cz+d d (ac + cd )z + bc + dd c z + d

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5 T (z) = az + b cz + d ; a, b, c, d C, ad bc 0 ( ) a b M T (z) = (z) az + b c d cz + d (T T )(z) = T (T (z) (T T )(z) = az+b a + cz+d b c az+b + = (aa + cb )z + a b + b d a z + b cz+d d (ac + cd )z + bc + dd c z + d a = aa + cb, b = a b + b d, c = ac + cd, d = bc + dd

6 a d c b = (a d b c ) (ad bc) GL(2, Z) 2 2 T M T T M T ( ) ( ) ( ) a b M T M T = a b aa c d = + b c a b + b d c d c a + d c c b + dd T 1 (z) = dz b cz + a T (T ( 1) z) = a dz b + b cz+a (bc ad)z = + d (bc ad) = z c dz b cz+a ( ) ( a b M T = M 1 1 d b c d T = ad bc c a ) SL(2, C) M T = 1 ad bc = 1 SL(2, C) D = {z : z < 1} D w = T (z) = e iθ z z 0 1 z 0 z T (z) = z z 0 1 z 0 z = z z 0 z z 1 z 0 z = e iθ z = 1 z 1 = z z 1 z 0 = z z 0 T (z) = z z 0 z z 0 = 1

7 z < 1 T (z) < 1 D T 1 (w) = w + eiθ z 0 z 0 w + e iθ = e iθ w w 0 1 w 0 w w 0 = e iθ z 0 D z 1, z 2, z 3, z 4 w i = T (z i ), i = 1, 2, 3, 4 (w 4 w 1 )(w 2 w 3 ) (w 1 w 2 )(w 3 w 4 ) = (z 4 z 1 )(z 2 z 3 ) (z 1 z 2 )(z 3 z 4 ) w 2 w 1 = (az 2 + b) (cz 2 + d) (az 1 + b) (cz 1 + d) = (z 2 z 1 ) (cz 2 + d)(cz 1 + d) z 2 z 1 z 1, z 2, z 3 T (z) T (z 1 ) = 0, T (z 2 ) = 1, T (z 3 ) = T (z) = (z z 1)(z 2 z 3 ) (z 1 z 2 )(z 3 z) T (z 1 ) = 0, T (z 2 ) = 1, T (z 3 ) = a = z 2 z 3, b = z 1 (z 3 z 2 ), c = z 2 z 1, d = z 3 (z 1 z 2 ) ad bc 0 a, b, c, d z T (z) = (ad bc) cz + d 2

8 az + b cz + d = z az + b = cz 2 + dz cz 2 + (d a)z b = 0 z 1,2 = a d ± (a d) 2 + 4bc 2c = a d ± (a + d) 2 4 2c ad bc = 1 a + d = 2 z 0 = a d, αν a + d = 2 2c z 1,2,3 w 1,2,3 z 1 z 2 z 3 z 1 w 1 w 2 w 3 w 1 w 1 = z 2 w = T (z) w w 1 w w 2 w3 w 2 w 3 w 1 = z z 1 z z 2 z3 z 2 z 3 z 1 w = T (z) U(z) = z z 1 z z 2 z3 z 2 z 3 z 1 z 1,2,3 w 1 = 0, w 2 =, w 3 = 1 w 1 = U(z 1 ) = 0, w 2 = U(z 2 ) =, w 3 = U(z 3 ) = 1 w = Q(z) = az + b cz + d a, b, c, d az 1 + b cz 1 + d = 0, az 2 + b cz 2 + d =, az 3 + b cz 3 + d = 1

9 a = b z 1, c = d z 2, b d = z 1 z 2 z 3 z 2 z 3 z 1 Q(z) = az + b cz + d = b z 1 z + b d z 2 z + d = = z z 1 z3 z 2 z z 2 z 3 z 1 Q(z) U(z) d z dω = (dθ) θ (dϕ) 2 z = θ 2 eiϕ 1 dz = 2 2 θ/2 eiϕ dθ + ie iϕ dϕ dz d z = θ/2 dθ2 + 2 θ 2 dϕ2 dω = (dθ) θ (dϕ) 2 = z = az + b cz + d 4 (1 + z z ) 2 dz d z z, z dz = dz (cz + d) 2, d z = 4 (1 + z z) 2 dzd z d z ( c z + d) 2

10 dz d z = dzd z (cz + d) 2 ( c z + d) 2 (1 + z z ( ) = 1 + az + b ā z + b ) cz + d c z + d = (cz + d)( c z + d) + (az + b)(ā z + b) (cz + d)( c z + d) dω = 4K 2 dz d z (z, z) (1 + z z ) 2 K(z, z) = (cz + d)( c z + d) + (az + b)(ā z + b) 1 + z z

11 A(z) = az + b cz + d ; ad bc = 1, a, b, c, d Z Γ ( a b A = c d ), ad bc = 1; a, b, c, d Z P SL(2, Z) SL(2, Z) {1. 1} A, A T : z z = z + 1 S : z z = 1 z 2 2 T = ( ) ( 0 1, S = 1 0 T : z z = z + 1 = z + 2, T n : z z (n) = z + n ) 2 2 ( ) 1 n T n = 0 1 S 2 = (ST ) 3 = ( ) A Γ Γ S T A = T n 1 ST n 2 S T n r

12 F F Γ A F τ, τ H τ = Aτ F Γ A F G G H τ H τ G τ L = {mω 1 + nω 2, m, n R} (ω 1, ω 2 ) ( ) ( ω2 a b = c d ω 1 ) ( ω 2 ω 1 ) ω 1 < ω 2 < ω 1 ± ω 2 L 0 < z 1 < z 2 < < z k < z 1 = ω 1 ω 2 ω 1 0, ω 1, ω 2 L (ω 1, ω 2 ) a, b, c, d ad bc = ±1 L = {mω 1 + nω 2, m, n R} (ω 1, ω 2 ) ( ) ( ω2 a b = c d ω 1 ) ( ω 2 ω 1 ) ad bc = 1 c = c, d = d, ω 1 = ω 1 a d b c = 1 ω 1 = z 1 ω 1 < ω 2 ω 2 < ω 1 ± ω 2

13 ω 1 = 1, ω 2 = τ ω 2 = aτ + b, ω 1 = cτ + d ω 2 /ω 1 = τ τ = aτ + b cτ + d ( a b τ = c d ) τ 1 τ τ ± 1 τ + τ < 1 F = {τ H : τ > 1 > τ + τ } Γ τ H τ F τ τ τ τ F (τ (τ) ) = (ac bd) cτ + d = 2 (τ) cτ + d 2 D = cτ + d 2 = c 2 τ 2 + cd τ + τ + d 2 c 2 cd + d 2 a, b, c, d c, d c 0 d = 0 D = c 2 τ 2 > c 2 1

14 d 0 D c 2 cd + d 2 = ( c d ) 2 + cd cd 1 c 0 cτ + d > 1 A Γ τ, τ F τ = aτ + b cτ + d, c 0 τ < τ τ = aτ b cτ + d c 0 τ < τ c = 0 ad = 1 a, d = ±1 ( ) ±1 b A = = T ±b 0 ±1 b = 0 τ = τ F y -½ 0 ½ x