(ω
1
, ω
2
)
g
2
, g
3
B
k
ζ(s)
σ
τ
∆ = g
3
2
− 27g
2
3
2
(ω
1
, ω
2
), g
1
, g
2

.
E
E.
G
k
,
ζ(2k), δ
k
(n), τ(n).
(ω
1
, ω
2
)
ω
1
ω
2
ω
1
/ω
2
nω
1
+ mω
2
m, n ω
1
ω

2
Ω (ω
1
, ω
2
)
(ω
1
, ω
2
)

ω
′
1
, ω
′
2

ω
1
/ω
2
ω
′
1
/ω
′
2
Ω (ω

1
, ω
2
) = Ω

ω
′
1
, ω
′
2

(ω
1
, ω
2
) (ω
′
1
, ω
′
2
)
2 × 2

a b
c d

ad − bc = ±1,


ω
′
2
ω
′
1

=

a b
c d

ω
2
ω
1

,

ω
′
2
= aω
2
+ bω
1
ω
′
1
= cω

2
+ dω
1
.
(ω
1
, ω
2
) (ω
′
1
, ω
′
2
)
Ω (ω
1
, ω
2
) = Ω

ω
′
1
, ω
′
2

.
a, b, c, d, a

′
, b
′
, c
′
, d
′

ω
′
2
= aω
2
+ bω
1
ω
′
1
= cω
2
+ dω
1
(ω
1
, ω
2
)

ω
′

2
ω
′
1

=

a b
c d

ω
2
ω
1

.

ω
2
ω
1

=

a
′
b
′
c
′

d
′

ω
′
2
ω
′
1

.

ω
′
2
ω
′
1

=

a b
c d

ω
2
ω
1

=


a b
c d

a
′
b
′
c
′
d
′

ω
′
2
ω
′
1

.
A =

a b
c d

, A
′
=


a
′
b
′
c
′
d
′

, X = A.A
′
=

x y
z t

,

ω
′
2
ω
′
1

= X

ω
′
2

ω
′
1


ω
′
2
= xω
′
2
+ yω
′
1
ω
′
1
= zω
′
2
+ tω
′
1
⇔

ω
′
2
(1 − x) = yω
′

1
ω
′
1
(1 − t) = zω
′
2
ω
1
/ω
2

x = t = 1
y = z = 0
⇒ X =

1 0
0 1

= A.A
′
,
det A = ±1 a, b, c, d, a
′
, b
′
, c
′
, d
′

2×2,

a b
c d

ad − bc = ±1

ω
′
2
= aω
2
+ bω
1
ω
′
1
= cω
2
+ dω
1
,
(ω
1
, ω
2
) (ω
′
1
, ω

′
2
)
α Ω = Ω(ω
1
, ω
2
)

ω∈Ω,ω=0
1
ω
α
α > 2
(ω
1
, ω
2
)
ω
1
± ω
2
, −ω
1
± ω
2
.
R r O
ω = 0 ω

1
ω
2
r ≤ |ω| ≤ R.
ω
1
ω
2
2r ≤ |ω| ≤ 2R.
3r ≤ |ω| ≤ 3R.
S(n) =





1
ω
α




,
8(1 + 2 + 3 + + n)
8

1
R
α

+
2
(2R)
α
+ +
n
(nR)
α

≤ S(n) ≤ 8

1
r
α
+
2
(2r)
α
+ +
n
(nr)
α

,
8
1
R
α
n


k=1
1
k
α−1
≤ S(n) ≤ 8
1
r
α
n

k=1
1
k
α−1
.
(ω
1
, ω
2
)
S (n) 8
1
R
α
ζ (α − 1)
8
1
r
α
ζ (α − 1) ,

8
1
r
α
ζ (α − 1) ,
α > 2.
8
1
R
α
ζ (α − 1)
8
1
r
α
ζ (α − 1) .
α > 2.
α ≤ 2.
α > 2, R > 0,

|ω|>R
1
(z − ω)
α
|z| ≤ R.
M R, α
α ≥ 1
1
|z − ω|
α

≤
M
|ω|
α
,
|ω| ≥ R, z |z| ≤ R.




z − ω
ω




α
≥
1
M
.
M, ω |ω| > R.
|ω| = R + d, d > 0.
|z| ≤ R |ω| ≥ R + d,




z − ω
ω





=



1 −
z
ω



≥ 1 −



z
ω



≥ 1 −
R
R + d
.





