ) = 1. ( # 6 g * # (ω1) (ω2) )7 D, ) ω1, ω2 ∈ Ω(DQ), i(DQ) = 1 ω1 = cω2. K ) i( ωD )
i( (ωD ) ) = r(D−1) = g − 2 − g + 1 + i(D) = 1. q = 2, θ1, ..., θg + ) Ω1(1). F 2g 1 2
2
2
2
ω1 θ1, ..., ω1θg , ω2θ1 , ..., ω2θg
Ω2(1). g g C. A1 A2 + Ω2(1), )
% E dimA1 ∪ A2 = dimA1 + dimA2 − dim(A1 ∩ A2 )
dimA1 ∩A2 = 1. ( * # ) + (ωj ) = DDj , j = 1, 2, # D1, D2 + g, (ω(ω )) = DD . 1 η ∈ A1 ∩ A2, η = ζj ωj , j = 1, 2, * ζj ∈ Ω1(1), ) θ1, ..., θg. + 1 * ζζ = ωω . 5 # (ζ1) = ⊕DD2 (ζ2) = DD 2
1
1
2
1
1
2
2
(ζ2) = (ω(ω )) (ζ1) = D1D (ζ1) = (ζ2) (ω(ω )) = D2D, ζ ζ ω = ω , D g − 2. E i(D1 ) = 1 = i(D2 ), ωζ ωζ ∈ C. A1 ∩ A2 ω1ω2. ( # 1 = i( (ωD ) ) = i(D2) = i(D1 ), (ω1) ∼ (ω2)
DD1 ∼ DD2 , D1 ∼ D2 . / ζ2 ω1 ∈ Ω(D1), ζ1 ω2 ∈ Ω(D2), ζ2 = c1 ω1 , ζ1 = c2 ω2 . A # η = ζ1 ω1 = ζωω ω1 = c1 ω2ω1 . 4 # dimA1 ∪ A2 = 2g − 1 2g Ω2(1) 2g − 1. ( ) ω3θ1, ..., ω3θg 7 # + A3 ⊂ Ω2(1), % 4
∈ Fg−2, D 2
1
1
2
2
1
1
2
1
2
2
1
2
2 2 1
dim(A1 ∪ A2) ∪ A3 = dimA1 ∪ A2 + dimA3 − dim(A1 ∪ A2) ∩ A3 = 2g − 1 + g − dim(A1 ∪ A2 ) ∩ A3 .
# dim(A1 ∪ A2) ∩ A3) = 2 ω1ω3 ω2ω3 . 1 η ∈ (A1 ∪ A2 ) ∩ A3, η = ζ3 ω3 = ζ1 ω1 + ζ2 ω2 . 0 * deg D = g. ) 7 D, (ζ3) = DD, 0 ) # ζ3 = x1ω1 + x2ω2, x1, x2 ∈ C, <# η = ζ3ω3 = x1ω1ω3 + x2ω2ω3. 4 # dimA1 ∪ A2 ∪ A3 = 3g − 3 = dimΩ2(1). ( q = 2. q = 3. 4 ) ω1, ω2 ∈ Ω1(1) # ω1, ω2 )7 P0 , ω3 ∈ Ω1(1) # ω3
)7 ω1ω2. f1, ..., f3g−3 ) ) Ω2(1), * fj + 2− Ω1(1). Aj =< ωj f1, ..., ωj f3g−3 >, j = 1, 2.
- @# dim(A1 ∪ A2) = dimA1 + dimA2 − dim(A1 ∩ A2). 1 ω1η1 = ω2η2 ∈ A1 ∩ A2, ωη = ωη + ) 66 ) @ P0, # * 6 P0. η2 = ω1ω, ω ∈ Ω1(1). 8 # dimA1 ∩ A2 = g, η = ω1η1 = ω2η2 = ω2ω1ω. 0 ) # dimA1 ∪ A2 = 5g − 6. B # ω33 = 0 P0 , ) A1 ∪ A2 ) P0 . 4 # ω33 ∈ / A1 ∪ A2 . ( ) * 7 # 5g − 5 = dimΩ3(1), 3− ) Ω1(1). 4 q = 3 % 2
1
1
2
( q ≥ 4 )
% f1, ..., f(2m−1)(g−1) + ) Ωm(1), m ≥ 3, 7 m− +
Ω1(1). Aj + # ωj f1, ..., ωj f(2m−1)(g−1), j = 1, 2, * ω1, ω2 Ω1(1) ) )7 % 0 * dimA1 ∪ A2 = 2(2m − 1)(g − 1) − dimA1 ∩ A2. 4 # dimA1 ∩ A2 = (2m − 3)(g − 1) = (2(m − 1) − 1)(g − 1)
% 4 # dimA1 ∪ A2 = (2m + 1)(g − 1) = (2(m + 1) − 1)(g − 1) = dimΩm+1(1),
) 4m − 2 − 2m + 3 = 2m + 1 = 2(m + 1) − 1. ( K m m+1 ) g − 1.
§
Cg ;
J(F ) ϕ(F ) ⊂ J(F ). g ≥ 1
g ! σg , " g × g !
#$ % ! σg ! X ! X g(g+1) 2 − $
&'()$ * ! σg X σg $ + , "
!
, ! $ -
! Jmτ τ τ0 X. ./ 0
Θ(z; τ ) =
1 exp2πi( (t N )τ N + (t N )z), 2 g
(1)
N ∈Z
z ∈ Cg , τ ∈ σg . 1 {(z, τ ) ∈ Cg × σg : ||z||Cg ≤ M, λ(τ ) ≥ λ0 > 0}
/ # $ λ(τ ) Jmτ. - 2 3 / Θ Cg × σg . 4 " 3 Θ(z; τ ) = Θ(z), τ $ , !
g = 1 / 1 2 Θ(z; τ ) = Σn=∞ n=−∞ exp2πi[ n τ + nz], z ∈ C, τ ∈ σ1 , 2
# #$
μ, μ ∈ Zg
1 Θ(z + Iμ + τ μ; τ ) = exp 2πi(−(tμ)z − (t μ)τ μ)Θ(z; τ ) 2
# z ∈ Cg , τ ∈ σg , I = Ig − g.
567 7.89: .25$ + (tτ ) = τ, (tN )μ ∈ Z, Θ(z + Iμ + τ μ; τ ) = 1 exp 2πi( (t N )τ N + (t N )(z + Iμ + τ μ)) = 2 g
N ∈Z
1 1 exp 2πi( (t(N + μ))τ (N + μ) + (t(N + μ))z − (tμ)τ μ + (t N )μ − (tμ)z) = 2 2 N ∈Zg 1 1 exp 2πi(−(tμ)z− (t μ)τ μ) exp 2πi( (t(N +μ))τ (N +μ)+(t(N +μ))z) = 2 2 g (μ+N )∈Z
1 exp 2πi(−(tμ)z − (t μ)τ μ)Θ(z; τ ). 2
! "
;< Θ(z + e(j); τ ) = Θ(z; τ ), =< Θ(z + τ (k); τ ) = exp 2πi(−zk − τ2 )Θ(z; τ ), e(j), τ (k) (j), (k) Ig , τ > ?< Θ(−z; τ ) = Θ(z; τ ) # z ∈ Cg , τ ∈ σg . , @ ;< μ = e(j), μ = 0; =< μ = 0, μ = e(k) (te(k))z = zk , (te(k))τ e(k) = (te(k))τ (k) = τkk . 2 3 Θ(z) ! Θ(z + e), e ∈ Cg , kk
Θ(z + e + e(k) ) = Θ(z + e), τkk Θ(z + e + τ (k) ) = exp 2πi(−zk − ek − )Θ(z + e). 2
! / Cg , # e, d ∈ Cg 3 / f (z) =
Θ(z + e)Θ(z − e) . Θ(z + d)Θ(z − d)
- !, f (z) Cg A e(k), τ (k), k = 1, ..., g. 1!
τ (k). , z ∈ Cg f (z + τ (k) ) = exp 2πi[−zk − ek − exp 2πi[−zk − dk −
Θ(z + e + τ (k) )Θ(z − e + τ (k) ) = Θ(z + d + τ (k) )Θ(z − d + τ (k) )
τkk 2 ]Θ(z τkk 2 ]Θ(z
+ e) exp 2πi[−zk + ek − + d) exp 2πi[−zk + dk −
τkk 2 ]Θ(z τkk 2 ]Θ(z
− e) = f (z). − d)
, τ ∈ σg " z , Θ(z; τ ) = 0, # t n∈Z exp(πi( n)τ n)exp(2πi(tn)z) B Θ, / exp(πi(tn)τ n) = 0. #
- τ ∈ σg . . f (z) Cg L(τ )− l, g
f (z + n) = f (z), f (z + τ n) = exp(−πil(tn)τ n − 2πil(tz)n)f (z)
n ∈ Zg . 5
! / "# ! Cg /L(τ )
$ , f0, ..., fm # # L(τ )− ! l, # a ∈ Cg " fj (a) = 0, z → (f0(z), ..., fm(z))
! Cg /L(τ )
Pm. - (Ig , τ )
(F, {a, b}) g ≥ 1. . / R, , # e ∈ Cg e = I ε2 + τ 2ε , ε, ε ∈ Rg . , Θ(z + e) = Θ(z + I ε2 + τ 2ε ). % #" Cg × σg ε, ε ∈ Rg :
Θ[ε, ε ](z; τ ) =
ε ε 1 t ε ε t exp 2πi[ ( (N + ))τ (N + ) + ( (N + ))(z + )]. (2) 2 2 2 2 2 g
N ∈Z
C , Θ[0, 0](z; τ ) = Θ(z; τ )
Θ[ε, ε ](z; τ ) =
ε 1 ε ε 1 t ε ε t exp 2πi[ ( N )τ N + ( N )(z + + τ ) + (t )τ + (t )z+ 2 2 2 2 2 2 2 g
N ∈Z
ε 1 t 1 t ε ε tε +( ) ] = exp 2πi[ ( ε)τ ε + ( )z + ( ε)ε ]Θ(z + I + τ ; τ ) = 2 2 8 2 4 2 2 1 ε 1 exp 2πi[ (tε)τ ε + (t )z + (tε)ε ]Θ(z + e; τ ). 8 2 4 1 ! , εk , εk ∈ Z. ε, ε ∈ Zg Θ[ε, ε](z; τ ) / [ε, ε]. . # #" @ εk Θ[ε, ε](z + e(k) ; τ ) = exp 2πi[ ]Θ[ε, ε](z; τ ), 2 tε
τkk εk Θ[ε, ε ](z + τ ; τ ) = exp 2πi[−zk − − ]Θ[ε, ε](z; τ ), 2 2 (tε)ε ]Θ[ε, ε](z; τ ), Θ[ε, ε](−z; τ ) = exp 2πi[ 2 (tε)ν ]Θ[ε, ε](z; τ ), Θ[ε + 2ν, ε + 2ν ](z; τ ) = exp 2πi[ 2 g g ν, ν ∈ Z , z ∈ C , τ ∈ σg . . # (tε)ε = 0(mod2) (tε)ε = 1(mod2) $
$ &'()$ # " 22g
(k)
/ $ + 2g−1(2g + 1) , 2g−1(2g − 1) $ % 22g # 22g Cg /L(F ), L(F ) = Ig n + τ m, n, m ∈ Zg ,
Cg , ! (Ig , τ ). %
./ 0 Θ(z; τ ) = Θ[0, 0](z; τ )
( = D < &'(, A$ ?(E)$ τ ∈ σg ! / # Θ[2ε, 2ε ](z; τ ) ,
#
Θ[2ε, 2ε ](z; τ ) = exp(πi(tε)τ ε + 2πi(t ε)(z + ε ))Θ(z + τ ε + ε ; τ ) =
1 exp2πi[ (t (n + ε)τ (n + ε)) + (t(n + ε))(z + ε )], 2 g
n∈Z
z ∈ Cg , ε, ε ∈ Qg . . , / / / Θ(z; τ ) , ! / &'()$ - # τ ∈ σg " L(τ )− Cg L(τ )− , / [22]. - $ B
n j=1 Θ(z + aj ; τ ) , f (z) = n Θ(z + b ; τ ) j j=1 n j=1
aj ≡
n j=1
bj modZg ,
( )
aj , bj ∈ Cg , j = 1, ..., n, Cg /L(τ ). , ! #, ( ) L(τ )− @ exp(− j [πi(t m)τ m + 2πi(t m)(z + aj )]) f (z + τ m) = f (z) = f (z). exp(− j [πi(t m)τ m + 2πi(t m)(z + bj )])
%
/ &==, A$ ;;F)$ 2 $ B f (z) = [
∂Θ ∂zj (z
+ a; τ )
Θ(z + a; τ )
−
∂Θ ∂zj (z
+ b; τ )
Θ(z + b; τ )
]=
Θ(z + a; τ ) ∂ ln ∂zj Θ(z + b; τ )
Cg /L(τ ), !
∂ Θ[2ε, 2ε ](z; τ ) , j = 1, ..., g, ln ∂zj Θ[2ε1, 2ε1](z; τ )
z z + τ n + m 3 / ! $ . $ B ∂ 2Θ ∂Θ ∂Θ ∂2 2 f (z) = (Θ − )/Θ = lnΘ ∂zi∂zj ∂zi ∂zj ∂zi∂zj
Cg /L(τ ),
z z + τ m + m
∂2 lnexp(−πi(tm)τ m − 2πi(t m )z) = 0. ∂zi∂zj
- g = 1 / − 2 3 , C1/L(τ ). 8 F g ≥ 1, {a, b} F, ζ Ω1(1), (Ig , Π) = (Ig , τ ), τ ∈ σg , e(j) = a ζ, π (j) = b ζ. -
" !
Θ(z; τ ) Θ[ε, ε](z; τ ). 8 F {a , b} ζ = (ζ1, ..., ζg) Ω1(1), (Ig , Π) Θ(z; Π). 1 , j
j
a b
=
A B C D
a A B · , ∈ Sp(g; Z)− b C D
a = Aa + Bb. 5 # a ζ = = (A + BΠ)−1ζ Π = b ζ = −1 −1 Ca+Db (A + BΠ) ζ = (C + DΠ)(A + BΠ) . 2 3 {a, b}
F g ≥ 1. 13 "# / Cg , , Cg /L(F ) = J(F ),
F
g, Aa+Bb ζ = A + BΠ. , ζ
$ 5 ! C ϕ : F → J(F ) F
P0 ∈ F. ϕ(P ) = PP ζ P ∈ F. 0 Θ ◦ ϕ D Θ(z; τ ) Θ[ε, ε](z; τ ) < # F. 2
Θ ◦ ϕ : F → Cg → C, # ϕ : F → Cg , / P0 P F. 5
! ϕ : F → J(F )
$ - / Θ ◦ ϕ !
> , " , " ϕ(P ) = z ∈ Cg . 1
,
! Θ[ε, ε] ◦ ϕ al D bl < P0 ∈ F, ! 0
ε
ρ(al ) = exp 2πi[ ε2l ] (ρ(bl ) = exp 2πi[− 2l −
τll 2
− ϕl (P )]), l = 1, ..., g. 1 ! f (P ) = Θ[ε, ε] ◦ ϕ F. G Θ[ε, ε](z; Π) J(F ) = Cg /L(F ), !
! J(F ). - # τ ∈ σg (Jmτ > 0) / ! ! Cg ;$ 2 , τ = Π Θ[ε, ε](z; Π)
# ! ; J(F ). . , ! Θ[ε, ε] ◦ ϕ F ϕ(F ) J(F ) ! , " Θ[ε, ε](z; Π). . F , " ! @ Θ[ε, ε] ◦ ϕ ≡ 0 F, Θ[ε, ε] ◦ ϕ F. H " ! , f (P ) = Θ[ε, ε]◦ ϕ ! .
f = 0
! 1 2πi
1 d log Θ[ε, ε] ◦ ϕ = 2πi ∂
g
df 1 = 2πi ∂ f k=1
− ak +bk +a− k +bk
df = f
g
1 [ 2πi
k=1
df df − ( − −)+ f ak f
df df − ( − − )], f bk f
f − f a−k b−k . -
" , P ∈ ak , εk τkk f (P ) = exp 2πi[− − − ϕk (P )]f (P ); 2 2 −
P ( dff
∈ bk , f (P ) = exp 2πi[ ε2k ]f −(P ). , P ∈ bk , − − − dff − )(P ) = 0; P ∈ ak , ( dff − dff − )(P ) = 2πiϕk (P ). 5 # 1 2πi
g
df 1 = 2πi ∂ f k=1
ak
2πiϕk (z)dz =
g k=1
ak
ζk = g.
#"
! F. 1 ϕ(D) / D f
! C df 1 ϕ : Fg → J(F ). / 2πi ∂ ϕ f , ϕ = (ϕ1, ..., ϕg ). +
D = P1...Pg , g
g
1 ( ϕ(D) = ( ϕ1(Pj ), ..., ϕg (Pj )) = 2πi j=1 j=1
df ϕ1 , ..., f ∂
∂
ϕg
df )= f
g 1 df df ϕ = ϕ = − − f 2πi f ∂ k=1 ak +bk +ak +bk g − 1 df df df − − df [ (ϕ − ϕ − ) + (ϕ − ϕ− − )]. 2πi f f f f ak bk
1 2πi
P
k=1
∈ ak
ϕ− = ϕ + τ (k); P g
1 ϕ(D) = [ 2πi k=1
g
1 2πi k=1
ϕ ak
∈ bk , ϕ = ϕ− + e(k) .
5 #
df df − (ϕ + τ (k) )( − 2πiϕk (z)dz)+ f f
− df − − df (ϕ + e ) − − ϕ − ] = f f bk
ak
[−τ
(k) df
f
+
−
(k)
2πiτ (k) ϕk (z)dz
+
2πiϕϕk (z)dz]
g
1 + 2πi k=1
bk
− (k) df e . f−
1 a dff b dff , k = 1, ..., g. + f (P2) = exp[2πi( ε2 )]f (P1), ε P1 P2 ak ; f (P2) = exp 2πi[− 2 − τ2 − ϕk (P1 )]f (P1) bk , k = 1, ..., g. - / k
k
k
k
g
1 εk ϕ(D) = [−τ (k) (2πi + 2πink ) + τ (k) 2πi + 2πi 2πi 2 k=1
ak
kk
ϕϕk (z)dz]+
g
1 (k) εk τkk − ϕk (P1)) + 2πimk ], e [2πi(− − 2πi 2 2 k=1
nk , mk $ ,
ak
1 ϕ(D) = 2πi ϕϕk dz − e(k) (
g
df (k) εk (k) εk −e + τ (k) (1 − nk )+ ϕ = [−τ f 2 2 ∂ k=1
ε τkk ε + ϕk (P1)) + e(k) mk ] = −Π − I + Πn + Im − K, 2 2 2
g τkk + ϕk (P1 ))] K=− [ ϕϕk dz − e(k) ( 2 ak k=1
0$ - P1 = P0 KP ; F. . , & &'()$ / Θ[ε, ε](z, Π), (F, {a, b}) , Θ[ε, ε] ◦ ϕ ! F, g F $ - P1 ...Pg $ . 0
ε ε ϕ(P1 ...Pg ) = −Π − I + Πn + Im − KP0 , 2 2
m, n ∈ Zg , ϕ(P1...Pg ) + KP = −Π 2ε − I ε2 J(F ). %
g = 1, ε = 0 = ε , F = C/{1, τ } K
0
− 12
+
τ 2.
=
0 #P → Θ(ϕ(P ) − e) F e ∈ Cg . 8 = 0 F, g P1 , ..., Pg ϕ(P1...Pg ) + K = e J(F ). , # e ∈ Cg ! e = −Π 2ε − I ε2 ε, ε ∈ Rg , ε, ε # Rg . +
"
ε ε Θ[ε, ε ](z, Π) = CΘ(z + Π + I ; Π) 2 2
C = 0. 8 Pj , Θ[0, 0](ϕ(Pj ) − e) = 0, j = 1, ..., g, Θ[ε, ε](ϕ(Pj ) − e − Π 2ε − I ε2 ) = 0. -
"
ε ε ε ε ϕ(P1...Pg ) − e − Π − I = −K − Π − I 2 2 2 2
ϕ(P1...Pg ) − e = −K J(F ). ! Θ[ε, ε](z, Π) J(F ) z → Θ(z−e), e Cg , ! !$ n ≥ 1,
! ϕ : Fn → J(F ), ! Θ
# # # Fn. ' &'()$ - e ∈ Cg . . Θ(e) = 0, e ∈ Wg−1 + K.
567 7.89: .25$ $ - !, e ∈ Wg−1 + K ⊂ J(F ), Θ(e) = 0. 2 # D = P1 ...Pg ∈ Fg
i(D) = 0, $$ D $ . # D , D Fg , i(D) = g−rank(ζk (Pj )), i(D) = 0. - ! e = ϕ(D) + K. 0 ψ(P ) = Θ(ϕ(P ) − e) # P F. " # ! @ ψ ≡ 0, ψ = 0 F. 2
! k = 1, ..., g 0 = ψ(Pk ) = Θ(ϕ(Pk ) − (ϕ(P1...Pg ) + K)) = Θ(−ϕ(P1...Pˆk ...Pg ) − K) = Θ(ϕ(P1...Pˆk ...Pg ) + K).
/ , aˆ
/ a, " / $ 2 ψ g Q1, ..., Qg F
" ϕ(Q1...Qg ) + K = e J(F ). + D = P1...Pg , 0 0 7 , P1...Pg = Q1...Qg Fg . , ϕ(P1...Pg ) = ϕ(Q1...Qg ) J(F ), , P1 ...Pg ∼ Q1...Qg F, $$ " f ∈ L( QP ...P ...Q ) P1 ...Pg # Q1...Qg F. 1 i(Q1...Qg ) = 0 r((Q1...Qg )−1) = g−g+1+i(Q1...Qg ). 1 - / f ∈ L( QP ...P ) ⊂ L( Q ...Q ) f ! F. ...Q , P1...Pg = Q1...Qg Fg . 5 # # k = 1, ..., g " l = 1, ..., g , 1
g
1
g
1
1
g
1
g
g
0 = ψ(Ql ) = Θ(ϕ(Ql ) − (ϕ(P1...Pg ) + K)) = Θ(ϕ(Pk ) − (ϕ(P1...Pg ) + K)) = Θ(ϕ(P1...Pˆk ...Pg ) + K).
D # ! Fg . , P1 ...Pˆk ...Pg Fg−1. 5 # Θ ! # Wg−1 + K. 1 $ - Θ(e) = 0, s 3 , # " #, Θ(Ws−1 − Ws−1 − e) ≡ 0,
(∗)
Θ(Ws −Ws −e) = 0. +
" 1 ≤ s ≤ g. - / " # / P1...Ps Q1...Qs Fs, " F, , Θ(ϕ(P1...Ps) − ϕ(Q1...Qs) − e) = 0.
0 # P → Θ(ϕ(P ) + ϕ(P2...Ps) − ϕ(Q1...Qs) − e). 5 ! ! # F, P = P1 . , P = Qj , j = 1, ..., s, (∗). -
" " T1...Tg−s ∈ Fg−s , ϕ(Q1...Qs) − ϕ(P2...Ps) + e = ϕ(Q1...QsT1...Tg−s) + K
e = ϕ(T1...Tg−sP2 ...Ps) + K ∈ Wg−1 + K.
. $ + / 3, P1...Ps Q1...Qs Fs !
P1 ...Ps Q1 ...Qs Fs. 5 # , e1 = ϕ(T1...Tg−sP2 ...Ps) ∈ Wg−1 s−1 P2, ..., Ps. - D = P2 ...Ps
P2...Ps Fs−1. . ,
" , , e = ϕ(T1...Tg−sP2 ...Ps) + K = ϕ(T1 ...Tg−s P2 ...Ps) + K.
- 7 T1...Tg−sP2...Ps ∼ T1 ...Tg−s P2 ...Ps F. ,
r(1/T1...Tg−sP2...Ps) ≥ s 0 0 i(T1...Tg−sP2...Ps) ≥ s. ,
!, e = ϕ(D1) + K, D1 ∈ Fg−1 i(D1) = s. . D1 s − 1 $ # / X ∈ Ws−1 − Ws−1 − e X = ϕ(P1...Ps−1) − ϕ(Q1...Qs−1) − e, e = ϕ(D1) + K = ϕ(P1...Ps−1δ) + K δ ∈ Fg−s. ,
X = ϕ(P1 ...Ps−1) − ϕ(Q1...Qs−1) − ϕ(P1...Ps−1δ) − K =
−ϕ(Q1...Qs−1δ) − K ∈ −(Wg−1 + K),
Θ(X) = 0.
(D0 / < &'(, $ ?;;)$ e ∈ Cg ,
Θ(e) = 0, e ∈ Wg−1 +K. 8 e ∈ Wg−1 +K, e = ϕ(D)+K D ∈ Fg−1, i(D) = s(≥ 1), Θ(Ws−1 − Ws−1 − e) ≡ 0. 5 , s 3 , Θ(Ws−1 − Ws−1 − e) ≡ 0, Θ(Ws − Ws − e) = 0, e = ϕ(D) + K D ∈ Fg−1, i(D) = s.
567 7.89: .25$ 5 3 !$ 5 ! i(D) ≥ s. B ! i(D) = s, , i(D) > s Θ(Ws − Ws − e) ≡ 0, /
# s. . $ 1 $ #
I D g F g ≥ 1 , r(D−1) > 1. % , " f ∈ L(D−1), f = const ⇔ " ω = 0 , (ω) ≥ D ⇔ i(D) > 0. % 0 0 , r(D−1 ) = degD − g + 1 + i(D) > 1.
#
" I D ,
" D∗ , DD∗ ∼ Z. D∗ D. I D , i(D) > 0. 2 ,
D ≥ 1 degD = g r(D−1) > 1 ⇔ i(D) > 0. 7 !,
/ i(D) = 0 ⇔ r(D−1) = 1.
) &'(, A$ ?;?)$ < 8 e ∈ J(F ) ψ(P ) = Θ(ϕ(P )−e) ≡ 0,
e = ϕ(D) + K D ∈ Fg i(D) = s ≥ 1
/ s 3 Θ(Ws+1 − Ws − e) = 0; < 8 e = ϕ(D) +K, D ∈ Fg , ψ = 0 ⇔ i(D) = 0 D ψ. 2 , ψ ≡ 0 ⇔ i(D) > 0, $$ D $ 567 7.89: .25$ - s 3 Θ(Ws+1 − Ws − e) = 0. . 0 ≤ s ≤ g − 1. 2 P1 , ..., Ps+1, Q1, ..., Qs , Θ(ϕ(P1...Ps+1) − ϕ(Q1...Qs) − e) = 0. 0 # P → Θ(ϕ(P ) + ϕ(P2...Ps+1) − ϕ(Q1...Qs) − e). 5 #, P1. - D s) g Q1, ..., Qs, T1, ..., Tg−s Θ(Wk+1 − Wk − e) ≡ 0
k < s. 5 # ϕ(Q1...QsT1...Tg−s) + K = ϕ(Q1...Qs) − ϕ(P2...Ps+1) + e
e = ϕ(T1...Tg−sP2 ...Ps+1) + K ∈ Wg + K.
D = T1...Tg−sP2...Ps+1 s $ . , r(D−1) ≥ s + 1 i(D) ≥ s. % <, ψ ≡ 0 s ≥ 1.
< , < i(D) = 0 ψ = 0. 5 ,
!,
i(D) > 0 ψ = 0. 2 ## Q ∈ F ψ(Q) = 0. - " D ∈ Fg−1 , ϕ(D) = ϕ(QD ). .! ψ(Q) = Θ(ϕ(Q)−e) = Θ(ϕ(D)−ϕ(D )−e) = Θ(−ϕ(D )−K) = 0. %
#, , ψ ≡ 0. ! 3
, e = ϕ(D) + K, D ∈ Fg , ψ = 0 i(D) = 0, D ψ. 8 D ψ, ϕ(D) = ϕ(D ). 1 i(D) = 0 7 D = D . . , D = D , ϕ( DD ) = 0 deg( DD ) = 0, , " f ∈ L( DD ). . , r( DD ) > 1 r( D1 ) ≥ r( DD ) > 1,
r(
1 ) = degD − g + 1 + i(D) = g − g + 1 + 0 = 1. D
- $ , D = D D ψ F. . $ %
" 6 ! " C @ # a ∈ J(F ) " Da ∈ Fg , ϕ(Da) = a J(F ). , a ∈ Cg # a ∈ J(F ) Cg . - ! e = a + K # ψ : P → Θ(ϕ(P ) − e). 8 ψ = 0, Da ∈ Fg ϕ(Da) + K = e = a + K. 8 ψ ≡ 0, e = ϕP (Q1...Qg−1) + K = ϕP (P0 Q1...Qg−1) + K ! Da = P0 Q1...Qg−1. * D0< &'(, $ ?;')$ - s 3 , Θ(Ws−1 − Ws−1 − e) ≡ 0, Θ(Ws − Ws − e) = 0. . e = ϕ(D) + K D ∈ Fg−1 i(D) = s. 6 ,
Θ 3 s ( e, s−
Θ # e. 5 , e ,
Θ 3 s ( e, " s−
, ( e, e = ϕ(D) + K, D ∈ Fg−1 i(D) = s.
0
0
# # K 0 (F, {a, b}) g ≥ 1. + &'()$ - D 2g − 2 F. . D F, ϕ(D) = −2K J(F ). 567 7.89: .25$ - !, −2K F. 2 D0
g − 1. . e = ϕ(D0) + K Θ , (−e) ! / , $ . , −e = ϕ(D1) + K, D1 ∈ Fg−1. , ϕ(D0D1 ) = −2K. 5
, D0D1 F. , D0 D1 g − 1 , , r( D 1D ) ≥ g i(D0D1 ) ≥ 1. . , " ω, (ω) ≥ D0D1 ≥ 1, , (ω) = D0D1 / $ # ω1 (ω1) ∼ (ω) ϕ((ω1)) = ϕ((ω)) = −2K. 5 ,
!, D 2g − 2 F ϕ(D) = −2K. . # ω D D ϕ( (ω) ) = ϕ(D) − ϕ((ω)) = 0 deg (ω) = 0. - 7 " D . , (f ω) = D, D f F, (f ) = (ω) f ω F. . $ ! 2 K 0 = J(F ), P02g−2 , P0 ϕ : F → J(F ). , K =, −2K = 0
1
0 = ϕ(P02g−2).
* ! / Θ J(F ) Wg−1 + K, P0. 1 , ! Θ P0. &'(, A$ ?;J)$ 5 !,
Fg−1, D ϕP (D) + KP , P0 F. 567 7.89: .25$ , Θ(e) = 0, ψ(P ) = Θ(ϕP (P ) − e) = 0, , e = ϕP (P1 ...Pg ) + KP , D = P1 ...Pg ψ F. 2 0
0
0
0
ϕP1 (P ) = ϕP1 (P0 ) + ϕP0 (P )
0
# P0, P1, P
∈ F.
5 #
Θ(ϕP0 (P ) − e) = Θ(ϕP1 (P ) − ϕP1 (P0 ) − e),
/
ϕP1 (P0 ) + e = ϕP1 (P1...Pg ) + KP1 .
, e
ϕP1 (P0 ) + ϕP0 (P1 ...Pg ) + KP0 = ϕP1 (P1 ...Pg ) + KP1 .
K ϕP (Pj ) = ϕP (Pj ) − ϕP (P0), j = 1, ..., g, , 0
1
1
ϕP1 (P0) + ϕP1 (P1 ...Pg ) − gϕP1 (P0 ) + KP0 = ϕP1 (P1...Pg ) + KP1
KP
0
= ϕP1 (P0g−1) + KP1 .
1 , # D ∈ Fg−1
ϕP1 (D) + KP1 = ϕP1 (P0g−1) + ϕP0 (D) + KP1 = ϕP0 (D) + KP0 ,
ϕP1 (Pj ) = ϕP1 (P0 ) + ϕP0 (Pj ), j = 1, ..., g − 1.
. $
! "
ϕQ(P
g−1
).
# P Q F KP
= KQ +
5
, &'(), ! ! J(M) : ;< ! Θzero = Wg−1 + K / Θ > 1 =< ! Θsing = Wg−1 + K, " e, Θ
# e > ?< ! Θsuperzero = Wg1 + K, " e, ψ(P ) = Θ(ϕ(P ) − e) ! # F. *! 3 Θzero ⊃ Θsuperzero ⊃ Θsing . 1 2# Θzero ⊃ Θsing Wg−1 ⊃ Wg−1 . 1 1 2# Θsuperzero ⊃ Θsing , Wg + K ⊃ Wg−1 + K. 2# Θzero ⊃ Θsupersero , e ∈ Θsupersero, ψ(P ) = Θ(ϕP (P ) − e) ≡ 0 P ∈ F, , 0 = Θ(ϕP (P0) − e) = Θ(e), e ∈ Θzero . - / ! @ ;< Θsuperzero g ≥ 2; 0
0
=< Θsing g ≥ 4 / g = 3; ?< dim Θzero = g − 1; E< dim Θsuperzero = g − 2; L< g − 4 ≤ dim Θsing ≤ g − 3; F< * ! Θzero Θsing Π (F, {a, b}) g ≥ 1, P0 ! ϕ : F → J(F ); # Ee (P, Q) = Θ(e + ϕ(Q) − ϕ(P )),
e ∈ Cg − , Θ(e) = 0, P, Q ∈ F. % , ,
a− ! / !
b−$ 5 #" , $&==, A$ ;==)$ - e ∈ Cg , Θ(e) = 0, Ee(P, Q) = 0. . " # 2g − 2 R1 , ..., Rg−1, S1, ..., Sg−1 ∈ F , Ee(P, Q) = 0, @ < P = Q, < P = Ri , < Q = Si , i = 1, ..., g − 1.
.
# Ee F × F 7 $ D7 < &==, A$;;E)$ 1 F
g ≥ 2 D = QP ...Q ...P ( " f (f ) = D, ϕ(D) = 0 J(F ). 567 7.89: .25$ 1 $ # t ∈ C D(t) ! D <, f
t, $ $
! f : F → P1 d t. - P0 − ! C ϕ. 0 ! ϕ(D(t)/P0d) # t. - D(t) t ,
1
d
1
d
D(t) P0d
(ϕ1, ..., ϕg)
! $ , ! t → ϕ(D(t)/P0d) ! δ : P1 → J(F ). 2 P1 / ! ! δ : P1 → Cg . 1 ! P1 $ 2
δ δ ! $ - / δ(0)
= δ(∞),
,
0 = δ(0) − δ(∞) = ϕ(D)
C $ $ + ϕ(P1...Pd) = ϕ(Q1...Qd) J(F ), # # f F (f ) = D. / e ∈ Cg , Θ(e) = 0, Ee(Pj , Q) = 0,Ee(Qj , Q) = 0, j = 1, ..., d. 0 # f (Q) =
d j=1 Ee (Qj , Q) d j=1 Ee (Pj , Q)
( )
F. K Ee / $ 2 σi P0 Pi τi P0 Qi , d i=1
σi
(ϕ1, ..., ϕg ) =
d i=1
τi
(ϕ1, ..., ϕg )
Cg . ,
τi, σi
# / 3 L(τ ). 2
! ϕ(P1...Pd) = ϕ(Q1...Qd) J(F ) ! , Cg . 2 # d
j=1 Θ(e
f (Q) = d
+ ϕ(Q) − ϕ(Qj ))
j=1 Θ(e + ϕ(Q) − ϕ(Pj ))
,
Pj D Qj ) Q
τj D σj ), P0 Q "
$ 2 , /
! Q F
, #" P0 A Q. 8 ak −, , k = 1, ..., g. 8 ! bk −, ! d
j=1 exp[−πiτkk − 2πi(ϕk (Q) − ϕk (Qj ) + ek )] d j=1 exp[−πiτkk − 2πi(ϕk (Q) − ϕk (Pj ) + ek )]
=
d d (ϕk (Q) − ϕk (Pj )))] = 1. exp[−2πi( (ϕk (Q) − ϕk (Qj )) − j=1
j=1
- / f F. -
" (f ) = D. . $ %
$ B Ee F ! , x − y P1, , ## # # P1 ! ! !
! (y − Qi) , f (y) = c i (y − P ) i i
( ) " / ! # g ≥ 2. B Ee / D
< $ %
& "# , / , ! F, ! &==, A$ ?=L ?=F)$ G$ *$ B 3 +$ 6 &FJ>'(),
! 2 3 / 0 $ G 3 , # F g ≥ 2 ! W 2 3 2g + 2 ≤ |W | ≤ g3 − g. - ! , F / $ - 2 3
D < F. 8
d, , ## P F, ! /
# # P
d − 1. . , " /
d / , 2 3 /
$ 2 F g ! g θ1, ..., θg F ⎛
⎜ ⎜ W (θ1, ..., θg) = det ⎜ ⎝
θ1 θ1 ... (g−1)
θ1
... θg ... θg ... ... (g−1) ... θg
⎞
⎟ ⎟ ⎟. ⎠
. F , " , #" g / $ * ! ,
, , W (θ1, ..., θg) g3 − g , , " 2g + 2 , ! # 3 g(g−1) 2 . 8 2 3 , F / 2g+2 2 3 , ! g(g−1) 2 . #
$ &'()$ . P F g ≥ 1 q− 2 3 , τ (P ) > 0
Ωq (1) q− F. .! ; 2 3
2 3 D 2 3 <$
& &'(, A$M')$ . P
F g ≥ 2 q− 2 3 , " D ! < q− ω F ≥ dimC Ωq (1) P. 1 F g = 1 q− 2 3 # q ≥ 1.
' &'(, A$ M')$ - g ≥ 2, q ≥ 1, τ (P ) P F, Ωq (1), Wq Ωq (1), d = dq = dimΩq (1). . Wq m(= mq )− , m = d2 (2q − 1 + d),
P ∈F τ (P ) = (g − g(g+1) 2 −
1)d(2q − 1 + d).
567 7.89: .25$ 1!
, Wq
m− F. 8 ξ1 , ..., ξd Ωq (1), z z z = f (z) , ξj = ϕj (z)dzq = ϕ j ( z)d zq , ϕ j (f (z))f (z)q = ϕj (z), j = 1, ..., d. 6 2 3 D $ ;$'<, , (det[ϕ1, ..., ϕd])dz m = (det[ ϕ1, ..., ϕd])d zm.
