C(i)

P1

Ψ

∆I1 . . . ∆Ir OoA11 . . . OoAss Ck Cn HPnr

HPnr

Gk [Gk ]

Gk

CH(Ck )

Gk

A(Ck )

i) ⇒ ii)

(r − 2) gdr

r

C(d−2r+1) C(i) ˜ d, g) C(r,

2r − 2 gdr

C(i+1) C(r, d, g) P1 Ig Ig

P1 gd2

Ig

g ≤ 10 P3

gd3

g ≤ 10

Ig

Q

k∗

k∗

k : x $→ kx k∗

k∗

Q k∗

k : x $→ kx k∗

k∗

k∗

C

g n

Cn

JC

un : Cn −→ JC k

Cn JC

k

Z

k : JC −→ JC x $−→ kx A(JC)

CH(JC) Q

A(JC) p Ap (JC) =

p !

Ap(i) (JC)

i=p−g

α Ap(i) (JC) k∗ α = k 2g−2p+i α

k

C Ag−1 (JC)

k ∗ α = k 2p−i α

JC

C C = C(0) + . . . + C(g−1)

C(i) ∈ Ag−1 (i) (JC) A(JC)

R Ai (X) i

CH (X)

C k∗

CHi (X)

k∗

R Ai (X)

C(i)

R R=

Q[C(0) , . . . , C(g−1) ] R

R

C(i) Q[C(0) , . . . , C(g−1) ]

C(i) C(1)

d 0≤i≤d−3

i

C(2)

C(i)

Ig g

P1

C i≥d−1

d

C(i)

G ⊂ Cd 1≤n≤d

n Gn = {D ∈ Cn | ∃ E ∈ Cd−n , D + E ∈ G} Gn

C # $ n " (−1)i−1 d un ∗ [Gn ] = i∗ C i n−i i=1

Gd ud

Gd

r

ud ∗ [Gd ]

An−r (Cn )

gdr "

1≤i1 ≤...≤ir

C

gdr

d

C [Gn ] =

G

#

n−

d %r

u=1 iu

$& ' r

v=1

" (−1)iv −1 ( [δi1 ,...,ir + (n − iu )o] iv r

u=1

o

C

CP iu

δi1 ,...,ir

δi1 ,...,ir = {i1 x1 + . . . + ir xr | xi ∈ C}

[Gd ] R

gdr

C

Ag−r (s) (JC) "

0≤a1 ,...,ar a1 +...+ar =s

β(d, a1 + 1, . . . , ar + 1) C(a1 ) ∗ . . . ∗ C(ar ) = 0

β(d, a1 , . . . , ar ) =

d "

...

i1 =1

[HPnr ]

Φ×n

s

d "

(−1)

i1 +...+ir

ir =1

#

$ d i1 a1 . . . ir ar i1 + . . . + ir [Gn ]

HPnr (Pr )n r gd

Φ : C −→ Pr [Gn ] [Gk ] k0

R ∩ A(i) =

g−1 !

d=i+1

d R(i)

pi i pa1 . . . pat d R(i)

R d R(i)

-g−1

i

d=i+1

i+1

R * (i) )) ) ×p1 ))) )) ' )) ×p2 i+2 i+2 )) R(i) R(i+1) ) )) * )) )) ) )) ×p1 ))) ×p1 )( ' )) ' )) ×p2 i+3 i+3 )) R(i) )) R(i+1) * )) )) ×p1 )) ×p1 ' ( ' i+4 R(i)

F

F

$

×p1

'

i+4 R(i+2)

×p1

i+4 R(i+3)

×p1

×p1

. .' .

. .' .

. .' .

. .' .

...

...

...

...

.) . .

...

...

...

×p1

'

'

g−2 R(i)

$

i+3 R(i+2)

i+4 R(i+1)

×p1

F

'

×p1

'

g−1 R(i)

×p1

'

g−2 R(i+1)

×p1

d R(i)

'

g−2 R(i+2)

×p1

'

g−1 R(i+1)

×p1

g−2 R(i+3)

×p1

'

g−1 R(i+2)

×p1

g−1 R(i+3)

pi d R(i) i+1 R(i)

i > 0

⊂

-g−1

d=i+1

-g−1

p1

d=i+1

d ⊂ Rd+k pk R(i) (i+k−1)

1 d R(i)

d R(i) d R(i)

pa1

j≥i j

g−1 R(i)

g−1 pi R(i)

d+1 R(i)

k>1

e

g−d+i R(i)

i+1 R(i) d p1 R(i)

F

p1

p1 pa s

e ≥ j +1 d − 2r + 2 ≤ j ≤ e − 1

-g−1

d R(i)

d=i+k+1

d R(i+k)

a1 + . . . + as − s = k d R(i) gdr

e R(j) e R(j)

r (r − 2)

2r − 2 gdr

C d−r+1 R(d−2r+1)

a R(i)

Ag−r (Cd ) a

d − 2r + 2 ≤ i ≤ a − 1

i

a ≥ i+1 C(d−2r+1)

p1 C(r, d, g) C(d−2r+1) = 0 C(r, d, g)

Q[d, g]

r

# $# $# $ r−1 " (−1)i d − r − i + 1 d − r − i g C(r, d, g) = r−i r−1−i r−1−i i i=0

Ig

C(i)

C

C(i) = 0

C(d−2r+1)

Ag−1 (i)

i ≥ d − 2r + 1. C(r, d, g)

Φ(C)

2r − 2

1 gd−2r+2

gdr

g

C(r, d, g) /= 0

r−2

Pr

C(d−2r+1)