z − ω
ω




α
≥

1 −
R
R + d

α
=
1
M
.
g
2
, g
3
M =

1 −
R
R + d

−α

,
g
2
, g
3
k k ≥ 2,
G
k
=

ω=0
1
ω
2k
g
2,
g
3
g
2
= 60G
2
, g
3
= 140G
3
.
∆
∆ = g
3

2
− 27g
2
3
.
g
2
, g
3
∆
ω
1
, ω
2
g
2
= g
2
(ω
1
, ω
2
) , g
3
= g
3
(ω
1
, ω
2

) , ∆ = ∆ (ω
1
, ω
2
) .
g
2
, g
3
λ = 0
g
2
(λω
1
, λω
2
) = λ
−4
g
2
(ω
1
, ω
2
) ,
g
3
(λω
1
, λω

2
) = λ
−6
g
3
(ω
1
, ω
2
) ,
∆ (λω
1
, λω
2
) = λ
−12
∆ (ω
1
, ω
2
) .
∆ λ =
1
ω
1
z =
ω
2
ω
1

,
g
2
(1, z) = ω
1
4
g
2
(ω
1
, ω
2
) ,
g
3
(1, z) = ω
1
6
g
3
(ω
1
, ω
2
) ,
∆ (1, z) = ω
1
12
∆ (ω
1

, ω
2
) .
g
2
, g
3
, ∆ ω
1
, ω
2
z =
ω
2
ω
1
H = {z| Im z > 0} .
k
z :
G
k
(z) =

(m,n)=(0,0)
1
(mz + n)
2k
, m, n ∈ Z.
C H,
H = {z ∈ C| Im z > 0} .

SL
2
(R)

a b
c d

ad − bc = 1,
SL
2
(R) =

a b
c d

|a, b, c, d ∈ R, ad − bc = 1

.
SL
2
(R)
∼
C
= C ∪{∞}
g =

a b
c d

SL

2
(R), z ∈
∼
C
. gz =
az + b
cz + d
,
Im gz =
gz − g
−
z
2
=
1
2


az + b
cz + d
−
a
−
z
+ b
c
−
z
+ d



=
Im z
|cz + d|
2
.
• z ∈ H, g =

a b
c d

∈ SL
2
(R) gz ∈ H, H
SL
2
(R) .
• −1 =

−1 0
0 −1

H.
SL
2
(Z) , SL
2
(R) ,
Z.
G = SL

2
(Z) / {±1}
g =

a b
c d

SL
2
(Z)
G.
S =

0 −1
1 0

T =

1 1
0 1

G,
z ∈ H
Sz = −
1
z
, T z = z + 1, S
2
= 1, (ST )
3

= 1.
D D =

z ∈ H




|Re z| ≤
1
2
, |z| ≥ 1

.
D G,
z ∈ H g ∈ G gz ∈ D.
z, z
′
D, z z
′
z ≡ z
′
(modG)
Rez = ± 1/2 z = z
′
± 1
|z| = 1 z
′
= −1/z.
z ∈ D I (z) = {g ∈ G| gz = z }. I (z) = {1}

z = i, I (z) S
z = ρ = e
2πi/3
, I (z) ST.
z = −
ρ = e
πi/3
, I (z) T S.
D → H/G
D
G S T.
G
′
G S T.
z ∈ H, g
′
∈ G
′
g
′
z ∈ D.
g =

a b
c d

∈ G
′
.
Im (g ) =

Im
|c + d|
2
.
R > 0, R,
cz + d. cz + d
|cz + d| < R.
cz + d, |cz + d|
g
′
∈ G
′
, (gz) T
n T
n
gz = z
′
−1/2 1/2,
z
′
∈ D.



z
′



< 1, z

′
= re
iϕ
, |r | < 1, Im z
′
= r sin ϕ.
Sz
′
=
−1
z
=
−1
r
e
−iϕ
=
1
r
e
(π−ϕ)i
,
Im Sz
′
=
1
r
sin ϕ > Im z
′
= Im (T

n
g
) = Im(gz).


z
′


≥ 1, g
′
= T
n
g, g
′
z ∈ D,
z ∈ D g =

a b
c d

∈ G, gz = z
′
∈ D.
Im(gz) ≥ Im z,
(z, g)

gz, g
−1


.
Im (gz) =
Im z
|cz + d|
2
≥ Im z,
|cz + d| ≤ 1.
z ∈ D, |z| ≥ 1 |cz + d| ≤ 1 ⇒ c ∈ {−1, 0, 1} .
c = 0,
d = ±1, a = ±1
g ±b, z, gz = z
′
D
b = ±1 ⇒ Re z = ±1/ 2.
c = 1, |z + d| ≤ 1 d = 0, z = ρ, d = 1, 0
z = −ρ d = 1, 0 d = 0 : |z| ≤ 1,
z ∈ D
|z| = 1, g =

a b
c 0

∈ SL
2
(Z) ,
b = −1, gz = a − 1/z. a = 0,
z = ρ, d = 0, −1 z = −ρ a = 0, 1
d = 1, z = ρ :
a − b = 1, gρ =
aρ + 1