, det[ϕ1, ..., ϕd] = det[(ϕ 1 ◦ f )(f )q , ..., (ϕ d ◦ f )(f )q ] = (f )m(det[ϕ 1, ..., ϕ d] ◦ f ).
- ! $
! (
g ≥ 2
" #
q−
2
3
# q ≥ 1. , # mq )− $
m(=
5 , ! W 2 3 , ! ! q− 2 3 F g ≥ 1 !
/ # 0$ 1 $ - Θ / 0 F, ϕP ! 7 C F C , KP P0 F. . Θ(gϕP (P ) + KP ) # 2 3
, ! 2 3 / F. P , P0 . 8 P0 ∈/ W, P → Θ(gϕP (P ) + KP ) P0 g g3 − g 2 3 $ % ! !$ - ! P → Θ((2q − 1)(g − 1)ϕP (P ) + (2q − 1)KP ), , # 2 3
q− F, q ≥ 2. 0 / , , α ∈ N, 0 < α < g, θ(αϕP (P ) + KP ) = 0 P ∈ F. .
, α = g , # 2 3 $ - / # α ≥ g, α ∈ N. - / 0 3# P → Θ(ϕ(P ) − e) F g ≥ 1
e ∈ Cg . 8 ! , g P1, .., Pg e = ϕ(P1...Pg ) + K J(F ). 0 "# # 0
0
0
0
0
0
0
0
0
0
ψ : P → Θ(αϕ(P ) − e) = Θ(ϕ(P α ) − e), α ≥ g.
- F. + ϕ(P ) = PP ζ, P P P αϕ(P ) = α P ζ = (α P ζ1 , ..., α P ζg ) = (αϕ1, ..., αϕg ). 5 # αe(j) = a αζ, απ(j) = b αζ, j = 1, ..., g. ,
! ;$;;$= 0
0
j
0
0
j
Θ(αϕ(P ) − e + αe(k) ) = Θ(αϕ(P ) − e), Θ(αϕ(P ) − e + απ (k) ) = α2 πkk = exp2πi[−α ϕk (P ) + αek − ]Θ(αϕ(P ) − e), k = 1, ..., g. 2 2
. , P
∈ b+ k,
ψ(P ) = ψ−(P ); P
∈ a+ k,
α2 πkk ]ψ(P ). ψ (P ) = exp2πi[−α ϕk (P ) + αek − 2 −
- / , P
2
∈ b+ k,
dψψ − dψψ
−
−
= 0;
P
∈ a+ k,
2
exp2πi[−α2ϕk (P ) + αek − α 2πkk ]2πi(−α2 )ϕk dz dψ dψ − 2 − − =− ϕk dz. = 2πiα 2 ψ ψ exp2πi[−α2ϕk (P ) + αek − α 2πkk ]
6 3,
D < 1 N= 2πi
g
1 dψ = 2πi ∂ ψ k=1
g
dψ dψ − − − ) = α2 ( ψ ak ψ
ak
k=1
ϕk (z)dz = α2 g.
8 Θ(αϕ(P ) − e) = 0, Θ(ϕ(P α) − e) = 0 P α
! 3 $ . , α. 4 α2g , ! , αg $ 1 3 ψ α. 8 Pj # Θ[0, 0](αϕ(Pj ) − e) = 0, Θ[ε, ε ](αϕ(Pj ) − e − I ε2 − Π 2ε ) = 0. N , ,
! ;$;;$L, , ϕ(P1...Pα g ) − αe − αI ε2 − αΠ 2ε = −α2K − αI ε2 − αΠ 2ε , α ϕ(P1α ...Pαg ) + α2 K = αe J(F ). , ψ ! F, α2g P1, .., Pα g , αe = ϕ(P1...Pα g ) + α2K J(F ),
ψ P0 , ! 7 C ϕ P0 F. #" ,
# / ! $ ) &'(, A$ ?;=)$ - P0 ϕ F g ≥ 1 ψ(P ) = Θ(αϕ(P ) − e) ≡ 0 F, e = ϕP (Q1...Qg−1) + KP i(P0α Q1...Qg−1) > 0. 2 ,
α = 1 , e = ϕP (D) + KP , D ∈ Fg ψ ! #, i(D) = 0 D ψ. 567 7.89: .25$ 8 ψ ≡ 0 D , Θ(e) = 0), ! P ∈ F
e = ϕP (Q1...Qg−1) + KP , αϕP (P ) − e Θ, , %
'
2
2
2
0
0
0
0
0
0
ϕP0 (P α ) − ϕP0 (Q1...Qg−1) − KP0 = −ϕP0 (R1 ...Rg−1) − KP0 ,
0
D
/ <$ 5 # ϕP (P αR1...Rg−1) = α ϕP (P0 Q1...Qg−1). + 7 , # P ∈ F , ,
P = P0, " 1 L( P Q ...Q ), P 3 α. + 0 0 , i(P0α Q1 ...Qg−1) > 0, 1 r( P Q ...Q ) ≥ α+g−1−g+1 = α, L( P QP...Q ) ⊂ L( P Q 1...Q ) 0
0
α 0
1
g−1
α
α 0
1
α 0
g−1
α + g − 1 − g + 1 + i(P0α Q1...Qg−1) = r(
1
1
g−1
) P0α Q1...Qg−1
α 0
1
g−1
≥ α + 1,
, i(P0αQ1...Qg−1) ≥ 1. 5 , i(P0αQ1...Qg−1) > 0, 0 0 1 r( P Q ...Q ) ≥ α + 1. - / " /
, 3 α # P ∈ F. , α 0
1
g−1
Θ(ϕP0 (P α )−ϕP0 (P0α Q1 ...Qg−1)−KP0 ) = Θ(ϕP0 (P α )−ϕP0 (P α R1...Rg−1)−KP0 )
. $
= Θ(−ϕP0 (R1 ...Rg−1) − KP0 ) = 0.
! * 8 α ≥ g, P0 ,
F. 567 7.89: .25$ 8 ψ ≡ 0,
" i(P0α Q1...Qg−1) > 0,
α ≥ g ! i(P0g Q1...Qg−1) = 0 deg(P0g Q1 ...Qg−1) > 2g − 2. - $ $ %
( &'(, A$ ?;?)$ +
" , e = ϕP (Q1...Qg−1) + KP i(Q1...Qg−1) = 1, " ! P0, ψ ≡ 0. , " ω (ω) ≥ Q1...Qg−1 2g − 2 , (ω) = Q1...Qg−1R1...Rg−1. . R1 ...Rg−1 ! P0 , i(P0α Q1...Qg−1) = 1 > 0. 6 , i(Q1...Qg−1) > 1 α = 1, # P0 i(P0Q1...Qg−1) = i(Q1...Qg−1), i(P0Q1...Qg−1) + 1 = i(Q1...Qg−1) ψ(P ) = Θ(ϕ(P ) − e) ≡ 0 F.
+ &'(, A$ ???)$ - e ∈ J(F ). . ψ : P −→ Θ(ϕ(P ) ± e) ! F g ≥ 3
# P0 ! ϕ, e ∈ Θsing . ψ≡0
0
0
567 7.89: .25$ K e ∈ Θsing / D 1 Θsing = Wg−1 + KP ) " # P1 , ..., Pg−1 F , e = ϕP (P1...Pg−1) + KP i(P1...Pg−1) ≥ 2. - 0 / / Θ(W1 − W1 ± e) ≡ 0. - ! $ &'(, A$ ??;)$ 9# Q = P0 ψ : P → Θ(gϕP (P ) + KP )(e = −KP ) 2 3 F g ≥ 1. 5 , # 2 3 F / $ 567 7.89: .25$ - !, Q = P0 / $ . 0
0
0
0
0
0
gϕP0 (Q) + KP0 = ϕP0 (R1...Rg−1) + KP0 ,
,
ϕP0 (Qg ) = ϕP0 (R1 ...Rg−1) = ϕP0 (P0R1 ...Rg−1).
- 7 " L( Q1 ) F. - / Q 2 3 , g
g
i(Q ) > 0.
5 , Q(= P0) 2 3 F. . ≥ 2 " R1 ...Rg , !
! ! P0 / Qg . , (f ) ≥ Q1 , (f − f (P0)) ≥ QP , , " # R1, ..., Rg−1 , (f − f (P0)) = R ...RQ P . 5 # , gϕP (Q) + KP = ϕP (R1...Rg−1) + KP Q ψ $ 5 , P0 / , Θ(KP ) = 0. . $ 4
" ψ g3, / P1 , ..., Pg # # ϕP (P1...Pg ) = g(g+1) 2 (−2KP ), / g(−KP ) = ϕ(P1...Pg ) + g2KP . - !, P0 ψ 3 g. , # P F D , , P = P0) r( Q1g )
0 g
g
1
0
0
g−1 0 g
0
0
0
3
0
3
0
3
0
0
Θ(gϕP0 (P )+KP0 ) = Θ(ϕP0 (P )+KP ) = Θ(−ϕP (P0 )+KP ) = Θ(ϕP (P0 )−KP ),
0 = ϕP (P ) = ϕP (P0 ) + ϕP0 (P ), KP = KP0 + ϕP0 (P g−1) = KP0 + gϕP0 (P ) − ϕP0 (P ).
2 / ! P P0 F. 2 P #" 2 3 $ . P0 → Θ(ϕP (P0) − KP ) g P0 = P, Θ(gϕP (P ) + KP ) = 0. 8 P 2 3 , 3 ! # P0 F, 0
0
0 = Θ(gϕP0 (P ) + KP0 ) = Θ(ϕP (P0) − KP )
# P0 F. 5 # , P = P0 ψ 3 g. 3 ψ ! g(g + 1) (−2KP0 ), 2
ϕP0 (P0g P1 ...Pg3−g ) =
, , ϕP0 (P1...Pg3−g ) =
g(g + 1) (−2KP0 ). 2
&'(, A$ ??=)$ . P1, ..., Pg −g # g(g+1) 2 − F 2 3 F. 8
2 3 , g(g+1) − Φ 2 F. H , ! (Φ) = (ψ) ψ(P ) = Θ(gϕP (P ) + KP ). K! , / $ 1 " , / ! # $ " 2 3 #
, , / ! # # F. - / " # !, # D # <$ 1
, / F (Φ) = (P1...P2g+2) , P1 , ..., P2g+2 2 3 F. &FJ, A$'')$ - F
g = 1. 8 g = 1 = α, Θ(ϕP (P ) + KP )
P0 $ % , !
3
0
0
g(g−1) 2
0
0
" 2 3 g = 1. 8 α ≥ 2 > g = 1, α ∈ N, Θ(αϕP (P ) + KP ) α2 F / P0. 5 P D 7 < ! #, ( PP )α F, $$ " , P $ - 0
0
0
αϕP0 (P ) + KP0 = ϕP0 (R1 ...Rg−1) + KP0 = KP0 , ϕP0 (P α ) = 0 = ϕP0 (P0α ).
- , P / ,
, " F,
# P0 3 α, α P. % 1 = r( PP ) = i( PP ). i( P1 ) = r(P0α) + α = α. - / P 2 3
Ω( P1 ). 5 , / ϕ(P ) α ϕP (F ) ≡ J(F )
P = P0. 0 # F g ≥ 2 # # α 0 α
α
α 0
α 0
α 0
0
f : P → Θ((g + k)ϕP0 (P ) + KP0 )
/
k ≥ 0, k ∈ N. 2 k = 0, ! ,
# 2 3 $ " &FJ, A$'M)$ * ! f (P ) = Θ((g +k)ϕP (P )+KP ) P0 P ∈ F, " D g − 1 , PP D $ - , ! Θ((g + k)ϕP (P ) + KP ) P0 P ∈ F , P 2 3
F, # 3 k + 1 P0. 567 7.89: .25$ . 0 / ! , / 0 C , # g −1
!# ϕP KP . 5 # , KP ! / , , P0 f. 8 P ∈ F f, " D ≥ 1 g − 1 , 0
0
k+1 0 g+k
0
0
0
0
0
ϕP0 (P g+k ) + KP0 = ϕP0 (D) + KP0 .
5 # 7 D1 = PP
k+1 D 0 g+k
F.
% ! ! , P , D1 F. , ! # ,
k = 0 # 2 3 $ 8 , PP D , D g − 1, D ≥ 1, " η F, D. 8 f1 F (f1) = D1−1, f1η (f1η) = PP D D2 ≥ 1, degD2 = g − 1. . , / # P0 3 k + 1. .
/ g + k( 0 = r(P0k+1) = −(k + 1) − g + 1 + i( P 1 ) i( P 1 ) = g + k) " f1 η P = P0 3 g + k, # # P = P0 f 2 3
V = Ω( P 1 ). + " / , P → Θ((g + k)ϕP (P ) + KP ) (g + k)2g F, $ 8 Q1...Q(g+k) g , J(F ) g+k
k+1 0
g+k
k+1 0
k+1 0
2
k+1 0
k+1 0
0
0
2
ϕP0 (Q1...Q(g+k)2g ) + (g + k)KP0 = −(g + k)2KP0 ,
# ϕP0 (Q1...Q(g+k)2g ) =
(g + k)(g + k + 1) (−2KP0 ). 2
- −2KP ϕP , 0
0
Q1...Q(g+k)2g (g+k)(k+1)
P0
q(= (g+k)(g+k+1) )− , $$ 2 q− ω, (g + k)2g − (g + k)(k + 1) = q(2g − 2). . $ % q− ω q− Ω (g + k)−
V , # 3 k + 1 P0,
" $ 5 (ω) (Ω) # # (g − 1)(g + k)(g + k + 1). . P ∈ F \{P0} 2 3 V, P i( P ) > 0. %
P0 F g+k
k+1 0
F ,
# $ . , P0 ! 2 3 V, 2 3 F ( i(P0g ) > 0). . ! f 2 3 V, , f 3 g P0. 2 ! !, P0 2 3
V. $ &FJ, $'J)$ * f (P ) = Θ((g + k)ϕP (P ) + KP ) P0 3 g, 3 g((g + k)2 − 1), #
# P0 3
k + 1. %
) 1 , ω Ω # # , ordP ω = ordP Ω P ∈ F. 23 # # ! " , ,
, P0 P1 P2 F. + # g ≥ 1,
0
0
f (P ) = Θ((g + k1)ϕP1 (P ) + (k2 + 1)ϕP2 (P ) + KP1 )
!
k1
k2 .
% #
Θ((g + k1)ϕP1 (P ) + (k2 + 1)(ϕP2 (P1 ) + ϕP1 (P )) + KP1 ),
! Θ((g + k1 + k2 + 1)ϕP1 (P ) − (k2 + 1)ϕP1 (P2 ) + KP1 ).
C , P f, " D g − 1 , ϕP1 (
P g+k1 +k2 +1D P g+k1 +k2 +1D ) = −2K = ϕ ( ), P1 P1 P2k2 +1 P2k2 +1P1k1 +1
P 2 3
# 3 k1 + 1 P1 k2 + 1 P2. 0 /
g + k1 + k2 + 1, /
O
@ > # l, 2 ≤ l ≤ k1 + 1 P1;
# l, 2 ≤ l ≤ k2 + 1 P2
# P1 P2. . P ∈ F 2 3 /
, P g+k1 +k2 +1 i( k1 +1 k2 +1 ) > 0. P1 P2
+ " / / , f (g + k1 + k2 + 1)2g Q1 , ..., Q(g+k +k +1) g , #" 3# 1
2
2
(g + k1 + k2 + 1)((k2 + 1)ϕP1 (P2) − KP1 ) = ϕP1 (Q1...Q(g+k1+k2 +1)2g ) + (g + k1 + k2 + 1)2KP1 ,
ϕP1 (
Q1...Q(g+k1+k2 +1)2 g (g+k1 +k2 +1)(k2 +1)
P2
,
(g + k1 + k2 + 1) + (g + k1 + k2 + 1)2 (−2KP1 ). )= 2 Q1...Q(g+k1+k2 +1)2 g
(g+k1 +k2 +1)(k1 +1)
P1
(g+k1 +k2 +1)(k2 +1)
P2
+k +2) (g+k +k +1)(g+k − # 2 3 (g+k1 +k2 +1)(k1 +1) (g+k1 +k2 +1)(k2 +1) P1 P2 $ %
, # # P1 P2 k1 + 1 k2 + 1 $ 2
V, (P0, k)
P0 k ∈ Z+ ∪ {0}. P ∈ F 1
D0 =
P0k+1 , D1
2
1
2
P0k+1 P0k+1 P0k+1 , ..., Dj = = , ..., D2g+k = 2g+k . P Pj P
! Dj
L(Dj )
r(Dj ). 1 , r ≥ 1
, r(Dr ) − r(Dr−1 ) = 0;
/
$ + 0 0
0 = r(D0) = r(D1) = ... = r(Dk )
r(D2g+k) = g. 5 # # g = r(D2g+k ) − r(D1) = (r(D2g+k ) − r(D2g+k−1)) + ...
+(r(D2) − r(D1)),
( )
" g
$ , g + k
1, ..., 2g +k, 1, ..., k
$ . 0 0 !
, P0k+1 Pr r( r ) = r − (k + 1) − g + 1 + i( k+1 ), P P0
i(P0−(k+1)) = dimC V
= g + k. i(
6 , r
,
P r−1 Pr ) − i( ) = 1. P0k+1 P0k+1
8 P = P0 2 3 ,
≥ g +1. 6 , P0 2 3 V, 2 3 $ !, P = P0 2 3 V,
≥ g + k + 1. -
! 2 3 ,
" / , , ! # P0
V
3 $ &FJ, A$M;)$ - F / g ≥ 2, P0 ∈ F k = 1. 0 2 3
V, " # P0 3 $ " , , P0 2 3 $ - P0 2 3 F P = P0 # 2 3 F. 2 /
@ r(P02 )
P02 P02 P02 = 0 = r( ), r( 2 ) = 1 = r( 3 ), ... P P P
P02 P02 P02 P02 r( 2i ) = i = r( 2i+1 ), ..., r( 2g ) = g = r( 2g+1 ). P P P P . , !
P 1, 3, ..., 2g + 1 2 3 P /
0 + 1 + 2 + ... + g =
g(g + 1) . 2
" 2g + 1 $ , 2 3 , 2g + 1 2 3 P = P0 , g(g+1)(2g+1) . 2 6 , 2 3 P0 2 3
V. 2 / ,
!, P = P0, / 2 3 , # 3 2 P0. 2 / (−2) + (−1) + 0 + ... + (g − 2) =
(g − 4)(g + 1) . 2
5" V 2g + 2 2 3 (g + 1)(g2 + g − 2). 1 , " V (g − 1)(g + 1)(g + 2). - / 2 3 /
$ , ! 2 3 V ! 2 3 , / $ 8 P0 2 3 , #" P = P0, 2 3 , r(P02 )
P02 P02 P02 = 0 = r( ) = r( 2 ) = r( 3 ), P P P
P02 P02 P02 P02 r( 4 ) = 1 = r( 5 ), ..., r( 2i ) = i − 1 = r( 2i+1 ), ... P P P P P02 P02 P02 P02 ) = g − 2 = r( 2g−1 ), r( 2g ) = g − 1, r( 2g+1 ) = g. r( 2(g−1) P P P P ,
D 2 3 < P
V ! 1, 2, 3, 5, 7, ..., 2g − 1, (g−2)(g−1) . - / 2 3 V, 2 g ≥ 3. " 2g + 2 , / "
V (g − 2)(g − 1)(g + 1).
- ! 2 3 , !
P0, ! $ - r(P02 ) = 0 = r(P0 ), r(1) = 1 = r(P0−1 ) = r(P0−2) = ... = r(P0−g ), −(g+1)
r(P0
−(g+2)
) = 2, r(P0
) = 3, ..., r(P0−2g) = g + 1,
P0
V 1, 3, 4, ..., g + 2,
−(g + 2). . / ! P0, / 2 3 V. , " 4g 2 + g − 2 2 3 V.
2 3
$ 8
, Ω = cω , c ∈ C ∗, "
2 3 $ F g = 2 # f (P ) = Θ(3ϕP (P ) + KP ). - P0 2 3 $ B f ? 3 2 3 F,
! 2 3
V F. 2 V ! 2 3 P = P0 # P0, / / 2 3
V. 8 P0 2 3 ,
;$;;$=E f @ 2 P0 3 ;F $ 2 &FJ, '()
/ 0 @ #
, !# 3 ! $ 0
0
§
! " #$% & % f (z) = u(z) +iv(z) # % $ ' ( D ⊂ C, % ) f = fxx +fyy = ∂ f = 0 D, ! ! u v % %)% * #$ $ ' 0 ∂z∂z D. + ,! ! -)% #$% ' $ % & % f = u + iv D . /0 #1 f = h + g, h g ' $ & + 2%% ' $/) ( ( ! 1 34"56578-9:7 4! ; & % ∂f ∂z = 2 (fx − ify ) % ' ∂ f ) "< '= + ) ∂(∂z ) = ∂z∂z = 0. > ∂f ∂z $ ( D. 4 / #) h + $ h = ∂f ∂z . ? %+ $ h $/) ( ! 6 & % g = f − h 1 % ) "< '= + = ∂f − ∂h = ( ∂f − ∂h ) = 0 h. > g ' ∂g ∂z ∂z ∂z ∂z ∂z $ % & % f = h + g, g 1 % $/) ( ! @ + f = (h + c) + (g − c) = h + g. > ' 1 #! ?# / #1 % f = h+g %# 2 %2 $ % / #1 f = h + g, */) $ 1 % % / 2 & ( h g. . % * 2 $ 2
& (0! 8 D ' $ % / $' ( & ( ∂D, & u1 u2, D $ D, $A u1 = u2 ∂D, u1 = u2 D. .4*A% 0 + ,! B ! >/ u ' $ % & % # D, u ' $ D ∂D, $ $ $ ! >/ u = 0 ∂D, # )$ $ $ $ ! 7 u = 0 D. 0 8 / $ / & u D = {|z| < 1}, *% ! @ + $ %
& % u = Re z+i i−z D u(i) = 0 ) ∂D, D, $ z = i, u = 0 D + !B ! 2
∂f ∂z
2
0 >/ f #$% $ % & % D ⊂ C | f | ' %% & % D, f ( D. 0 >/ + $ f (z) zf (z) ' #$ $
& D ⊂ C, f ' % & % D. ! 8 f ' #$% & % C ' %+ $ f f 2 $ + f ' $+ f ' $! 34"56578-9:7 4! 8 f = u + iv ' $ % & %+ Δu = 0 = Δv C. 8 f 2 = u2− v 2 +i2uv ' $ %+ Δ(u2 −v 2 ) = 0 = Δ(uv). > {(ux)2 + (uy )2 = (vx)2 + (vy )2, uxvx + uy vy = 0}. ?# A (uy )2 = (vx)2 (ux)2 = (vy )2. > uy = ±vx ux = ±vy . > 1 #! " >/ w = f (z) ' #$% ' 1 $ % & % D f (z) = 0 D. 8 f (z) ' $ %
& %+ f (z) $ % & %+ ' $ % & % D. ∂ f 34"56578-9:7 4! 3+ $ ∂z∂z = 0. 7 # % 2
1 ∂ −1 ∂f ∂ 1 ∂f ∂2 ( )= [ 2 + ( ) ]= 0= ∂z∂z f (z) ∂z w ∂z ∂w w ∂z −1 ∂ 2f 2 ∂f ∂ −1 ∂w ∂f 2 ∂f ∂f ∂ −1 ∂f ( 2 )) +( 2) =[ 3 + ( 2) ] = 3 . ∂z w ∂z w ∂z∂z w ∂z ∂w w ∂z ∂z w ∂z ∂z ∂f 4) ∂f ∂z = 0, ∂z = 0. > 1 #! # >/ f : D → D1 ' $ % & %! 8 g ' $ D1, g ◦ f 1 $ % D. 8 g ' $
D1 .# D1 1 ) z → z), g ◦ f $ D. 34"56578-9:7 4! > w = f (z) (
∂ ∂f ∂f ∂ ∂f ∂f (g ◦ f ) = (gw ◦ f ) + (gw ◦ f ) , (g ◦ f ) = (gw ◦ f ) + (gw ◦ f ) . ∂z ∂z ∂z ∂z ∂z ∂z
7
∂ ∂ ∂ ∂ ∂f ( (g ◦ f )) = [(gw ◦ f )fz + (gw ◦ f ) ] = [( (gw ◦ f ))fz + (gw ◦ f )fzz ] = ∂z ∂z ∂z ∂z ∂z ∂f ∂f ∂ 2g ∂f ∂f ∂ 2g +( ◦ f) = 0, ( 2 ◦ f) ∂ w ∂z ∂z ∂w∂w ∂z ∂z #$ + g ◦ f ' $ D. 8 w = f (z) ' $ D, ∂ ∂g ∂g ∂g ∂w ∂w ∂f (g ◦ f ) = ( ◦ f) +( ◦ f) =( ◦ f) , ∂z ∂w ∂z ∂w ∂z ∂w ∂z
∂ ∂ ∂f ∂g ∂ 2f ∂f ∂g ∂ ∂g ( (g ◦ f )) = ( ◦ f )z +( ◦ f) = ( ◦ f) = ∂z ∂z ∂w ∂z ∂w ∂z∂z ∂z ∂w ∂z ∂ 2g ∂ 2g ∂w ∂f ∂w ∂f ◦ f) + ( 2 ◦ f) = 0. ( ∂w∂w ∂z ∂z ∂z ∂z ∂w > g ◦ f 1 $ D. 7 #!
.> & % * 2 $ ' 2 & (0 + ! ! 0 >/ u ' $ % & % D z0 ∈ D + u D; 0 8 u ' D, $ D D, maxD u % / $' ( ∂D. 34"56578-9:7 4! :$ 1 + $ u U (z0, r) ⊂ D. :* f = u + iv $ % & % ! 7 ef (z) ' $ % |ef (z)| = eu(z). > / & % ex #)*% x, % u % eu(z). : /+ % |ef (z)|. > & ' % $ ( & ef (z) $ + $ ef = c eu 1 ! 7 #+ u ( ! 4/ #/ 1 ( D. >/ S ' 1 $ z ∈ D 2+ $ u = c ( $ z #$ u(z0). 7 z0 ∈ S S = ∅, 1 + S 1 ! 7 / $ #/+ $ S ' # D. 3 ( /+ / z1 ∈ S, z1 ∈ D. 7 u D, u(z1) = u(z0 ) ' + $ * ) $ # S $ # z1. > u 1 z1, ( $ u z1 . 6$ + z1 ∈ S 0 #! > 0 1 # & % % ' $ 2 & ( D. 7 #! % + ,! ! >/ u ' $ D∗ = U (z0, r)\z0 u $ D∗ , u 1 % $ (
& D = U (z0, r). & + ,! ! >/ D ' %#% / C, z1 , ..., zn ' # $ $ D, D∗ = D\{z1 , ..., zn}. 8 u '
* % $ % D∗, * ) * $ a1 , ..., an $ % & % f D∗ %+ $ $
u(z) −
n
ak log |z − zk | = Ref
k=1
D∗.
8 u ' $ % f ' $ %
& D, u = Ref U (z0, r) ⊂ D, u = Ref D. 8 {un}∞ n=1 ' // $ ' 2 & ( D, % 2 % 2 2 # D, u = limn→∞ un ' 1 $ % & % D. .> & 0 + ,! B ! >/ D+ = {|z| < 1; Imz > 0}, u ' D+ ∪ (−1, 1), $ D+ u = 0 (−1, 1). 7 u 1 % $ ( & u {| z |< 1}, u (z) = −u(z), z ∈ D+ . + ,! CC ! >/ u ' $ % & % ' D γ ' % $ % + 1*% ∂D, $A D 1 γ. > 1 + $ u 1 % ' ( & D ∪ γ, % #$ 0 γ. 7 u 1 % $ ( & ( + 1' * ( D ∪ γ. > 1 $ ( & % $ ( & f, 1
% 2 % & % f $ z0 , % % $ z0, % ! 6 $ + %+ =< D & %2! 8 f = u + iv ' #$% $ % & % ' %#( D+ 1 ) * ' ) $ & g+ h D + $ f = h + g D. E
& f Jf = uxvy − uy vx = |h |2 − |g |2. ( ! D $ % & % f # % ' 2%)* ( & ) z0, h (z) 1 ) ω(z) = hg (z) $ z0 . #1 ( ( $ ( z0), (z) |ω(z0)| < 1. ?# % + $ Jf (z0) > 0, $% h (z0) = 0. F /+ $ f % & ) z0, f = g +h 2% & ) z0. 7$ z0 # A %( $ ( & f, f 2% & ) % & ) z0. $ + Jf (z0) = 0 )( %( $ ! 4 ! 8 f ' %% $ % & % D, f / $ ( 2% & ) )( $ # D .# / g = 0 ω = 0 < 1). '
>/ f = αz n + βz m {|z| < 1}, n ≤ m, m|β| < n|α|,
f $ (+ % 2% & ) )( $ # {|z| < 1}. 3 ( /+ h (z) = nαzn−1 1 )
βmz m−1 βm m−n g ω= = = z h αnz n−1 αn
$ )!
{|z| < 1},
|ω| =
|β|m m−n | |α|n |z
≤
|β|m |α|n
< 1
{|z| < 1}
-)% $ z0 ∈ C % $ ( & f (z) = z + z %(+ h = 1 = 0 ω = 1, |ω(z0 )| = 1. $ + Jf (z) = 0 C. %( $ z0 % f Jf (z0 ) > 0, Jf (z0 ) < 0, $ h (z0 ) = 0 g (z0 ) = 0. @ Jf (z0 ) = 0 % % % 2 % % f z0. 4 $ % % & f z0. 3% 2%)* 2 & ) $ 2 & ( f $ z0 % % % % z0 2 / 2 #' 1 ( f = h + g. > 1 + $ f (z0) = 0 $ z0, f 2' % & )! 6 < #1 % % 7 ( % h g z0 : ∞ ∞ h(z) = a0 + k=1 ak (z − z0 )k , g(z) = b0 + k=1 bk (z − z0 )k . 7 %+ $ b0 = −a0. ?# % h (z) + $ ak .k ≥ 10 1 / $ ! >/ an ( (
& . n ≥ 1). 7 bk = 0 % 1 ≤ k < n, & % ω = hg $ %
z0, |bn| < |an |,
|ω(z0)| < 1. 3 ( /+
h(z) = a0 + an (z − z0 )n + an+1(z − z0 )n+1 + ..., g(z) = b0 + b1(z − z0 ) + b2(z − z0 )2 + ... + bn (z − z0 )n + ...,
h (z) = nan (z − z0 )n−1 + (n + 1)an+1(z − z0 )n + ...,
g (z) = b1 + 2b2(z − z0 ) + ... + nbn (z − z0 )n−1 + ....
@
g (z) b1 + ... + (n − 1)bn−1(z − z0 )n−2 + nbn (z − z0 )n−1 + ... = h (z) nan (z − z0 )n−1 + ...
1 / $ ( z0, #1
0
n
0
n
3% 2 ( $ ( & f ( # 2 (! 3 ( /+ f (z0) = 0, |ω(z0)| < 1, {0 < |z − z0| < δ} f (z) = h(z) + g(z) = an (z − z0 )n (1 + ψ(z)),
bn (z − z0 )n(z − z0 )−n + O(z − z0 ), z → z0 . an E+ $ |ψ(z)| < 1 % z, # 2 z0, | abnn | < 1. 4) f (z) = 0
{0 < |z − z0| < δ} z0 # A % f. 8 f % & ) z0, f (z0) = 0, ) f 2%' & ) z0 % ! $ /+ $ f / % (−n). >% % ' % %! 7 1 # %+ $ %)* & ) % f 1 # % f ! 6 + $ % ' ψ(z) =
$ ( & 1 # ! @ + $ %
& % f (z) = z + z = 2x ) y. ! .> & % $ 2 & (0 ! >/ f ' $ % & % 1 ( D ' & ( C, f D f (z) = 0 C. > 1 + $ f % 2 ( D N / % ( % f D. 7 * ΔC arg f (z) = 2πN, 2 C 1 /' ! 34"56578-9:7 4! >/ $ f ( D, N = 0 $ w = 0 1 # f (D ∪ C) 1 ) w = f (z). ?# + $ ΔC arg f (z) = 0, $ ! 3' 1 $ ( ! 3 ( /+ / φ # # # $ S D ∪ C, $A φ : ∂S → C / #! > F = f ◦ φ 1 # S w− /+ )* (! > 1 + $ Δ∂S arg F (z) = 0. @$A # % S $ $ 2 Sj , 1 # 2 π % F (z) # % % /< 2 . 7 Δ∂S arg F (z) = 0, #$ + Δ∂S arg F (z) = j Δ∂S arg F (z) = 0, %#' # * / & ∂Sj , $ 1* 2 ∂S. > 1 /+ $ f D. > / # '
f = 0 C, * / $ $ # $ 2 (
D. 4#$ 2 zj , j = 1, ..., ν. >/ γj ' 1/ δ > 0 j
j
& zj , δ $ + $ γj 1 D / ! : 1) 1/ γj C ( 1 ( ( λj D. = # ( / Γ, ' # ( 1 C 1 / 1 % A # λj γj , 2 γj & / ' λj C. G % Γ 1 ( % f, #$ + ΔΓ arg f (z) = 0, # $ < # / ! λj , j = 1, ..., ν, # *)%! : /+ ' ν ΔC arg f (z) =
Δγj arg f (z),
j=1
1% 1/ γj 2 % # 1 / ' ! G /) #$ /(! >/ / f / % n > 0 z0 . 7 + + f /) f (z) = an (z − z0 )n [1 + ψ(z)], an = 0, |ψ(z)| < 1 $ ( 1 γ : |z−z0 | = δ. G # + $ Δγ arg f (z) = nΔγ arg(z − z0 ) + Δγ arg(1 + ψ(z)) = 2πn.
5 $ $ %+ $ Δγ arg f (z) = 2πn, f / z0 % n < 0. 7 #+ f zj % nj , j = 1, ..., ν, ν ν ΔC arg f (z) =
7 #!
Δγj arg f (z) = 2π
j=1
nj = 2πN.
j=1
:* $ ( ' $( $ %#( 1 & ! ?# & $)% 1 %! ) * " .5 =< % $ 2 ' & (0! 8 f f + g 2%)* & ) $ ' & D, D, |g(z)| < |f (z)| C, f f + g ) $ ( D. 34"56578-9:7 4! " % $% $ 2 & (+ ' C $A+ $ f, f +g ) ( C. 4) # C & % ) $ * % ' w = 0 w. : #! ! : # / % ' + 2 % & % f f + g # /
+ $ ) % 2 (+ $ ( ' / 1 / 2 & / 2 % (! 8 {fk }∞ k=1 ' // $ 2 & ( D, ' % 2 % / + 2 /% & % f $ D. ) * # .5 D & % $ 2
& (0! 8 f fk 2%)* & ) $
& D, f / z0 % n D / +
1 z0 . $ ( + $ 1 2 ( & f ) 1 % $ n (+ $A + % fk $ /< 2 k. 3 # / $ %+ $ $ + =< . !!B0 % f g = fk − f. :2%)* & ) $ & ) ( 1 %+ ! ! % 1 ! ) * $ 8 f ' 2%)*% & ) $ %
& % ( U (z0) $ z0, f (z0) = w0, f (z) − w0 / z0 % n ≥ 1, % 1 $ ε > 0 * δ > 0 )* ( H % 1 α ∈ U (w0, δ) = {w : |w − w0 | < δ} & % f (z) − α $ n (+ $A + U (z0, ε). 3 # / % !!B . =' < 0 % & ( f − w0 g = w0 − α. & + % + A #' 1 $ 2 & (! # 2 1 ( # %+ $ % )( # ( & μ, %)* ( ) μ ∞ < 1, F / fz = μfz < F ) < f = ψ ◦ F % ( $ ( & ψ. " #$% $ % & % 1 / %
D, # + % ) F / fz = ωfz ' + ω ' $ % & % ( |ω(z)| < 1 D. : ( + ) < % $ (
& ( D. " + D ' %# ω ∞ < 1 D, < ! @ %) ' #$) $ ) & ) f # / 2%)* ( & )+ % ) fz = ωfz % ( $ ( & ω ( |ω(z)| < 1.
7 $/ 2% % # & $ (
& (+ # )*% #$+ $ )% 2%)*% ' & ) #$% $ % & % f %/%
f = F ◦ ϕ % ( ( $ ( & F ( $ ( & ϕ. $ #+ $ )% #$% $ % & % f / ' % % f = h + g % 2 $ 2 & ( h, g. 3% f % Jf = |h |2 − |g |2, ω %' ) g = ωh . > %% $ % & % f 2%/ & ) D, / Jf (z) ≥ 0
( ! > - % ( $ (
& * % /+ #$ + Jf (z) > 0 % 2%' )* ( & ) #$( ( $ ( ' & ! :$ A + / 1) /
/ (< ! ! ! >/ f ' $ ( f (z) = z 2 + 23 z 3 . 7 f ω(z) = z, f 2%/ & ) $ D = {z : |z| < 1}. > 1 + $ f #1 % )( $! > 1 + $ f = F ◦ϕ, ϕ ' $ % & % $ F ' % $ % & % # & ϕ. 7 F 2% ' & )+ f ! F # $ % * $ /+ $ ϕ(0) = 0. ; & % F F = h + g $' + h g $ ) #1 % %
n ∞ n h(ζ) = Σ∞ n=1An ζ , g(ζ) = Σn=1Bn ζ ,
|A1 | > |B1 | ≥ 0. 7 $ % $/ % f (z) h(ϕ(z)) = z 2 , & % ϕ 1 / #1
ϕ(z) = c2 z 2 + c3 z 3 + ..., c2 = 1/A1.
> & % g ◦ ϕ #1 + $ )* % $A( z. @ $ )+ $ g(ϕ(z)) = 23 z 3 . 7 #+ f #1 % f = F ◦ ϕ )( $! " ! >/ f = z 2 + 12 z 4 . 7 f ω(z) = z 2 , #$ + 2% & ) $ D = {z : |z| < 1}. > < & % f #1 f = F ◦ ϕ D, F (ζ) = ζ + 12 ζ 2 ϕ(z) = z2 .
