4 1. ELLIPTI C CURVE S
2. Th e weak Mordell-Weil theorem.
THEOREM
1.4. For any integer n 1,£A e growp E(F)/nE(F) is finite.
These tw o ingredient s ar e combine d i n th e followin g descent lemma o f Ferma t
of which th e Mordell-Wei l theore m i s a direct consequence .
LEMMA
1.5 (Fermat) . Let G be an abelian group equipped with a height function
satisfying the properties of Theorem 1.3, and assume that the quotient G/nG is
finite for some n 1. Then G is finitely generated.
It i s not ou r intentio n t o focu s an y furthe r o n height s (whic h ar e discusse d a t
length in [Si86 ] for example), or on the (elementary ) descen t lemma , whose proof is
relegated t o th e exercises . Th e wea k Mordell-Wei l theore m i s the mos t interestin g
ingredient fro m th e poin t o f vie w o f a stud y o f th e Birc h an d Swinnerton-Dye r
conjecture, sinc e i t i s the sourc e o f the non-effectivity i n th e proo f o f the Mordell -
Weil theorem .
The proo f o f Theore m 1.4 begin s wit h th e observatio n tha t thi s theore m i s
trivially tru e ove r a n algebrai c closur e F o f F, sinc e th e multiplicatio n b y n ma p
is surjectiv e o n E(F). Recal l tha t E
n
: = E n(F) denote s th e kerne l o f thi s map .
Hence th e sequenc e
(1.10) 0 En E(F) - ^ E(F) 0
of module s equippe d wit h thei r natura l continuou s actio n o f GF '•= Gal(F/F) i s
exact. Followin g th e usua l convention s o f Galoi s cohomology , denot e b y
H\F,M) :=IT{G F,M)
the grou p o f continuou s i-cocycle s modul o th e grou p o f continuou s z-coboundarie s
with value s i n th e G^-modul e M. (For detail s o n thes e definitions , se e [CF67] ,
Chapter IV , o r [Si86] , appendi x B) . Takin g th e Galoi s cohomolog y o f th e exac t
sequence (1.10 ) give s rise to th e lon g exact cohomolog y sequenc e
0 E
n
(F) E{F) - ^ E{F) - ^ H\F,E
n
) H\F,E) - ^ H l{F,E)
from whic h ca n b e extracte d th e so-calle d descent exact sequence
(1.11) 0 E{F)/nE(F) -U H\F,E n) H\F,E)
n
0.
The connectin g homomorphis m S embeds th e grou p E(F)/nE(F) int o a n objec t
of Galois-theoreti c nature , sinc e
H 1 (GF,
E
n
) depend s onl y o n th e structur e o f
GF
and o f the Gi?-modul e E
n
, no t o n the ellipti c curve E itself . Fo r example , i f all th e
n-division point s o f E(F) ar e define d ove r F s o that GF act s triviall y o n E
n
, the n
elements i n
H1(F,En)=Rom(GF,En)
are indexe d b y pair s (L , y) wher e L i s a finite Galoi s extensio n o f F an d (f i s a n
identification o f Gal(L/F) wit h a subgroup of En. I f H l(F, E
n
) wer e a finite group,
the mer e existenc e o f th e exac t sequenc e (1.11woul ) d b e enoug h t o conclud e th e
proof o f Theore m 1.4. However , thi s i s never th e cas e whe n F i s a numbe r field
and n 1. I t i s therefor e necessar y t o exploi t loca l informatio n t o pi n dow n th e
image o f 5 in H X{F^ E
n
) wit h greate r accuracy . Mor e precisely , fo r an y plac e v o f
F (archimedea n o r not ) th e embeddin g o f F int o th e completio n F
v
a t th e plac e
v, extende d t o a n embeddin g o f F int o F
v
, induce s a n inclusio n Gp
v
C GF- Th e
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