ρ + 1
= a −
1
ρ + 1
= a + ρ,
a = 0, 1. d = −1, z = ρ + 1,
c = −1, c = 1 a, b, c, d.
z ∈ D, I (z) = {g ∈ G| g = z} .
z gz z D
Re z = ±1/2 ⇒ gz = z + 1,
g /∈ I (z) , g = 1.
|z| = 1
gz = z = −1/z,
z = i, I (z) S
z = ρ = e
2πi/3
, I (z) ST,
z = −
ρ = e
πi/3
, I (z) T S.
g ∈ G, z
0
D
z = gz
0
. g
′
∈ G
′

, g
′
z ∈ D.
z
0
g
′
z = g
′
gz
0
D G, z
0
= g
′
gz
0
⇒ g
′
g = 1 ⇒ g = g
′ −1
∈ G
′
G ⊂ G
′
. G = G
′
S, T
k f H
2k f H,

f (z) = (cz + d)
−2k
f

az + b
cz + d

.
g =

a b
c d

∈ SL
2
(Z) , gz =
az + b
cz + d
.
d (gz)/dz = (cz + d)
−2
,
f (gz)
f (z)
= (cz + d)
−2
=

d (gz)
dz


−k
,
f (z) (dz)
k
= f (gz) (d (gz))
k
.
f (z) (dz)
k
G. G
S, T, S, T.
f H
2k f z ∈ H :
1)f(z + 1) = f (z)
2)f(−1/z) = z
2k
f (z).
f q = e
2πiz
f
1
. f
1
0 < |q| < 1
f
f
1
(q) =
+∞


n=−∞
a
n
q
n
.
f
1
q = 0 a
n
= 0 n < 0 f
∞ ∞.
∞
f ∞, f(∞) = f
1
(0)
f ∞.
f H ∪ {∞}
f(∞) = 0 f
2k
f(z) =
+∞

n=0
a
n
q
n
=

+∞

n=0
a
n
q
2πizn
,
|q| < 1 Im(z) > 0
f(−1/z) = z
2k
f (z).
f a
0
= 0.
f
1
, f
2
2k
1
, 2k
2
,
f
1
.f
2
2k
1

+ 2k
2
.
k k ≥ 2, z ∈ H,
G
k
(z) =

(m,n)∈Z
2
|(0,0)
1
(mz + n)
2k
2k, G
k
(∞) = 2ζ (2k) .
G
k
(z + 1) =

(m,n+n)∈Z
2
|(0,0)
1
(m (z + 1) + n)
2k
=

(m,n+m)∈Z

2
|(0,0)
1
(mz + (n + m))
2k
= G
k
(z) .
G
k
(−1/z ) =

(m,n+n)∈Z
2
|(0,0)
1
(−m/z + n)
2k
= z
2k

(m,n)∈Z
2
|(0,0)
1
(nz − m)
2k
= z
2k
G

k
(z).
G
k
(z) H ∪{∞} . G
k
(z)
H k ≥ 2 H. G
k
(z) D
m = 0 , Im(z) → ∞, lim 1/(mz + n)
2k
= 0.
m = 0 , 1/n
2k
,
lim G
k
(z) =
+∞

n=−∞
1/n
2k
= 2
+∞

n=1
1/n
2k

=2ζ (2k) .
G
k
(z) ∞ G
k
(∞) = 2ζ (2k) .
∆(z) = g
3
2
(z) − 27g
2
3
(z), z ∈ H
g
2,
g
3
∆(z) H ∪ {∞} .

a b
c d

∈ SL
2
(Z) , z ∈ H.
∆

az + b
cz + d


=

g
2

az + b
cz + d

3
− 27

g
3

az + b
cz + d

2
=

(cz + d)
4
g
2
(z)

3
− 27

(cz + d)

6
g
3
(z)

2
= (cz + d)
12

g
3
2
(z) − 27g
2
3
(z)