7 / 1 ) #1 + ' % A 2 $ % % #1 ' $ ( & ! % + ,! ! >/ f #$( ' %( $ ( & (+ A( D ⊂ C, ω ' A ! 7 % + $ f #1 f = F ◦ ϕ % ( $ ( & ϕ D ( ( ' $ ( & F ϕ(D), 2 $+ $ |ω(z)| = 1 D ω(z1) = ω(z2), f (z1) = f (z2). > 2 %2 ' $/) 1 % H ) f = F ◦ ϕ F = F ◦ ψ−1 ϕ = ψ ◦ ϕ % 1 % ψ, A ϕ(D). > 1 $ # / + # $ %! !! + 2% f % % % )' ( $! !!B / F #' $ + $ f (z1) = f (z2), / z12 = z22, / ω(z1 ) = ω(z2).
34"56578-9:7 4! > 1 $+ $ f = F ◦ϕ. 4#$ $ # A(ζ) % & F. 7 ω(z) = A(ϕ(z)) % 2 z # D - |ω(z)| = 1. " + f (z1) = f (z2) $A ϕ(z1) = ϕ(z2), F ! 4) # + $ ω(z1 ) = ω(z2).
4+ 1 + $ |ω(z)| = 1 D ω(z1) = ω(z2), f (z1) = f (z2). @ 1 / & F ϕ ' */) # ( ( F / ! F # $ % * 1 1 /+ $ |ω(z)| < 1 D, $ 1 ( %1 ( & f . @ /+ $ < #$ % 21 ) (
& G Ω = f (D), % ( # & % G ◦ f $ D. > / ) (G ◦ f )z = 0, (Gw ω + Gw )fz = 0.
: /+ (G ◦ f )z = 0 D, G / %)* ( ' ) F / Gw = μGw , μ(w) = −ω(f −1(w)). > ) f %# / (+ f −1(w) 1 / #$' ( & (! 4 + $ ω(z1) = ω(z2), f (z1) = f (z2),
$A+ $ # & % ω ◦ f −1 ' #$! 7 #+ & % μ + |ω(z)| < 1 $A |μ(w)| < 1 Ω. "
+ supw∈E |μ(w)| < 1 % ) 1 E ⊂ Ω. = $ {Dn}∞ n=1 % D, ,%* # ' 2 1 ( /)! >1 Ωn = f (Dn) ' & μn(w) = −ω(f −1(w)) w ∈ Ωn. >1 μn ' C + $ μn (∞) = 0 maxw∈C |μn (w)| = maxw∈Ωn |μn (w)| = maxz∈Dn |ω(z)|.
?# * ( # 2 1 ( + $ F / Gw = μnGw < Gn C + $ Gn (∞) = ∞. ; $ z0 z1 D1 + $ f (z0) = f (z1). G
#1+ f D1. #/A w0 = f (z0) w1 = f (z1), & ) Hn (w) =
Gn (w) − Gn (w0) . Gn (w1) − Gn (w0)
7 Hn 1 < % F / + % Hn(w0) = 0, Hn(w1) = 1 Hn(∞) = ∞. I ( / 1 / ) /' /+ 2%*)% 2 1 2 Ω ' ( & H(w), % % ) F / Hw = μHw f (Ω). 4) + $ ϕ = H ◦ f % ' ) "< '= ϕz = 0, #$ + $ ( & ( D.
D $/ & F = H −1 ϕ(D) # $
& f = F ◦ ϕ D. 3 ( /+ )( $ ζ = ϕ(z), ϕ (z) = 0, $ + $ F = f ◦ ϕ−1 $+ ϕ−1 / A% % & %! @ F / $ + # $ 2 $ % ϕ 1 % F. : /+ F $ % & % ϕ(D). G A #1 f = F ◦ ϕ. 3% # / #1 %+ 1 + $ % *A #1 f = F ◦ ϕ 1 ( ! >1 G = F−1 # + $ G % & %+ A ' % % % % $ ( & (! " + # & % G ◦ f %( $ ( & (+ #$ + G 1 %/ ) F / Gw = μGw . # 1 % #( ( 2 ( (+ 1 # )$ /+ $ G = ψ ◦ H % ( '
& ψ, % ϕ(D). 7 #+ F = F ◦ ψ−1, 1/! 7 #! > *% $ & + ' ) / #1 ! % % % /( #1 $ 2 & (! J1 # + $ f /
$ z0, / Jf (z0) = 0. $ + f / / #1 f = F ◦ ϕ F = f ϕ(z) = z. > 1 + $ Jf (z0) = 0. 8 |ω(z0)| = 1, ' * ( f #1 % $ z0 . : /+ #$/ $( |ω(z0 )| = 1, + 2%+ 1' + %1 ( & f , $ /+ $ |ω(z0)| < 1. 7 % ' Jf = (1 − |ω|2)|h |2, ) ) h (z0) = g (z0) = 0. F # $ % * $ + $ z0 = 0 f (z0) = 0. > f ∞ ∞
f (z) =
an z n +
n=m
bn z n , |am | > |bm | ≥ 0,
n=m
% % m ≥ 2. > 1 + $ f #1 f = F ◦ϕ, ϕ $ % F $ ϕ(0) = 0. 7 F /) F (ζ) =
∞
n
An ζ +
n=1
∞
Bn ζ n , |A1| > |B1 | ≥ 0.
n=1
: /+ ϕ 1 / ϕ(z) =
∞
cn z n , cm = am /A1.
n=m
: %
& + $ an = A1cn bn = B1cn m ≤ n < 2m, # 2 ) bm an = am bn % m < n < 2m. 8 + $ bm = 0, bn = 0 % m < n < 2m, !! ! 7 #+ # )*% ! & + ,! ! >/ f 2%)*% & ) $ % & % ( $ z0, A % Jf (z0 ) = 0. > 1 + $ f / #1 f = F ◦ ϕ % 2 & ( H ϕ ' $ ( $ z0 F ' $ ( ( $ ζ0 = ϕ(z0), f, F ϕ ) + A) < % m ≥ 2. 7 bm an = am bn % m < n < 2m.
6 + $ % * ( % %)% 2 % * % / #1 %! 4 + # ' )* ( + % %)% $ ! # ! >/ f (z) = 2z 2 + z 4 + z 5 + z 2 + z 4 + z 5 . 7 m = 2 a2 = 2, a3 = 0, b2 = 1, b3 = 0. 7 #+ b2a3 = a2 b3 = 0. K 2
& A $ < % A1 c2 = 2B1c2 = 2, c3 = 0,
A1c5 = B1c5 = 1. G # + $ f / #1 %
%! L1 1 / !! % # )$ %! 6 / |ω(z)| < 1
2z + 4z 3 + 5z 4 ω(z) = 4z + 4z 3 + 5z 4 %! J f (z1) = f (z2)
z12 − z22 = 2Re{z22 − z12 + z24 − z14 + z25 − z15 },
% % + z1 = −z2 = it, t > 0. : ( ' + ω(z1) = ω(z2) $A+ $ 4(z12 − z22 ) = 5(z23 − z13 ), % % z1 = −z2 = 0. 7 #+ f (z1) = f (z2) $A ω(z1 ) = ω(z2 ). : /+ !! f #1 % %! F ) & ) $ ( $ % $ ( & 1 ( !
§
ω
C 1 F ! " γ ω = 0 # $! γ, % & $ γ ## # % % % % ! D % % ' F. ( & " '
ω ) 2 ∗ ||ω|| = F ωω . * F ) # '
ω
C 1 F $ &$ +!"& & " Γ1, Γ1c , Γ1e )
' " ' & '
) ω
C 1 F , $& #
Γ1e ⊂ Γ1c ⊂ Γ1 ⊂ L2 (F ), Ω1 ⊂ H(F ) ⊂ Γ1 ⊂ L2 (F ). 1∗ 1 ,- & " Γ1∗ c Γe !"& Γ , #. " '
#- '
" Γ1c Γ1e ) /" & H(F ) = Γ1c ∩ Γ1∗ c . # ω1, ω2 ∈ Γ1
(ω1, ω2) =
F
∗
ω1 ω2 =
F
(a1 a2 + b1 b2 )dx ∧ dy
# # " + # )
0 |(ω1, ω2)| ≤ ω1ω2 ||ω1 +ω2|| ≤ ω1 + ω2 . 1 " 23)4# # ) " - # 5 ||ω1 + ω2 ||2 = ω1 2 + ω2 2 + 2Re(ω1, ω2) ≤ ω12 + ω22 + 2ω1ω2.
( % % ' F ) 1∗
## # # Γ1∗ e 6 Γc ## # ) # Γ1e ) Γ1. /" & " . # & ' %
% ' ) ' % ! " 7! % ' % Γ1c
7 % 1" # & " . # ) & '
# " " ! ' ( ω # Γ1 &
# - ω1 ω2 ω1 − ω2. + Γ1 % % ' F ! % +!"& & " Γ " Γ1 % 2 ! & " ) '
! - # # '
) &$ # - $. $ 8) $ ! 1 * ω ∈ Γc , γ ω # #
' ' γ, "
% [γ]. 1 -
$ 5$ Γ1c , " - $ )
" Γc # Γ1c . 9 - Γ1c $ Γc . 1 Γ0 ) Γ. % ) l Γ0 " # "&# # Γ0 # & l(c1 ω1 + c2 ω2 ) = c1 l(ω1) + c2 l(ω2),
c1, c2 ∈ C. : l " # & . ) M # & |l(ω)| ≤ Mω, ω ∈ Γ. ; & ) & % % % ! |l(ω) − l(ω0)| ≤ Mω − ω0 , ω, ω0 ∈ Γ. +! l ! . δ > 0 & ω < δ &5 |l(ω)| ≤ 1. # $! ω = 0 |l(δω−1ω)| ≤ 1, $ |l(ω)| ≤ 1δ ω. , !) " % % ) & + & % % % - # Γ0 " Γ. 1 1 γ ω Γ c ## # % " & | γ ω| ≤ M(γ)ω, M(γ) ) "
) % % γ. < " # " % & -
% - # % ## # % & & " ' & ' ' 6=
):3 > $!% & % % % l ! Γ l(ω) = (ω, σ) # ) σ ∈ Γ. +2<,*?, + % % ω → (ω, σ) ! ) & 0 (ω, σ) ≤ ωσ, M = σ. 1 - & l ) & % % % Γ. ,
l # Γ0 # l ! " % ) Γ. * l ## # - $ ) . ω % & l(ω) = 0. 9 ω "- ω = ω0 + ω1 , ω0 ∈ Γ0 , ω1 ⊥ Γ0. , l(ω) = l(ω1) = 0, &5 ω1 = 0. 1- σ = l(ω1)ω1−2ω1 .
; & σ ⊥ Γ0 l(σ) = σ2. 9 l(ω)σ − l(σ)ω ∈ Γ0 # $! ω. ! σ. + $ l(ω)σ2 − l(σ)(ω, σ) = 0, l(ω) = (ω, σ). * l = 0, "5 σ = 0. * " " % # " # , ) " 1 =
):3 Γc &
γ ω, % - Γ1c Γc. & ! σ σ∗, σ ∈ Γ∗c
γ
ω = (ω, σ ∗), ω ∈ Γc .
, $ # $! &
ω, σ∗ ⊥ Γe, "& σ∗ ∈ Γ∗c σ ∈ Γc. 1 σ ∈ H(F ). # $! γ . % . )
% & %
σ % & γ ω = (ω, σ∗) # ' " '
ω % % ')
F. +2<,*?, + + " &
σ ! .
@& # γ ω = γ ω, &) (ω, σ ∗) = (ω, σ ∗) = (ω, σ∗ ).
σ = σ. , " 4 & σ ## # "#.
)
γ, σ = σ(γ). ( & σ(γ1 + γ2) = σ(γ1) + σ(γ2) # $! ' ' 1 %5 "& $ & ' & ' )
! # % % ' F. . # % - ! #) '
% % ' F. 9 % " % ' + # "&
" '
! # # % % ' F # "& ) & &
' ) ! # 1 P0 ) & F. 4 &
θ0 # ! P0, 5 % )
& P0. 7# & %
θ, - 5 % % & P0 , ! θ0 6 # $ ! & θ0), " θ −θ0 ! % & P0. * θ0 % & P0 , P0 ) # !# & # θ0. 7& # ! # # & .$ &
θ0 % & 4 ! & % ! θ0 ) & %
2- % & %
θ0 θ0 = ϕ0 + ψ0 , ϕ0 ψ0 ) &
% & ) % ' , θ02 = ϕ02 + ψ02, θ0 $ !$ & ϕ0 ψ0 $ & $ !$ & (3 ) ! 5 % & %
) " & ! # # ) & &
A ! $ ϕ0, % ! $ ψ0 . , !" - ! " & # !. - & θ0 ) & %
) % & P0 F. , ' ) z = z(P ) θ0 = f dz, f ) & # # {z : 0 < |z| < 1}. ="- f # f (z) =
∞
m
am z +
m=0
∞
bn z −n
n=1
{z : 0 < |z| < 1}. ( {z : r < |z| < 1}, & # # ∞
f dz2,
"##
∞
1 1 1 (1 − r2m+2)|am |2 + |b1 |2 log + (r2−2n − 1)|bn|2 ]. 4π[ 2m + 2 r 2n − 2 m=0 n=2
+ $ & P0 ) # !# & # θ0,
bn = 0, n = 1, 2, ... * & & bn $ P0 " # & $ # θ0 . 1 θ ) " %
C 1, & & & ' ! % % ) % ' F. , . % & % )
τ % & " τ − θ ∈ Γe. +2<,*?, + 1 $ θ
C 1 dθ = 0, ! ' &
τ ! & ' - ! % & θ.
τ - - & θ. * " " & τ1 τ2 '
) τ1 −τ2 ! & & "& τ1 − τ2 = 0 F. # " . # " & θ − iθ∗ ! B #' ! ' & "- # θ − iθ∗ = ωh + ωe + ωe∗ ,
& % - H(F ), Γe, Γ∗e ) 1 # θ − ωe = iθ∗ + ωh + ωe∗ ,
%5 &
% ! "
) " $!% ! F, -. % ! ' & # θ. 1 τ = θ − ωe ! & F. , " # "& # ! '
-
) $ θ0 =
dz , m ≥ 0, (z − ζ)m+2
(1)
' z, z(P0 ) = ζ. 4 " & # !. - & ζ = 0. , "& "
ζ & ζ = 0. 1 r1, r2 & |ζ| < r1 < r2 < 1. ! $ $ e(z)
C 2 {z : |z| < 1}, # C {|z| < r1} B {|z| > r2}. 1- e(z) = 0 Δ = {z : |z| < 1}, - 5 $ ' F. + vm (z) = −
1 1 , z ∈ Δ, (m + 1) (z − ζ)m+1
- θm (z) = d(evm ) = vm de +
edz , z ∈ Δ, (z − ζ)m+2
F \Δ. , θm ) " %
C 1 ) & % ! $ 6C> + - & !% & 1 ) . % . % & %
τm (τm − θm ∈ Γe ). "5 τm = ϕm + ψm , ϕm ) & ! 6C> ψm ) & $ +)
$ ϕm +ψm & ϕm −ψm . D! " ζ ! θm = 0
ϕm = hm (z, ζ)dz, ψm = km (z, ζ)dz.
( hm km " # ! - #
τm, ϕm ψm " # ! e(z). % " e e , θm " # θm = d(e vm), " θm − θm = d((e − e)vm) ∈ Γ1e . 1 α = a(z)dz ) & %
% ) % ' F. 2π EBC (α, ψm) = (m+1)! a(m) (ζ).
& m = 0 5
(α, ψ0) = 2πa(ζ),
' ' i 2π
( )
F
a(z)k0(z, ζ)dzdz = a(ζ).
4 & k0(z, ζ) ) "#. # # & '
6 ) # 4 > & ζ ) # & F, "# a(z) = k0(z, ζ ), &
1 π
F
k0(z, ζ )k0 (z, ζ)dx ∧ dy = k0(ζ, ζ ).
1 % ζ ζ & 3 k0(ζ , ζ) = k0(ζ, ζ ), ! ! ' !"& #' k0(z, ζ) = k0(ζ, z). 9 " " ) & k0(z, ζ) ) & ζ. 2 k0(ζ, z)dζ " ! & ζ, - # k0(z, ζ)dζ. , !" %%
k0(z, ζ)dzdζ
(2)
! " # !' ' < ) - 68> ## #
# ! - 3 "
# 1 ## ( ) # α = ψm = km(z, ζ )dz %5 & (ψm, ψ0) = 2πkm (ζ, ζ ). % " !. 5% α = ψ0, &
(ψ0, ψm ) =
2π dm m k0 (ζ , ζ). (m + 1)! dζ
# - # " ## !"& # & & 1 dm km (ζ, z) = k0(z, ζ), (m + 1)! dz m
km & # " k0
& ) km(z, ζ) & ζ. & " &$ # # ϕm = hm(z, ζ)dz. & " & (α, ϕm) = (α, ϕm + ψm) = (α, τm) = (α, θm) = 0. < . )
& # " # # "& # < ζ ζ " ϕm ' ϕ0. % (ψm, ϕ0) = (ψ0 , ϕm) = 0 &5 - ' 6 F EB > m
hm (z, ζ) =
d 1 h0 (ζ, z). (m + 1)! dζ m
& m = 0 & # & h0, h0(z, ζ) = h0 (ζ, z). # " m hm (z, ζ) " & ζ h0 (z, ζ)dzdζ ! " # !' ' , !" " ! # $!% ! θ0 =
dz , m ≥ 0, (z − ζ)m+2
(1)
. $.% # " %
θm % % ' F, $.% $ ! ) 2 . % & %
) τm(τm −θm ∈ Γe). * τm = ϕm +ψm , ϕm = hm (z, ζ)dz, ψm = km(z, ζ)dz ) &
dm dm 1 1 hm (z, ζ) = h0 (z, ζ), km(ζ, z) = k0(z, ζ), (m + 1)! dz m (m + 1)! dz m
h0(z, ζ) = h0(ζ, z) k0(z, ζ) = k0(ζ, z).
ψm ) 2π "#. % (α, ψm) = (m+1)! a(m) (ζ) # ' α = a(z)dz ) & '
F, - (α, ψm) = 0, &5 # " # # "& # $. %3' ! % ## # ! (
1 1 − )dz, z − ζ2 z − ζ1
ζ1, ζ2 - & {z : |z| < 1}, |ζ1|, |ζ2| < r1 < r2 < 1. ! '#.$ $ e(z). + ) "&$ v(z) = log z−ζ z−ζ {z : r1 < |z| < 1}. / "# 1 1 - θ = ( z−ζ − z−ζ )dz, |z| ≤ r1; θ = d(ev), r1 < |z| < 1; θ = 0, |z| ≥ 1. &
θ !
C 1 dθ = 0, $&# ζ1 , ζ2. 1 CCE . % & %
) τ, τ − θ ∈ Γe. # $!% "% % F 2 1
2
1
γ
τ=
γ
θ = 2πi[n(γ0, ζ2) − n(γ0, ζ1 )],
n(γ0, ζ) ) & ζ γ0, γ = γ0 + γ1, γ0 - & {z : |z| < 1}, γ1 F. % n(γ0, ζ2) − n(γ0, ζ1) = c × γ0 = c × γ, # c
# & ζ1 ζ2 {z : |z| < 1}. + $ & γ τ = γ θ = 2πi(c × γ) # $! γ F \{ζ1 , ζ2 }, & "
% [γ] [c] 6 ' ζ1, ζ2) F \{ζ1, ζ2}. @& # " c, ! τ = τ (c) = ϕ + ψ, ϕ = ϕ(c) = h(z, c)dz, ψ = ψ(c) = k(z, c)dz. , τ & ! ψ & ζ1, ζ2. 1 % !- c → τ (c) - - " & c, τ (c1 + c2) = τ (c1 ) + τ (c2). & # ϕ(c) ψ(c). ,- % - θ(c) # c & $
" Γ1e . + ! " # ∂c,
θ, τ, ϕ $ $ &
∂c.
! & c ) & (α, ϕ) (α, ψ) # & '
α. ) &
&% c -. % ) & Δ F. ( & (α, ψ) = 2π c α F EBG 9 "#. % # ψ.
! α = ψ0 = k0 (z, ζ)dz, & $ (ψ0 , ψ) = 2π c k0 (z, ζ)dz. = " & (ψ, ψ0) = 2πk(ζ, c). 1 k(ζ, c) = c k0 (z, ζ)dz. /" ## !"& # ! ! & k(z, c) = k0 (z, ζ)dζ. c
k(z, c) & # " k0(z, ζ) & ) - km &
< - & ) # " ' & ' % & # (α, ϕ) & A (α, ϕ) = (α, τ ) = (α, θ) = −2π
α. c
2!# %5 (ω, τ ∗) # $! &
ω. , ω = α + β, (ω, τ ∗) = (α + β, −iϕ + iψ) = i(α, ϕ) − i(β, ψ) = −2πi
ω, c
τ = τ (c). ( %5 # h(z, c). / )
(τ0, τ ∗) = (ϕ0 + ψ0 , iϕ − iψ) = i(ϕ0, ψ) − i(ψ0, ϕ) = 2πi
c
ψ0 .
% (τ0 −θ0, τ ∗ −θ∗) = 0 Γc Γ∗e . 1 # & θ0 θ $ "& ! & (θ0, θ∗) = 0. + $ ∗
(τ0, τ ∗ ) = (τ0, θ ) + (θ0, τ ∗) = −i(ϕ0, θ) − i(ψ0, θ) − i(ϕ, θ0) + i(ψ, θ0) = = −2πi ϕ0 + 2πi ψ0 − 2πih(ζ, c) + 0. c
c
1 '# ! &
h(z, c) = −
c
h0 (z, ζ)dζ.
, !" " " # $!% &% c . " %
θ = θ(c), % $ &
∂c. + # # &)
$
" Γ1e . 2 . %
& %
τ = τ (c), τ − θ ∈ Γe . < # τ (c) = ϕ(c) + ψ(c), ϕ(c) = h(z, c)dz, ψ(c) = k(z, c)dz, & h(z, c) = − h0 (z, ζ)dζ, k(z, c) = k0(z, ζ)dζ, c
(α, ψ) = −(α, ϕ) = 2π
c
α c
# &
α % % ') F. # $! &
ω (ω, τ ∗) = −2πi
c ω.
=
! ! &% c ) , ) $.
ϕ(c) ψ(c) ! & F. #
# $! &
ω )
=
):3 c ω = (ω, σ(c)∗), "& τ (c) = 2πiσ(c). ( σ(c) ) . %
1) τ (c) = ϕ(c) + ψ(c) ! & Re(ϕ(c) + ψ(c)) = 0. , ϕ(c) + ψ(c) & %
ϕ(c) = −ψ(c) 1 1 σ(c) = π Imϕ(c) = − π Imψ(c). 2 γ τ (c) = 2πi(c × γ). * ) - c = c1, γ = c2, & (σ(c1), σ(c2)∗) = c1 × c2. , !" " # * c ) & % % % 1 ' F, ϕ(c) = −ψ(c) σ(c) = 2πi τ (c) = π1 Imϕ(c). 1 # $!% & ' (σ(c1), σ(c2)∗ ) = c1 × c2 . & $ & # σ(c), γ σ(c) = (c × γ) ∈ Z. , !" # & & )
!" {aj , bj }gj=1 F g, g ≥ 1, " % & σ(aj ) C bj , σ(bj ) )C aj ,
# ' g B 2 . & %
) τ = j=1[yj σ(aj ) − xj σ(bj )] " " ) x1, ..., xg, y1, ..., yg a1, ..., ag , b1, ..., bg $ H c % % ' F ! "!$. σ(c) ) & %
F. % & ( $!% % % ' F g, g ≥ 1, . % & %
( % % ' F g, g ≥ 1, . % & %
&
& & ' $ " & # ) % & % $ " !" %5 3 # - a− b− # & '
' ! '
) % % ' F g, g ≥ 1. 1 Ω = (πjk ) ! " b− # & !" α1 , ..., αg " ' ! '
F, % !" {aj , bj }gj=1. , Ω - # # b− $! &
& a− " %)
a− # &
ϕ x1, ..., xg , xj = a ϕ, j = 1, ..., g, "- ϕ !" α1 , ..., αg & ϕ = Σgj=1cj αj , xk = a ϕ = Σgj=1cj a αj = ck , k = 1, ..., g. 1) ϕ = Σgj=1xj αj . < " # b− ! "#
$ Ω, & j
k
t
(
ϕ, ..., b1
bg
k
ϕ) =t (Σgj=1xj πj1 , ..., Σgj=1xj πjg ) = Ωt (x1, ..., xg).
+!"& & " α =t (α1, ..., αg ) & % !" Ω1 ' ! '
F, & " σ(a) =t (σ(a1), ..., σ(ag )), σ(b) =t (σ(b1), ..., σ(bg)),
#$ & % !" H(F ) ) & '
F. , α = −σ(b)+Ωσ(a). % ) & " & # # &
$ !" ( " & αj = −σ(bj ) + πj1 σ(a1) + ... + πjg σ(ag ), j = 1, ..., g.
/
δjk = πjk =
bk
ak
αj = −
αj = −
bk
ak
σ(bj ) + πj1
σ(bj )+πj1
bk
ak
σ(a1) + ... + πjg
σ(a1 )+...+πjg
bk
ak
σ(ag ) = −(−δjk ),
σ(ag ) = πjk , j, k = 1, ..., g.
< α∗ = −iα, $ & −σ(b)∗ + Ωσ(a)∗ = iσ(b) − iΩσ(a).
1- Ω = X + iY . & &
&5 . σ(a) σ(b), −σ(b)∗ + Xσ(a)∗ = Y σ(a), Y σ(a)∗ = σ(b) − Xσ(a).
1 σ(a)∗ = Y −1σ(b) − Y −1Xσ(a), σ(b)∗ = XY −1σ(b) − (XY −1X + Y )σ(a).
9 3 # "#$ # #- %
ω∗ # $! &
ω. % ω a− ξ = (ξ1, ..., ξg) b− η = (η1, ..., ηg), ω = −ξσ(b) + ησ(a). + $ ω ∗ = −ξσ(b)∗ + ησ(a)∗ = = −ξ(XY −1 σ(b) − (XY −1 X + Y )σ(a)) + η(Y −1 σ(b) − Y −1Xσ(a)) = = −(ξX − η)Y −1σ(b) + [ξ(XY −1 X + Y ) − ηY −1X]σ(a).
, !"
a− b− # ω ∗ (ξX − η)Y −1 ξ(XY −1X + Y ) − ηY −1X ' I Ω " b− # & !")
'
% % ' F g, g ≥ 1, - % % ' F. + & - 12 g(g + 1) ' ) 1 g > 1 # ' " 3g − 3 " ' ' # g > 3 Ω - # 3 #
§
! G(w, z) = 0,
(1)
G(w, z) ! " z w. # $ % &' ' $ " F, $ ! ( ( )& F $ * "& g ≥ 1, * * ! {(z, w) ∈ C2 : G(w, z) = 0}, z w $ + F. )& ω $ !
+$ F, dzω * * * + F, '
+* +& % * z w, dzω = R(w, z). )$ ! !, !
+ F & R(w, z)dz, R(w, z) $ +&* +* z w. -! *
R(w, z)dz =
(w,z) (b,a)
R(w, z)dz + C,
(2)
R(w, z) $ +&* +* z w, . (w, z) $
*( ( / 0' C $ ' *, &$ ! ! (b, a). 1 % 2g−1 " F, ! 2 w = z(z − 1) k=1 (z − λk ), * * ! "
+ ." " ! & !,( ( "& F, ( ! / 0 2 ' W $ * * +* "3 F, W = R(w, z) $ +&* +* z w. 4$ % ' ' N * " ! " " Φ(w, z) = 0 F (w, z) = 0 .* !+& " ! *"' ! *" z = g(w, z) w = h(w, z) ∂(g,h) +& +* g, h, ∂(w,z) = 0 ' 3 5 ! $ *' (!( ( ! (
( % ! & ( ( ! $ ( ( G(w, z) = wm + A1(z)wm−1 + ... + Am(z),
(3)
G(w, z) $ Ak (z) $ 6 k, k = 1, ..., m. 7, ( ! * ! $
' 3 85 9 ! / 0 ! % &
Q(w, z) dz, Gw (w, z)
(4)
Q $ 6 m − 3 / % ' 5: 0 9 " F g > 0, * / 0 / 0' . {ak , bk }gk=1 ! ( *( "& F1. ;. + ∂F1 a+1b+1a−1 b−1 ...a+g b+g a−g b−g . )& w = u + iv $ *' (,( !+ a1 . . . ag b1 . . . bg u α1 . . . αg β1 . . . βg v α1 . . . αg β1 . . . βg
. u|a
(5)
= u|a−k + αk , u|b+k = u|b−k + βk , k = 1, ..., g. ! " ; w = u + iv $ ' $ *' * u v, !+ /0' + k
0<
∂F1
g udv = (αk βk − αk βk ). k=1
(6)
<=->-4;?@74 9. ' &* $ + * F1. 2
∂F1
g
udv =
g k=1
k=1
a+ k
bk
−
(u − u )dv +
(αk
+
dv − βk
ak
b+ k
(u+ − u−)dv =
g dv) = (αk βk − αk βk ). k=1
7 ' ** A * '
∂F1
udv =
F1
{(
F1
* =6$
∂u 2 ∂u ) + ( )2}dx ∧ dy > 0. ∂x ∂y
) % ' & " !% ! F1. )& φ, ψ $ (! ! F ( !+ $ 4
a1 . . . ag b1 . . . bg φ α1 . . . αg β1 . . . βg , ψ α1 . . . αg β1 . . . βg
∂F1
φdψ =
g
(αk βk − αk βk ),
(7)
k=1
! * φ ψ % ∂F1. 7 ' " l ∂F1
φdψ = 2πi
resPj φdψ,
(8)
j=1
P1, ..., Pl $ ! * & $ % *' %, F1. = ' * ! " 6 ! " # ; w, w $ F !+
a1 . . . ag b1 . . . bg w ω1 . . . ωg ωg+1 . . . ω2g , w ω1 . . . ωg ωg+1 . . . ω2g g (ωk ωg+k − ωk ωg+k ) = 0. k=1
(9)
< & /B0 /C0' . ' $ & % ! " F. $ 2 w1, ..., wn (* F, , ( c1, ..., cn, c, $ ( ' ' c1w1 + ... + cnwn = c F.
4% ' ' /10 * ! " % )& w1, ..., wg $ g $ " ! " "$ F g > 0 !+ a1 w1 ω11 w2 ω21 ... ... wg ωg1
a2 ω12 ω22 ... ωg2
... ... ... ... ...
ag b1 ω1g ω1,g+1 ω2g ω2,g+1 ... ... ωgg ωg,g+1
. . . bg . . . ω1,2g . . . ω2,2g , ... ... . . . ωg,2g
(10)
4 det(ωij )gi,j=1 = 0 det(ωi,g+j )gi,j=1 = 0. ( % & ! u1, ..., ug $
!+ a1 a2 . . . ag b1 u1 πi 0 . . . 0 a11 u2 0 πi . . . 0 a21 ... ... ... ... ... ... ug 0 0 . . . πi ag1
b2 a12 a22 ... ag2
... ... ... ... ...
bg a1g a2g ,
(11)
agg
aij = aji πiwk = ωk1u1 + ... + ωkg ug , i, j, k = 1, ..., g, ! v1, ..., vg !+ a1 a2 . . . ag b1 v1 1 0 . . . 0 τ11 v2 0 1 . . . 0 τ21 ... ... ... ... ... ... vg 0 0 . . . 1 τg1
b2 τ12 τ22 ... τg2
... ... ... ... ...
bg τ1g τ2g ,
(12)
τgg
τij = τji * Ω = (τij ) ImΩ > 0, ImΩ * * * %$ & . , + ; ξ1, ..., ξg $ & , ' $ ( ' τij = τij + iτij , i, j = 1, ..., g, (! v = ξ1v1 + ... + ξg vg = v + iv *% + v , v !+
v v
a1 . . . ag b ... b g 1 g g ξ1 . . . ξg τj1ξj . . . τjg ξj . j=1 j=1 g g 0 ... 0 j=1 τj1 ξj . . . j=1 τjg ξj
) ! * ! ' & ( * !&6 +
F. )& F
$ * "&' * ! f (x, y) = 0,
(13)
& m y. <* (a, b) ∈ F ! ' (, & ( (a, b) F, Qm−2(x, y)dx , [y − b − b (x − a)]fy (x, y)
Za (x) =
(14)
Qm−2(x, y) $ 6 m−2. 2 &
+ a ! Za(n)(x) = A (x−a) + r(x) ( * n a r(x) )& P1 = (a1, b1) P2 = (a2, b2) $ F, L(x) = (a2 − a1 )(y − b1 ) − (b2 − b1 )(x − a1 ). 4 n
=Π P P (x) = Π 1 2
Qm−2(x, y)dx , L(x)fy (x, y)
Qm−2 $ 6 m − 2, ! ! & * ! P1 P2 F. ) = Clog(x − a1 ) + r1(x), Π = −Clog(x − a2 ) + r2(x), Π
" *" P1 P2, r1(x) r2(x) P1 P2 ' C = 0. * Za(x) Π P P (x) "*, !+ $ v1, ..., vg, Ya(x) ΠP P (x) & ' ( a− : b− * " " ! " & ! 4 !' !+ 1 2
1 2
Yξ (x)
a1 . . . ag 0 ... 0
(n)
b1 −2ϕ1(ξ) (n−1)
ϕ
(ξ)
1 Yξ (x) 0 . . . 0 −2 (n−1)! Πξη (x) 0 . . . 0 2uξη 1
uk (x) = xx ϕk (x)dx + ck , uξη k
& F1, k = 1, ..., g. 0
... ...
bg −2ϕg (ξ)
= uk (ξ) − uk (η),
(n−1)
,
uk (x)
$ *
(ξ) g . . . −2 ϕ(n−1)! ... 2uξη g
(15)
)& & * "& F g > 0 *$ * & ! G(w, z) = 0 m n & w z 4 ! $ *(*
Q(w, z)dz , Gw (w, z)
(16)
Q $ 6 m − 2 n − 2 & w z ' 3 B: 9' + * " F g ≥ 2 , Γ = L1, ..., Lg , Lg+1, ..., L2g :
g
[Lj , Lg+j ] = 1,
j=1
* F U, F U/Γ. = ' &( !& F1 * Γ % !& D. g k=1[ak , bk ] z0 ∈ U, %, ! O * * ) + − + Lj : a− j → aj , Lg+j : bj → bj , j = 1, ..., g.
E * * +* ψ F * π $ + U, *(, * ψ(Lt) = ψ(t), t ∈ U, L ∈ Γ.
9 ! F ' $
uj |a+j = uj |a−j + πi, uj |a+k = uj |a−k , k = j, k, j = 1, ..., g,
(* + U * uj (Lj (t)) = uj (t) + πi, uj (Lk (t)) = uj (t), k = j, k, j = 1, ..., g, uj (Lg+k (t)) = uj (t) + ajk , k, j = 1, ..., g.
)
uj (t) =
t c
φj (t)dt =
a
z
ϕj (z)dz, j = 1, ..., g, t ∈ U,
φj (Lk (t))Lk (t) = φj (t), φj (Lg+k (t))Lg+k (t) = φj (t), k, j = 1, ..., g, U. F * ustj = uj (s) − uj (t) = ts duj , j = 1, ..., g, ' ustj, +* s, (,( !+ a1 a2 . . . ag b1 ust πi 0 . . . 0 a11 1 st u2 0 πi . . . 0 a21 ... ... ... ... ... ... ust 0 0 . . . πi ag1 g
b2 a12 a22 ... ag2
... ... ... ... ...
bg a1g a2g .
(17)
agg
- ' ! Yξ (z) (m) Yξ (z) (* " " + Yτ (t) (m) Yτ (t) U ( τ & ! F1 , Yξ (z) = Yτ (t) dzdt (ξ) U. ( ' ( (,( !+ Yτ (t) (m) Yτ (t)
a1 . . . ag 0 ... 0
b1 −2φ1 (τ )
0 ... 0
(τ ) 1 −2 φ(m−1)!
(m−1)
... ...
bg −2φg (τ ) . (m−1) (τ ) g . . . −2 φ(m−1)!
(18)
9 ! & Πxy ξη = Πξη (x) − Πξη (y)
(* Πstστ U, Πst στ = log(s − σ) + r(s) = −log(t − σ) + r(t) = log(t − τ ) + r1(t) = −log(s − τ ) + r1(s)
(," *" ) * * $ στ Πst στ = Πst .
(19)
! " G+* Πst στ − log
(s − σ)(t − τ ) = G(s, t, σ, τ ), (s − τ )(t − σ)
& % & ! F1, ! $ F14. = ' Πstστ $ +*
& " "' (,( + s t σ τ
a1 0 0 0 0
... ... ... ... ...
ag b1 0 2uστ 1 0 −2uστ 1 0 2ust 1 0 −2ust 1
... bg . . . 2uστ g . . . −2uστ g . st . . . 2ug . . . −2ust g
+( t+ t,τ + τ X(t, τ, t, τ ) = t τ exp(−Πt,τ )=
(t − τ + t)(τ − t + τ )exp(−G(t + t, τ + τ, t, τ )).
(21)
4 . t → 0, τ → 0 ! +* X(t, τ ) = −(t − τ )2exp(−G(t, τ, t, τ )) = (t − τ )2 Ψ(t, τ ),
(22)
* * * * F12. ( X(t, τ ) Ψ(t, τ ) $ + )& c, c + c ∈ b−j , cj , cj + cj " ! Lg+j (t) b+j . 2&* !+ ' 6 * c+ c,τ + τ j ,τ + τ Πccjj + c = Πcτ + 2ujc+ c,τ + τ + 2ucτ ,τ j + 2ajj .
(23)
7 ' /5 0 ( j ,τ + τ X(cj , τ, cj , τ ) = cj τ exp(−Πccjj + c ), ,τ
c+ c,τ + τ ). X(c, τ, c, τ ) = c τ exp(−Πcτ
)
X(cj , τ, cj , τ ) =
cj X(c, τ, c, τ )exp(−2ujc+ c,τ + τ − 2ucτ j − 2ajj ).
c
) & " c → 0, τ → 0, 6 *
X(cj , τ ) = Lg+j (c)exp(−4ucτ j − 2ajj )X(c, τ ),
6 (
X(t , τ ) = Lg+j (t)exp(−4utτ j − 2ajj )X(t, τ ), t = Lg+j (t), j = 1, ..., g.