C(d−2r+1)

2r − 2

1 gd−2r+2

C(i)

C(i+1)

r−2 C(d−2r+1)

D G(−D)

(r − 2)

r

2r − 2

r−2

C(r, d, g) 2r − 2 A(Cd )

gdr

gdr G p1 , . . . , pl

gdr

V p

G(−p1 . . . − pl ) r−2

r−2

Pr

r−1 C G(−p) gd−1 r−l gd−l k = r−1 P1

G(r − 1, r + 1)

Pr G(−p1 . . . − pl )

k ≤ r−1 C r−k−1 gd−l

l

l

1 gd−r+1

2r − 2 1

2r−2 # $# $# $ r−1 " (−1)i d − r − i + 1 d − r − i g C(r, d, g) = r−i r−1−i r−1−i i i=0

Cd gdr

1 gd−2r+2

gdr

gdr C(d−2r+1)

r−2 C(j)

C(d−2r+1) Ad−r+1 (d−2r+1) (JC)

gdr "

1≤a1 ≤...≤ar a1 +...ar =d−r+1

a1 ! . . . ar ! pa . . . par γ(a1 , . . . , ar ) 1

2r − 2 j≥i

C(i)

gdr

pg+r−d−2 1

Ag−1 d−2r+1 (JC)

"

1≤a1 ≤...≤ar a1 +...ar =d−r+1

t=0

p1 a1 = . . . = at = 1

1

ai γ(a1 , . . . , at ) = t! r−1 "

a1 ! . . . ar ! pg+r−d−2 pa1 . . . par γ(a1 , . . . , ar ) 1

"

1 1

(−1)r+t

10 g≥0 6 d g √ 24g − 15 + 3 ≤d 2 d

g

gd3

Ig

g d

g F

pd!

d R!I

I

d−4

g √

C Q[p1 , . . . , pg ]

I

P d−3

Ig

d−2 R(d−5)

d%

d

24g − 15 + 3 g ≤d≤ +3 2 2

g ≤ 10

gd3

g 3 g+3≤d 4 P3 3

0

6

gd3

gd1!

gd3

g=6 g=7 g=8

g73 g83 g83

g31 g41 g41

p1 p22 = 0 9p1 p2 p3 + p32 = 0 9p1 p2 p3 + p32 = 0

g=9

g93

g51

8p1 p2 p4 + 3p1 p23 +2p22 p3 = 0

g83

g41

9p1 p2 p3 + 2p32 = 0

3 g10

g61

g93

g61

20p1 p2 p5 + 12p1 p3 p4 +4p22 p4 + 3p2 p23 = 0 10p21 p5 + 8p1 p2 p4 +3p1 p23 + 2p22 p3 = 0

g = 10

g ≤ 10

gd3

g

p2 p3 = 0 p32 = p1 p2 p3 = 0 p1 p2 p4 = p1 p23 = p22 p3 = 0 p2 p4 = p23 = 0 p2 p3 = 0 m p1 p2 p3 = 0 m ∈ {0 . . . 2} 3 = 0 pn p n ∈ {0, 1 1 2 m 2 20pm+1 p5 + 4pm 1 1 p2 p4 + p1 p3 = 0 m ∈ {0 . . . 3} n+2 n+1 10p1 p5 + 8p1 p2 p4 + 3pn+1 p23 1 n 2 +2p1 p2 p3 = 0 n ∈ {0 . . . 2}

10 gd3

Ig

gd3

gd3

Ig

gd1!

C

g

g

C G

r G

d Cn n σn

σn : C n → Cn

un

D

un : Cn → JC

Cn

gdr G Gn Φ

d

Φ : C → Pr

gdr

G

Gn = {D ∈ Cn | ∃ E ∈ Cd−n , D + E ∈ G}

k

r≤n≤d

Z

x $→ kx

k

JC A1 (JC)

θ D ( = CHi (X) )

0

CH 1 (JC) D

X

JC g

i CH(X)

gdr

G

Θ

CH g−i (X)

G

Φ : C → Pr

k

D

r

X

Q

i (X) CH(s)

CH i (X) k ∗ α = k 2p−i α

α

Ag−i (X)

i (X) CH(s)

( = Ai (X) )

k k∗ α =

k 2g−2p+i α

X

Q

g i

A(X)

X Q

Ai(s) (X) α Ai (X) k ∗ α = k 2p−i α

Ai(s) (X)

k k∗ α =

[V ]

k 2g−2p+i α

CH(X) V

A(X)

X A = I1 5 . . . 5 In

5

A

R

A(JC) CH(JC) k∗

k∗ p R(i)

R

Ap(i)

Ig

R

∆I1 . . . ∆Ir . . . OoAss

Cn j

δi1 ,...,ir

x

It xi = xj n = i1 + . . . ir

t A(Cn )

t {1, . . . , s} a

CH(Cn )

ΨP

At

{1, . . . , r} i xt = ot

{i1 x1 +. . .+ir xr | xi ∈ C}

ΨP : C k → C n (Pn )r

HPnr JC

g

Pr C

∗ F

CH(X) S

A(X

g−1 CH(s) (J)

C(i) C

k∗ C(s) = k 2g−2−s C(s) pi

i≥0

[a]

pi

Ni Ni

k ∗ C(s) = k 2+s C(s)

pi+1

[a]

pi

Ag−1 (s) (J)

pi a!

C(i) [a]

a ≥ 0 pi

Ni = −F(C(i−1) ) = −pi

i≥1

0

a