= (cz + d)
12
∆(z).
∆(z)
∆(∞) = g
3
2
(∞) − 27g
2
3
(∞)
= [60 G
2

(∞)]
3
− 27[140G
3
(∞)]
2
= [60 .2ζ (4)]
3
− 27[140.2ζ (6)]
2
=

120.
8
4!
.
1
30
π
4

3
− 27

280.
32
6!
.
1
42

π
6

2
= 0.
∆(z)
f H, f = 0, p H.
n
f (z)
(z − p)
n
p, f p, υ
p
(f) .
f(z) p ⇔ υ
p
(f) = 0.
f(z) m p ⇔ υ
p
(f) = m > 0.
f(z) m p ⇔ υ
p
(f) = −m < 0.
g =

a b
c d

∈ G, f 2k,
f (z) = (cz + d)

−2k
f

az + b
cz + d

= (cz + d)
−2k
f (gz) .
z ∈ H cz + d = 0, f p f
g(p). υ
p
(f) p H /G.
υ
p
(f) υ
0
(f
1
(q)) . e
p
p
I(p)
I (p) = {g ∈ SL
2
(Z)| g (p) = p} .
e
p
=


2 , p ≡ i (modG)
3 , p ≡ ρ (modG)
1 , p ≡ i, ρ (mo dG)
.
f 2k, f = 0.
υ
∞
(f) +

p∈H/G
1
e
p
υ
p
(f) =
k
6
.
f
G. f
1
(q)
r > 0, f
1
(q)
0 < |q| < r.
f(z)
Im (z) >
1

2π
log
1
r
.
D ∩

Im (z) ≤
1
2π
log
1
r

,
f(z)
υ
∞
(f) +
1
2
υ
i
(f) +
1
3
υ
p
(f) +


p∈
H
/
G
∗
υ
p
(f) =
k
6
,

p∈H/G
∗
υ
p
(f)
p ∈
H
/
G


{i, p} .
1
2πi

df
f
,

D.
f
D, i, ρ, ρ + 1.
T, f G
i, ρ.
1
2πi

T
df
f
=

p∈
H
/
G
∗
υ
p
(f).
T :
ABB
′
CC
′
DD
′
EA
1

2πi

T
df
f
=
1
2πi




AB
df
f
+

BB
′
df
f
+

B
′
C
df
f
+


CC
′
df
f
+

C
′
D
df
f
+

DD
′
df
f
+

D
′
E
df
f
+

EA
df
f




1
2πi

EA
df
f
:
q = e
2πiz
= e
−2πy
e
2ix
, z = x + iy, −1/2 ≤ x ≤ 1/2.
w q = 0, |q| = e
−2πy
, arg q : π →
−π.
1
2πi

EA
df
f
=
1
2πi


w
df
1
f
1
= −υ
0
(f
1
) = −υ
∞
(f) .
1
2πi

BB
′
df
f
,
1
2πi

DD
′
df
f
,
1
2πi


CC
′
df
f
.
ρ
f (z) = (z − ρ)
υ
p
(f)
g (z) ,
g (ρ) = 0. υ (f) > 0, ρ f. υ
ρ
(f) < 0,
ρ f.
BB
′
,
z − ρ = re
iϕ
,
r α ≤ ϕ ≤ π/2, α r.
f
′
(z)
f (z)
=
υ
p

(f)
z − ρ
+
g
′
(z)
g (z)
.
1
2πi

BB
′
df
f
=
1
2πi

BB
′
f
′
(z)
f (z)
dz
=
1
2πi
α


π/2

υ
p
(f)
re
iϕ
+
g
′

ρ + re
iϕ

g (ρ + re
iϕ
)

rie
iϕ
dϕ
= −
υ
p
(f) (π/ 2 − α)
2π
+
r
2π

α

π/2

g
′

ρ + re
iϕ

g (ρ + re
iϕ
)

dϕ.
r → 0 ⇒ α → π/2 ,
r
2π
α

π/2

g
′

ρ + re
iϕ

g (ρ + re
iϕ

)

dϕ → 0.
1
2πi

BB
′
df
f
= lim
r→0
1
2πi

BB
′
df
f
= −
υ
ρ
(f)
6
.
1
2πi

DD
′

df
f
= lim
r→0
1
2πi

DD
′
df
f
= −
υ
ρ
(f)
6
.
1
2πi

CC
′
df
f
.
ρ
f (z) = (z − ρ)
υ
i
(f)

h (z) , h (i) = 0.
1
2πi

CC
′
df
f
= −
υ
i
(f)
2
.
T AB ED,
f (T z) = f (z) ,
1
2πi

AB
df
f
+
1
2πi

D
′
E
df

f
= 0.
1
2πi

B
′
C
df
f
+
1
2πi

C
′
D
df
f
.
S :
S (z) = −1/z , i → i, ρ → 1 + ρ,
B
′
C DC
′
, f (Sz) = z
2k
f (z) .
df (Sz)

f (Sz)
=
df (z)
f (z)
+ 2k
dz
z
.
1
2πi

B
′
C
df
f
+
1
2πi

C
′
D
df
f
=
1
2πi

B

′
C

df (z)
f (z)
−
df (Sz)
f (Sz)

=
1
2πi

B
′
C

−2k
dz
z

= −2k

−1
12

=
k
6
.