(24)
- * 6
X(t , τ ) = Lj (t)X(t, τ ), t = Lj (t), j = 1, ..., g.
(25)
) ( $ +( /= 0 Ω(t, τ ) = C
X(t, τ ) = (t − τ )Q(t, τ ), C = 0,
(26)
Q(t, τ ) $ * +* F12, !,(,** & t τ, Q(t, τ ) = Q(τ, t). 4 & $ % Γ & ! F1 U . + Ω(t, τ ) Q(t, τ ) F12. > ' /580' /50'/5:0 * *(* " ' &(' $ &( % * ( ' *( $ +( 7 ,&( + *(* ! ."
& ! F1 : Πst στ = log
Ω(s, σ)Ω(t, τ ) tt , Πστ = 0, Πst τ τ = 0; Ω(s, τ )Ω(t, σ)
Yτst =
Ω(t, τ ) Ωτ (t, τ ) Ωτ (s, τ ) ∂ st ∂ Πστ = log = − , ∂τ ∂τ Ω(s, τ ) Ω(t, τ ) Ω(s, τ )
Yτ(m)st
∂ m st ∂m 1 1 Ω(t, τ ) . = Π = log στ (m − 1)! ∂τ m (m − 1)! ∂τ m Ω(s, τ )
2
(
(27)
st st Πst σg+j ,τ = Πστ + 2uj
1 Ω(s, σg+j )Ω(t, σ) ; ust j = log 2 Ω(s, σ)Ω(t, σg+j ) φj (t) =
Ω(t, σg+j ) 1∂ log ; 2 ∂t Ω(t, σ)
Yτ (tg+j ) = Yτ (t) − 2φj (τ ), j = 1, ..., g.
(28)
7 -! * * ! " ." )& H(t) $ ** * & Γ $ * +* U, * α1 , ..., αm ( β1, ..., βm F1 / . 0 & ! w(t) $
U !+ a1 . . . ag b1 . . . bg w ω1 . . . ωg ωg+1 . . . ω2g . 1 m1 . . . mg 2πi logH(t) −n1 . . . −ng
1 ** 2πi ∂F w(t)dlogH(t), ' + ' $ "' 1
wα1 β1 + ... + wαm βm = m1 ω1 + ... + mg ωg + n1ωg+1 + ... + ng ω2g ,
(29)
mj , nj ∈ Z, j = 1, ..., g. ' w1, ..., wg $ ! " ! " ' m
wjαk βk ≡ 0(modωj1, ..., ωj,2g ), j = 1, ..., g,
(29 )
k=1
( + 6 2g, %. !+ t
(ω11, ..., ω1g ), ...,t (ω2g,1, ..., ω2g,g ).
% ! % , $ + * α1 , ..., αm ( β1, ..., βm F1, αj = βk , j, k = 1, ..., m, * m
uαj k βk = mj πi + n1 aj1 + ... + ng ajg , j = 1, ..., g,
k=1
* * * (29 ). 4 + % &
Ω(t, α1)...Ω(t, αm) tτ exp(−2(n1utτ 1 + ... + ng ug )), C = 0. Ω(t, β1)...Ω(t, βm) 2πi1 Πtcστ dlogH(t), .* + ∂F1 ' *(, σ τ !"*, σ τ F1. ) *' H(t) = C
m
Παστk βk
k=1
g H(σ) +2 = log nj uστ j . H(τ ) j=1
(30)
> & nj , j = 1, , , ., g, + ' * ! ! & ! σ τ. - * (* 6 * m k=1
Yταk βk
g H (τ ) −2 =− nj φj (τ ); H(τ ) j=1
m k=1
Yτ(l)αk βk
g
1 2 dl (l−1) =− logH(τ ) − nj φj (τ ); l (l − 1)! dτ (l − 1)! j=1
(31)
> ' +& !* ' *(, αk βk , k = 1, ..., m, % &' " /510'/ 0 + $
( 76 * /510 $ / 0 * *(* !" * -! * * , * + $ * ( " g > 0. ) (29 ) * * * % / ! " 0 " ! " " -! * * (, ! & /-! *0 5' 5' 58 7 (! ! $ R(x, y)dx C, F (x, y) = 0, * " !, & N * C $ C , F (x, y) = 0 m, +& +
+ F , % $ +& " + " %
+$ ' . % % * % & & /-! *0 )& ** * C * $
* C m. 4 , ! R(x, y)dx C & " ' " $ * C C , ! (' Nj=1 (x(x ,y,y )) R(x, y)dx
+ F * C . & /-! *0 B' 3 11 )& D $ & $ " F g > 0. 4 D * * * $ + F, & F , 1−+ & γ *' . + ∂γ = D
γ
j
j
0
0
− → φ =0
(∗)
! H! J(F − ) * (! ! ! "
+ → t φ1 , ..., φg ' φ = (φ1 , ..., φg ). <=->-4;?@74 A+ ∂γ = D + γ * * * $+ &( 7$ , r ≥ 1, Pk = Qj , k, j = 1, ..., r. r = 0 )& D = QP ...P ...Q → φ , 7 Qj Pj . γj F. ϕ(D) = rj=1 γ − 1
r
1
r
j
− t → φ = (φ1 , ..., φg ) φ1 , ..., φg $ (! ! " ! "
$ + F, ϕ $ !% H! 2 * (∗) ϕ(D) = 0 J(F ). → − → φ = γ0 φ , γ0 $ !' ϕ(D) = 0 J(F ), ϕ(D) = rj=1 γj − + F. )% γ = γ1 + ... + γr − γ0, ∂γ = D − ∂γ0 = D. 4
-! * ) . & ! " 6 $ 6 7 ,&( " ! " Yτts
$ . ( +(
Yτts Yτts1
φ1 (τ ) φ1 (τ1)
... ...
φ (τ ) φ (τ ) g g 1
. . . Yτtsg . . . φ1 (τg ) ... ... . . . φg (τg )
= F (t, s; τ, τ1, ..., τg ).
(32)
! τ1, ..., τg ∈ F1 ' !
φ1 (τ1) . . . φ1 (τg )
Δ =
... ... ...
φg (τ1) . . . φg (τg )
= 0.
(33)
% & / 50 !+ Yτts = φ1 (τ )Y1ts + ... + φg (τ )Ygts +
' ' Y1ts
1
=
Δ
Yτts1 φ2 (τ1) φ3 (τ1) ... φg (τ1)
. . . Yτtsg . . . φ2 (τg ) . . . φ3 (τg ) ... ... . . . φg (τg )
F (t, s; τ, τ1, ..., τg ) , Δ
Yτts1
ts −1 φ1 (τ1)
, Y2 =
φ3 (τ1)
Δ
...
φ (τ ) g 1
. . . Yτtsg . . . φ1 (τg ) . . . φ3 (τg ) ... ... . . . φg (τg )
, ...
) ' + t, ( (,( !+ a1 0 0 ... ... Ygts 0
Y1ts Y2ts
... ... ... ... ...
ag b1 0 −2 0 0 ... ... 0 0
. . . bg ... 0 ... 0 . . . . ... . . . −2
H' Y1ts, ..., Ygts C. = ' 2g + ts ts ts uts 1 , ..., ug , Y1 , ..., Yg % C F1 . ; w1, ..., wg $ & ' g
wj =
)& " ( a1 w1 ω11 ... ... wg ωg1
λjk uk , j = 1, ..., g.
(33 )
k=1
. . . bg . . . ω1,2g . . . . ... . . . ωg,2g ( λkj = ωπikj πiwj = gk=1 ωjk uk , j, k = 1, ..., g, . +(ωkj )gk,j=1 (ωk,g+j )gk,j=1 ( = &6 ' ! & wj = ψj (t)dt, ' ψj (t) = gk=1 λjk φk (t), j = 1, ..., g. . ,. Pτtsσ
... ... ... ...
=
ag b1 ω1g ω1,g+1 ... ... ωgg ωg,g+1
Πts τσ
−
g
τσ ckj uts k uj .
j,k=1
<* " cjk , j, k = 1, ..., g. ( Zτts
Pτtsσ = Ptsτ σ ,
& ckj
g ∂Pτtsσ ts ts , Zτ = Yτ + =− ckj uts k φj (τ ). ∂τ
=
(34)
j,k=1
> %
+(
Zτts Zτts1
ψ1 (τ ) ψ1 (τ1)
... ...
ψ (τ ) ψ (τ ) g g 1
. . . Zτtsg . . . ψ1 (τg ) ... ... . . . ψg (τg )
= ΛF (t, s; τ, τ1, ..., τg ),
* . % Zτts = ψ1(τ )Z1ts + ... + ψg (τ )Zgts +
F (t, s; τ, τ1, ..., τg ) , Δ
(35)
' '
1 Z1ts =
D
Zτts1 ψ2 (τ1) ψ3 (τ1) ... ψg (τ1)
. . . Zτtsg . . . ψ2(τg ) . . . ψ3(τg ) ... ... . . . ψg (τg )
Zτts1
ts −1 ψ1 (τ1)
, Z2 =
ψ3 (τ1)
D
...
ψ (τ ) g 1
. . . Zτtsg . . . ψ1(τg ) . . . ψ3(τg ) ... ... . . . ψg (τg )
, ...
> &
D = ΛΔ, Λ $ + (λkj )gk,j=1. )& ηkτ , k = 1, ..., 2g, $ " ' + t. 4 / 80
% * * ηjτ
= πi
g
τ cjk φk (τ ), ηg+j
=
k=1
g
clk ajl φk (τ ) − 2φj (τ ), j = 1, ..., g.
(36)
l,k=1
)$ ' 6 * 6 ηkτ = μk1 φ1 (τ ) + ... + μkg φg (τ ) = νk1ψ1 (τ ) + ... + νkg ψg (τ ),
+ * τ, τ1, ..., τg, k = 1, ..., 2g. = '
Zτts1 . . . Zτtsg
ψ (τ ) . . . ψ2 (τg ) Z1ts =
2 1 ... ...
...
ψ (τ ) . . . ψ (τ ) g 1 g g
ψ1 (τ1)
ψ2 (τ1)
...
ψg (τ1)
. . . ψ1(τg ) . . . ψ2(τg ) ... ... . . . ψg (τg )
,
"** & + Z2ts, ..., Zgts. = + t, ( (,( !+ a1 Z1ts η11 ... ... Zgts ηg1
... ... ... ...
ag b1 η1g η1,g+1 ... ... ηgg ηg,g+1
. . . bg . . . η1,2g . . . . ... . . . ηg,2g
<* "% * " ηjk , ηj,g+k / 0 $ 6 * Zτts|ak = ηkτ = ψ1 (τ )η1k + ... + ψg (τ )ηgk , k = 1, ..., g.
7 ' / :0 ηjτ
= πi
g
cjk φk (τ ), j = 1, ..., g.
k=1
φk (τ ) ψk (τ ) ,&( ! ! *
φ1 (τ ) = λ11ψ1 (τ ) + ... + λ1g ψg (τ ), ...
φg (τ ) = λg1 ψ1(τ ) + ... + λgg ψg (τ ). ( ηjτ = πi gk,l=1 cjk λkl ψl (τ ). ψl (τ ), l = 1, ..., g, ηlj = πi
g
cjk λkl , l, j = 1, ..., g.
k=1
)%* *' .' * Zτts|b g
ηl,g+j =
j
τ = ηg+j
ckm ajk λml − 2λjl , j, l = 1, ..., g.
k,m=1
< . *& % * Zτts Pτtsσ . 7 ' $ ' * * ηkτ = ψ1 (τ )η1k + ... + ψg (τ )ηgk , k = 1, ..., 2g,
/ 0 Pτtsσ
=
−w1τ σ Z1ts
− ... −
wgτ σ Zgts
−
τ σ
F (t, s; τ, τ1, ..., τg ) dτ, Δ
. & γ, *(, σ τ, % F1. 4 F1\γ + Zτts t Zτts = 1 1 3 t−τ − s−τ + r(t, s, τ ), r(t, s, τ ) ! F1 . 4% $ , F (tj , s; τ, τ1, ..., τg ) = F (t, s; τ, τ1, ..., τg ) = F (tg+j , s; τ, τ1, ..., τg ), j = 1, ..., g,
'
τ σ
F (tj , s; τ, τ1, ..., τg ) dτ = Δ
σ
τ
F (t, s; τ, τ1, ..., τg ) dτ Δ
* (! j = 1, ..., 2g. 7 &' * Pτtsσ , + t, (,( !+ Pτtsσ −
a1 ... ag g τσ . . . − k=1 ηkg wkτ σ k=1 ηk1 wk
g
... g b1 g bg . τσ − k=1 ηk,g+1wk . . . − k=1 ηk,2g wkτ σ
Pτtsσ
(37)
9 + %
ts wg+j = Zjts, ωg+j,k = ηjk , j = 1, ..., g, k = 1, ..., 2g.
4 ' + t, ( !+ w1ts
... wgts ts wg+1 ... ts w2g
a1 ω11 ... ωg1 ωg+1,1 ... ω2g,1
. . . ag b1 . . . ω1g ω1,g+1 . . . ... ... . . . ωgg ωg,g+1 . . . ωg+1,g ωg+1,g+1 . . . ... ... . . . ω2g,g ω2g,g+1
... bg . . . ω1,2g ... ... . . . ωg,2g . . . . ωg+1,2g ... ... . . . ω2g,2g
(37 )
4 + w1, ..., w2g C, $ & * + (ωjk )2g j,k=1 * ) . ! 6 * 6 <* + G(t, τ ) U 2, * * $ 6 *
tk G(tk , τ ) = G(t, τ ), τk G(t, τk ) = G(t, τ ), k = 1, ..., 2g,
* * g j,k=1
∂ 2Pτtsσ ∂ 2Πts τσ − . ckj φk (t)φj (τ ) = G(t, τ ) = ∂t∂τ ∂t∂τ
(38)
G = ' ! & % $ & . &
% 4 $ +( /= 0 Πts τ σ = log −Yτts
Ω(t, τ )Ω(s, σ) , Ω(t, σ)Ω(s, τ )
Ω2 (t, τ ) Ω2(s, τ ) ∂Πts τσ = − , = ∂τ Ω(t, τ ) Ω(s, τ )
∂ Ω12(t, τ ) Ω1(t, τ )Ω2(t, τ ) ∂ 2Πts τσ ts (−Yτ ) = = − . ∂t ∂t∂τ Ω(t, τ ) Ω(t, τ )2
) ** +* ! t τ, 1 Ω(t, τ ) = −Ω(τ, t). ) * * (t−τ ) + r(t, τ ), r(t, τ ) F12. )! ! * Pτtsσ . 4 2
∂ 2Ptsτ σ ∂ 2Pτtsσ = , G(t, τ ) = ∂t∂τ ∂τ ∂t
.
∂ ∂ 1 (−Zτts) = (−Ztτ σ ) = G(t, τ ) = + R(t, τ ) ∂t ∂τ (t − τ )2
+ G R ! t τ. <* "% * ! & 6 % C dτ C G(t, τ )dt, C dt C G(τ, t)dτ, *
!(* ! )& c ∈ a−j , cj $ (,* a+j . &. % s, t, ! c, cj $ ! C (' %,( F1 *(,( + s t. )! ! &. γ ∈ a− l , γl ∈ al ( Γ F1 , $ *(,( σ τ (,(* C. 4 * & * Pτtsσ , * * *
Pτtsσ =
dt C
Γ
G(t, τ )dτ =
dτ Γ
G(τ, t)dt C
(39)
F1. ) ' +* t, !+ / B0 $ & " lim(τ,σ)→(γl ,γ) [lim(t,s)→(cj ,c) Pτtsσ ] = lim(t,s)→(cj ,c) [lim(τ,σ)→(γl ,γ) Ptsτ σ ].
) !+ / B0 % * " !" ( $
(, −
g
ηkj wkτ σ , −
k=1
g
ηkl wkts
k=1
( ! " 6 6 g (ηkj ωkl − ηkl ωkj ) = 0, j, l = 1, ..., g. k=1
" 6 ! g(g−1) 2 .
(40)
4 & % &. c ∈ b−j γ ∈ b−l ! 6 6 <* * & " 6 &. c ∈ a−j γ ∈ b−l . ; j = l, / 10 6 )& j = l, * C % & Γ. ) C ' ! ! ( Γ. )%
P1 =
τ
t
dτ
σ
s
G(t, τ )dt, P2 =
t
τ
dt
s
σ
G(t, τ )dτ.
( 6 * P1 = (Pτtsσ )1 + 2πi, P2 = (Ptsτ σ )2 + 2πi,
(Pτtsσ )1 ! * & + (t, s) $ (τ, σ). ). ( t = s. & (Ptsτ σ )2 $ * * ! P1, P2 (* ' ( 2πi. > & P1 − P2 , '
τ
t
dτ
σ
s
)
P1 − P2 =
τ
R(t, τ )dt =
t
dτ
σ
s
t
τ
dt
s
dt − (t − τ )2
R(τ, t)dτ.
σ
t s
dt σ
τ
dτ . (t − τ )2
7 & "'
lim(τ,σ)→(γg+j ,γ)
τ
dτ
σ t
s
t
dt = log(τ −cj )|γγg+j −log(τ −c)|γγg+j = −χi−λi, 2 (t − τ )
τ
dτ = log(t − γg+j )|ccj − log(t − γ)|ccj = μi + νi, 2 s σ (t − τ ) c, γ, cj , γg+j !( ." & λ, ν, χ, μ $
7 " 2π. 4 !' lim(t,s)→(cj ,c)
dt
lim(τ,σ)→(γg+j ,γ) [lim(t,s)→(cj ,c) Pτtsσ ] − lim(t,s)→(cj ,c) [lim(τ,σ)→(γg+j ,γ) Ptsτ σ ] = −2πi.
( & ! 6 6 −
g k=1
ηkj ωk,g+j +
g
ηk,g+j ωkj = −2πi, j = 1, ..., g.
k=1
(41)
7 &' ! " 6 6$ g
(ωkj ηkl − ωkl ηkj ) = 0,
k=1
g
(ωkj ηk,g+l − ωk,g+l ηkj ) =
k=1 g
0, j = l; −2πi, j = l;
(ωk,g+j ηk,g+l − ωk,g+l ηk,g+j ) = 0, j, l = 1, ..., g.
(W )
k=1
I % ! * ! " 6 $ % ! " <* * $ (, ' + t :
i) ∂F1
wjts dwlts ; ii)
∂F1
wjts dZτts ; iii)
∂F1
Zτts dZτts .
i) ' ' ( =6' $ * * + ' * ! 6 g (ωjk ωl,g+k − ωj,g+k ωlk ) = 0, j, l = 1, ..., g. k=1
<* * ii) 3 * 6 g
τ (ωjk ηg+k − ωj,g+k ηkτ ) = −2πiψj (τ ).
k=1
< &' (, dZτts = (−
1 + r(t, τ ))dt, wjts = wjτ s + ψj (τ )(t − τ ) + ..., 2 (t − τ ) wjts dZτts = (−
ψj (τ ) + r1 (t, τ ))dt. t−τ
= ' ! 6
ηlτ = ψ1 (τ )η1l + ... + ψg (τ )ηgl , l = 1, ..., g.
2 ( g
(ωjk ηl,g+k − ωj,g+k ηlk )ψl (τ ) = −2πiψj (τ ), j = 1, ..., g.
k,l=1
( ! 6 g
0, j = l; j, l = 1, ..., g. −2πi, j = l;
(ωjk ηl,g+k − ωj,g+k ηlk ) =
k
9 +' iii) & g τ τ τ τ (ηk ηg+k − ηk ηg+k ) = k=1
Zτts dZτts
(τ )
+
(τ )
Zτts dZτts = 0.
< &' '
(τ )
ZτtsdZτts = 2πi
∂ τs Z . ∂τ τ
7 ' d(ZτtsZτts ) = Zτts dZτts + Zτts dZτts ,
'
Zτts dZτts +
(τ )
< &
(τ )
(τ )
Zτts dZτts = 0.
Zτts dZτts = 2πi
∂ τs Z ∂τ τ
∂τ∂ Zττ s = ∂τ∂ Zττ s. 4 & & & ηkτ * ηij & . 4 !'
g g g g g [ ηjk ψj (τ ) ηl,g+k ψl (τ ) − ηlk ψl (τ ) ηj,g+k ψj (τ )] = 0, k=1 j=1
l=1 g
l=1
j=1
(ηjk ηl,g+k − ηj,g+k ηlk )ψj (τ )ψl (τ ) = 0.
k,j,l=1
( & ! 6 g
(ηjk ηl,g+k − ηj,g+k ηlk ) = 0, j, l, = 1, ..., g.
k=1
7 &' ! " 6 g
(ωjk ωl,g+k − ωj,g+k ωlk ) = 0;
k=1
g (ωjk ηl,g+k − ωj,g+k ωlk ) = k=1 g
0, j = l; −2πi, j = l;
(ηjk ηl,g+k − ηj,g+k ηlk ) = 0, j, l, = 1, ..., g.
(R)
k=1
) . % ! ! " 6 6 ) !+ (37 ) $ ."! +
A B C D
.
I !& A +' ( + A. I 6 * 6 ! $ " + & (,
g ( (ωkj ηkl − ηkj ωkl )) = A C − C A = 0; k=1
(
g
(ωkj ηk,g+l − ηkj ωk,g+l )) = A D − C B =
k=1
0, j = l; −2πi, j = l;
g ( (ωk,g+j ηk,g+l − ηk,g+j ωk,g+l )) = B D − D B = 0;
(W )
k=1 g ( (ωjk ωl,g+k − ωj,g+k ωlk )) = AB − BA = 0; k=1
(
g
(ωjk ηl,g+k − ωj,g+k ωlk )) = AD − BC =
k=1
0, j = l; ; −2πi, j = l;
g ( (ηjk ηl,g+k − ηj,g+k ηlk )) = CD − DC = 0. k=1
(R)
F * * +
* ! * (ajk ) = a, (cjk ) = c λ $ + ! * (33 ), % !* + Λ, %
& + A = πiλ, B = λa, C = [πicΛ] , D = [acΛ + 2Λ] . ) + a, c * *(* 4 $ & " " 9 ' * / 0
A C − C A = [πiλ] [πicΛ] − [πicΛ][πiλ] =
(πi)2λ Λ c − (πi)2cΛλ = −π 2 c + π 2 c = 0.
I 6 * % ! . ! 6 * 6 ' $ ' 6 6 g
(ωkj ηk,g+l − ωk,g+l ηkj ) =
k=1
6 g
(ωjk ηl,g+k − ωj,g+k ηlk ) =
k=1
0, j = l; −2πi, j = l;
0, j = l; −2πi, j = l;
)% ωg+k,ν = ηkν , k = 1, ..., g, ν = 1, ..., 2g. 4 ωμν * " μ, ν = 1, ..., 2g. ) ! 6 6 ! & g
(ωkμωg+k,ν − ωg+k,μωkν ) =
k=1
0, ν = g + μ, μ ≤ ν; −2πi, ν = g + μ;
! 6 $ g
(ωμk ων,g+k − ωμ,g+k ωνk ) =
k=1
0, ν = g + μ, μ ≤ ν; −2πi, ν = g + μ;
μ, ν = 1, ..., 2g. I * !, & g = 2. <* ! ( G = (x1y3 − x3y1 ) + (x2y4 − x4y2 ).
! * √
2πixμ = ω1μ x1 + ω2μ x2 + ω3μ x3 + ω4μx4 ,
√ 2πiyμ = ω1μ y1 + ω2μy2 + ω3μ y3 + ω4μy4 , μ = 1, ..., 2g = 4.
(42)
& ! * * ) % " ! ' G ! '
+ *( !$ 6 ( < &'
G =
1 [(x1y4 − x4y1 )(ω11ω43 + ω12ω44 − ω41ω13 − ω42ω14 )+ 2πi
(x1y2 − x2y1 )(ω11ω23 + ω12 ω24 − ω21ω13 − ω22ω14)+ (x2y3 − x3y2 )(ω21ω33 − ω22ω34 − ω31ω23 − ω32ω24)+ (x1y3 − x3y1 )(ω11ω33 + ω12 ω34 − ω31ω13 − ω32ω14)+ (x3y4 − x4y3 )(ω31ω43 + ω32 ω44 − ω41ω33 − ω42ω34)+ (x2y4 − x4y2 )(ω21ω43 + ω22 ω44 − ω41ω23 − ω42ω24)].
9 &' !" %* % * * !$ 6 * 4 !'
G = (x1y3 − x3y1 ) + (x2y4 − x4y2 ) = G .
4 & ! ! ! ( /850' √
√ √ √
2πix1 = ω33x1 + ω34x2 − ω31x3 − ω32x4,
2πix2 = ω43x1 + ω44x2 − ω41x3 − ω42x4,
2πix3 = −ω13x1 − ω14x2 + ω11x3 + ω12x4,
2πix4 = −ω23x1 − ω24x2 + ω21x3 + ω22x4.
- * yj , j = 1, ..., 4. ) % $ * G % " % * G . < &'
G =
1 [(x1y2 − x2y1 )(ω44ω23 + ω34ω13 − ω14ω33 − ω43ω24 )+ 2πi (x2y3 − x3y2 )(ω34ω11 − ω31ω24 + ω44ω21 − ω41ω24)+ (x1y3 − x3y1 )(ω33ω11 − ω31ω13 + ω43ω21 − ω41ω23)+ (x1y4 − x4y1 )(ω33ω12 − ω32ω13 + ω43ω22 − ω42ω23)+ (x2y4 − x4y2 )(ω34ω12 − ω32ω14 + ω44ω22 − ω42ω24)+ (x3y4 − x4y3 )(ω42ω21 − ω41ω22 + ω32ω11 − ω31ω12)].
7 ' % * !" % ' ! $ 6 6 (
G = (x1y3 − x3y1) + (x2y4 − x4y2 ) = G.
4 !' ' " G G * !$ 6 * , ** % ! "' *$ ! 6 * 6 7 &' % $ & & ' I 6 * $ 6 * (! " F g ≥ 2.
§ ! " #$%& ' ( ) F g > 1, ) 3g−3 ) * + ) , ) ) )- . / #00%& 1 ) )2 3 4 F 3 4 1 )2 2g− ) 4 4 3 + 3 T1, ..., Tg, 4 3 5 T1, ..., Tg ) 3g − 3 1 2 6 7 5 .7 #89& 1 0:0% 2 3 + ) g. ) 4 ) )3 4 ; 4 ) + - ≈, ∼ ∼ = ) 3 ) ) 5 4 F 4 g, g > 1, {ak , bk }gk=1 F, ) aj bk 4 + 4 O
4 g π1 (F, O) = a1 , ..., ag , b1, ..., bg : [aj , bj ] = 1, j=1
[a, b] = aba−1 b−1 4 a b F. ' )4 [F, {ak , bk }gk=1] 4 , ) ) Ng 4; 4 π1(F, O), 3 + b1, ..., bg, ) (F, π) + ) 7 + 4 F, π - , ) + [F, {ak , bk }gk=1]. 5 4 F 4 0AD , )
.7 )- 3
- f 4 D. 4 D ∈ 0AD ,
- D 3 4 , ) ) G ) 3 F. <) + G ) 3 ) F. = G = {f Tf −1 : T ∈ G} 3 D G ∼ = π1 (F, O)/Ng g. , g. z0 ∈ D π(z0) = O, π = πf −1. ( 4 γ = gj=1[aj , bj ] F. 5 4 γ = α1+ β1+ α1− β1−...αg+βg+αg− βg−
>7 γ ) z0. , ) ) F 4 ) F b1, ..., bg, ) H 4 D, βj+, βj−, j = 1, ..., g, βj− b−j , j = 1, ..., g, + ) z0 ∈ D, βj+ b+j , j = 1, ..., g. 4 H, F 4 π, 4
4 G. ? + ) G :
T1 , ..., Tg,
(1)
Tj (βj−) = βj+ , j = 1, ..., g. 5 G = T1 , ..., Tg g ) + T1, ..., Tg. . T G, T (H) ) 4 D ) {T (H) : T ∈ G} 4 ) D. . Tj 3 ext(βj−) int(βj+) Tj w − ξ1j z − ξ1j = λj , w − ξ2j z − ξ2j
λj 3 4 Tj , 0 < |λj | < 1, ξ1j , ξ2j + + 3 Tj , 7 ξ1j ∈ int(βj+) ξ2j ∈ int(βj−), ξ1j , ξ2j , j = 1, ..., g, j = 1, ..., g. ) @ D = Ω(G), Λ(G) = C\D AD 43 A 4 ) D, + )) 3 ξ ∈
Λ(G) W T1, ..., Tg , T1−1, ..., Tg−1 limk→∞ Wk (z) = ξ D. @ 4 Wk + W,
k ) )
4 ) 4 3 Λ(G). ) {ξ1j , ξ2j , j = 1, ..., g} ⊂ Λ(G). 4 D
7 ξ11 = 0, ξ21 = ∞, ξ12 = 1. B (D, π, T1, ..., Tg) . 4 H 3 ,.!@!CDE* , 5 3 + (D1, π1, K1, ..., Kg). ) + ) ϕ : D → D1 + : 1)π1ϕ = π; 2)ϕTk = Kk ϕ. <) 02 ϕ ) 3 ( Kk = ϕTk ϕ−1 7 Kk 3 4 λk , + 3 ϕ(ξ1k ) + 3 ϕ(ξ2k ). ϕ(0) = 0, ϕ(1) = 1, ϕ(∞) = ∞, ) ϕ = id. 5 3 + 4 H1. * 7 F H1 = L(H) L ∈ G. * H1 3 LTk L−1, k = 1, ..., g. , Tk = LTk L−1, k = 1, ..., g, )3 ;4 L = 1. D ) [F, {ak , bk }gk=1] + (D, π, T1, ..., Tg) 3
Φ : [F, {ak , bk }gk=1] → (T1, ..., Tg ).
5 3 (T1, ..., Tg) (ξ13, ..., ξ1g , ξ22, ..., ξ2,g , λ1, ..., λg ) ∈ C3g−3,
3 4
Φ ) C3g−3. 0, 1, ξ13, ..., ξ1g , ∞, ξ22, ..., ξ2,g )
5
5 7 ) ) 5 4 F0 F g > 1. 5 4 ) [F0, {ak , bk }gk=1] [F, {ak , bk }gk=1] O0 = ak ∩bk , O = ak ∩bk . + ) f : F0 → F f (O0) = O , f (ak ) = ak ,
f (bk ) = bk , k = 1, ..., g.
5 ) f + +4 ) ) F0 F ) + 7 C γ1, γ2 ) F, f1, f2 ) F F 02 γ1 γ2, 4 γ1 ) γ2; 92 f1 f2, 4 f1 ) f2 . ! 3 ) ) 1 4 2 6 ) G #:H& 6; #I$& @ O F. .3 ) ψ F 1 + - 2 O, ) Tψ ∈ Aut[π1(F, O)]. #:H& #H8& , 3 + ψ Tψ , 3 JJ - )) 3 ) ) Aut[π1(F, O)]. = ) f F ) - 4 f ) id. #I$& 5 4 [F, {ak , bk }gk=1] [F, {ak , bk }gk=1] bk ≈ bk 1 ) 4 O), k = 1, ..., g. + - ξ F ξ(bk ) = bk . 5 3 [F, {ak , bk }gk=1] 4 ak ∩ bk = O, ξ - F ξ(O) = O, ξ(bk ) ) bk 4 O, k = 1, ..., g. ξ ) id, 4 O. ,.!@!CDE* , 5 4 {ξt : 0 ≤ t ≤ 1} ) ) + id ξ. , ) ) σ ) 4 {ξt(O) : 0 ≤ t ≤ 1}. , ξ(ak ) ≈ σ−1ak σ, ξ(bk ) ≈ σ−1bk σ, k = 1, ..., g, ) 4 O. * 7 σ−1bk σ ≈ bk , k = 1, ..., g, ) π1(F, O) σ = 1 π1(F, O). ) ξ(ak ) ≈ ak , ξ(bk ) ≈ bk , ) 5 9 0 % ξ ) ) id. D ) 5 4 F ) 4 {p(t) : 0 ≤ t ≤ 1} K p(0) = O. + ) {ξt : 0 ≤ t ≤ 1}, ) + id ξ ξt(O) = p(t). 1! = -2 #$:& 5 4 [F, {ak , bk }gk=1] 4 ϕ ) F ϕ(bk ) ∼ bk , k = 1, ..., g. ϕ = id.
) 4 ϕ
- F
- ) b− #$8& @ 4 [F0, {ak , bk }gk=1] g > 1 O0 = ak ∩ bk . ( 3 Fg
[F, f ], F 4 g > 1 f ) ) F0 F. ak,f = f (ak ), bk,f = f (bk ) Of = f (O0). 5 9 0 9 Fg 3 g.
<4) 9 0 0 F [F0, {ak , bk }gk=1] (D, π, G = {Tk }gk=1, αk+, βk+, H, z0);
(2)
[F, f ] + + (Df , πf , Gf = {Tk,f }gk=1, αk,f , βk,f , Hf , zf ).
= ) f
: F0 → F
) f : D
0 ) = zf , πf f = f π, fTk = Tk,f f, f(z + ) = α+ , f(β +) = β + . f : H → Hf , f(α k k,f k k,f
(3) → Df
(4)
5 3 4 Tg , Ug Vg g, 4 4 4 ) g, )47 3 Fg . B 3 3 4 g. [F, f ] [F1, h] ) + ) ϕ : F → F1 ϕf h F0. ) ; ) F 4 ) 3 @ 4 4 F ) 4 b1, ..., bg. 5 4
) H 3 T1, ..., Tg, 3 5 3 4 Ug Fg + ; 3 + ) + ) 3 3 B [F, f ] [F, h] f (bk) = h(bk ), k = 1, ..., g. B - 3 Gf = T1,f , ..., Tg,f 3 )4 ) + ) )
4 Hf . ,
- )3 , ) ) Σg 3 ) ψ F0 ψ(O0) = O0, ψ(bk ) ∈ Ng , k = 1, ..., g. 5 4 + + − − + + − − ξ1,ψ η1,ψ ξ1,ψ η1,ψ ...ξg,ψ ηg,ψ ξg,ψ ηg,ψ
Πgk=1[ψ(ak ), ψ(bk )], + z0 , ψ ∈ Σg . * + − + T1,ψ , ..., Tg,ψ ) + G Tk,ψ (ηk,ψ ) = ηk,ψ , k = 1, ..., g. + . ηk,ψ , k = 1, ..., g, 4 Hψ D, 4 - π F0, 7 4 ψ(bk ), k = 1, ..., g.
, 3 Θg = {ψ ∈ Σg : Tk,ψ = Tk , k = 1, ..., g};
(5)
Bg = {ψ ∈ Σg : ψ(bk ) ≈ bk ∈ π1 (F0, O0 ), k = 1, ..., g}.
(6)
< 1 ≤ Bg ≤ Θg . ,.!@!CDE* , B 3 4 )4 Bg ≤ Θg Θg 4 )- 5 3 ψ ∈ Bg . < 4) 9 0 L 9 0 I 7 ) ξ F0 O0, ) ) id ξ(bk ) = bk,ψ , k = 1, ..., g. 5 4 {ξt : 0 ≤ t ≤1} ) ) ) + id ξ. , 4 γt = gk=1[ξt(ak ), ξt(bk )], 4 γt γt (D, π), + z0 . ) G, γt 3 T1, ...,Tg. <4) 4 7 γ1, 4 >7 gk=1[ak,ψ , bk,ψ ]. , Bg ≤ Θg .
3 Θg 5 4 ψ, ϕ ∈ Θg . 7 Hψ , 3 4 7 9 0 0 7 F 2 Φ[F0, ψ] = (T1, ..., Tg); 2 {T (Hψ ) : T ∈ G} 4 ) D; 2 πψ = π. <4) 4 ) {T (H)} {T (Hψ )}, ) ψ : D → D + k = Tk ψ; H2 ψ(H) 0) = z0 . ! F 02 πψ = ψπ; 92 ψT = Hψ ; %2 ψ(z 53 E = ψϕ. <) 3 4 ϕ. + ) F 2 πE = ψϕπ; 2 ETk = Tk E; 2 E(H) = Hψϕ; 2 E(z0) = z0. , ψϕ ∈ Θg . 5 ) ) ψ −1 ∈ Θg . * 4 3 Θg D ) + ∼ βk+ D bk,ψ ∼ bk F0 5 4 ψ ∈ Θg . βk,ψ k = 1, ..., g. ,.!@!CDE* , ) 4 9 0 : 4 ) 4 + 4 3 Λ(G), 7 βk,ψ βk+ ) + - 3 Λ(G). 5 βk,ψ ∼ βk+ D. 5 - π = πψ bk,ψ ∼ bk F0 k = 1, ..., g. D ) 5 !" #$%& 5 [F, f ] = [F, h]modE1, E3, E5, 4 + - ξ F ξ(Of ) = Oh h−1 ξf ∈ Θg , Bg , id @ [F, f ] = [F1, h]modE2, E4, E6, 4 + ) ϕ : F → F1 [F1, ϕf ] = [F1, h]modE1, E3, E5 , + F Tg = Fg /E6;
(6)
Ug = Fg /E4;
(7)
Vg = Fg /E2.
(8)
B 4 Ej , j = 2, 4, 6, ; Fg . C [F, f ] = [F1, h]modE2, E4, E6, Φ[F, f ] = Φ[F1, h]. ,.!@!CDE* , 4 Φ[F, f ] = Φ[F, h] [F, f ] = [F, h]modE1.
