2 1. ELLIPTI C CURVE S
2. I f F i s th e field R o f rea l number s o r th e field C o f comple x number s (o r an y
locally compac t field), the n E{F) inherit s fro m th e topolog y o f F th e structur e o f
a compac t abelia n group . Fo r example , th e grou p E(M) i s abstractl y isomorphi c
either t o a circl e group , o r th e produc t o f a circle an d Z/2Z , an d E(C) i s topolog-
ically isomorphi c t o a torus . A s a comple x analyti c manifol d i t i s isomorphi c t o
the quotien t o f C b y a lattic e A C C generate d b y th e period s o f a holomorphi c
differential LJ against th e integra l homolog y o f E(C). T o mak e th e isomorphis m
C/A E(C) explicit , le t
PA(JZ )
b e th e Weierstras s p-functio n attache d t o A ,
defined b y
(1.4)
where
(1.5)
and th e ma p
(1.6)
92
'-=
y
2
=4x
3
-g 2x- g
3i
60 ^ ^ , £ 3 = 140
A€A-{0}A
®w(z) = (PA(Z),PA(*) )
AGA-{0}
The A-periodi c function s x PA(^) an d y = p'j^(z) satisf y th e algebrai c relatio n
l 2
A
6
AGA-{0}G
gives an isomorphism (o f groups as well as complex analytic varieties) betwee n C/ A
and th e ellipti c curv e wit h equatio n (1.4), whic h i s isomorphi c t o E ove r C . Fo r
more details , se e [Si86] , Chapte r VI .
The isomorphis m betwee n E(C) an d C/ A make s i t transparen t tha t th e grou p
En o f point s o n E o f orde r n , wit h coordinate s i n C o r i n an y algebraicall y close d
field of characteristic zero , is isomorphic t o Z/n Z x Z/n Z a s a n abstrac t group .
3. I f F i s a p-adic field (Q p, say, or a finite extension of Qp) the n E(F) i s a compac t
p-adic Li e group, henc e a n extensio n o f a finite grou p b y a pro-p grou p Ei(F) (cf .
[Si86], Chapter s I V an d VII) .
One ma y assume , afte r a chang e o f variables , tha t E i s give n b y a n equatio n
of th e for m (1.2) i n whic h th e coefficient s ai belon g t o th e rin g o f integer s OF o f
F, an d fo r whic h th e associate d discriminan t A min = A ha s minima l valuatio n i n
Op. I f A belong s t o Op, the n th e equatio n obtaine d b y reducin g (1.2) modul o a
uniformiser
TT
G
OF
define s a n ellipti c curv e ove r th e finite field k =
OF/{K).
I n
this cas e one say s that E/F ha s "goo d reduction" .
An important rol e is played in this monograph by elliptic curves having a special
type of bad reduction, referre d t o as multiplicative reduction. Thi s is the case where
A
m
i
n
£ OF is not a unit an d where the equation obtained b y reducing (1.2) modul o
n ha s a n ordinar y doubl e poin t a s it s onl y singularity . I n tha t cas e ord 7r(jf') 0 ,
and ther e is a q G Fx whic h can be obtained b y formally invertin g th e powe r serie s
expressing j i n term s o f q
j = - + 74 4 + 1968847 + -
Q
to expres s q as a powe r serie s i n 1/j wit h integra l (an d henc e p-adically bounded )
coefficients. Th e curv e E i s isomorphi c ove r F (mor e precisely , ove r a quadrati c
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