υ
∞
(f) +
1
2
υ
i
(f) +
1
3
υ
p
(f) +

p∈
H
/
G
∗
υ
p
(f) =
k
6
.
f
D. AB,
λ, T λ = 1 + λ DE.
B
′

C, ρ
1
, Sρ
1
= −1/ρ
1
C
′
D.
T λ, Sρ
1
,
λ, ρ
1
.
f D,
υ
∞
(f) +
1
2
υ
i
(f) +
1
3
υ
p
(f) +


p∈
H
/
G
∗
υ
p
(f) =
k
6
.
k ∈ Z, M
k
C
2k. M
0
k
2k,
M
0
k
⊂ M
k
,
M
0
k
= ker {M
k
→ C, f → f (∞)} .

dimM
k
/M
0
k
≤ 1.
k ≥ 2, G
k
G
k
(∞) = 2ζ (k ) = 0,
k ≥ 2
G
k
∈ M
k
\ M
0
k
,
M
k
= M
0
k
⊕ CG
k
.
g
2

= 60G
2
, g
3
= 140G
3
,
∆ = g
3
2
− 27g
2
3
M
0
6
.
M
k
= 0 k < 0 k = 1.
k = 0, 2, 3, 4, 5 M
k
1, G
2
, G
3
, G
4
, G
5

M
0
k
= 0.
f → ∆.f M
k−6
M
0
k
.
f ∈ M
k
, f = 0,
υ
∞
(f) +
1
2
υ
i
(f) +
1
3
υ
p
(f) +

p∈
H
/

G
∗
υ
p
(f) =
k
6
.
f ∈ M
k
, f H, f H.
k ≥ 0. k
k = 1
n
1
+ n
2
/2 + n
3
/3 = 1/6,
n
1
, n
2
, n
3
∈ N, k < 0, k = 1 M
k
= 0.
f → ∆.f M

k−6
M
0
k
,
f = G
k
. k = 2,
n
1
+ n
2
/2 + n
3
/3 = 2/6,
n
1
= n
2
= 0 n
3
= 1 ⇒ υ
p
(G
2
) = 1,
υ
p
(G
2

) = 0, p = ρ (modG) .
k = 3,
n
1
+ n
2
/2 + n
3
/3 = 3/6,
n
1
= n
3
= 0, n
2
= 1 ⇒ υ
i
(G
3
) = 1,
υ
p
(G
3
) = 0, p = i (modG) .
f = ∆, k = 6
n
1
+ n
2

/2 + n
3
/3 = 1,
∆ (∞) = 0 ⇒ υ
∞
(∆) ≥ 1 υ
∞
(∆) = 1 υ
p
(G
3
) = 0
p = ∞. ∆ H ∞.
f ∈ M
0
k
f (∞) = 0. g = f/∆, g 2k − 12.
υ
p
(g) = υ
p
(f) − υ
p
(∆) =

υ
p
(f) , p = ∞
υ
p

(f) − 1, p = ∞
.
υ
p
(g) ≥ 0 ρ g H∪{∞} ⇒ g ∈ M
k−6
.
∆ M
k−6
M
0
k
.
k ≤ 5 ⇒ k − 6 < 0 M
k−6
= 0,
M
k−6
≃ M
0
k
dim M
k
≤ 1. 1, G
2
, G
3
, G
4
, G

5
= 0
M
0
, M
2
, M
3
, M
4
, M
5
dim M
k
= 1 k = 0, 2, 3, 4, 5.
dim M
k
=




k
/
6

k ≡ 1 (mod6) k > 0

k
/

6

+ 1 k ≡ 1 (mod6) k > 0
0 , k < 0
k.
0 ≤ k < 6 :
k < 0 :
k ≥ 6, M
k
= M
0
k
⊕ CG
k
⇒ dim M
k
= dim M
0
k
+ 1
M
k−6
≃ M
0
k
,
⇒ dim M
k+6
= dim M
0

k+6
+ 1 = dim M
k
+ 1
=

k
/
6

+ 1, k ≡ 1 (mod6)

k
/
6

+ 2, k ≡ 1 (mod6)
=
 
k+6
/
6

, k ≡ 1 (mod6)

k+6
/
6

+ 1, k ≡ 1 (mod6)

k + 6.