* 3 4 3 F ξ - F, Φ[F, ξf ] = Φ[F, f ]. [F, f ] ) 1H2 4 {ξt : 0 ≤ t ≤ 1} ) ) + id ξ. 53
γt = gk=1[ξt (ak,f ), ξt(bk,f )]. , γt γt (Df , πf ) γ0 zf . D 4 γt 3 Tk,f , k = 1, ..., g, γt (g + 1)− ) Rt, πf F, 7 4 ξt(bk,f ), k = 1, ..., g. , F 02 Hf = R0; 92 {T (Rt) : T ∈ Gf } 4 ) Df . 5 9 0 0 ) ; ) 4 4
3 )+ 4 ) 4 ; 3 4 ) + 34 h = f ψ ψ ∈ Θg . 7 ) 4 9 0 : , f, 1%2 ψ,
) E = fψ : D → Df . * F 2 πf E = f ψπ = hπ; 2 ETk = Tk,f E; 2 E(z0 ) = zf . <4) 9 0 0 ) F 02 Φ[F, f ] = Φ[F, h]; 92 E = h; H2 Hh = E(H); %2 πh = πf . D ) 53 Qg = Φ(Fg ) ⊂ C3g−3. 5 9 0 00 3 ΦT : Tg → Qg , ΦU : Ug → Qg ,ΦV : Vg → Qg , 7 3 3 JJ G3 Qg C3g−3. ,.!@!CDE* , )4 x0 ∈ Qg 4 Φ[F, f ] = x0. ( x = (Tk,x) ∈ C3g−3 − − W = W (x0). . βj,f Tj,x(βj,f ) 4 (g + 1)− ) 4 Hx, 4 3 Tk,x, k = 1, ..., g. 5 4 Gx 37 3 = ) {T (Hx) : T ∈ Gx } 4 ) Dx Gx g ) + Tk,x, k = 1, ..., g. , 3 Dx = Df , Gx = Gf , Tk,f = Tk,x k. 5 4 Fx 4 Dx/Gx πx : Dx → Fx - +4 πx (Hx) Fx. ) 4) 9 0 9 x ∈ Qg . ) 3 Qg C3g−3. D ) −1 −1 * + ) ) 7 Φ−1 T (x), ΦU (x), ΦV (x) x ∈ Qg . ; - ) 4 7 W = W (x0) ⊂ Qg . 5 3 4 {[Fx, fx] : x ∈ W } ) 0
0
0
+ Tg , Ug Vg . . ) 4 Φ[Fx, fx] = x. ) 4 9 0 09 4) 4 )
-
) ϕx,
+ + F 02 ϕx : Df → Dx; 92 ϕx(z) ≡ z − ) βj,f ; H2 ϕ x Tk,f = Tk,x ϕ x; %2 ϕ x (Hf ) = Hx; L2 ϕ x (z) ≡ z. * ) ϕx F 2 ϕx : F → Fx; 2 ϕx(Of ) = πx(zf ) = Ox; 2 ϕxπf = πxϕx. ( [Fx, ϕxf ] ) ) ak,x = ϕx(ak,f ), + + + + bk,x = ϕx(bk,f ), αk,x =ϕ x(αk,f ), βk,x =ϕ x(βk,f ). <) 9 0 0 + + F 02 Φ[Fx, ϕxf ] = x; 92 (Dx, πx, Gx, αk,x , βk,x ) ϕxf ; H2 ϕ xf. ) xf = ϕ {[Fx, ϕxf ] : x ∈ W }. 5 3 Φ[F1, h] = x. A3 ) Φ[Fx, ϕxf ] = x. < 4) 4 ) 4 ) Dh = Dx. πh (z1) = πh (z2), 4 πx(z1) = πx(z2), + ) μ : F1 → Fx μπh = πx. , F 02 [F1, h] = [Fx, μh]modEj , j = 2, 4, 6; 92 Dx = Dh = Dμh ; H2 πx = πμh ; %2 Hh = Hμh . ) + + Hh πx (βk,h ) = μπh (βk,h ) = μ(bk,h) = bk,μh . * 4 πx ) ) Hh Fx, 7 4 bk,μh. * 9 0 0 πx = πμh Hh = Hμh. 5 ) + 3 4 F1 = Fx πx = πh , μ = id. * 4 [Fx, h] [Fx, ϕxf ]. * +
4 Gx Hh Hx , )3 Hh Hx zh = zf . 9 0 $ ) 4 9 0 00 ) h ξh + - ξ Fx. 5 3 ) + 3 4 zh = zf Ox = Oh. −1ϕ , ψ = h−1ϕxf E = (h) xf. ? ψ ) F0 + O0, E ) D + z0. , ) F 2 ETk = Tk E; 2 πE = ψπ. 53 + ±1 ψ ∈ Θg . ) πE(αk+) = ψ(a±1 k ) πE(βk ) = ψ(bk ). ) π ) ) E(H) F0, 7 + + 4 ψ(bk ), ξk,ψ = E(αk+), ηk,ψ = E(βk+). * 4 Tk,ψ = Tk ψ ∈ Θg . 3 ) 4 0
[Fx, ϕxf ψ], ψ ∈ Θg . 4 ; F 3 ΘT , + ) ψ ∈ Θg , id; 3 ΘU , + ) ψ ∈ Θg , ) b, b ∈ Bg . B 4 ΘT ≤ ΘU ≤ Θg . 5 3 ψ1, ψ2 ∈ Θg x ∈ W. F 02 [Fx, ϕxf ψ1 ] = [Fx, ϕxf ψ2]modE1 ; 92 [Fx, ϕxf ψ1] = [Fx, ϕxf ψ2]modE4 , E6, 4 ψ1−1ψ2 ∈ ΘU , ΘT
,.!@!CDE* , B 3 )4 4 3 92 4 ) ) 5 3 [Fx, ϕxf ψ1] = [Fx, ϕxf ψ2]modE4. 5 4 q ) Fx [Fx, qϕxf ψ2] = [Fx, ϕxf ψ1]modE3. - ξ Fx ξϕx f ψ1 = qϕx f ψ2ψ ψ ∈ Bg . 5 9 0 0M 7 q(bk,x) ∼ ξ(bk,x) ∼ bk,x , k = 1, ..., g. = - 1 9 0 8 2 7 q = id. , ψ1 ≈ ψ2ψ ψ1−1ψ2 ∈ ΘU . B - ψ = id, ψ1−1ψ2 ∈ ΘT . D ) @ + )3 3 F Θg =
χα ΘU ;
(9)
α
ΘU =
kβ ΘT ;
(10)
χα kβ ΘT .
(11)
β
Θg =
α,β
)3 |Θg : ΘU |, |ΘU : ΘT |, |Θg : ΘT | . )3 id 3 3 Θg ) 7 ) ) )3 1:2 10M2 1002 7 <) + 3 x ∈ W + > F Φ−1 T (x) =
[Fx, ϕxf χα kβ ] ⊂ Tg ;
α,β
Φ−1 U (x) = Φ−1 V (x)
[Fx, ϕxf χα ] ⊂ Ug ; α
= [Fx, ϕxf ] ⊂ Vg .
, 3 ΦV : Vg → Qg )) , + 1092 Tg@
@ @ @ πT U @ πT V ? @ΦT Ug @ PP @ π Φ P U V U PP PP @ PP R ) q@ P V g Qg (⊂ ΦV
C3g−3).
π− 3 - , Tg , Ug , Vg . Tg 1 2N Ug Vg + ) g. 5 ) ) ))4 3 ΦV . . ) Ug 3 ) + * + 3 F 02 [F, f ] = [F, h]modE3; 92 + - ξ F ξ(bk,f ) = bk,h , k = 1, ..., g.
,.!@!CDE* , 5 9 0 L ) 02 92 C 92 b = h−1ξf. , ) F0 + O0 b(bk ) = bk , k = 1, ..., g. 5 b ∈ Bg . D ) Tg 7 dT , 3 dT ([F, f ], [F1, f1]) inf lnK(h), K(h) 4 1 2 h 7 ) 3 h : F → F1 3 f1f −1 F. , 3 ΦT : Tg → Qg ,.!@!CDE* , 5 4 [Fm, hm] → [F, h] dT Tg m → ∞. + ) )
qm : F → Fm - ξm Fm qm h = ξm hm lnK(qm) → 0 m → ∞. ' ) + ξm = id, [Fm, hm ] = [Fm, ξm hm ] Tg . , (Dh , πh, Gh , Hh , Tk,h) (Dm, πm , Gm , Hm , Tk,m) [F, h] [Fm, hm ] 5 qm ) ) qm : Dh → Dm F 02 πm qm = qm πh ; 92 qm (Hh) = Hm ; H2 qm Tk,h = Tk,m qm ; %2 lnK(qm) = lnK( qm). 54 Dh ∈ 0AD , Dm ∈ 0AD ,
) 3 + 3 q m qm 4 C lnK( qm) = lnK(qm) 0, 1, ∞. 5 #HN 9& ) lnK(qm) → 0 q m(z) → z m → ∞ ρ C. ) Tk,m → Tk,h m → ∞. D ) <) ) Tg 3 ΦT ) 4 .7 #89& G3 Qg 4 C3g−3. 5 4 ψ ∈ Θg W0 3 W. 3 {[Fx, ϕxf ψ] : x ∈ W0} Tg. ,.!@!CDE* , x ∈ W0. B 3 )4 + δ > 0 dT ([F1, h], [Fx, ϕxf ψ]) < δ 7 F 2 y = Φ[F1, h] ∈ W0; 2 [F1, h] = [Fy , ϕy f ψ]modE6. 5 2 ) ) 9 0 0I 5 3 dT ([F1, h], [Fx, ϕxf ψ]) < δ δ. + ) ) q : F1 → Fx K lnK(q) < δ q ≈ ϕxf ψh−1 . [F1, h] = [Fy , ϕy f ψ]modE6, 4 [F1, hψ−1] = [Fy , ϕy f ]modE6, 3 ) + 34 ψ = id. 5 qhξ = ϕxf, ξ - F0. 54 [F1, hξ] = [F1, h] Tg, ) + 3 34 ξ = id, qh = ϕxf. + + 7 1 2 (Dh, πh, Gh, Hh, Tk,h, αk,h , βk,h ) + + (Dx, πx, Gx, Hx, Tk,x, αk,x, βk,x) [F1, h] [Fx, ϕxf ] , 3 q ) ) q : Dh → Dx F 02 πxq = qπh ; 92 q(Hh) = Hx; H2 qTk,h = Tk,xq; %2 qh = ϕxf; L2 lnK( q) = lnK(q) < δ. <) y = Φ[F1, h] + (Dh, πh , Gh , Hh, Tk,h) = (Dy , πh , Gy , Hh , Tk,y ).
3 4; q ) 3 q C, 0, 1, ∞ lnK(q) =
lnK( q).
)4 ε > 0. * ) 4 9 0 0I δ ρ( q(z), z) < ε z ∈ C. B q(Hh) = Hx. * 7 ε δ, 3 4 - η F1 [F1, ηh] + + (Dy , πh , Gy , Hy , Tk,y , αk,y , βk,y ). +
) μ : F1 → Fy μπh = πy [Fy , μηh] + + (Dy , πy , Gy , Hy , Tk,y , αk,y , βk,y ). * 4 ak,μηh = ak,y , bk,μηh = bk,y k, ) μηh ≈ ϕy f. ) [F1, h] = [F1, ηh] = [Fy , ϕy f ] Tg . D ) , 3 ΦT : Tg → Qg + 3 ,.!@!CDE* , )47 )4 x0 ∈ Qg ) 9 0 0% , Pαβ = {[Fx, ϕxf χαkβ ] : x ∈ W } ⊂ Tg . <) 9 0 08 3 Pαβ ) + ) <4) 3 ϕx, 4 3 Pαβ 3
ΦT : Pαβ → W ) Φ−1 (W ) = αβ Pαβ , T ΦT + 3 D ) 4 3 7 #$%& 5 4 3 Qg ) C3g−3, Tg τg , - dT . + ) (Ug , τU ) (Vg, τV ) 1092 ) + 3 5 3 ΦV
) x ∈ Qg 3 ΦU ΦT 7 1 2 ) |Θg : ΘU | |Θg : ΘT | ,.!@!CDE* , )47 τU τV ; 4 3 πT U πT V 1092 ) + 3 ) ) 9 0 0% 9 0 0I 9 0 08 9 0 0: C 4 ) <) 9 0 0% ) 3 ΦU ΦT |Θg : ΘU | |Θg : ΘT | ,4 )4 |Θg : ΘU | |Θg : ΘT | 7 4 3 G 9 0 % * + ) ψn ∈ Σg +
) θn ∈ Aut[π1(F0, O0)] ) F θn(ak ) cna1 c−n k = 1 ak k = 1; θn(bk ) cn b1c−n k = 1 bk k = 1, c = [a1, b1]. = ) ψn 4 c = [a1, b1]. B 4 ψn ∈ Θg . 5 3 4 ψm−1ψn ≈ b ∈ Bg 3 + 4 σ ∈ π1 (F0, O0) σ −1ψm (bk )σ ≈ ψn (bk ) π1 (F0, O0) k. 5 σ m = n. * 4 |Θg : ΘU | = ∞. ) #$ Tg ) ) 9 0 9M 4 + (Ug , τU ), (Vg, τV ) Qg g > 1.
#$ * )4 (Vg , τV ) Qg 3 )4 + 4 9 9 #$ D 4 ΘT 4 Θg ΘU = ΘT Bg . * 4 ΘU /ΘT ∼ = Bg /(Bg ∩ ΘT ). 5 9 0 I ΘU /ΘT ) ) ) ) Bg . @ 9 0 % ΘU /ΘT ∼ = {θ ∈ Aut[π1(F0, O0 )] : θ(bk ) = bk , k = 1, ..., g.}
6 ) ) ) 7 3 πT U |ΘU : ΘT | = ∞. 4 7 ; )47 [F, f ] ∈ Fg . 5 4 ζ1, ..., ζg )
- F :
f (bj
)−1
ζk = δjk ,
f (aj )
ζk = pjk .
(13)
G- P = (pjk ) 3 4 7 4 5 4 (Df , πf , Gf ) [F, f ]. @; ζk = ϕk (z)dz, ϕk (z) Df ϕk (T z)T (z) = ϕk (z) T ∈ Gf . <) - P = (pjk ) ) 3 Tg . * + 3 ) ) 4 D !4 Tg . #9N $%& ) (Tg , τT ) 4
- P = (pjk ) 6 Qg ⊂ C3g−3 3 ΦT . ,.!@!CDE* , * 3 4 * + 3 Tg , 3 P = (pjk ) )) 5 7 ) 4 [F, f ] ∈ Tg , 4 AutF ) 4 ? 3 ! G #9%N 9L& (
{[Fx, ϕxf ] : x ∈ W }. , ) 4 x ∈ W. , 3 4 4 + ) 3 Fx ) x W. 3 AutFx 4 x. 4 ) ; ) 4 4 5 4 3 (z1, ..., zn) → (w1, ..., wm), m ≥ n, )) {|z1| < 1} × ... × {|zn | < 1}. ) n × n - ? 3 3 ) 4 - m n, 4) )
- #%9N 0MH& ) 4 4 σ1 σ2 (Tg, τT ), 4 t s 5 9 0 99 9 0 9H 3g − 3 p = (p1, ..., p3g−3),
-4 Ω=
∂(p1, ..., p3g−3) dt1 ∧ ... ∧ dt3g−3 ∂(t1 , ..., t3g−3)
3 Tg . - P = (pjk ), t ↔ p ↔ s ) ) divΩ 5 #0MH K 0:0& divΩ. * 4 ,4 )4 3 5 3
Tg Qg ⊂ C3g−3 3 ΦT . 5 4 [F, f ], Tg , ΦT [F, f ] = x0. ΦT 4 ) N W [F, f ] x0 3 ΦT ) . 4; [Fx, fx] = Φ−1 T (x) ∈ N + + (Dx, πx, Gx, Hx, Tk,x, αk,x, βk,x), x ∈ W. ) 4 - P = (pjk ), 4 N. W 3 4) + 3 (ξ11, ξ21, ξ12) 4 )4 3 4 ∞ ∈ Hx x ∈ W. * +4 - + ) 4
|T (z)|2
T ∈Gx
x ∈ W, z ∈ Hx. 4; - m 5
(T (z))2/(T z − w)m
Q(z; w, x) =
T ∈Gx
x ∈ W, z ∈ Dx, w ∈ Hx ∩ C. @ 4 Q(z; w, x) z m w Gx. . T ∈ Gx
Q(T z; w, x)T (z)2 = Q(z; w, x),
Q(z; w, x)dz2
- Fx. C ) 4 4g − 4. @ ) s1 t1 ) Hx ∩ C. 6 3 4 ) Q(z; s1, x0)dz2 Q(z; t1, x0)dz2 4 ξ = Q(z; s1, x0)/Q(z; t1, x0)
- Fx 4g − 4 + 2m. 3 3 4 s2, t2 ) Hx ∩ C c ∈ C F 02 ξ −1(c) ) 4g − 4 + 2m ) z1 , ..., z4g−4+2m Gx N 92
- η = Q(z; s2, x0)/Q(z; t2, x0) η(zk ) ) N H2 ) ξ η W,
- 0
0
0
0
ξ(z; x) = Q(z; s1, x)/Q(z; t1, x),
η(z; x) = Q(z; s2 , x)/Q(z; t2, x),
x ∈ W, z ∈ Dx. / - ξ(z; x) η(z; x) (z, x) 02 H2 5 4 ; ) ) 4 #%9& 7 x, 7 m n P (ξ; η; x) = j=1 k=1 cjk (x)ξ j ηk
- x 3 x ∈ W : 2 cmn(x) = 1; 2 P (ξ; η; x) N 2
- ξ = ξ(z; x), η = η(z; x) ) {P (ξ; η; x) = 0} 4 [Fx, fx]. 6 ) Fx P (ξ; η; x) = 0. B - )
- ϕk (z; x)dz [Fx, fx] + ) x. < 4 αk,x 7 )4 P = (pjk ) N. ) #$ 3 x ∈ Qg , Gx = T1,x, ..., Tg,x g. * +
- ϕk (z; x)dz, k = 1, ..., g, [Fx, fx] ) x. 6 7 ) 4 ) D ' #L9& ) 4 + 5 .7 #89& 5 7 4 ) 4 ' , )4 - P = (pjk ) + Tg 3 Tg . @ ) ) Tg , 3 4 - P = (pjk ) 7 Tg . + 4 - P = (pjk ) ) , #0MH K H$:H:9& - -B 4 + - 5 3 0 < r < 2. B ) ) 4 ' [59, c.55 − 57] #00% K 9H09HI& 3 4 [F, f ] ∈ Fg N 1 2
|T (z)|r
T ∈Gx
z ∈ Hx, x ∈ W. * 3 = ' #%8 K H%:HL0& , ) 5 G #0MM O $L& 3 g > 2 + T ∈G |T (z)| ≡ ∞. 5 4 r = 1. / x ∈ W 5
x
ηa(z) =
T (z)/(T (z) − a)
T ∈Gx
z ∈ Dx, a ∈ Hx ∩ C.
- ηa(z)dz ±1 z = a(modGx ), z = ∞(modGx),
- Fx. 5 7 F
+ βk,x
ηa(z)dz = 0, k = 1, ..., g.
5 ηa(z)dz
- 4 [Fx, fx]. , ) ) uk (z; x) duk = ϕk (z; x)dz. <) ; 3
2πi[uk (a, x) − uk (∞, x)] =
b
Tk,x (b)
ηa (z)dz,
− b ∈ βk,x 4 3 4 -4 ) ,
1 a − T Tk,x(b) ), uk (a, x) − uk (∞, x) = log( 2πi a − T (b) T ∈Gx
+ ) * 4 ϕk (a; x) =
1 2πi
(
T ∈Gx
1 1 − ). a − T Tk,x(b) a − T (b)
@ b ) 3 4 7 a. <) ϕk (a; x) ) x. , - P = (pjk ) N ⊂ Tg. ' 3 pjk =
1 (T Tk,x(b) − Tj,x (a))(T (b) − a) , log 2πi (T Tk,x(b) − a)(T (b) − Tj,x (a)) T ∈Gx
+ ) 4 + 4 4 ; ) Qg 4 C3g−3, (Tg , ΦT ), (Ug , ΦU ) (Vg, ΦV ) Qg #%9& 5 4 dT , dU , dV
- 3 3 dQ(x) x - ∂Qg. 5 - ΦT , ΦU , ΦV + 3 5 dT = dQΦT , dU = dQΦU dV = dQΦV . 4 ' 6 #LH& 9 0 90 4 (Tg, ΦT ) #$%& ( (Tg, ΦT ), (Ug , ΦU ), (Vg , ΦV ) 4 Qg ,.!@!CDE* , . #%9 K 9HM 9$:& 3 Tg 4 4 5 #%9 K 9II& logdT (ξ)−1
- Tg. dT = dQΦT , logdQ(x)−1 3
- Qg . ) Qg , Ug Vg )
§ (g, s, m)
m m T (z) = (az + b)/(cz + d), a, b, c, d ∈ C, ad − bc = 1, C MC .
G MC EST − (g, s, m)
! T1, ..., Tg , U1, V1, ..., Us, Vs , W1, ..., Wm
!
γ1 , γ1, ..., γg , γg , Λ1, ..., Λs, Δ1, Δ1, ..., Δm, Δm,
Λi 1, ..., s,
"
Ωi , Σi , Ωi , Σi i =
#$ ! % Δt, Δt ξt , t = 1, ..., m, & (2g + s + m)− D, " Tj (D) ∩ D = Ui(D) ∩ D = Vi(D) ∩ D = Wt(D) ∩ D = ∅ Tj (γj ) =
γj , Ui(Ωi) = Ωi, Vi (Σi) = Σi, Wt(Δt) = Δt , j = 1, ..., g; i = 1, ..., s; t = 1, ..., m; '$ i Ui, Vi UiVi (z) = Vi Ui(z) z ∈ C {Ui, Vi}, " Ui, Vi ,
i = 1, ..., s; ($ Wt ξt, t = 1, ..., m.
) *
G = T1 , ..., Tg , U1, V1 , ..., Us, Vs, W1, ..., Wm.
+ Ω(G) G ! 0AD . , Ω(G)/G g + s 2m +
g + s 2m - EST − G (g, s, m) G = {T1, ..., Tg , U1, V1, ..., Us, Vs , W1, ..., Wm : Ui Vi Ui−1Vi−1 = 1, i = 1, ..., s},
G g + m & s ' + EST − (g, s, m) "
! ! " . T1, ..., Tg, U1, V1, ..., Us, Vs W1, ..., Wm, " (Ui, Vi) (Vi, Ui) i = 1, ..., s. ! / EST − G = T1 , ..., Wm
G = T1, ..., Wm
" * ! B ∈ MC
Tj = BTj B −1, j = 1, ..., g, Ui = BUiB −1, Vi = BVi B −1, i = 1, ..., s,
Wt = BWt B −1, t = 1, ..., m.
+ [G] * ! G. ) " Q(g, s, m), *
[Gn] → [G], < T1n, ..., Wmn >∈ [Gn ] < T1, ..., Wm >∈ [G] Tjn (z) → Tj (z), j = 1, ..., g, Uin(z) → Ui(z), Vin(z) → Vi (z), i = 1, ..., s, Wtn (z) → Wt(z), t = 1, ..., m, C n → ∞. 0 Q(g, s, m) " 1
(g, s, m). ) 2 g, s, m & &
2g + s + m ≥ 3. !
G =< T1, ..., Wm > ∞, G . (p1, q1, r1eiθ1 , ..., pg , qg , rg eiθg , p11, q11, p12, ..., ps1, qs1, ps2, p1, q1, ..., pm, qm), (1) (ξ11, ξ12, k1, ..., ξg1, ξg2, kg , p11, q11, p12, ..., ps1, qs1, ps2, p1, q1, ..., pm, qm ),
& 3 rj Tj ; θj ! rj2eiθ = (1/c2j ); kj Tj ; kj , |kj | = 1, k + (1/k) = (aj + dj )2 − 2
|k| < 1; (ξj1, ξj2) Tj , |ξj1 − qj | < rj , |ξj2 − pj | < rj (Tj (z) = (aj z +bj )/(cj z +dj ), aj dj − bj cj = pj , qj , pi1, qi1, pi2, pt , qt −1 Tj , Tj , Ui, Ui−1, Vi , Wt, Wt−1
j
1, cj = 0), j = 1, ..., g; i = 1, ..., s; t = 1, ..., m.
4#$ ! - Tj (z) = [qj z − (pj qj + rj2 eiθj )]/(z − pj ), Ui(z) = {qi1z − [(pi1 + qi1)2/4]}/(z − pi1), Vi (z) = {(pi1 + qi1 − pi2)z − [(pi1 + qi1)2/4]}/(z − pi2), Wt (z) = { qt z − [( pt + qt )2/4]}/(z − pt ).
* ! T1, ..., Tg, U1, U2, ..., Us−1, Us, W1, ..., Wm ! ) * ! 4
! ξ3, ξ2, ξ1) ! ξ3, ξ2, ξ1 1, 0, ∞ 5 (p1, q1, p2) (p, q) & U = (a1b1|c1d1), V = (a2b2|c2d2) & (ab|cd), ai = 2qi/(qi − pi), bi = (pi + qi)2/(2pi − 2qi), ci = 2/(qi − pi), di = a = 2 q /( q − p), b = ( p + q )2/(2 p −2 q ), c= 2pi/(pi −qi ), i = 1, 2, q2 = p1 +q1 −p2 ; 2/( q − p), d = 2 p/( p − q); 4 $
T T (∞) = ∞, T (0) = 0, k T (k 2 $3 (U, V )
W, ∞ . & (1b1|01), (1b2|01) & (1b|01)
0 ! " ! (U, V ) ! W, ∞ (b1, b2), bi = 0, i = 1, 2, b, b = 0; 4 ∞) (a1, c1 , c2)(a2 = [c2 (a1 − 1)/c1] + 1, U V = V U, di = 2 − ai , bi = (ai di − 1)/ci, ci = 0, i = 1, 2) ( a, c), d = 2 − a, b = ( ad − 1)/ c, c = 0. 6 A− (g, s, m) : (p1, q1, r1eiθ1 ), ..., (pg−1, qg−1, rg−1eiθg−1 ), (pg , qg , rg eiθg ), (p11, q11, p12), ..., (ps−1,1, qs−1,1, ps−1,2), (ps1, qs1, ps2), ( p1, q1), ..., ( pm−1, qm−1), ( pm, qm),
B− (g, s, m) : (ξ11, ξ12, k1), ..., (ξg−1,1, ξg−1,2, kg−1), (ξg1, ξg2, kg ), (a11, c11, c12), ..., (as−1,1, cs−1,1, cs−1,2), (as1, cs1, cs2 ), ( a1 , c1), ..., ( am−1, cm−1), ( am, cm ),
m ≥ 1 Wm (pm, qm)((am, cm)) bm 3 m = 0, s ≥ 1 (Us, Vs) (ps1, qs1, ps2)((as1, cs1, cs2)) (bs1, bs2); m = 0, s = 0, g ≥ 2 Tg (pg , qg , rg eiθ )((ξg1, ξg2, kg )) kg ; " 4 $ 7 g ≥ 1, s = 1, m = 1 Q(g, 1, 1) . g
rg2eiθg = (pg − 1)(1 − qg ), q11 = −p11; ξg2 = 1, a11 = 1.
0
(p1, q1, r1eiθ1 , ..., pg−1, qg−1, rg−1eiθg−1 , pg , qg , p11, p12, b1) (ξ11, ξ12, k1, ..., ξg−1,1, ξg−1,2, kg−1, ξg1, kg , c11, c12, b1)
* 3g + 2 ) Q(g, s, m) 3(g + s) + 2m − 3 7" A− B− (g, s, m). 8 A− B− (g, s, m) Q(g, s, m) Q∗A (g, s, m) Q∗B (g, s, m) W0 ! Δ0, Δ0 (p0, q0) ∈ C2, (U0, V0) ! Ω0, Ω0, Σ0, Σ0 (p01, q01, p02) ∈ C3 . 0 ! ε0 > 0 ε1 > 0
(p, q)((p1, q1, p2)) U ((p0, q0), ε0) C2(U ((p01, q01, p02), ε1) C3) W (p, q) 4 (U, V ) (p1, q1, p2)) ! ! Δ, Δ (Ω, Ω , Σ, Σ ).
/+9:;:0<=>60)+ / W0 ! - U (ξ0, r), ξ0 = (˜p0 + q˜0)/2, Δ0, Δ0 IW , IW + IW (IW ) & U (ξ0, r) a, b(c, d) " W0(a) = c, W0(b) = d, l a c. / (˜p, q˜) ∈ U ((˜p0, q˜0), ε0), ε0
W (a) c 2 l/2, a ξ = (˜p + q˜)/2 b ξ [a, ξ] [b, ξ]. 0 arg W (ξ) = 0, 2 ε0, Δ = Δ0\(Δ0 ∩ U (ξ0, r)) ∪ [a, ξ] ∪ [b, ξ] Δ = W (Δ) ξ, ! W. / (U0, V0) Ω0 Σ0, Ω0 Σ0, Σ0 Ω0 Σ0 Ω0 a, b, c, d r a, b, c, d, ! ε1 > 0 (p1, q1, p2) ∈ U ((p01, q01, p02), ε1) U (a) ∈ U (c, ε2), V (a) ∈ U (b, ε2), V U (a) = U V (a) ∈ U (d, ε2)(ε2 < r/2). + b c Ω0 ∩ {z ∈ C : |z − b| = ε2} Σ0 ∩ {z ∈ C : |z − c| = ε2}, γ1 , γ2 ! c U (a) U (c, ε2), b V (a) U (b, ε2) Ω = Ω˜ 0 ∪γ2, Ω = U (Ω), Σ = Σ˜ 0 ∪γ1, Σ = V (Σ), Ω˜ 0(Σ˜ 0) Ω0(Σ0) ! U (b, ε2)(U (c, ε2)) 0 2 ε1 γ1 U (c, ε2), γ2 U (b, ε2), Σ Ω U V (a) = V U (a). 0 Ω, Ω , Σ, Σ ! U, V. = 1
Q(g, s, m) -
Q∗A(g, s, m)(Q∗B (g, s, m)) C3(g+s)+2m−3. /+9:;:0<=>60)+ ? * A(B)− ϕ, ! * [G] A(B)− [G], 7 [Gn] → [G] A(B)− [Gn] 4 C3(g+s)+2m−3) A(B)− [G]. 6 ϕ - Q(g, s, m) Q∗A(g, s, m)(Q∗B (g, s, m))). Q∗A = Q∗A(g, s, m) C3(g+s)+2m−3 4 Q∗B (g, s, m) $ τ0 ∈ Q∗A Φτ - !
0
0
0
0
0
γ1 , γ1, ..., γg, γg , Λ1 , ..., Λs, Δ1, Δ1, ..., Δm, Δm
! T1 (τ0, ·), ..., Tg(τ0, ·), U1(τ0, ·), V1(τ0, ·), ..., Us(τ0, ·), Vs(τ0, ·), W1 (τ0, ·), ..., Wm(τ0, ·)
Gτ A− τ0. 6! ε0 = ε0(τ0) > 0 τ ∈ U (τ0, ε0) 4 C3(g+s)+2m−3) 0
γ1 , T1(τ, γ1), ..., γg , Tg (τ, γg ), Λ1τ , ..., Λsτ , Δ1τ , W1(τ, Δ1τ ), ..., Δmτ , Wm(τ, Δmτ ),
Λiτ (Δtτ ) Λi(Δt), i = 1, ..., s(t = 1, ..., m), ''# !
! T1 (τ, ·), ..., Tg(τ, ·), U1(τ, ·), V1(τ, ·), ..., Us(τ, ·), Vs(τ, ·), W1(τ, ·), ..., Wm(τ, ·)
EST − Gτ (g, s, m). τ0, τ1 ∈ Q∗A(τ0 = τ1), # @ABC ! wμ −- - C Gτ = wμGτ (wμ)−1. τ (s), 0 ≤ s ≤ 1, Q∗A , τ (s) A− [wsμGτ (wsμ)−1] Gτ (0) = Gτ , Gτ (1) = Gτ . 6 Q∗A 0
9 A(B)− (g, s, m) Q(g, s, m) 3(g + s) + 2m − 3. T Q∗A(T Q∗B ) Q∗A(g, s, m)(Q∗B (g, s, m)) (τ, z) ∈ C3(g+s)+2m−2, τ ∈ Q∗A(g, s, m)(Q∗B (g, s, m)), z ∈ 1
0
Ωτ = Ω(Gτ ).
C3(g+s)+2m−2.
0
0
1
T Q∗A(T Q∗B )
/ ''' τ ∈ Q∗A (g, s, m)(Q∗B (g, s, m)) Ωτ / - 1
(g, s, m). 6 EST − G =< T1 , ..., Tg, U1, V1, ..., Us, Vs , W1, ..., Wm >
(g, s, m) " ! * !
! = * G
- G, ! * G. +! - !
@D# E #'AC < G =< T1 , ..., Tg, U1, V1, ..., Us, Vs , W1, ..., Wm >
EST − (g, s, m), - G,
* & ! * - . #$ Tj → Ti, Ti → Tj , i = j; '$ Ti → Ti−1; ($ Ti → Tk Ti, i = k; B$ Wj → Wi, Wi → Wj , i = j; A$ Wi → Wi−1; D$ Wi → Wk Wi, i = k; F$ Uj → Vj , Vj → Uj ; G$ Uj → Uj−1; H$ Ui → UiVi; #I$ Uj → TiUj Ti−1, Vj → TiVj Ti−1; ##$ Uj → WiUj Wi−1, Vj → WiVj Wi−1; #'$ Uj → ViUj Vi−1, Vj → ViVj Vi−1, i = j; #($ Ti → Vj Ti; #B$ Ti → Wj Ti; #A$ Uj → Uk , Uk → Uj , Vj → Vk , Vk → Vj , j = k. ; - 4 $ ! 0 @D#C - G
!
0 & - Q∗ = Q∗(g, s, m), Q∗(g, s, m) Q∗A (g, s, m), Q∗B (g, s, m). +
AutQ∗(g, s, m). AutQ∗(g, s, m) Q∗(g, s, m).
/+9:;:0<=>60)+ AutQ∗(g, s, m)(= AutQ∗)
Q∗, τ ∈ Q∗ !
∗ ∗ 3(g+s)+2m−3
- {fn∗}∞ ), n=1 AutQ fn (τ ) → τ 4 C ∗ ∞ ∗ " ! {fn }l=1, fn (τ ) = τ l, fn∗ (τ ) = τ l. + G =< T1, ..., Wm > τ : - fn∗ &
- fn G, < T1, ..., Wm > < fn(T1), ..., fn(Wm) > 4 $
* + " < fn(T1), ..., fn(Wm) > ξ3n, ξ2n, ξ1n Bn ! ξ3n, ξ2n, ξ1n 1, 0, ∞ 0 fn∗(τ ) < Bnfn(T1)Bn−1, ..., Bnfn(Wm)Bn−1 > . 0 4 Q∗) * AutQ∗ 2 # ! τ ∈ Q∗ ε > 0 U (τ, ε) ⊂ Q∗ * 4 $ AutQ∗. / * τ < Bn fn(T1)Bn−1, ..., Bnfn (Wm)Bn−1 >, !
< T1, ..., Wm >, " l
l
l
fn∗(τ ) = τ, n ∈ N.
(2)
∞ {fn (T1)}∞ n=1, ..., {fn(Wm )}n=1 ∞ {Ln,1}∞ n=1 , ..., {Ln,g+2s+m}n=1, T1 , ..., Wm L1 , ..., Lg+2s+m.
#$ J! k = k0(k = 1, .., g + 2s + m) ∞ {Ln,k }∞ n=1 " {Ln ,k }l=1, Ln ,k = Ln ,k l ? Bn Ln ,k Bn−1 → Lk l → ∞ Bn = Bn l, {Bn }∞ n=1 4 $ * Bn → B ∈ MC l → ∞. 9
! k = k1(k1 = k0) {Ln,k }∞ n=1
4 $ * 4'$ + Ln ,k → Bn−1Lk Bn , Ln ,k → B −1Lk B, ! * G, ! * MC . 6 G
τ ∈/ Q∗. '$ 9 g + 2s + m 4 $ * $ < ! {fn∗ }∞ l=1 (ξ3,n , ξ2,n , ξ1,n ) (ξ3,n , ξ2,n , ξ1,n ), Bn = Bn 0
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l. 0 Ln ,1 → Bn−1L1Bn τ ∈/ Q∗. $ {(ξ3n, ξ2n, ξ1n)}∞ n=1 4 $ ! 3 {(ξ3,n , ξ2,n , ξ1,n )}∞ ! C (ξ30, ξ20, ξ10). 0 {Bn }∞ l=1 l=1 B ∈ MC , B ξ30, ξ20, ξ10 1, 0, ∞ Bn 7 {Ln ,1}∞ l=1 G Ln ,1 + Bn Ln ,1 Bn−1 Bn Ln ,1Bn−1 → L1 ∈ MC . 0 Bn → B ∈ MC Ln ,1 → B −1 L1B ∈ MC . 6 τ ∈ / Q∗. 0 ; EST − G MC " " EST −
! T1, ..., Tg , ..., U1, V1 , ..., Us, Vs , ..., W1, ..., Wm, ..., " T1, ..., Tg, U1, V1, ..., Us, Vs, W1, ..., Wm EST − (g, s, m), G 9 @(DC " " EST − (g, s, m). ) * l
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G =< T1, ..., U1, V1, ..., W1, ... > .
" " EST − Gn =< T1n, ..., U1n, V1n , ..., W1n, ... > 4 !$ G, " T1, ..., U1, V1, ..., W1, ..., #$ T1, ..., U1, V1, ..., W1, ... MC , '$ Tjn(z) → Tj (z), Uin(z) → Ui(z), Vin(z) → Vi(z), Wtn(z) → Wt(z) i, j, t n → ∞ C.
/ (1, 0, 0), (0, 1, 0) (0, 0, 1) & * " EST − (g, s, m) 2 3(g + s) + 2m − 3 > 0. ) @DIC ! (g, 0, 0)(g > 1). Gn =< T1n, ..., U1n, V1n , ..., W1n, ... >
" " EST − G, " T1, ..., U1, V1, ..., W1, ..., n → ∞. 0 G
K ! Q(g, s, m), @#'IC 0 L 2 / 2 ! &- EST − = EST − (0, 0, m) 1
m. /+9:;:0<=>60)+ W ! Σ Σ , (p, q, r) ξ
W, ξ = Σ∩Σ . M ! U (ξ, ε) Σ Σ IW IW h IW ∩ {z ∈ C : |z − ξ| = ε} W, Tn (p, q, r−(1/n)) n > n(h)
Σ\{Σ ∩ IW } W Tn 2 h/2, IT 4 Tn$ Σ U (ξ, ε) 0 Σ Tn(Σ) ! Tn Tn(z) → W (z) n → ∞ C. / (0, 0, m) 2 ! W1, ..., Wm n ! Σ1, T1n(Σ1), ..., Σm, Tmn(Σm) 3 Tin Wi, i = 1, ..., m. = /+9:;:0<=>60)+ 4 ''F$ 6 {Gn }∞ n=1 EST − (g, s, m). G !
