Elliptic curves 9780691085593, 0691085595

436 34 79MB

English Pages 443 Year 1992

Polecaj historie

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Elliptic curves
9780691085593, 0691085595

Author / Uploaded
Anthony W. Knapp

Commentary
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Citation preview

I

E(Q)/2E(Q), E(Q)/2E(Q),

E(Q) E(Q) p

L

L L

L

(P P')(QQ')

(PQ)(P'Q')

113

E

E(Q)

L L

L L L L

L L L

L L

L L

L

E) E) E) E) E) L

L

L

A

0(1) 0(1)

a

a a

I

SL(2,

S£(2,

A A

[A: B] Autk(I M > 0 contribute

m

^

\muji +

ncj r" 2

< c~

s

|m|>|n|>0

^2

| m | " = 4c" 5

^

8

|m|>|n|>0

m"

5

+

1

< oo,

m=l

and the t e r m s w i t h \ n \ > \ m \ > 0 s i m i l a r l y m a k e a finite c o n t r i b u t i o n . T h e l e m m a follows. P r o p o s i t i o n 6 . 8 . I f F is a n y finite subset o f A a n d i f the t e r m s corresponding to F are o m i t t e d i n ( 6 . 1 ) , then the r e s u l t i n g series converges absolutely u n i f o r m l y on a n y compact subset of C — ( A — F ) . Consequently p { z ) is m e r o m o r p h i c i n C , its o n l y poles are double poles at the points o f A , a n d p ' { z ) m a y be compute d t e r m by t e r m . P R O O F . W e m a y assume F contains UJ — 0. T h e s u m for A — F is ,2

]L

_ W A z~— w , ,) A 2 2

Z

-

) ~ XI „ OJ(Z — UJ)

,,.2 J UJ 2

2 z -

_

UJ

2

'

wz-w

(z)

fJ(-z) fJ(-z)

fJ(z) fJ(z). fJ(z)

fJ'(Z)

w fJ'(z)

fJ(z

fJ(z

fJ(z)

(z

fJ(z w) w) fJ(z)

fJ(z).

fJ'(z) fJ(z)

fJ(z

fJ'(z

fJ(z).

w) -

fJ(z)

fJ(z)

f(z) f 1 -2

fJ(z). fJ'(z)h(fJ(z))

g(fJ(z))

(f(z)

fe

fo

f( -z))

fe(z)

f 9

h fo(z) =

f( -z)) fo

fo

fo

=

U

z

z

I

art

I

I

I =

I

I g

g

=

=

+

+ p

p

z

(k

P2(C)

E(C),

E

P (C)

:j::. I.

E(C).

:j::.

:j::.

:j::.

E(C)

:j::.

Z

X

1).

4

E

(x,

E(C)

oE

oE =

w

h( x) E(

P2(C) P2 (C)

E(C).

E(C) E(C),

E(C) (x, j j

z

w) z