* Sl = Sl (T1, ..., Wm) = 1 G, ! 1 l → ∞. D @DIC Ψn : Gn → G
n
Ψn (Tjn) = Tj , Ψn (Uin) = Ui , Ψn(Vin) = Vi , Ψn (Wtn) = Wt, j = 1, ..., g; i = 1, ..., s; t = 1, ..., m, - Gn G. ? ! Gn ! G
!
{Sln = Sln(T1n, ..., Wmn)}∞ n=1,l=1
$ Sln ∈ Gn n, $ Sln ! Gn , ! Sl ! G, $ Sln = 1 l, n 4 D @DIC$ $Sln(z) → Sl (z) C n → ∞. ) N N
! 1 l → ∞. / l " Pn
! Gn G˜ n = {Sl,n , Pn } ! Sl,n Pn . ) {Pn }∞ l=1 " ! Gn , ! ! G. * {T1,n }∞ l=1 4 $ ! ! ! T1 G. ' @D#C l G˜ n EST − G˜ n
! An , Bn . 0 G˜ n =< An , Bn > " . (2, 0, 0), (1, 0, 1), (0, 0, 2). ''G ˜ n =< A , B > ˜ n 1
G G n n iθ (p, q, re ) An An , Bn Bn Sl,n Sl,n , T1,n T1,n 2 1/nl , Sl,n , T1,n ! An , Bn , ! Sl,n , T1,n ! An , Bn . 0 l Sl,n , T1,n ! G˜ n ,
' @DIC G˜ n =< Sl,n , T1,n > . 6
1
G˜ n =< Sl,n , T1,n > T1,n (z) → T1(z) Sl,n (z) → 1 l → ∞, * B @DIC / " " EST − #$ {Pn }∞ l=1 !
! '$ l G˜ n = {Sl,n , Pn } EST − / '$ " " Gn , " Pn ! Gn , Sl,n . 0 G˜ n "
" EST − ' @D#C " EST − / #$ {Sl,n }∞ l=1 . $ $ * ! Tj,n , j = 1, 2, ..., Wk,n , k = 1, 2, ..., Ui,n , Vi,n , i = 1, 2, ... ) $ * {Pn }∞ l=1 ! {Sl,nl }∞ l=1,
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) $ ! Tj,n , Wk,n , Ui,n , Vi,n {Sl,n }∞ l=1. * " ! ! ! * a (z) → ! T1,n a b b c d c d T1 (z) = 1, ( W1,n (z) → W1 (z) = 1, U1,n V1,n (z) → U1 V1 (z) = 1) & a, b, c, d(a · b · c · d = 0), Sl,n (z) → 1 l → ∞ 6 $ 0 l
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§
! ! " #$%&' ( " R3 )
! *!+ #,$' -" ( ( ) ! ! . #$$,' / / ( / /
0 ! *! 1 #$' ( ! 2! #34' 5
! 6!7! 0 #48' 5
)
!
6 ! " ! . ( ( )
! 9 F ) ( R3 : C ∞) " ( C, ( )
! ( F. ; F, ! 5 ( ( ) F R3 F ! 9 ( ( ( ( ! < " (
(! 2 ( ! 1! 7 #=$' ! " #$%>'
( R3. ? - @! 2 " $>>A ! 2 A F ( ! < 0)
AutF F h, h ≥ 2, ! < ( A F " )
< A >, O(A) A. #$%&'! B F, A, ( e : F → R3 A, ( eAe−1 e(F ) R3. 9 ξ ( A, ( (U, ϕ) ϕ(ξ) = 0 ϕAϕ−1(z) = zexp(iα) z ∈ ϕ(U ),
α = α(A, ξ) −π < α ≤ π, ! ! : ! " #$%&'C! B F h, h ≥ 2, A, F, ( D $! E( Φ(A) ( A ( Φ(< A >) ( )
< A >, O(A)−1
Φ(< A >) = ∪m=1
Φ(Am);
A! F N (A) = 2n(A) ( A " G 8! |α(A, ξ1)| = |α(A, ξ2)|, ξ1, ξ2 ∈ Φ(A); 4! * α(A, ξ) = π :! ! A2 = id), ξ∈Φ(A) α(A, ξ) = 0. ? ( ! 6 Φ(< A >) (! 5( A ) 7) " ( !
9 F A R3, Φ(A) F ! < N (A) " F
! H "! < A F h, F = F/ < A > h . 9 Φ(A) = Φ(< A >), 7) h − 1 = O(A)(h − 1) + (O(A) − 1)n(A).
9 Φ(< A >) = ∅, O(A) h − 1 h = O(A)(h − 1) + 1. 9 A " ( h = h O(A) + (O(A) − 1)(n(A) − 1).
< F (
< A > F ( ! ; - ) F
) π : F → F/ < A > ( O(A) F . ; ! 6 A!8!A A!8!8! " #$%&'! < ( A ( F, D C γ F , π−1(γ) O(A) F O(A) G C ) π ( ( F \γ ( O(A) G C B ∈< A > ( ( ! <" ! ! 1 #4,'
! < O(A) pq, p 1 < Aq > πq : F → F/ < Aq > . ; γ F/ < Aq > p πq−1 (γ) F p
R1, ..., Rp. < γ0 I" γ πq . 5 Am(γ0)
An(γ0) π(γ0), π : F → F " Am(γ0) An(γ0) m = n O(A). ; π−1(π(γ0)) = ∪An(γ0) pq = O(A) q − 1 ( ( Rj , j = 1, ..., p. < " ( q − 1 γjk Rj , γjk Aq (γjk ) F Aq (γjk ) ∩ Rj = ∅. . π−1(π(γ0)) F O(A) ( γ0. * ! " #$%&'! < ( A N (A) = 2n(A) = 0, D C ) ξ1, ..., ξ2n π : F → F ( n = n(A) γj , ) ξ2j−1 ξ2j , j = 1, ..., n, F O(A) G C ) π ( ( F \ ∪ γj ( n " ( ( ( G C ( B ∈< A > ( ! <" "! A!8!A F ( O(A) h −1 = (h−1)/O(A) ! 1 B ) ( ( ( ! 9 Φ(A) A!8!8 " F O(A) n(A) h B ) ( ! J ( ( #$%&'! 5 ! 7
G, F : C ( R3, : C ( e F F1 ege−1 R3 g ∈ G. 5 ( e : C (
G. 9 G (
)
3 R ,
5
R3,
! < )
A, B : A2 = B n = 1 , n ∈ N, D
! . (
(
! . 7)
! 6 @!*! * #&%' )
( " ( R3. B 5 #34'! F ( φ F φ2 = id. + φ F. B h (
0 h, " ! H ( ! 7 w = w(z) Δ Δ1 " ( z ∈ Δ + wz = μ(z)wz , μ(z) ∈ M0(Δ), ! ! μ(z) ) Δ esssup|μ(z)| ≤ k < 1 z ∈ Δ. B μ(z) ∈ M0(C) ( wμ(z) C + wμ(0) = 0 wμ(1) = 1 #,$'! 9 μ(z) ∈ M0(C) " ! ! ( T D F1 .
μ(T (z)) =
Tz μ(z) Tz
" ( S(z) = (az + b)/(cz + d) D μ(S(z)) =
Sz μ(z), Sz
T μ = wμT (wμ)−1 S μ = wμS(wμ)−1 " " ( ! 9 G
0 ("
T1, ..., Th μ(z) T1, ..., Th, T1μ , ..., Thμ (
0 Gμ . B - ! " # #$$,'! < M N " (! 9 μ(z) M MN, N. " $ #$$,'! 9 μ(z) ∈ M0 (C) " ( R, ( ( Ca,ρ ρ ) z = a, ! ! R(z) = a + ρ2 /z − a, wμ(Ca,ρ) ( ) wμ(a) Rμ (wμ (z)) = wμ (a) + λ2/wμ (z) − wμ (a),
( wμ(Ca,ρ) B;21J159*K.56;! <( ψ(z) = wμ(z) + ρ2/wμ(R(z)) − wμ(a). 5 ) ψ(z) − wμ(a) wμ(z) − wμ(a) ( + 0 z = a ∞ z = ∞. 6 ( θ. < μ
(w (z) − w
μ
(a))(wμ(R(z))
−
wμ (a))
ρ2 . = θ−1
B z ∈ Ca,ρ R(z) = z |wμ(z) − wμ(a)| = ρ(θ − 1)− = λ, λ ( ! . wμ(Ca,ρ) ( ) wμ(a) λ. * ! % 9 Ca,ρ ( Rμ = wμR(wμ)−1 ( (! 9 μ " ( T S, T = RSR, T μ = RμS μRμ. " & #$$,'! < μ(z) ∈ M0 (C) " ( Q(z) = a − ρ2/z − a, ) ( ( Ca,ρ ρ ) z = a, z = a π, wμ(Ca,ρ) / (/ ! ! w1 wμ(Ca,ρ), w1 /)/b = wμ(a) wμ(Ca,ρ) w2 (w2 −b)(w1 − b) ) ! 1 " ( Qμ(z) - wμ(Ca,ρ) " wμ(Ca,ρ) ( " / (/ ! B;21J159*K.56;! 2 1 2
(wμ (z) − wμ (a))(wμ(Q(z)) − wμ (a))
c. < z = a + ρ z = a − ρ
c
! < ( c > 0. 5 √z0 √ μ μ
w (z0) = w (a) + c wμ(Q(z0)) = wμ(a) + c. < z0 = Q(z0). H Q(z) ( ! . c = −λ2 Qμ (wμ (z)) = wμ (Q(z)) = wμ (a) − λ2 /wμ (z) − wμ (a).
< ( w0 ∈ wμ(Ca,ρ). <( w0 = wμ(a) + ρeiθ . 5 Qμ (w0) = wμ (a) − λ2 /w0 − wμ (a) = wμ (a) − (λ2/ρ)eiθ .
; Qμ(w0) ∈ wμ(Ca,ρ) ( w0. 5 wμ (Ca,ρ ) (! * ! H ( f
F F , ( f F F , ! ! ( f ( f. 9 F F ) φ φ, f φ = φf, f ( - fφ = φf. < μ(z) ∈ M0(C) T1, ..., Th
0 G, wμ : C → C ) ( Ω(G)/G Ω(Gμ)/Gμ. 2 F, F F f μ : F → F g μ : F → F : C ( + F, gμ(f μ)−1 ( F F . J ! 9 φ F ( ( ( ! 5 ! 9 τ ≥ 0
F h : C " (h, +τ ) φ. 6 : C (h, −τ ). 6 F/φ τ (h − τ + 1)/2 ! 6 F/φ τ k = h − τ + 1 " E" ! 5 F (h, ετ ), ε = ±1, ε = +1 h − τ + 1
" 0 ≤ h − τ + 1 ≤ h, ε = −1 0 ≤ τ ≤ h.
B ε = ±1 h > 0 τ ≥ 0 / / (h, ετ ). . ( ε = +1, τ > 0. < Cs, s = 1, ..., τ − 1, Γr , r = 1, ..., h − τ + 1, h ( ( - ( C0, ) (as ar C ! ; Rs (z) ( ( Cs, s = 0, 1, 2, ..., τ − 1. < Ar " ( - Γr Γr+1 , r = 1, 3, ..., h − τ. ;( ( C1, ..., Cτ −1 Γ1 , ..., Γh−τ +1 C0, ( C1, ..., Cτ −1 Γ1, ..., Γh−τ +1. 6- 2h ( H;
0 G, (" h " (
A1 , A3, ..., Ah−τ , A1, A3, ..., Ah−τ , R0R1 , ..., R0Rτ −1 ,
Ar = R0Ar R0 r. ; R0Rs ( Cs Cs Cs Cs, C0, ( R0Rs
G. ; π : Ω(G) → Ω(G)/G, H/G = Ω(G)/G (h, +τ ) R, Rπ = πR0 . ? " / / (h, +τ ). < H/{G, R0} τ (h − τ + 1)/2 ! B F (h, +τ ) f : H/{G, R0} → F/ φ . <( f ( H/G F f R = φf. <( (
( ! ? ( ( f : H/G → F ( ! ; ( f H ) μ(z) = fz∗/fz∗, f ∗(z) = ζf z −1 (z ζ p0 H/G f (p0) F C! 6 μ(z) R0. <( ) μ C
G μ ∈ M0(C). < wμ(z) ( C + wz = μ(z)wz wμ(0) = 0, wμ(1) = 1. ;
wμ ) ( Ω(G)/G Ω(Gμ)/Gμ. H H μ = wμ(H)
0 Gμ. ; g = f (wμ)−1 ( Ω(Gμ)/Gμ F. 7
0 Gμ
H μ
F. H μ. A!8!,
D C H μ ( ( gRμ = f (w μ)−1Rμ = f R(w μ)−1 = φf (w μ )−1 = φg,
φ F ( (G C Aμr Arμ ( - ( ( Γμr Γrμ ( μ ( Γμr+1 Γr+1 r = 1, 3, ..., h − r. J μ μ μ Γ = w (Γ) Γ, Ar = R0μ Aμr R0μ ; C A!8!4 A!8!, Csμ Csμ ( " ( (R0Rs)μ = R0μRsμ - Csμ Csμ ( Csμ Csμ (
Gμ, R0μRsμ. < " ) H μ, ( ( : ( ( C0μ) ( ! ! ( C0μ, H μ/Gμ. 5 ' ( #$$,'! . F (h, +τ ) φ
0 ( ( C, D C 2(τ − 1) ( ( ( C1, C1, ..., Cτ −1, Cτ −1, (G C h−τ +1 ( ( - C h − τ + 1 ( (
( C. . φ F
( C, τ F C τ − 1 ( C1, C1, ..., Cτ −1, Cτ −1. B τ = 0, ε = −1. B (h, −τ ), h (
C1, C1, ..., Cτ −1, Cτ −1, K1, K1, ..., Kh−τ +1, Kh−τ +1
( ( C0, ! ; Qi ( Ki, ) bi Ki π. ; Rs, " H, - 2h (
0 G = R0 Q1 , ..., R0Qh−τ +1, R0R1 , ..., R0Rτ −1 .
. R0Rs, Cs Cs ( R0Qi ( ( P Ki Ki , ( R0(P ) :( P C0). < H/G, (h, −τ ), h R, " τ ! < (H/G)/R τ h − τ + 1 ( ( ! ! " E" ! J F (h, −τ ), τ = 0, f : H/G → F f R = φf. 2 - (
0 Gμ H μ, F, C C! 2 A!8!4 A!8!3 C ! ! Kiμ Kiμ ( " ( (R0Qi)μ - Kiμ Kiμ ( P ( ( P ( ( R0μ(P ), ( P. 5 H μ, ( ( (
Gμ, " ( ( ! . ' ) #$$,'! . F (h, −τ ), τ = 0, φ
0 ( ( C, D C 2(τ − 1) ( ( ( C1, C1, ..., Cτ −1, Cτ −1; C 2(h − τ + 1) ( ( ( K1, K1, ..., Kh−τ +1, Kh−τ +1. . φ F
( C, τ F C τ − 1 ( C1, C1, ..., Cτ −1, Cτ −1. < (
Ki, Ki , i = 1, ..., h − τ + 1, h − τ + 1 " E" F/ φ . H ) (h, 0),
! ? ( R0 ( (! < K0, K1, ..., Kh ( Qj , j = 0, 1, ..., h, ( Kj ) Kj π. ; Kj ( Q0(Kj ), j = 1, ..., h, H - 2h ( K1, K1, ..., Kh, Kh. 5 H
0
G = Q0 Q1, ..., Q0Qh ,
H/G Q, Qπ = πQ0 π )! ; h ! B P Kr Q0(P ) Q0Qr (P ) ( Kr . < (H/G)/Q = H/{G, Q0} h + 1 " E" ! 9 F (h, 0), f H/G F φf = f Q. 5 H μ
0 Gμ, F. 2 D C φ F Qμ0 , : A!8!3C Qμ0 (w) = b − λ2/w − b; C A!8!3 Kjμ Kjμ ( " ( (Q0Qj )μ - Kjμ Kjμ ( P ( ( P (
( Qμ0 (P ). . H μ, ( ( (
Gμ, ( ! + ) K0μ λ = 1. < Qμ0 (w) = −1/w. 5 ' * #$$,'! . F (h, 0) φ
0 2h ( ( K1, K1, ..., Kh, Kh, ( = −1/w. . φ F ( Q(w)
. ( ( K, ( Q.
( ( K K1, K2, ..., Kh. ; ( Q ( ( Ki Ki
P Ki ( Ki , ) K ( P ( Q(P i
G. 5 ( K h ( Ki, Ki , i = 1, 2, ..., h, h + 1 " E"
F/ φ .
2 ' + :
n ( γ1, ..., γn,
(! B
F. ; n − 1, φ, " γ1, ..., γn
! 2 F/ φ = D. J A!8!= D n ( ! 5 ! " B A
O(A) F Φ(AO(A)−k ) = Φ(Ak ),
(1)
k = 1, ..., O(A) − 1. B;21J159*K.56;! 9 p ( Ak F, ! ! Ak (p) = p, p ∈ F, A−k (p) = p. ; AO(A)−k (p) = A−k (p) = p, ! ! Φ(AO(A)−k ) ⊇ Φ(Ak ). ; AO(A)−k (p) = p, A−(O(A)−k)(p) = p. . A−(O(A)−k)(p) = A−O(A) ◦ Ak (p) = Ak (p), Ak (p) = p. ; Φ(AO(A)−k ) ⊆ Φ(Ak ). * ! " 6 Φ(Ak ) ⊆ Φ(AmO(A)−nk ),
k = 1, ..., O(A) − 1, m ∈ Z, n ∈ Z.
(2)
B;21J159*K.56;! < Ak (p) = p. 5 n ∈ Z, Ank (p) = p A−nk (p) = p. AO(A) = id " AmO(A) = id, m ∈ Z, ( AmO(A)−nk (p) = AmO(A) ◦A−nk (p) = A−nk (p) = p. ; n, m ∈ Z Φ(Ak ) ⊆ Φ(AmO(A)−nk ), k = 1, ..., O(A) − 1. * ! ,
Φ (< A >) = Φ(A) ∪ Φ(A2) ∪ ... ∪ Φ(At),
t = O(A)−1 " O(A) t = O(A) 2 2 " O(A). B :$C! - . # 6 A!8!$$ Φ(< A >) = ∪s Φ(Ans ),
(3)
1 < s ≤ t, 1 < ns ≤ t, ( s, ns - (1, t], s. B;21J159*K.56;! B D Φ(A) ⊆ Φ(An), Φ(A2) ⊆ Φ(A2n), Φ(A3) ⊆ Φ(A3n), n ∈ N, k ∈ N ( Φ(Ak ) ⊆ Φ(Akn), n ∈ N, ( ( D L ⊆ N, L ∪ N = N. <( ! ' $ 9 A O(A), F, O(A) Φ (< A >) = Φ(A).
(4)
B;21J159*K.56;! < O(A) = 2 ( ! 9 O(A) A "! < ( 2
Φ(< A >) = Φ(A) ∪ Φ(A ) ∪ ... ∪ Φ(A
O(A)−1 2
O(A)−1 2
) = ∪s=2 Φ(Ans ),
ns ( ( " 2 ≤ ns ≤ O(A)−1 . 2 5 O(A) ns < O(A) s = 2, ..., O(A)−1 , 2 O(A) ns m(s), n(s) ∈ Z m(s)O(A) − n(s)ns = 1. A!8!$$ O(A)−1
O(A)−1
Φ(< A >) = ∪s=22 Φ(Ans ) ⊆ ∪s=22 Φ(Am(s)O(A)−n(s)ns ) = Φ(A).
; Φ(A) ⊆ Φ (< A >) ! 5 Φ (< A >) = Φ(A). 5 ! J ( )
: C
! B"
2" ( R3 ) ! 6
2" ! . A!A 4!$ #$A,' : C
2" (
2" ! 7
G
2" (g, s, i1, ..., ip) 2 EST −
Gg,s (g, s) p
Gi , ..., Gi i1, ..., ip, ( Γi , ..., Γi #$A$G $A,'! 6 A!A Gg,s. 6
Gi " (
Gi 3ik − 3
Fi = IntΓi /Gi
( 5 Ti C3i −3, k = 1, ..., p, (IntΓi ( Γi ) #$G A'! <(
2" G (g, s, i1, ..., ip) ( 1, 0, ∞, (
C3h−3, h = g + s + i1 + ... + ip. < [F0, {ak , bk }hk=1] h, h ≥ 2, O0 ∈ F0, F0 = Δ0/G0, Δ0
2" G0 , F0 Δ0. 5 hk=1[ak , bk ] Δ0, -
Ng,s,i ,...,i , ( 1
1
p
p
k
k
k
k
k
k
k
k
k
1
p
b1, ..., bg , [ag+1, bg+1], ..., [ag+s, bg+s], ..., i1 j=1
[ag+s+j , bg+s+j ], ...,
ip
[ah−ip+j , bh−ip+j ],
j=1
z0 O0, #$A$' T1, ..., Tg , U1, V1, ..., Uh−g, Vh−g
(1)
2" G0 (g, s, i1, ..., ip). J [a, b] = aba−1b−1. 7
Θg,s,i ,...,i ϕ : F0 → F0 ϕ(O0) = O0 , ) ϕ Tϕ Ng,s,i ,...,i 1
1
p
p
T1,ϕ , ..., Tg,ϕ, U1,ϕ, V1,ϕ , ..., Uh−g,ϕ, Vh−g,ϕ
Gϕ, hk=1[ϕ(ak ), ϕ(bk)]
z0 Δ0, :$C! 6 - ! ; M0(C) ) μ(z) C, esssupz∈C|μ(z)| < 1. H
T ∈ MC μ(z) ∈ M0(C), ! ! μ(T (z))T (z) = T (z)μ(z), T μ = wμ T (wμ)−1 ∈ MC . J wμ ( C - +
wz − μ(z)wz = 0
C, wμ : {0, 1, ∞} → {0, 1, ∞}. 9
G < MC , μ(z), " T1, ..., Tn, ..., T1μ , ..., Tnμ, ...
Gμ = wμG(wμ)−1 < MC [51]. (F, W ) " F h, h ≥ 2, W : C F :
C : C N, N ≥ 2, (
F ; C N, N ≥ 2, ( F ; C A 2k ( F, k ≥ 2. < (F, W ) C! 5 A #$%&' (F, W ) ( R3. < $ #$%&' γ " )
< W >, (" W, F N " (
h1 h = h1N + 1. < C! < W1(z) = αz, z ∈ C, 0 < α < 1. J ) K1 : α < |z| < 1 ( Ki+1 = W1i(K1), i = 1, ..., N − 1. 6 ) K1 " ) D 2g ( Γ1, Γ1, ..., Γg , Γg ; s " Λ1, ..., Λs, p 4i1, ..., 4ip− Λs+1, ..., Λs+p, g, s, i1, ..., ip ≥ 0, ik = 1, k = 1, ..., p,
g + s + i1 + ... + ip = h1 .
2 : ! #$A$'C
T1, ..., Tg , U1, V1 , ..., Uh1−g , Vh1 −g ∈ MC
(1)
Tn(Γn) = Γn, Tn(ExtΓn) = IntΓn, n = 1, ..., g; Uj , Vj ( ( ( Λj , j = 1, .., s,
Us+i1+...+il +j , Vs+i1+...+il +j ∈ MC , j = 1, .., il+1,
(2)
( Λs+l+1, (
Gi ( γl+1 ⊂ IntΛs+l+1, ExtΛs+l+1 Extγl+1, l = 0, ..., p − 1. 5 ( i Ki Γi1 , Γ1i , .., Γig , Γgi , Λi1 , .., Λis+p ( l+1
−(i−1)
Tki = W1i−1Tk W1
−(i−1)
, k = 1, .., g, Uji = W1i−1Uj W1
−(i−1)
Vji = W1i−1Vj W1
, j = 1, .., h1 − g, i = 2, .., N.
,
< F/ < W > h1 + 1, ( D ( K1 :$C ( |z| = 1 |z| = α
W1. ; H ( ( K1 W1i, i = 1, .., N − 1, ( ( |z| = 1 |z| = 1/αN . 5 H
2" G, W1N , T1, .., Tg , U1, V1, .., Uh −g , Vh −g , T12, ..., Uh −g,N , Vh −g,N , D = ∪T ∈GT (H)[51; 96]. L D/G = H/G F. ; H " (F, W ) C! <
) #,$' f K1/G {F \ ∪Ni=1W i(γ)}/ < W >,
G T1, .., Uh −g , Vh −g f−1 η W (η), η ∈ γ ( |z| = 1 |z| = α, (
W1. <( f f : H/G → F W f = f W1. B ( μf H μ C μ G W1. B ( μf Intγl+1, l = 0, 1, .., p − 1. B ( l, l = 0, 1, .., p − 1, " K1 γl+1 1
1
1
1
1
1
Intγl+1 ⊃ Λs+l+1 K1
Λs+l+1, ( Int(K1\Intγ ). ; G, γl+1 l+1 s+l+1 γ Λs+l+1 γl+1. < Λ l+1 s+l+1 μ ) Λ γl+1, Intγl+1 ( μ ! 5 :AC μ Intγl+1. <
G μ Ω(G) = C\Λ(G) W1. H Λ(G) ( μ #&3'! ( wμ : D/G → Dμ/Gμ, Gμ = wμ G(wμ )−1, Dμ = wμ (D), ) wμ #,$G &3'! 5 H μ = wμ (H)
Gμ ∈ C0, ! ! Gμ ("
#&3'! ; f (wμ)−1 H μ/Gμ F. 5
Gμ H μ F Dμ. B W F W1μ = wμ W1(wμ )−1 H μ/Gμ; μ 1 N H μ /Gμ , ) W μ , W 1 μ μ W1 π = πW1 . < ( 4 #,$' W1μ : C ( ( 0, ∞, ! ! W1μ(z) = α(μ)z, 0 < |α(μ)| < 1, ) α(μ). B ( i Kiμ ( wμ ( |z| = αi−1, |z| = αi, Kiμ = (W1μ )i−1(K1μ), i = 2, .., N. J K μ K wμ. < (
Gμ ∈ C0 (
2" F. < $ #&3' f1
Gμ
2" f1Gμ f1−1. 6 ( f1 W1μ f1−1 f1, ! J
W1μ Dμ. 7
Gμ
(W1μ )N , T1μ , W1μT1μ (W1μ )−1, ..., (W1μ)N −1T1μ (W1μ)−(N −1), ... U1μ , V1μ , .., (W1μ)N −1V1μ (W1μ)−(N −1), ...
( W1μ
W1μGμ(W1μ)−1 (W1μ )N , W1μ T1μ (W1μ)−1, ..., (W1μ)N T1μ (W1μ )−N , .., W1μ U1μ (W1μ)−1, W1μV1μ (W1μ)−1, .., (W1μ)N V1μ (W1μ )−N , ...
6 (W1μ)N T1μ(W1μ)−N ( Gμ, T1μ (W1μ)N , ( ( Gμ. 1 ! 5 ( W1μ " Gμ W1μ Gμ ! ; ( f1 ) : C
Gμ
f1Gμf1−1. 6
f1Gμf1−1 ( f1W1μf1−1 (
Gμ ( W1μ. <
f1Gμf1−1, (
f1W1μf1−1,
f1Gμf1−1, ( f1 W1μ ! ; f1W1μf1−1
2" f1Gμf1−1 ! < 4 #&3' f1W1μf1−1 ! 7
2" f1Gμf1−1 f1(Dμ ) F. 2 W F ( f1W1μf1−1 f1(Dμ)/f1Gμf1−1. 5 ' & B (F, W ) C (g, s, i1, .., ip) ) ) g + s + i1 + ... + ip = [(h − 1)/N ], ik = 1, k = 1, .., p, D ( W1(z) = αz, z ∈ C, 0 < |α| < 1; γ, 0 ∈ Intγ;
2" G H " D $C F/ < W > [(h − 1)/N ] + 1 : C ) γ W1(γ) ( W1, 2g + s + p ( T1, .., Tg , U1, V1, .., U[(h−1)/N ]−g, V[(h−1)/N ]−g ∈ MC .
?
2" W1, T1, .., V[(h−1)/N ]−g (g + 1, s, i1, ..., ip); AC
G "
W1N , T1, ..., V[(h−1)/N ]−g , T12, ..., V[(h−1)/N ]−g,N ,
−(i−1)
Tki = W1i−1Tk W1
−(i−1)
, k = 1, ..., g, Uji = W1i−1Uj W1
,
−(i−1)
Vji = W1i−1Vj W1
, j = 1, ..., [(h − 1)/N ] − g, i = 2, .., N,
G F G 8C W F 1 H/G, ) W1 ∈ MC . W J (F, W ) C ( R3 #$%&'! ; a, b ( W F. < A #$%&' γ, a b F " )
< W >, (" W, F N ! 2( h1 ( a b, h = h1N. 1 A!8!$3 - (F, W ). <( W1(z) = ze2πi/N , z ∈ C, ) " N Kl : 2π(l − 1)/N < argz < 2πl/N, l = 1, ..., N. <( ( ' ( B ( R3 (F, W ) C (g, s, i1, ..., ip) ) ) g + s + i1 + ... + ip = h/N, ik = 1, k = 1, .., p,
γ, 0 ∞,
2" G H " D $C F/ < W > h/N / / γ W1(γ), ( W1(z) = ze2πi/N , z ∈ C, 2g + s + p ( T1, ..., Tg, U1, V1, ..., U(h/N )−g, V(h/N )−g ∈ MC , (
2" (g, s, i1, ..., ip); AC
G " T1, ..., V(h/N )−g, T12, ..., V(h/N )−g,N , −(i−1)
−(i−1)
Tki = W1i−1Tk W1 , Uji = W1i−1Uj W1 1, ..., (h/N ) − g, i = 2, ..., N, k = 1, ..., g,
−(i−1)
, Vji = W1i−1Vj W1 ,j =
G F
G 8C W F 1 H/G, ) W1 ∈ MC . W B (F, W ) C! 5 $ #$%&' (F, W ) ( R3 . < A #$%&' ( η1 , η2, ..., η2k W F ( k γj , η2j−1 η2j , j = 1, ..., k.
2 γj ∪ W (γj ), j = 1, ..., k, F ! 2( h1 k γj ∪ W (γj ), j = 1, ..., k, h = 2h1 + k − 1. < - (F, W ). 6 " W1(z) = −z, z ∈ C. <" ( ( D k − 1 ( C1, ..., Ck−1 ) y; 2g ( Γ1, ..., Γ2g, s " Λ1, ..., Λs, p 4i1, ..., 4ip− Λs+1 , ..., Λs+p, g, s, i1, ..., ip ≥ 0, in = 1, n = 1, ..., p, g + s + i1 + ... + ip = h1 . < ( :$C Tj (Γ2j−1) = Γ2j , j = 1, ..., g. 6 (
Ci , Γj , Λn, ( W1 Ci, Γj , Λn, i = 1, ..., k − 1, j = 1, ..., 2g, n = 1, ..., s + p. . ( Ai Ai(Ci) = Ci , Ai(ExtCi) = IntCi , i = 1, ..., k − 1. H ( Tj , Un, Vn, j = 1, ..., g, n = 1, ..., s + i1 + ... + ip, ( W1 :$C! ; H - Ci, Ci , Γj , Γj , Λn, Λn, i = 1, ..., k − 1, j = 1, ..., 2g, n = 1, ..., s + p. 5 H
2" G A1, ..., Ak−1, T1, ..., Tg, U1, V1, ..., Uh −g , Vh −g , T1, ..., Tg , U1, V1 , ..., Uh −g , Vh −g , ( " D. B W F 1 H/G, ) W1. J ( W i, i = 1, ..., k − 1, Ci Ci y, ( 1 H/G. Ai, ( W 5 0 ∞ 2k ( 1 H/G. H" H W (F, W ) C! 1 A!8!$3 A!8!$= ( f H/G F, f W1 = W f. B H μ = wμ(H)
Gμ ∈ C0. ; H μ W1μ 0, W1μ : Lμ1 → Lμ1 , Lμ1 x wμ, W1μ = W1. F H μ/Gμ. B W F 1 H μ /Gμ . J ( Aμ = wμ Ai(wμ )−1 W i W1, ! ! ξ1iaμ = −ξ2iaμ, ξ1ia , ξ2ia ( Ai ξliaμ = wμ(ξlia ),
1
1
1
1
l = 1, 2; i = 1, ..., k − 1. 5 Ciμ Ciμ Lμ2 , wμ y, ( Aμi, i = 1, ..., k − 1, 0 ∞ 2k 1 H μ /Gμ . ( W 2 A!8!$3 Gμ ∈ C0
2" f1Gμ f1−1, f1 (Dμ ) F. ;
( W1
Gμ ! 7
Gμ
Aμ1 , ..., Aμk−1, T1μ, ..., Tgμ, ..., W1T1μ W1−1, ..., W1Tgμ W1−1, ...
( W1
W1GμW1−1, " Aμi, i = 1, ..., k − 1, W12 = 1. ; aμi, aiμ Aμi Lμ2 Aμi(aμi) = aiμ. < ξ1iaμ, ξ2iaμ ( Aμi. 5 aμi = (Aμi)−1(aiμ), Aμi(ξliaμ) = ξliaμ, l = 1, 2. W1AμiW1−1 aiμ, ξliaμ, l = 1, 2.
aμ aμ aμ aμ aμ W1Aμi W1−1(ξ1i ) = W1 (ξ2i ) = ξ1i , W1Aμi W1−1(ξ2i ) = ξ2i ,
W1 AμiW1−1(aiμ ) = W1Aμi (aμi ) = W1(aiμ ) = aμi ,
aμ aμ aμ aμ aμ W1(ξ2i ) = ξ1i = −ξ2i , W1(ξ1i ) = ξ2i , W1(aμi ) = −aμi = aiμ , i = 1, ..., k − 1.
; W1AμiW1−1 = (Aμi)−1, i = 1, ..., k − 1. . ( W1 )
Gμ , W1 Gμ ! 5 ( f1W1f1−1 ! 5 ' ) B (F, W ) C (g, s, i1, ..., ip) ) ) g + s + i1 + ... + ip = (h − k + 1)/2, in = 1, n = 1, ..., p, D L1 L2, 0 ∞, 0 ∞;
2" G H " W1(z) = −z, z ∈ C, z = 0 D $C ( C\{L1 ∪ W1(L1)} ( D Ci, L2∪W1(L2) i = 1, ..., k−1; 2g ( Tj ∈ MC , j = 1, ..., g; s " p
( Ui, Vi ∈ MC , i = 1, ..., s + i1 + ... + ip, L2 ∪ W1(L2). 5 L2 ∪ W1(L2) Ci W1(Ci), ( Ai ∈ MC , ( Ai W1 0 ∞ 2k ( 1 H/G, ) W1; W AC
G " A1 , ..., Ak−1, T1, ..., Tg, U1, V1, ..., Us+i1+..+ip , Vs+i1+..+ip ,
T1 , ..., Tg, U1, V1 , ..., Us+i1+..+ip , Vs+i1+..+ip ,
Tj = W1Tj W1−1, j = 1, ..., g, Ui = W1UiW1−1,
Vi = W1Vi W1−1, i = 1, ..., s + i1 + ... + ip ,
G F G 8C W F 1 H/G, ) W1 ∈ MC . W % 5 A!8!$3 A!8!$>
F W C C C ! < " ( ( W F F
F ! B
2" F, EST −
(g, s, m)
(g, n; ∞, ..., ∞) : ! #$A%G &='C! % 7 : C
G, F C C C ! ! - G A!8!$3 A!8!$>! 6 -
2" G - G. % # A!A 4!$ #$A,'
2" G - AC A!8!$3 A!8!$> G (F, W ) C C C ! % $ < (F, W ) C N ( R3 A!8!$, ! " #$%&'!
./
6 ( ("
) MNO
! B" MNO
("
) ! 6 MNO
! . A!A : C MNO
( MNO
! (F, W ), (F, P ), (F, M) C C C F h, h ≥ 2, W, P, M F : C W N, N ≥ 2, ( F ; C P N1, N1 ≥ 2, ( F G C M A 2k, k ≥ 2, ( F. ( ! C < A #$%&' (F, W ) ( R3. . $ #$%&' γ " )
< W >, (" W, F N
" ( h1 h = h1N + 1. C < ( (F, P ) ( R3. ?
N1. ; ξ1, ξ2 ( P F. . A #$%&' γ, ξ1 ξ2 F " )
< P > F N1 ! 2( h1 ( ξ1 ξ2, h = h1 N1. C < (F, M) $ #$%&' ( R3. . A #$%&' ( ξ1, ξ2, ..., ξ2k M F ( k γj , ξ2j−1 ξ2j , j = 1, ..., k. 2 γj ∪ M(γj ), j = 1, ..., k, F ! 2(
h1 k γj ∪M(γj ), j = 1, ..., k, h = 2h1 +k −1. 5 (F ; W, M) : C " (F ; W, M), D W M = MW m ∈ N k = N m,
(F, W ) (F, M) C C ! J ( ( M F W :! ! )
< W >) M(ξl ) = ξl , M(W j (ξl )) = W j (M(ξl )) = W j (ξl ), l = 1, ..., 2k, j = 1, ..., N − 1,
W j (ξl) ( M. 6 " ( M, ξ1, [ξ1] = {ξ1, W (ξ1), ..., W N −1(ξ1)} < W > . 5 2k > N, " ( ξi, [ξ1]. 6 ( ( M
i = 2. 5 [ξ2] = {ξ2, W (ξ2), ..., W N −1(ξ2)}. < ( [ξ1] ∩ [ξ2] = ∅. < ( j1 , j2, j1 = 0, ..., N − 1, j2 = 0, ..., N − 1, W j (ξ2) = W j (ξ1 ). 5 ξ2 = W j −j (ξ1). < |j2 − j1 | = 0, ..., N − 1 1
2
2
1
ξ2 ∈ / [ξ1]. 9 2k > 2N, ! ! m > 1, ξn, [ξ1], [ξ2]. < ( - ( n = 3. 5 ( 2k ( M N
( ( 2m = 2kN
) < W > . . ( ξl , l = 1, ..., 2m, ( Int(γ ∪ W (γ)) W j ({ξ1, ..., ξ2m}) ∈ Int(W j (γ) ∪ W j+1(γ)), j = 1, ..., N − 1. B ( ( ( γl, ( ξ2l−1 ξ2l M F, l = 1, ..., k, ( γl ∪ M(γl ), l = 1, ..., k,
< W > . B l = 1, ..., m, j = 1, ..., N − 1,
W j (γl ∪ M(γl )) = W j (γl ) ∪ W j (M(γl )) = W j (γl ) ∪ M(W j (γl )).
F/ < W > . H ) M ◦ π. < ( M F, ! ! π : F → F/ < W > π ◦ M = M " M 2 = id M
M 2 = id. ◦ π ◦ M M (! π ◦ M 2 = M
B p1 p2 F ( π(p1) = π(p2 ), ! ! < W > p1
< W > p2. 5 M M ( ! ( M < ) ( ξ1, ..., ξ2m M π : F → F/ < W > < A #$%&' ( ξ1, ..., ξ2m M. γl, ξ2l−1 ξ2l, l = 1, ..., m, F/ < W >, γl ), l = 1, ..., m, F/ < W > γl ∪ M(
! < F Int(γ ∪ W (γ)) = F0 < W > / )/W j (Int(γ ∪ W (γ))), j = 1, ..., N − 1,
F γl, l = 1, ..., k. < F0 / / M. <" δ1 F0, γl ∪ M(γl ), l = 1, ..., m, γ W (γ) p W (p) ! 5 δ = δ1 ∪ W (δ1) ∪ ... ∪ W N −1(δ1) M(δ) F. F F0 F 2(g + s) + m γ W (γ), M. 5 F h = [2(g + s) + m]N + 1. < (F ; W, M) : C! J α, α > 1. < ) K 1 = 1 " {1 < |z| < α, Imz > 0}. 6 K
( - D 2g ( Γ1 , Γ1, ..., Γg , Γg ; s " Λ1 , ..., ΛsG m ( C1, ..., Cm ) y, m = Nk . <( M1 (z) = −z W1(z) = 1 , ) K1 " αz, z ∈ C. < M1 K D ( Γg+i = M1(Γi), Γg+i = M1(Γi), i = 1, ..., g; " Λs+j = M1(Λj ), j = 1, ..., s, (
Cl = M1 (Cl ), l = 1, ..., m. B K1
(
T1 , ..., Tg, U1, V1, ..., Us, Vs
(1)
Ti(Γi) = Γi, Ti(ExtΓi) = IntΓi, Uj , Vj (Uj Vj = Vj Uj ) ( (
( Λj ; ( (
y
6
Λ1
n C2 Γ1 T?1 ˜ n iC 1 '$ n K1 s Γ1
K1 α W1(z) = αz n T &% ] Yg+1 M1 (z) = −z Γg+1 n iC 1 Γg+1 nC2 Λs+1 0 1
-
x
Tg+i = M1 Ti M1 , i = 1, ..., g, Us+j = M1 Uj M1 , Vs+j = M1 Vj M1 , j = 1, ..., s,
) ( A1 , ..., Am
(2)
Al(Cl) = Cl , Al(ExtCl) = IntCl , l = 1, ..., m. ; G1 EST −
(g, s), (" :$C! ) K1 Kj+1 = W1j (K1), j = 1, ..., N − 1. B ( j = 2, ..., N Kj
j Γj1 , Γ1j , ..., Γj2g, Γ2gj ; Λj1 , ..., Λj2s; C1j , ..., Cm , C1j , ..., Cmj
( −(j−1)
Tij = W1j−1TiW1
−(j−1)
, Ukj = W1j−1Uk W1 −(j−1)
Alj = W1j−1Al W1
−(j−1)
, Vkj = W1j−1Vk W1
,
(3)
, i = 1, ..., 2g, k = 1, ..., 2s, l = 1, ..., m.
6 " F01 F0 ( ) 1, ..., C m}, δ1 ∪ W ( γ ) ∪ M(δ1 ) ∪ γ ∪ {C
γ γ, δ1 M(δ1); C1, ..., Cm ( F0, / / l M δ1 ∪ M(δ1 ), " C γl−1 ∪ M(γl−1) γl ∪ M(γl ) ξ2l−2 ξ2l−1, l = 2, 3, ..., m,
γ1 ∪ M(γ1) γm ∪ M(γm) ξ1 ξ2m ! 5 F01 f ( K 1/G1 ( )
1 C
[1, α] ∪ {|z| = α, Imz > 0} ∪ [−α, −1] ∪ {|z| = 1, Imz > 0} ∪ {∪m l=1Cl }.
< f ) F01 ) K 1/G1; W M, W1 M1 ( M ) F01 ( 1 /G1 C1, ..., Cm y. J K ) K 1, Γi, Γi, i = 1, ..., g, Λj , j = 1, ..., s, ( :$C! < H ( " ( K1, W1j , j = 1, ..., N − 1, ( y ( {|z| = 1} {|z| = αN }.
6
n s
i n
n ? n
n i ? s'$ n
0 1
n Y&% i ] n n Y n n ] i
α
α
2
α
N
H
-
x
n
5 H, ( EST −
G, D G =< W1N ; T1, ..., Tg , ..., T2g; U1, V1, ..., Us, Vs , ..., U2s, V2s ; A1, ..., Am; T12, ..., T2g,N ; U12, V12, ..., U2s,N , V2s,N ; A12, ..., Am,N : [Uj , Vj ] = 1, [Ujk , Vjk ] = 1, j = 1, ..., 2s, k = 2, ..., N >,
(4)
[U, V ] = U V U −1V −1. ?
D = ∪T ∈GT (H). 5 D/G = H/G F. ; H "
(F ; W, M) : C! <
+
) #,$' f, f, f ≡ f )! ; ( f ( f F01 F0 " f M = M1f. 5 f
M F0 M1 K1. <( f f : F → H/G W1f = f W, M1f = f M. 6" μ = μf = (f(f )) H. <( μ H μ C μ G, W1, M1. <
G μ Ω(G) = C\Λ(G), W1 M1 , Λ(G) ( μ ≡ 0. < ( ( Λ(G) : * C ! < ( wμ : H/G → H μ/Gμ ) ( wμ : C → C, wμ + wμz − μ(z)wμz = 0. 5 Gμ = wμG(wμ)−1 < MC H μ = wμ(H) EST −
Gμ. ; + −1
−1
−1
z
z
f −1 ◦ (w μ )−1 : H μ /Gμ → F
Gμ H μ F Dμ = wμ (D). 2 W, M F ( W1μ = wμW1(wμ)−1 M1μ = wμM1(wμ)−1 H μ , ) μ M μ H μ /Gμ . L j = 2, ..., N W 1 1 Kjμ = (W1μ)j−1(K1μ) ( wμ ( {|z| = αj−1} {|z| = αj }; K1μ K1 wμ . ; H μ ) M1μ = M1 . 5 Clμ, Cl μ Cljμ, Cl jμ Lμ2 y wμ , (
Aμl Aμlj , l = 1, ..., m, j = 2, ..., N, 2k ( μ H μ /Gμ . M 1 ' * B (F ; W, M) : C (g, s) N (2(g + s) + m) = h − 1, D L1 L2, % ∞, % ∞; γ, 0 ∈ Intγ, L1 ∪ L2; EST −
G H, ) M1, M1(z) = −z,
/ / W1, W1(z) = αz, α ∈ C, |α| > 1 D $C 2 F/ < W > 2(g + s) + m + 1
/ ) / M1 γ W1(γ), ( W1, 2(2g + s + m) D Cl, M1(Cl) : ( L2 ∪ M1(L2) C ( Al ∈ MC , l = 1, ..., m;
Γ1, Γ1, ..., Γ2g, Γ2g " Λ1, ..., Λs, ..., Λ2s, (
T1 , ..., Tg, Tg+1 = M1 T1M1 , ..., T2g = M1Tg M1 ; U1, V1 , ..., Us, Vs, Us+1 = M1 U1M1 , Vs+1 = M1 V1 M1 , ..., U2s = M1 UsM1 , V2s = M1 Vs M1 ∈ MC .
5 L2 ∪ M1(L2) Cl M1(Cl), ( 1 ,
Al, l = 1, ..., m, 2m ( M ) M1. AC 7
G =< W1N ; T1, ..., Tg , ..., T2g; U1, V1, ..., Us, Vs , ..., U2s, V2s ; A1, ..., Am; T12, ..., T2g,N ; U12, V12, ..., U2s,N , V2s,N ; A12, ..., Am,N :
[Uj , Vj ] = 1, [Ujn, Vjn ] = 1, j = 1, ..., 2s, n = 2, ..., N >, −(n−1)
Tin = W1n−1TiW1
−(n−1)
Aln = W1n−1Al W1
−(n−1)
, Ujn = W1n−1Uj W1
−(n−1)
, Vjn = W1n−1Vj W1
,
, i = 1, ..., 2g, j = 1, ..., 2s, l = 1, ..., m, n = 2, ..., N,
F D = ∪T ∈GT (H). 8C B W M F 1 M 1 H/G, ) W W1, M1 ∈ MC ! - < F = F/ < W, M > ( ("
< W, M > F ( ! H ) D F ,
) π : F → F F N / /
W, / / M. H ( ! ( R3 #=$'! < : C F h R3, h ( D $C k − 1 L : ( M); AC k − 2 ( ( L, θt, θ = 2π , t = 1, ..., N2 − 1, ( L; N 8C mN M W ! ; h = k−2 2 N + mN + 1, 2k ( M, k = 2k1, k1 ∈ N; N " W ; m ∈ N. ; ξ1, ..., ξ2k ( M F,
( L. < A #$%&' ) ξ1, ..., ξ2k π1 : F → F/ < M > ( k γj0, ξ2j−1 ξ2j , j = 1, ..., k, F ! 2 ) π1 ( ( (F/ < M >) \ ∪kj=1γj0 ! < γj , j = 1, ..., k, I" γj0 π1 . 5 γj ∪ M(γj ), j = 1, ..., k, ( ( ( M. <( k
k
2 γ0 = ∪j=1 γj ,
−1
2 σ = ∪j=1 σj ,
σj ξ2j ξ2j+1, j = 1, ..., k2 − 1; δ = σ ∪ γ0. 2 δ ( / -/ δ ( ( δ) ξ1 ξk . 5 δ∪M( ( ξ1 ξk , " ( M. . $ #$%&' F/ < W > γ, ) π2 : F → F/ < W > F N ! 5
γ F/ < W > ( π2(δ ∪ M(δ)). J F ( R3 W M " N = 2
! 5 ("
: C P N1, N1 ≥ 2, ( p q F ; M A 2k, k ≥ 2, ( F, p, q; P l = M (2l = N1 ) m ∈ N 2k − 2 = mN1. < A #$%&' ) ξ1, ..., ξ2k π1 : F → F/ < M > ( k γj0, ) ξ2j−1 ξ2j , j = 1, ..., k, F ! <( ξ2k−1 = p, ξ2k = q. 2 ) π1 ( ( (F/ < M >) \ ∪kj=1γj0. 2( k ( ( ( ( M. < γj I" γj0 π1 ξ2j−1, j = 1, ..., k. γj = π(γj ), j = 1, ..., k, π : F → F/ < P, M > . 5 P s γj P s Mγj ( I" l−1 n γj π 1 ≤ s ≤ l, π −1 ( γj ) = ∪n=0 P (γj ∪ Mγj ) l j = 1, ..., k. < ∪kj=1π−1(γj ) F 2l = N1 ( γk ∪ P γk ∪mj=1γj ∪ Mγj , P. J ( ( ( {ξ1, ..., ξ2k} M F
< P > . E( 2k ( M l + 1
l
( 2m ( (
) < P >, l + 1
( ( p q; 2k = 2ml + 2. ; γ γk , p q. J l
ξ1, ..., ξ2m ( F0 = Int(γ ∪ P (γ)) ∪ Int(M(γ) ∪ M(P (γ))), P i ({ξ1, ..., ξ2m}) ⊂ P i (F0), i = 1, ..., l−1. 1 : C
( γj ∪M(γj ), ( ξ2j−1 ξ2j , j = 1, ..., m, ( F0, ( γj ∪ M(γj ), j = 1, ..., k,
< P > . F F0 F 2(g + s) + m ) γ ∪ P (γ) ∪ M(γ) ∪ MP (γ). ; / / M. < F h = [2(g + s) + m]l. < (F ; P, M) ), z ∈ C, ) : C! <( P1(z) = z exp( 2πi N N >2
1
" N1 Ki :
2π(i − 1) 2πi < arg z < , i = 1, ..., N1. N1 N1
6 K1 ( - D 2g ( Γ1, Γ1, ..., Γg, Γg ,
(
T1, ..., Tg ∈ MC ;
" Λ1, ..., Λs, ( s
U1 , V1, ..., Us, Vs ∈ MC , [Uj , Vj ] = 1, j = 1, ..., s;
( C1, ..., Cm ) K1. < M1z = −z, z ∈ C, K1 Kl+1 D 2g ( Γg+1, Γg+1, ..., Γ2g, Γ2g , ( m
Tg+1 = M1 T1M1 , ..., T2g = M1 Tg M1 ;
" ( s
Λs+1, ..., Λ2s,
(5)
Us+1 = M1U1 M1 , Vs+1 = M1 V1 M1 , ..., U2s = M1Us M1 , V2s = M1 Vs M1 ;
(6)
( C1, ..., Cm ) Kl+1. 2 (
m
A1, ..., Am ∈ MC ,
(7)
Ai(Ci) = Ci , Ai(ExtCi) = IntCi , i = 1, ..., m. B n = 2, ..., l Kn ∪ Kn+l D 4g ( Γ1n, Γ1n, ..., Γ2g,n, Γ2g,n ( (
2s
T1n, ..., T2g,n;
" Λ1n, ..., Λ2s,n, ( ( U1n, V1n, ..., U2s,n, V2s,n;
( C1n, ..., Cmn, C1n, ..., Cmn, ( ( 2m
A1n, ..., Amn.
J
−(n−1)
Tin = P1n−1 TiP1 −(n−1)
Ujn = P1n−1 Uj P1
, i = 1, ..., 2g, −(n−1)
, Vjn = P1n−1Vj P1
−(n−1)
Aen = P1n−1 AeP1
, j = 1, ..., 2s,
, e = 1, ..., m, n = 2, ..., l.
(8)
y 6 " " " " " " " K2 " s " n
" " " Γ " n R 1 " Γ1 n T1 ( " ( ( " ((v( C K1 " (u ( 2 1 ( n 1 ( C " " M ((((1(( " ( (((1 s ( 2π/N " M1 (γ) (
( (
" ( ( ( ( " x ( γ (" 0 Kl+1 (( ( ( C ( " ( K C2 ( 1( " (( n " (((( ( ( " (((( " " Γg+1 n " n " " Γg+1 " n " " Kl+2 K " " " " " n n " " " " R n n
2 - " F01 F0 ( ) 1, ..., C m}, γ ∪ P (γ) ∪ {C
C1, ..., Cm ( F0, / / M γ ∪ M(γ) ∪ P (γ) ∪ MP (γ), " l γl−1 ∪ M(γl−1) γl ∪ M(γl ) C 1 γ1 ∪ M(γ1 ) ξ2l−2 ξ2l−1, l = 2, 3, ..., m, C γm ∪ M(γm ) ξ1 ξ2m ! ; H ( " ( I K1 ∪ Kl+1 ∪ K2 ∪ Kl+2 ∪ ... ∪ Kl ∪ K2l .
; H, (
G, (" :,C :>C ( T1, ..., Tg; U1, V1, ..., Us, Vs. <( μ : C P1μ, M 1 μ μ ) P1 = P1 , M1 = M1 ' + B (F ; P, M) : C (g, s) l[2(g + s) + m] = h, D L1 L2, 0 ∞, 0 ∞; EST −
G H, / / M1(M1(z) = −z) P1(P1(z) = z exp( 2πi N )) D $C H F γ, ( M P, < P > F N1 ! 2 EST −
1
F0 = {Int(γ ∪ P (γ)) ∪ Int(M(γ) ∪ MP (γ))}
2(g + s) + m / / L1 ∪ P1(L1) ∪ M1 (L1) ∪ M1 P1 (L1), 2(2g + s + m) D
Ci, M1(Ci), ( L2 ∪ M1(L2) ( Ai ∈ MC , i = 1, ..., m;
Γ1, Γ1, ..., Γ2g, Γ2g
" Λ1, ..., Λ2s, (
T1 , ..., Tg, M1 T1M1 , ..., M1Tg M1; U1, V1, ..., Us, Vs , M1U1 M1 , M1V1 M1 , ..., M1Us M1 , M1Vs M1 ∈ MC .
5 L2 ∪ M1(L2) Ci M1(Ci), (
Ai, i = 1, ..., m, 2m ( 1 , ) M1 . M AC 7
G =< A1, ..., Am, T1, ..., T2g, U1, V1, ..., U2s, V2s, A12, ..., Aml, T12, ..., T2g,l, U12, V12, ..., U2s,l, V2s,l : [Uj , Vj ] = 1, [Ujn, Vjn] = 1, j = 1, ..., 2s, n = 2, ..., l >,
- :,C :3C :>C F D = ∪T ∈GT (H).
8C B P M F 1 H/G, ) P1, M P1 , M1 ∈ MC ! % & 5 A!8!$& A!8!A% F ( ) ! < " ( ( F ∗ F ∗
F ! B EST
F, EST −
(g, s, m), m = 0 #$A%'! 5 ( ) F ( : +! E C
2" #&3' #$A$' $ ! % ( 7 : C
G, F ) ! .- G A!8!$& A!8!A% - EST −
G - G. % ) A!A EST −
G - AC A!8!$& A!8!A% G, : C : C !
+! E #&4' 1!B! E #A4G A,' @!*! * #&%' ! P #=,==' .!E! H #A=A>' QRSTUTVWX M! B! J #,>' !
§
!" ! # $% & ! '( ! !" " !" ! ) * " " " ! ! $ # ! ! ) ! " !" ! # ) " " ! " + ! ! ! , # ) ! ! * " #!"
$ " $% & ! '( - F ! ! g > 1, Ω2 ! " ! " * ! F, W "# ! F N. . ! W ! # ! ! Ω2, + W [φ] = φW −1, φ ∈ Ω2. / ) ! " W ! " * 3g − 3. 0 * ! W ! ! " ! N − " *" ! ! ! W N = 1. / nk ! ! εk # *" k = 0, 1, ..., N − 1, ε !"# N − "#
*" $ 12 ! 34 " 56 7 ν ! ! ! # *
F → F/W ! F/W ! g1 > 0, n0 ! ΩW (F ), W − ! " " ! " * ! F, ! 3g1 − 3 + ν; 6 nk = 0 k ≥ 1, ! " ! ! 3g1 − 3 + ν
2(N − 1) N −1 ≥ nk ≥ 3g1 − 3 + ν ; N N
!6 ! k∗, 1 ≤ k∗ ≤ N − 1, nk = 0; 6 g1 > 0, n0 ≤ 3g − 5, ! # ! y2 = x6 + Ax4 + Bx2 + 1, g = 2, g1 = 1, n0 = 2 = 3g − 4 = 3g1 − 3 + ν. 1 ! # " 8 8 8 %! * . + !2) #9 ∗
:12 ! 34 6 - g # !# ! F, t ! 2" ! W # ! N 8 ! ΩW (F ) W − ! " " ! " * ! F ! 3g − 3 + t, g ! F/W . 1 ! # " 8 8 !# ! " π : F → F/W . ; + "! 2 F/W W − ! ! * φ(z)dz2 2 9 " ) " ! ! 2" ! W. / Mg∗,t 2 ! " !" ! # ! g > 1, ) "# ! W N t ! 2" Mg,t 2 ! ! !) ! ! , # ) Tg . " < '( 1! " " !" ! [F, α] [F ∗, α∗] g !+ W − ! ! " F, F ∗ ∈ Mg∗,t f : [F, α] → [F ∗ , α∗] *
W ∗f W −1, f ∈ α∗ α−1 , α α∗ " ! ) * ) !# # ! !
F0 .
:$% & ! '( 6 02 ! Mg,t ! ! , # ) Tg # 3g −3+t ≤ 2g−1, !" ! " " ! ) ! # " ModTg ! , # ) Tg . 0 ) Mg,t ! !) ! # 2 # ! ! 8 Tg /ModTg ! ! , Tg /τg , τg , !) # # ! ModTg . - ! + ) ! # " ! '( $ ! + " # ! !" ! # . "! W − ! ! ! # ! ! - 2 ! Mg,t " W − ! ! ! ! , # ) Tg . = "! 2"# # W − ! ! ! !
! , # ) Tg . > ! ! " !) ! 2 ! Mg,t, 2 ! Mg,t " !" - #
# " ! 2 ! Mg,t # ! " ! ! Tg , ! !) # [F, α] [F, β], ! F ! " " ? " 8 %! * 2g − 2 = N (2 g − 2) + t(N − 1)
π : F → F/W ! ! 3g − 3 + t ≤ 2g − 1, N ≥ 2. ; @ ! , # ) Tg,t, ! ! g t " (3g − 3 + t > 0)
! ) ) - "! ! !2 Tg,t !) 2 ! Mg,t, ! ! # 3g − 3 + t. $! !" Mg,t " ! ! Tg + ! 2 ! Mg,t : " ! " !) " ! Tg ). A * "! Mg,t ! 2 ! !) # " ! Tg , Mg,t ! Tg . . ) " # " )
! , # ) '( - ! , # ) Tg 2 ! ! !! ! ! " !) ! ) Mg,t. .# # [F, α] " 2 ! " " !" ! # " [F, α] " ! " 2 B!" ! " * φ = φ(z)dz2, " 2 ! ΩW (F ), " k 2 [0; 1). " ; ! ! F g " ! # "# ! J, J 2 = id. - F
! J n ! 2" ! !" # / g ! F/ < J > . , g = 2g + n − 1 ) ! F 2 ! F/ < J > . %"# * φ = φ(z)dz2 # ! F "! * & φ(Jz)dJz2 = φ(z)dz2. / )
! 2# ! ! " ; " " !" ! # g, "# ! J " n ! 2" !" # Ug∗, Ug ! !) 2 ! ! ! , # ) . # ! ! " C'( 2 )) $ '( 02 ! Ug ! ! ! 6g − 6 + 3n ! ! , # ) 6 g −6+3n > 0, !" !" " ! ) " ! Tg . % & > ! 6 g − 6 + 3n > 0 ) # g = 1. / ! ! U0 ! : ! "!# " ! ! F ). A * 2 ! Tg , !" Ug. ' '( - ! , # ) Tg 2 ! ! !! ! ! " !) ! ) Ug. .# # [F, α] " 2 ! " " !" ! # " [F, α] " ! " 2 B!" " ! " * !) * & " k 2 [0; 1). % & % 8 DE 2 ! ! , # ) Tg , " !" " ! [F, α], ) $ # 9 !" !" n, 2 ≤ n ≤ g. F ) ) n + 2g − 3 ≥ 2g − 1. 0 !" ! # ) φ(z)
2g − 1.
( )*
$ #9 σ !"# "# * " * " g, s, i1, ..., ip, ! !) ! ) ik =
|σ| = g + s + i1 + ... + ip. 1 "! !! # " Qth 2 ! ! ! ;+ Qσ , ! !) " ;+ " σ = (0, 0, h, 0, ..., 0), ) " !" ! F h, h ≥ 2, h = |σ|, ) "# ! W N t ! 2" = h ! F/ < W > . ; 2 ! Qth. - 1, k = 1, ..., p,
G0 =< T1 , ..., Tg, U1, V1, ..., Uh−g, Vh−g >
! ;+ " σ. ;! "# ! f 9 # # C "! ! # * # " G0, μf f ! ! ! G μf |Λ(G0 ) = 0, μf (T (z))T (z)/T (z) = μf (z), T ∈ G0 , z ∈ Ω(G0), μf 2 " 9 M0 (C) L∞ (C) 3 1! ! " *
f f1 " G0 "!) ! ! " ! B ∈ MC # f1T f1−1 = Bf T f −1B −1 ! T ∈ G0. F ! ! ) f1|Λ(G0 ) = Bf |Λ(G0). / Vσ 2 ! ! ! ! [f ] ! ! " * # f " G0 < f T1f −1, ..., f Uh−gf −1, f Vh−g f −1 > # # ;+ " σ. = dT Vσ , 2 !
dT ([f1], [f2]) = inf lnK(f1f2−1),
K(f1f2−1) f1f2−1 < 3 + ! ! f1 ∈ [f1], f2 ∈ [f2]. , (Vσ , dT ) ! 1! " " ;+
< T1, ..., Tg , U1, V1, ..., Uh−g, Vh−g >
< T1 , ..., Tg, U1, V1 , ..., Uh−g, Vh−g >
" σ "!) # ! ! " ! B ∈ MC
Tj = BTj B −1, j = 1, ..., g, Uk = BUk B −1, Vk = BVk B −1, k = 1, ..., h − g.
- ! Vσ ! ! : ##6 ! ! [G], [G] 2 # ) ;+ G "
σ. $! ! ) τ, n → ∞, !) !
[Gn] → [G]
< T1n , ..., Tgn, U1n, V1n, ..., Uh−g,n, Vh−g,n >∈ [Gn ]
< T1, ..., Tg , U1, V1, ..., Uh−g, Vh−g >∈ [G]
Tkn(z) → Tk (z), k = 1, ..., g, Ujn(z) → Uj (z), Vjn(z) → Vj (z), j = 1, ..., h − g, ! ! C n → ∞. , 2 + dT Vσ , 2 ! dT ([G], [G ]) = inf lnK(f ), + ! ! " * f
Tk = f Tk f −1, Uj = f Uj f −1, Vj = f Vj f −1, k = 1, ..., g, j = 1, ..., h − g,
< T1, ..., Vh−g >∈ [G], < T1, ..., Vh−g >∈ [G ]. , (Vσ , dT ) ! ! + - ! (Vσ , dT ), (Vσ , dT ) (Vσ , τ ) " 1/;=,7 H.,$/ = 2 ϕ1 : V σ → Vσ , 2 ! ϕ1([f ]) = [< f T1f −1, ..., f Vh−gf −1 >]. I ϕ1 : 3< C CC 6 - 2 ! ) 2 ϕ1. - [f1] [f2] !" [< f1T1f1−1, ..., f1Vh−g f1−1 >] = [< f2T1f2−1, ..., f2Vh−g f2−1 >]. , ! B ∈ MC f1T f1−1 = Bf2T f2−1B −1, T = f1−1Bf2T (f1−1Bf2)−1, ! T ∈ G0 . , f ≡ f1−1Bf2 # ! Ω(G0), Ω(G0) ! C, 2 @0 J' ! ! "# ! "# ! f C # f|Ω(G0) = f f(z) = z, z ∈ Λ(G0 ). / ) f1−1Bf2(z) = z, z ∈ Λ(G0 ), f1|Λ(G0 ) = Bf2|Λ(G0 ), [f1] = [f2]. 1 2 "! ϕ1, dn = dT ([fn], [f ]) → 0 n → ∞, [Gn ] → [G] !
τ n → ∞. 1 # ! 2 n !) !" ! fn ∈ [fn], f ∈ [f ] ! ! ! ln K(fnf −1) ≤ dn + (1/n). , K(fnf −1) → 1, μf f → 0 ! ) ! C n → ∞. / ) !" 2 fμ id : 2 ! 2 C). . ! fn f −1 ≡ fμ id n → ∞ ! C. , fnf −1T (fnf −1)−1 → T n → ∞ ! T ∈ G, [Gn] → [G] !
τ n → ∞.
n
−1
fn f −1
fn f −1
/ + ! ϕ−1 1 "! $ ! # ϕ1 ! (V σ , dT ) (Vσ , dT ) " " - ϕ 1 : (Vσ , dT ) → (Vσ , τ ),
ϕ1 2 ! ! Vσ , 1 # ! [Gn] → [G] !
τ n → ∞, !) ! < T1n, ..., Vh−g,n >∈ [Gn], < T1, ..., Vh−g >∈ [G] Tjn → Tj , j = 1, ..., g, Uin → Ui, Vin → Vi, i = 1, ..., h − g, ! ! C n → ∞. - 2 dT ([Gn], [G]) = inf lnK(fn) → 0 n → ∞, fn ! ! " *
" G ! Gn . % ;+ G ! 3 * " ) " G Gn !) fn0 ! " *
" G Gn K(fn0) ! * - ! [Gn] → [G] !
τ n → ∞ ! + [Gn] → [G] ! dT n → ∞. . ! 2 ϕ−1 1 "! A " " 02 ! " !" ! # F h, ) "# ! W N t ! 2" ∗ ˜ Mh,t ˜ , h !# ! F/ < W >, Mh,t ˜ ! !) ! ! ! , # ) Th. / < [F, α] >W W − ! ! 2 # ) ! [F, α], ! ! Mh,t ˜ = ∪ < [F, α] >W = ∪ < [F1, α ] >W , ! K + ! ∗ F ∈ Mh,t ˜ ! α, ! α ! ! F1 < F1 ∗ ! ! Mh,t ˜ . $! + " 2" #9 ; " ! M0 M1 2 ! Mh,t ˜ !+ Θ∗σ − ! ! " ! [F, α] ∈ M0 ψ ∈ Θσ < C CC [F, αψ] ∈ M1. , " 2 ! Mh,t ˜ K ) ! " Θ∗σ − ! ! F ! ! ∗ K ) ! Θ∗σ − " # ! F ∈ Mh,t ˜ , [F, α] ! Θ∗σ − ! ! # [F, β], ! ψ ∈ Θσ β = αψ F0 : ! ! h). ?! ! ! Th # " ModTh ! [F, α] ! [F, β], α, β "
" ! : 6 F0 F. - ! ! Γβ ([F, α]) = [F, αβ] ! # α ! F, β ! F0, 2) ! ) ModTh. $ < C ! ϕ0 ! (Vσ , τ ) Qσ ⊂ C3|σ|−3 . - ( CC C C "! 2 Φσ : Th → Qσ , h =| σ | . = !) "!" 2
−1 Φσ : Th → Vσ , Φσ = ϕ−1 0 Φσ , Φσ : Th → Vσ , Φσ = ϕ1 Φσ ,
˜ = Φ (M˜ ). I Th = Th(F0). / V˜h,t ˜ = Φσ (Mh,t ˜ ) Vh,t σ ˜ h,t t Qh˜ = Φσ (Mh,t ˜ ). , 02 ! Qth˜ ! " ˜ 3 +t, # 3h− + !" " 1/;=,7 H.,$/ ? !# ! 2 Φσ " Θσ Φσ ([F, α]) = Φσ ([F, β]), [F, α] ! Θ∗σ − ! ! # [F, β]. - ! " Θ∗σ − ! ! 2 ! Mh,t ˜ 2) Φσ 2 2 ! ! V˜h,t ˜ . $ "! Φσ ! ∗ Mh,t ˜ , " Φσ ! Θσ − ! ! ˜˜ . 2 ! Mh,t ˜ ! Vh,t ; Θ∗σ − ! ! Mh,t ˜ , 2 # < [F, α] >W , < [F, α] >W,Θ . - 2 " ! V˜h,t ˜ !" - [G] [G ] " V˜h,t ˜ !
τ, ! dT !! " C'L , [G] [G ] 2 # ! V˜h,t ˜ . 1 # ! ! < [F, α] >W,Θ # [G] ∈ Φσ (< [F, α] >W,Θ ). - [F, α] 2 [G] f ! * G G K(f ) * ˜ = K(f ) , f˜ : [F, α] → [F , f˜α] K(f) * = f˜ * 2 f ! ) " G F = D /G D ! " G . F ! 2 f0 : [F, α] → [F , f˜α] ! ! ! ) 1 ≤ K(f0) ≤ K(f˜). . ! [F , f˜α] 2 [G ] ! dT [F, α]. ? '( [F , f˜α] ∈< [F, α] >W , [G ] ∈ Φσ (< [F, α] >W,Θ ). , " V˜h,t ˜ !"
∗ σ
∗ σ
∗ σ
∗ σ
1 ! V˜h,t ˜ ! Vσ . $ ! V˜h,t ˜ " C'L ! [Gn ] → [G] ! dT n → ∞, [G] ∈ Vσ , [Gn] ∈ Φσ (< [F, α] >W,Θ ) < [F, α] >W,Θ "# Θ∗σ − ! ! Mh,t ˜ . - 2 [G] ∈ Φσ (< [F, α] >W,Θ ). - [F , α ] 2 [G] fn ! " *
" G Gn K(fn) → 1 n → ∞. 8 Fn = Dn /Gn [Fn, αn ], αn = f˜nα , f˜n * ! 2 fn D, D Dn ! " " G Gn ! ! - ) Φσ (< [Fn , αn] >) Gn ! n. , ! [Gn] ! dT , ! [Fn, αn] ! dT n → ∞. $ ! Mh,t ˜ [Fn , αn ] ∈ M0 9 n, M0 ! < [F, α] >W,Θ . ? M0 [Fn, αn] ! dT [F0, α0] ∈ M0. A "! Φσ ! + [Gn] = Φσ ([Fn, αn]) ! dT n → ∞ Φσ ([F0, α0]) = [G] ∈ Φσ (< [F, α] >W,Θ ), V˜h,t ˜ !
∗ σ
∗ σ
∗ σ
∗ σ
Vσ .
∗ σ
-! Φσ M0 : Θ∗σ − ! ! < [F, α] >W,Θ ) Φσ (M0) ≡ Φσ (< [F, α] >W,Θ ). A2 Φσ ! M0 (Φσ )−1 "! - [F, α] = [F , α ] ! Th, + [F, α]
[F , α ] 2 M0, [G] = [G ], [F, α] [F , α ] 2 [G] [G ] ! ! , ! "# q : F → F. , [F , α ] W − ! ! [F, qα ], [F, qα ] ∈ M0 2 2 [G]. . ! qα ! αψ, ψ ∈ Θσ . 7 ψ 2 ! ! F0, [F, α] [F, qα ] 2 ! " < [F, α] >W,Θ . / ) [F , α ] = [F, qα ] = [F, α] ! Th. , "! (Φσ )−1 "! 2 ! V˜h,t ˜ . , Φσ M0 . , " Mh,t ˜ ) ! ModTh, )" ! " V˜h,t ˜ " 1 # ! V1 V2 " V˜h,t ˜ . $+ M1 M2 "
Mh,t ˜ , 2 V1 V2 ! ! . ! Γ ∈ ModTh , ! M1 M2. , Φσ Γ(Φσ )−1 2 V1 V2 ; (Φσ )−1 ! V1 M1 . $ '( M0 ! # 3h˜ − 3 + t. . !
∗ σ
∗ σ
∗ σ
Φ σ (M0) V˜h,t ˜ , ! ˜˜ . V˜h,t ˜ , ! + Vh,t 1 ! F + ! ∗ ∗ ˜˜ Θσ − ! ! # F ∈ Mh,t ˜ . . ! Vh,t + t $ " C'L ! ! 2 V˜h,t ˜ ! " Qh ˜ ! Qσ , 2 V˜h,t ˜ ! Vσ . , / Qnc,h˜ 2 ! ! ;+ Qσ ! !) " ;+ " σ, ) " !" ! F h, ) ) !)* ) J n ! 2" ) " !" = h˜ ! F/ < J >, < J > 2+ J, h = 2h˜ + n − 1, h ≥ 2. - 02 ! Qnc,h˜ ! ! ! " 6h˜ − 6 + 3n, + !" " 1 ! ! ! " C'3
! !)) C'E
# ( .* (
; " !" ! ) " ! " " !" G %! * L3< LJ 8 'E 8 8+ D3< DJ ; F 4(< 4' $% & ! '( ?! " !" ! ) ! " " ! " !) " ! ! 8 ! # ! ! " !" ! # ! = !) ) ) ! ! [F0, {ak , bk }hk=1] h ≥ 2, {ak , bk }hk=1 F0 ak ∩ bk = O0 ∈ F0, k = 1, ..., h. ;2 * @ μ = μ(z) dzdz F0 ! ! ! 2 f ! F0 ) ) ! ! F 2 h, ) ! {f (ak ), f (bk)}hk=1 F. - ) ) ) ! ! ! #9 [F, f ]. $ ) ! ! Th,
!" " " !" ! h ≥ 2, ) ! " ! + ! ! !) ! ! & - 2 ! * " ! 2+" " ! ! # # !# ! $! " !" " 2 ! !" ! # ) " " " ! ! @ Nσ 9) ) ! # π1 (F0, O0 ) = a1 , b1, ..., ah, bh :
h
[aj , bj ] = 1,
j=1
2) " # b1 b2, ..., bg, [ag+1, bg+1], i i [ag+2, bg+2], ..., [ag+s, bg+s] j=1 [ah−i +j , bh−i +j ], j=1 [ag+s+j , bg+s+j ], ..., −1 −1 [aj , bj ] = aj bj aj bj , σ = (g, s, i1, ..., ip) !"# "# * " * " ! !) ! G ik = 1, k = 1, ..., p g + s + i1 + ... + ip = |σ| = h. ; #! Gσ ∼ = π1 (F0, O0 )/Nσ , ! !) Nσ
) ! p
1
p
p
Gσ = T1 , ..., Tg , U1, V1 , ..., Uh−g, Vh−g : [U1, V1 ] = ... = [Us, Vs ] = =
i1
[Us+j , Vs+j ] =
j=1
i2
[Us+i1+j , Vs+i1+j ] = ... =
j=1
ip [Us+p−1 in+j , Vs+p−1 in +j ] = 1, = j=1
n=1
n=1
[Uk , Vk ] = Uk Vk Uk−1Vk−1, 2 ! ! C, # # ;+ " σ F ! !" ! # EST −" (g, s) p " !" Gσ = Gg,s ∗ Gi1 ∗ Gi2 ∗ ... ∗ Gip .
@ Θσ ! ψ ! F0 G 6 ψ(O0) = O0 , C6 ! π1 (F0, O0), * !"# ψ, ! ! Nσ , (6 " ;+ Gσ Gσ,ψ , ) ! F0 !) " "
Tj,ψ = Tj Ui,ψ = Ui, Vi,ψ = Vi , j = 1, ..., g, i = 1, ..., h−g, Gσ,ψ ) " ψ(Nσ ), Gσ Nσ .
1 # " !2 !" !" " ! ! ) " " !" ! ! ! ) " " EST −" "# # " ;+ " σ : p = 0). / ! " " " " # # - F ! ! h ≥ 2, ) ! " ! G W N ≥ 2,
) # ! 2" F ; M C 2k, k ≥ 2, ! 2 " F, + W M = MW N |k. - ! ! ! 2 ! ! 2" M !
! " # ! * # " < W > . 0 2 ! " !" ! # F h ≥ 2, ) ! " ! W M, ∗ Mh,2k , Mh,2k ! !) 2 ! ! ! , # ) Th = Th(F0), F0 ! ! h. = h ! F/ < W, M >, < W, M > 2+ ! W M. = h 2 9 8 %! * 2h − 2 = (2 h − 2)2 + 2m
" π1 : F/ < W >→ F/ < W, M >, h ! F/ < W >, "# ! 2(g + s) + m + 1, k = N m. / ) h = [(h − m − 1)/2] + 1 = g + s + 1. . # "! !! # " 2 ! Mh,2k Q2kh 2 ! ! Qσ , ! !) " EST − σ = (g, s), (g + s = h), ) " !" ! ;
) !# ! 2 ! Q2kh ) ! σ. , 2 ! " !! Mh,2k 9 W − ! ! 2 ! Mh,2k " W − ! ! " ! # , # ) = 2"# W − ! ! ! + " M− ! ! " ! # 1 (W, M)− ! !
" # 1!
" ! :! , # ) 6 [F, α] [F , α ] h "!) (W, M)− ! ! " ∗ F, F ∗ ∈ Mh,2k
f : [F, α] → [F ∗, α∗] ! * W ∗f W −1 M ∗f M −1, f ∈ α∗α−1, α α∗ " ! : 6 F0 F F ∗ ! ! A ! (W, M)− ! ! ! # ! ! ) " " !" ! # 1 # ! [F, α] [F ∗, α∗] ! (W, M)− ! ! " ! , # ) q (q∗) "# [F, α] !# [q(F ), qα] :[F ∗, α∗] [q∗(F ∗), q∗α∗]) : ∗
∗
[F, α]
q
-
α∗ α−1 f
[q(F ), qα] f1 ∈ q ∗ α∗ α−1q −1
?
[F ∗ , α∗]
?
q
∗
-
[q ∗ (F ∗), q ∗α∗ ]
- 2 " ! , # ) [q(F ), qα] [q ∗(F ∗), q ∗α∗ ] 2 (W, M)− ! ! " 1 ) * ) (q ∗W ∗ (q ∗)−1)f1(qW q −1)−1 ≈ q ∗W ∗ (q ∗)−1q ∗ α∗ α−1q −1 qW −1q −1 = = q ∗ W ∗ α∗ α−1W −1q −1 ≈ q ∗ W ∗f W −1q −1 ≈ q ∗ f q −1 = f1
: ≈ )6 / ) ! ) W − ! ! " ! # ! ! M. , ! + 2 ! Mh,2k " (W, M)− ! ! " ! # , # ) M" * ) (W, M)− ! ! ! ! Th !) ! !
, # ) ; F 12 ? 4(< 4E - X : C ∞) ! h ≥ 2. / Dif f +(X) ) ! ) * ) ! X C ∞− #< Dif f0(X) ! Dif f +(X), ! " " 2 ! X; M(X)
! " X, ! " # X * # $ " " " " " ! C ∞− X. $" ) ) μ0 X, ) ! ! Xμ . - z "# +"# ! U (p) p ∈ Xμ . , z = ϕ(q), q ∈ U (p) ⊂ Xμ , z ∈ D ⊂ C, 2) p ! ) ?! " !# ! ! ) X. - 2# μ ∈ M(X) !"! ! ! 2 +" ! " ! ds2 = |dz + μ(z)dz|2, ! " ! ! " * " = μ(z)
* ! " Xμ ! 0
0
0
0
μ∞ = esssup|μ(z)| < 1, z ∈ D.
(1)
; * μ = μ(z) dzdz ! # " Xμ "! * @ ! Xμ . - * ) μ(z), +) ! Xμ , # *
μ(z) U ! # "!) # Xμ . - ! # *
π : U → Xμ , * @ μ(z) dzdz z ! Xμ * @ μ(z ) d U, d z ! !) 9 ) 0
0
0
0
0
0
μ(γ z)
γ ( z) = μ( z ), γ ∈ Γ, z ∈ U, γ ( z)
Γ ! ! # !) ! ! U U/Γ = Xμ : ) # ! ! 6 > ! : 6 ! ! *
μ(z) ! " Xμ 2 !" + μ( z ) U, 2 " ! Xμ . 8 9 ! @ 0
0
0
wz − μ( z )wz = 0
(2)
! U, ! : ) *
# * #6 ! 2 wμ U !# wμ (U ). > ! : 6 ! !! 9 !
:C6 ! 2 * ) ; ! 2 wμ *
: ! ) * )6 2 !# # Γ,
! # !) # ! U, ! !# # Γμ, ! # !) # wμ(U ), Γ γ → wμ γ(wμ)−1 ∈ Γμ .
A! ! ! μ
μ
w (U )/Γ :
Xμ
!
U
w
-
μ
wμ (U )
π
πμ ?
? -
Xμ0
f
Xμ
- wμ ! Xμ , ! 2 f !" ! # Xμ → Xμ f π = πμwμ. , 2 * μ(z) Xμ ! ! 2 f ! Xμ Xμ, μ(z) = μf (z) = ff
w = f (z) 1 ! " X !) ) !) # |dz+μ(z)dz|2, 9 ! @ wz − μ(z)wz = 0 ! # "!) # Xμ
! 2 f ! Xμ !) ! ! Xμ : # μ = μf ). 7 ! # ! Dif f0(X) M(X) 4( G 0 0
0
0
z
z
0
0
μf · g = μf ◦g , g ∈ Dif f0(X).
8 ! " " H < Dif f +(X), " # H < Dif f +(X). - # " H ! " ! ! Xμ h, # H ! # " ! ! 8 2 ! ! " X, " H 2 # " ! ! !" ! F 2 ! H− ! " M(X)H , M(X)H = {μ1 ∈ M(X) : μ1 · h = μ1 , h ∈ H}.
@ ! M(X)H N2 N # μ. = ) ) μ ∈ M(X), ! ! μ, 2 ! H < Dif f +(X),
H, ! # " ! ! !# ! Xμ . , + 2 ! ! " " 2 " H :
M(X)H = {μ 1 ∈ M(X) : μ 1 · h = μ 1 , h ∈ H },
N2+ N # μ . , ! + ! ! M(X), ) " H < Dif f +(X), " " ! ! " " # # H < Dif f +(X), ! ! ! K 2 ! H − ! " " 8 2 ! ! 2" : 6 " H : M(X)H = {μ ∈ M(X) : μ · h = μ, h ∈ H}.
7 μ ∈ M(X)H , # !
Dif f0(X) μ !2" ! G μ·f0 ∈ M(X)H , μ·f0 2 M(X)H , f0 ∈ Dif f0(X). $ ! μ · f0 !#+ ! 2 " " ! 2 ! M(X)H *
# ! ) Dif f0(X). M" 2 ! Dif f0 (X) !" 2 f0 " " !"! " 2 ! M(X)H . 1 ! / 4( - f0 ∈ Dif f0 (X), μ ∈ M(X)H . , μ · f0 2 2 2 ! M(X)H , f0 ∈ C0(H) = Dif f0(X) ∩ C +(H), C + (H) = {f ∈ Dif f +(X) : f h = hf, h ∈ H} * " H ! Dif f +(X). 8 !G Th = Th(X) = M(X)/Dif f0(X) ! , # ) h T (X, H) = M(X)H /C0(H) ! , # ) / i ! ! 2 T (X, H) ! Th(X). 0 4( $2 i : T (X, H) → Th(X) K ! 1/;=,7 H.,$/ / ! - μ1, μ2 ∈ M(X)H . - 2 !) ! " 2 C0(H), [μ1 ]C0 (H) = [μ2 ]C0 (H) ,
(3)
i([μ1]C0 (H) ) = i([μ2]C0 (H) ) = [μ]Dif f0(X) , μ ∈ M(X). / ) μ1 = μ2 · f0 f0 ∈ Dif f0(X). , μ2 ∈ M(X)H , f0 ∈ Dif f0(X) μ1 = μ2 · f0 ∈ M(X)H , " # ) f0 ∈ C0(H), [μ1]C0(H) = [μ2]C0(H). - !
:(6 . ! 1 ! !2 i # *
Φ : M(X) → M(X)/Dif f0(X) = Th (X)
!"!)) G M(X)H μ
id
-
μ ∈ M(X) Φ
C0(H) ?
Dif f0(X) ?
-
T (X, H) [μ]C0 (H)
i
[μ]Dif f0(X) ∈ Th(X)
? ! # " Φ(M(X)H ) = i(T (X, H)).
/ +
Dif f (X)/Dif f0(X) :
! ) * )
Θ
(4) Dif f +(X)
Θ : f → f Dif f0(X), f ∈ Dif f +(X).
/ 2 ! Θ(H)− ! T (X)Θ(H) = {Φ(μ) ∈ Th(X) : Φ(μ) · Θ(H) = Φ(μ)},
! T (X)Θ(H) = {[μ]Dif f0(X) ∈ Th : [μ]Dif f0(X) · [h]Dif f0(X) = [μ]Dif f0(X) , h ∈ H}.
= μ ∈ M(X)H , Φ(μ) ∈ T (X)Θ(H), 2 ! M(X)H 2 ) Φ ! T (X)Θ(H) : Φ(M(X)H ) ⊆ T (X)Θ(H)
+ :'6
i(T (X, H)) ⊆ T (X)Θ(H) ⊂ Th(X).
02 ! T (X)Θ(H) ! K 2 ! Φ(M(X)H ) ! " H < Dif f +(X), " H, Θ(H ) = Θ(H). ; Φ(M(X)H ) Φ(M(X)H ) ) ) ! !2 H = f Hf −1 ! 2 f ∈ Dif f0(X) ! Xμ Xμ , μ μ ! ! " μ ∈ M(X)H , μ ∈ M(X)H . 1 # ! μ = μf = μ f, f ∈ Dif f0(X). A Xμ # ! " ! ! H, Xμ * ! H = f Hf −1 ! " ! ! 7 H 2 # " ! ! Xμ , ) μ · h = μ , h ∈ H , μ ∈ M(X)H . -! μ ∈ M(X)H , !" ! ! " H : μ · h = μ , h ∈ H. 1 H = f −1H f. / ) h ∈ H ! !
μ · h = μf · h = μf ◦h = μf ◦f −1h f = μh ◦f = μf = μ ,
μ ∈ M(X)H . :; F 4' 6 1 ) h ≥ 2 ! G 6 ! T (X)Θ(H) ! " !" 2 ! ! Th(X), " ! ! !< 6 ! T (X)Θ(H) ! Th(X) Φ : M(X)H → T (X)Θ(H)
! " 2 ? " !" Φ(M(X)H ) = i(T (X, H)) ! ! "" " ! T (X)Θ(H), ! ! 2 ! T (X)Θ(H) ) Φ(M(X)H ) = T (X)Θ(H). / ) 0 # 4' /2 Φ : M(X)H → T (X)Θ(H) )K ! ? ! + :'6 ! i(T (X, H)) = T (X)Θ(H). , ! 2 ! T (X)Θ(H) ! # Φ 2 ! M(X)H 2 ! ! H − ! " " H = f Hf −1 ! 2 f ∈ Dif f0(X).
. 6 ! " (W, M)− ! ! " " ! 2 ! Mh,2k , ! ) !" " 2 ! ! Th(F ), " ! ! ! " @ [F, α]W W − ! ! " ! # , # ) 2 # ! [F, α] : ! 6 [F, α](W,M ) (W, M)− ! ! 2 ! [F, α]. 02 ! Mh,2k ! ! ! ! K # Mh,2k = ∪[F, α](W,M ) = ∪[F1, α ](W,M ) , ! K + ∗
! α ! ! F ∈ Mh,2k α ! F1 < F1 ! ∗ ! Mh,2k . " ; " D0 D1 (W, M)− ! ! " ! # , # ) 2 ! Mh,2k "!) Θ∗σ − ! ! " ! [F, α] ∈ D0 ψ ∈ Θσ [F, αψ] ∈ D1. , ! " 2 ! Mh,2k K ) ! " Θ∗σ − ! ! F ! ! K ) ! ∗ Θ∗σ − " # ! F ∈ Mh,2k , [F, α] Θ∗σ − ! ! [F, β], ! ψ ∈ Θσ , β = αψ F0. $ C( ! ϕ0 ! (Vσ , τ ) ! : ##6 ! ! " ;+ σ = (g, s, i1, ..., ip) :! EST − i1 = i2 = ... = ip = 0) Qσ ⊂ C3|σ|−3 . , 2 "! 2 Φσ : Th → Qσ , h = |σ|. = ! "! 2 Φσ : Th → Vσ , Φσ = ϕ−1 0 Φσ , 2k Th = Th(F0). -2 Vh,2k = Φσ (Mh,2k ) ⊂ Vσ , Qh = Φσ (Mh,2k ) ⊂ Qσ . 6 02 ! Mh,2k ! " ! Th # 3h − 3 + 2m (m = Nk ), + !" " < C6 02 ! Q2kh ! " # 3h − 3 + 2m, + !" " 1/;=,7 H.,$/ ? !# ! 2 Φσ " Θσ Φσ ([F, α]) = Φσ ([F, β]), [F, α] ! Θ∗σ − ! ! # [F, β]. , ! " % &
#
Θ∗σ − ! ! 2 ! Mh,2k 2) Φσ 2 2 ! ! Vh,2k . $ "! Φσ ! Mh,2k " Φσ ! Θ∗σ − ! ! Mh,2k ! Vh,2k . ; Θ∗σ − ! ! Mh,2k 2 # [F, α](W,M ),
[F, α](W,M ),Θ∗σ .
-! " 2 ! Mh,2k : " (W, M)− ! ! 6 !" " - [F, α] ∈ Mh,2k
[F ∗, α∗] 2 ! Mh,2k , [F, α] ! , # ) dT . >! 2 [F ∗, α∗] (W, M)− ! ! ! [F, α]. , " [F, α]W 2 ! Mh,2k ∗ ) F ∈ Mh,2k
!" '( [F, α] [F ∗, α∗] 2 [F, α]W . - 2 2 2 [F, α](W,M ) , [F, α] [F ∗, α∗ ] M− ! ! " - f 2 , # ) ! [F, α] ! [F ∗, α∗], K0 , 1 ≤ K0 [F, α] [F ∗, α∗] , # ) dT . 8 2 f −1M ∗ f : [F, α] → [F, f −1M ∗ f (α)], ) K ≤ K02 : !# ! ! " 2 # K(f1f2) ≤ K(f1)K(f2), K(f −1) = K(f )). - K02 dT ([F, α], [F, f −1M ∗f (α)]) A ! , # ) !) ! ! F : " 6 !" ! , # ) K02 ≥ 1 ! ! [F, α] = [F, f −1M ∗ f (α)]. / ) f −1M ∗f 2 - ! ! ! M− ! ! ) ! [F, α] [F, f −1M ∗f (α)] : " # !) ! " M M ∗∗ = f −1M ∗f ! ! 6 M− ! ! " ? M− ! ! f −1M ∗ f ≈ M ∗∗ f −1M ∗ f M −1, ) f −1M ∗ f ≈ (f −1M ∗ f )f −1M ∗ f M −1 M ≈ f −1M ∗ f. $! ! f −1M ∗ f, %! * ) f −1M ∗ f = M. , [F, α] (W, M)− ! ! [F ∗, α∗], (W, M)− " 2 ! Mh,2k !" - 2 1 ! ) !) D0 = [F , α ](W,M ) ! , # ) [Fn, αn], 2) D0 )
, # ) [F, α] ∈ Th. 1 2 [F, α] 2 # 2 !# D0. = D0 = [F , α ]W ∩ [F , α ]M . - ! [Fn, αn] 2 [F , α ]W [F, α] ∈ Th, !! 2 ! [F , α ]W [F, α] ∈ [F , α ]W . . # " ! [Fn, αn] 2 2 2 ! [F , α ]M . . ! + :! , # ) 6 !) [F, α] ! ! 2 2 ! [F , α ]M . ,
[F, α] ∈ [F , α ]W ∩ [F , α ]M = D0.
- D0 2 ! ! , # ) ! Th. , ! ! ! !) ! Vh,2k Vh,2k ! Vσ ! 2 ! A ! ! ! Vh,2k !+ [G] [G ] " Vh,2k !
τ, ! , # ) dT !! " C'L , [G] [G ] 2 # ! Vh,2k . 1 # ! ! [F, α](W,M ),Θ # [G] ∈ Φσ ([F, α](W,M ),Θ ). - [F, α] 2 [G] :G [F, α]) f ! * G G , K(f ) * , f : [F, α] → [F , fα] = K(f ). = f * 2 f K(f) Ω(G) "! EST −" G F = Ω(G )/G , Ω(G ) "! " G . F ! 2 f0 : [F, α] → [F , fα] ! ! ! ) 1 ≤ K(f0) ≤ K(f). . ! ) [F , fα] 2 [G ] ! dT [F, α] : dT ([F, α], [F , fα]) = lnK(f0)). - " 2 ! Mh,2k : " (W, M)− ! ! 6
!" [F , fα] ∈ [F, α](W,M ), [G ] = Φσ ([F , fα]) ∈ Φσ ([F, α](W,M ),Θ ). , " Vh,2k !" 1 ! Vh,2k ! Vσ . $ ! Vh,2k " C'L ! [Gn], ) [G] ! , # ) dT n → ∞, [Gn] ∈ Φσ ([F, α](W,M ),Θ ), [G] ∈ Vσ [F, α](W,M ),Θ "# Θ∗σ − ! ! Mh,2k . - 2 [G] ∈ Φσ ([F, α](W,M ),Θ ). - [F , α ] 2 [G] fn ! "
∗ σ
∗ σ
∗ σ
∗ σ
∗ σ
∗ σ
*
" G Gn, K(fn) → 1, n → ∞. 8 Fn = Ω(Gn)/Gn [Fn , αn ], αn = fn α , fn 2 fn Ω(Gn )/Gn. ? ) Φσ ([Fn, αn]) = Gn ! n. , ! [Gn] ! dT , ! [Fn, αn] 2 ! dT Th n → ∞. $ ! Mh,2k [Fn, αn] ∈ M0 9 n, M0 2 ! Mh,2k ! [F, α](W,M ),Θ . ? M0 [Fn, αn] ! dT [F0, α0] ∈ M0. A "! Φσ ! + [Gn] = Φσ ([Fn, αn ]) 2 ! dT n → ∞ Φσ ([F0, α0]) = [G] ∈ Φσ ([F, α](W,M ),Θ ), Vh,2k ! Vσ . - 2 Φσ 2 " M0 :)# Θ∗σ − ! ! [F, α](W,M ),Θ ) Φσ (M0) ≡ Φσ ([F, α](W,M ),Θ ). A2 Φσ ! M0 (Φσ )−1 : Φσ (M0) → M0 "! - [F, α] = [F , α ] ! ! , # ) Th, [F, α] [F , α ] 2 M0 [G] = [G ], [F, α] [F , α ] 2 [G] [G ] ! ! , ! "# q : F → F. , [F , α ] ! ! [F, qα ], [F, qα ] 2 M0 2 [G]. . ! qα = αψ, ψ ∈ Θσ . 7 ψ 2 ! 2 ) F0, , # ) [F, α] [F, qα ] 2 ! " [F, α](W,M ),Θ . - ! ! ) 2 M0. / ) [F , α ] = [F, qα ] = [F, α] ! Th . , Φσ ! M0 . 1 ! "! (Φσ )−1 ! ! ! Vh,2k . / ) Φσ M0. ; " !"9 " Mh,2k : " (W, M)− ! ! ! # , # ) 6 " / ) ! # " ModTh :! " " W − ! ! 6 ) " ! " 2 # :, # ) 6 fφ, +" ! W − ! " " ! ∗ :! " * ! φ(z)dz2 F ∈ Mh,2k " 2 W − ! ! 6 / ) ) " ! " Vh,2k 2 " 1 # ! V1
V2 )" ! " Vh,2k . $+ M1 M2 " Mh,2k , 2 V1 V2 ! ! - g
∗ σ
∗ σ
∗ σ
∗ σ
∗ σ
2 " M1 M2. , V1 2 V2 ! Φσ g(Φσ )−1; (Φσ )−1 ! V1 M1 . 1 Vh,2k , ! !! # " Vh,2k . ; ! ! , # ) Th,2m, ! ! , # ) h 2m = 2kN " :!" " 6 ! ) ) , 2 ! '( 2 ! !2 im Th,2m !) M0 2 ! Mh,2k , ! ! 3 h − 3 + 2m. 1 # ! !2 ! ) - [F, α] ! , # ) Th,2m, # ξ1, ..., ξ2m !) " :!" " 6 α : F0 → F = F0 = F0/W0 , M0 0 ! ! Th,2m ξ10, ..., ξ2m , " !) * π0 : F0 → F0/W0, M0 ! 2" ! M0 F0. -! [F, α] ! ! ! ! ! [F , α], !)) ! # "!) # F ! ! ξ j , j = 1, ..., 2m, ! ξj , j = 1, ..., 2m. A ! # "!) # F # ! ! " ! M ! "# * !# " F . / α "!) # ! F ! π1α = απ10, π1 : F → F, π10 : F0/W0 → F0/W0, M0. , [F , α] ! !+) N − ) "!)) [F, α], # * ! ! M, # ! * W , 2+ ! W. ! W, !) # " ! N − "
F F , 2 " : N ) * !# " F. 02 ! ! 2" M F 2 " ! ! ) 2 ! 2k = N 2m G ξ1 ∈ π−1(ξ1), ..., ξ2m ∈ π−1(ξ2m), W l {ξ1 , ..., ξ2m}, l = 1, ..., N − 1, π : F → F. - ! ! ! 2 ! ! 2" M F # ! " W . - " ! " M W ) F ) / α F ! ! ! ) πα = απ0. / ) [F, α] ∈ Mh,2k . / ! [F, α] ! # !# " M0 = [F, α](W,M ) ! ! [F/W, M, π(ξj ), j = 1, ..., 2k; α1] Th,2m,
α1π0 = πα * # π : F → F/W, M π0 : F0 → F0/W0, M0; M(ξj ) = ξj , j = 1, ..., 2k, F. $ !# " Mh,2k !2 im ! 1 # ! !) ! " ! , # ) [F, α] [F1, α1] Th,2m, " im([F, α]) = im([F1, α1]) = [F, α], ) !2
! ! F = F/ < W, M >= F1. / " ! F F1 2 ! *
! 2" M F. - α F ! ! ! ! ! G πα = α π 0 , πα = α 1 π 0 .
/ ) α = α1 ! [F, α] = [F1, α1] ! Th. - ! !" " 2 ! Mh,2k " + ! 2 ! Mh,2k . ; Φσ M0, 2 Φσ (M0) Vh,2k ! ∗ Vh,2k , ! + Vh,2k . 1 ! F ∈ Mh,2k
+ ! Θ∗σ − ! ! # F. . ! Vh,2k + $ ! 2 Vh,2k ! " 2 Q2k ! Qσ ! ϕ0. h 1 ! Mh,2k ! 2 " ! Th, ! ! !) ! ! C B # # G ! ! ΩW,M (F ), (W, M)− ! " " ! " * ! F. - Mh,2k ! Th . , % & .! ! ! C' ' ! 2 2 ! 4' ! + !"9 / ! 2 2 ! T (X)Θ(H), !) ! 9 # " 2 ! Mh,2k H =< W, M >, ! " !" ! ! , # ) Th. % & $ 8 " "
Q2kh 2 2 ! " EST −" G, ) # ! [F, α] ∈ Mh,2k . ; ! C( G
) ! G G =< W1N ; T1, ..., T2g; U1, V1, ..., U2s, V2s ; A1, ..., Am; T1,2, ..., T2g,N ; U1,2, V1,2, ..., U2s,N , V2s,N ; A1,2, ..., Am,N : [Uj , Vj ] = 1, [Ujn, Vjn ] = 1, j = 1, ..., 2s, n = 2, ..., N >,
M1(z) = −z, W1(z) = αz, α ∈ C, |α| > 1, Tg+i = M1TiM1, i = 1, ..., g, −(n−1)
Us+j = M1 Uj M1 , Vs+j = M1 Vj M1 , j = 1, ..., s, Ti,n = W1n−1TiW1 ,i = −(n−1) −(n−1) n−1 n−1 , Vj,n = W1 Vj W1 ,j = 1, ..., 2g, n = 2, ..., N, Uj,n = W1 Uj W1 −(n−1) n−1 , l = 1, ..., m, n = 2, ..., N. 1, ..., 2s, n = 2, ..., N, Al,n = W1 Al W1 $ ## ! ! " EST − = BGB −1 , B [G] !" # ! G
# 2 ! # " ! 2" ) T1 g = 0 : ! ))6 ! ! 2" ) W1 ! D ∞ 2 ) B # ! D ∞ ! ! A !"# ) " G 2 2 ! ) " 3N (2(g + s) + m) " : # : " G) 1 2 1 2 , ξg+1 , kg+1, ..., ξ2g , ξ2g , k2g ; (α; ξ12, k1, ξ21, ξ22, k2, ..., ξg1, ξg2, kg ; ξg+1
1 1 , p2s+1, ..., p12s, q2s , p22s; ξˆ11, ξˆ12, kˆ1, ..., p11, q11, p21, ..., p1s , qs1, p2s ; p1s+1, qs+1 1 ˆ2 ˆ 1 2 1 2 1 , ξm, km ; ξ1,2 , ξ1,2 , k1,2, ..., ξ2g,N , ξ2g,N , k2g,N ; p11,2, q1,2 , p21,2, ..., ξˆm p1 , q 1 , p2 ; ξˆ1 , ξˆ2 , kˆ1,2, ..., ξˆ1 , ξˆ2 , kˆm,N ), 2s,N
2s,N
2s,N
1,2
1,2
m,N
m,N
(5)
* 2 W1< " " ! 2" Ti, Ti,k , Al, Al,k ! ! i = 1, ..., 2g, k = 2, ..., N, l = 1, ..., m, ) T1 g = 0 + ! ξ11 = 1; ki 2 Ti , i = 1, ..., 2g; ki , |ki| = 1, ! z+b k + k1 = (aj + dj )2 − 2 ! |k| < 1 :Ti (z) = ac z+d , ai di − bi ci = 1, ci = 0, i = 1, ..., 2g); kˆl , ki,n , kˆl,n 2 Al , Ti,n, Al,n ! ! 1 l = 1, ..., m, i = 1, ..., 2g, n = 2, ..., N ; p1j , qj1 , p2j , p1j,n , qj,n , p2j,n * " −1 2 # Uj , Uj−1, Vj , Uj,n, Uj,n , Vj,n ! ! α
∈
C, |α|
>
1,
1 2 1 ˆ2 (ξi1, ξi2), (ξi,k , ξi,k ), (ξˆl1, ξˆl2), (ξˆl,k , ξl,k )
i
i
i
i
j = 1, ..., 2s, n = 2, ..., N.
M" !" :E6 ! " ) " G !" ! " , !" W1; T1, ..., Tg ; U1, V1 , ..., Us, Vs ; A1, ..., Am.
. ! ! ! ! " # " G !" ) # 3g + 3s + 2m " (α, ξ12, k1, ξ21, ξ22, k2, ..., ξg1, ξg2, kg ; p11, q11, p21, ..., 1 ˆ p1s , qs1, p2s ; ξˆ11, kˆ1, ..., ξˆm , km).
(6)
, + Q2kh ! 3g + 3s + 2m = 3h − 3 + 2m. % & ' $ " : 6 ! :E6 ! ! :46 !"2) ! " $" 9 ! 9 2 g = 0 : EST −" G 1 2 6 ξg+i = M1 (ξi1) = −ξi1, ξg+i = M1 (ξi2) = −ξi2 , kg+i = ki , i = 1, ..., g; C6 " G N N 1 M1, p1s+j = −p1j , qs+j = −qj1 , p2s+j = −p2j , j = 1, ..., s; (6 ξˆl2 = −ξˆl1, l = 1, ..., m; k k '6 ξi,n = αn−1ξik , ξg+i,n = −αn−1ξik , i = 1, ..., g, n = 2, ..., N, k = 1, 2; ki,n = ki, i = 1, ..., 2g, n = 2, ..., N ;
E6 " G N N 2 1 1 W1, pkj,n = αn−1pkj , qj,n = αn−1qj1 , pks+j,n = −αn−1 pkj , qs+j,n =
−αn−1qj1 , j = 1, ..., s, n = 2, ..., N, k = 1, 2; 1 2 1 46 ξˆl,n = αn−1ξˆl1 , ξˆl,n = −ξˆl,n = −αn−1ξˆl1 , kˆl,n = kˆl , l = 1, ..., m, n = 2, ..., N.
- # :46 "# ! " ) " G ! ! ! ! " G W1N (z) = αN z, ξj1 (z − ξj2 ) − kj ξj2 (z − ξj1 ) , j = 1, ..., g, Tj (z) = z − ξj2 − kj (z − ξj1 ) 1
1 2
i) qi1 z − (pi +q 4 , i = 1, ..., s, Ui(z) = z − p1i 1
1 2
i) (p1i + qi1 − p2i )z − (pi +q 4 Vi (z) = , i = 1, ..., s, z − p2i ˆ1 ˆ ˆ1 1 z + ξl + kl (z − ξl ) ˆ Al (z) = ξl , l = 1, ..., m. z + ξˆl1 − kˆl (z − ξˆl1 )
0 Q2kh ! # 3h − 3 + 2m !
Qσ ⊂ C3h−3, σ = (N (2g + m) + 1, 2N s, 0, ..., 0), h = |σ|. 1/;=,7 H.,$/ , ! C' ' Q2kh ! 2 ! !) # " ! Qσ . A " : # 6 ! :E6 ) ) EST −" G ! Qσ . 02 ! Q2kh ! 2 !
Qσ , 9 2 EST −" G, !" " !
C'E ) ! ) *
# " Qσ , ! - :46 3h − 3 + 2m ! " " # ! # ! :E6 ! " ) ) Q2k ! Qσ . ; 2 !2" !"9 :! h C' '
C'E6 Q2kh ! 3h − 3 + 2m. , - F ! ! h ≥ 2, ) ) ! " ! G P N1 ≥ 2 ! ! 2" F ; M C 2k, k ≥ 2, ! 2" F, ! " !) ! 2" P, + P l = M ! N1 = 2l, l ∈ N. $ " ! " # " < W, M >, + # " " ! ! < P, M > . , 2 h ! 1 F/ < P, M > . ; ! ! 2 ! Mh,2k 2 ! ! ! , # ) Th, " " !" ! # h ≥ 2, ) ! " ! P M, ! !! + 9 P − M− ! ! , (P, M)− ! ! ! # ! ! ) " " !" ! 1 # : ! 6 ! + 2 ! Mh,2k + " (P, M)− ! ! " ! # , # ) 1 ' 6 02 ! Mh,2k ! " ! Th # 3h+2m− 1 = 3 h−3+(2m+2), m = k−1 l , + !"
$
" < 1 C6 02 ! Q2kh,1 = Φσ (Mh,2k ) ⊂ Qσ ! " # 3h + 2m − 1, + !" " 1/;=,7 H.,$/ 1 ! " ! # ) * !# ! !2+# " ! ! ! ! C' ' C' E - 9 " ! ! " "!) ) !# ! " ! ! P M. > ! !! 1 # " Mh,2k . ; ! ! , # ) Th,t, ! ! , # ) h t = 2m + 2 " :!" " 6 ! ) ) , 2 ! '( 2 ! !2 Th,t !) M0 1 2 ! Mh,2k : "# (P, M)− ! ! " ! # , # ) 6 ! ! 3h − 3 + t = 3h − 1 + 2m. - [F, α] ! , # ) Th,2m+2 2m + 2 , " α : F0 → ξ1 , ..., ξ2m, ξ2k−1, ξ2k α F. = F0 = F0/ < P0 , M0 > ! ! Th,2m+2. -! F ! ! ! ! ! F , !)) ! # "!) # F ! ! ξ1, ..., ξ2m, ξ2k−1, ξ2k ! ξ1, ..., ξ2m, ξ2k−1, ξ2k . A ! # ! F # ! ! M :! 6 !) # " "# * !# " , F l− ) "!)) F, π1 : F → F , ! ! ξ2k−1, ξ2k l − 1 ξ2k−1, ξ2k . 02 ! ξ1 ∈ π1−1 (ξ 1 ), ..., ξ2m ∈ π1−1 (ξ 2m ), P i{ξ1 , ..., ξ2m}, i = 1, ..., l − 1, ξ2k−1, ξ2k + 2ml + 2 = 2k ! 2" * ! ! M F. $! 2 ! ! 2" M F
! # ! # ! P, ! ) " ! l− "
F F . - ! " M P, " * !# " F, ! - + α ! F ! ! ! ) πα = απ0 , π0 : F0 → F0/ < P0, M0 >, π : F → F. / )
1 [F, α] ∈ Mh,2k .
/ ! [F, α] ! # !# " M0 = [F, α](P,M ) ! ! [F/ < P, M >, π(ξj ), j = 1, ..., 2k; α1] Th,2m+2, M(ξj ) = ξj , j = 1, ..., 2k, P (ξ2k−1) = ξ2k−1, P (ξ2k ) = ξ2k F α1 ! ! ! ) πα = α1 π 0 , π : F → F/ < P, M >, π0 : F0 → F0/ < P0 , M0 > . $ !# 1 " Mh,2k !2 ! > ! M0 ! 1 2 ! Mh,2k + ! 2 ! 1 Mh,2k . , 2 ! ! " C' ' 2 1 1 2 ! Vh,2k = Φσ (Mh,2k ), Φσ : Th → Vσ , +
!" " " 2 # 1 3 h−1+2m. $! 2 ! Vh,2k
Q2kh,1, ! ! 2 1 Vh,2k ! !" 2 Q2kh,1 ! Qσ . 8 Q2kh,1 2 2 ! ! ! " # 1 EST −" G, ) # ! [F, α] ∈ Mh,2k . , G ) !
G =< T1, ..., T2g; U1, V1, ..., U2s, V2s; A1, ..., Am; T1,2, ..., T2g,l; U1,2, V1,2, ..., U2s,l, V2s,l ; A1,2, ..., Am,l :
[Uj , Vj ] = 1, [Ujn, Vjn] = 1, j = 1, ..., 2s, n = 2, ..., l >, M1 (z) = −z, P1 (z) = z exp( 2πi N1 ), Tg+i = M1 Ti M1 , i = 1, ..., g, −(n−1)
Us+j = M1 Uj M1, Vs+j = M1 Vj M1 , j = 1, ..., s; Ti,n = P1n−1 TiP1 ,i = −(n−1) −(n−1) n−1 n−1 , Vj,n = P1 Vj P1 , j = 1, ..., 2s, At,n = 1, ..., 2g, Uj,n = P1 Uj P1 −(n−1) n−1 , t = 1, ..., m, n = 2, ..., l. P1 AtP1
$! + ! 2 ! # " ! 2" ) T1 g = 0 : !) )6 !# - ! 2" P1 2 !" D ∞. $ ! ) " G !" ! " T1, ..., Tg; U1, V1, ..., Us, Vs; A1, ..., Am. F 2 2 ! ) " 3g + 3s + 2m − 1 ! " " : #6 1 ˆ (ξ12, k1, ξ21, ξ22, k2, ..., ξg1, ξg2, kg ; p11, q11 , p21, ..., p1s , qs1 , p2s ; ξˆ11, kˆ1, ..., ξˆm , km).
(7)
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Sa
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n^bS\_b
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y s
b_b\`Z\U`` ]S_cZ`
p\Uc)^
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V R
W_`)ScZ_cYSY`
jg[[bZ\_)
o\SY]`
Uce
b[pbee_co
V_b[Ucc
Sa
`Y\aU)b`
V_b[Ucc Rcc
Sa
`Y\aU)b`
XUZ^
!!
jZYe_b`
< (&
1 j)^SZZwg t lbpb\ b_cb `]b)_bTTb tYcwZ_Sc ubT)^b pb_ b_cb\ pb`Z_[[Zbc T_cbU\bc n\Uc`aS\[UZ_Sc _^\b` R\oY[bcZ Ycmb\Ubceb\Z pTb_pZ !! d Vb_cb Rcobu XUZ^ 22( x C O **( *(* 4 j_pcb\ V d lc_aS\[_{UZ_Sc Sa `g[[bZ\_) V_b[Ucc `Y\aU)b` pg j)^SZZwg o\SY]` !! n\Uc` R[b\ XUZ^ jS) &'4 h ' < 1 O (& 24 ' U~c\gp x R `_[]Tb ]\b`bcZUZ_Sc aS\ Z^b [U]]_co )TU`` o\SY] Sa Uc S\_bcZUpTb `Y\aU)b !! `\ d XUZ^ &2, h 14 < * , O 4( (1 ( b_b\`Z\U`` hS\Tb`Ycobc Ybpb\ n^bS\_b eb\ RpbT`)^b n\Uc`)bcebcZbc rb` XUZ^ b\wb xUce 1 xb\T_c &C* ', ) 2 bgT y W_b ebb eb\ V_b[Ucc`)^bc tTUb)^b jZYZoU\Z$ nbYpcb\ &44 '* ) & F0I - @3 3 0 E !! 8 / 999: &(2 ; *1, < , O 422 4& *C F0I 6 Jo ` [K !! 9 0 &(& ; *C < , 9 ',* '1C * F0I 3 0 E !! 9 0 &2 ; ** < 4 9 &C *C4 ** F0I E !! 9 0 &2* ; *, < ' 9 &' &( 8 %/%;% *C 2* *
*, F0I @ @
+7 3 0 !! ?
$ # 0 07 0@ E
E?5 && O 2 , *1 F0I @ @
@0 @N 3 0 ! E - !! /?5 *CC* ; * , 9 *( *4 F0I #0 0+ @ + F * E $ E?5 *CC, *12 )
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