1.2. TH E MORDELL-WEI L THEORE M 5
exact sequenc e (1.11 ) ha s a local counterpart wit h F replace d b y F
v
whic h fits int o
the commutativ e diagra m
(1.12)
0 - ^ E(F)/nE(F) -*- ffi(F,£ n) H\F,E)
n
+ 0
1 res
v
i \d
v
i res
v
0 * E(F v)/nE(Fv) - ^ H^F^E*) —* H\F v,E)n + 0
in whic h th e vertica l arrow s correspon d t o th e natura l restrictio n maps . Sinc e
S(E(F)/nE(F)) i s containe d i n th e kerne l o f d
v
fo r al l v, i t i s containe d i n th e
so-called n-Selmer group of E ove r F define d a s follows .
DEFINITION
1.6. Le t E b e a n ellipti c curv e ove r a number field F.
(1) Th e n-Selme r grou p of E ove r F , denote d Se\ n(E/F), i s the se t o f classe s
c G
H1^,
E n) satisfyin g d v(c) = 0, for al l places v o f F.
(2) Th e Shafarevich-Tat e grou p o f E/F, denote d III (E/F), i s th e se t o f
classes c G
Hl(F,
E) satisfyin g res v(c) = 0 , for al l places v o f F.
The exact sequenc e (1.11 ) can now be replaced b y the exact sequenc e involvin g
the n-Selme r grou p an d th e n-torsio n i n th e Shafarevich-Tat e group :
(1.13) 0 E{F)/nE{F) - ^ Sel
n
(E/F) —- IU(E/F)n 0 .
The wea k Mordell-Wei l theore m i s the n a consequenc e o f th e followin g genera l
finiteness theore m fo r th e Selme r group .
PROPOSITION
1.7. The Selmer group Sel n(E/F) is finite.
SKETCH O F PROO F O F PROPOSITIO N
1.7. Th e proof ca n itself be divided int o
two stages : a loca l study , i n whic h i t i s shown tha t Sel
n
(E/F) i s contained i n th e
group H^A(F, E
n
) consistin g of cohomology classes c G Hl(F, E
n
) whos e restrictio n
to th e inerti a grou p I
v
a t v i s trivial, fo r al l places v no t dividin g nA . I t i s only a t
this stag e o f the argumen t tha t certai n fact s abou t th e geometr y an d arithmeti c o f
elliptic curve s (albeit , ove r local fields) ar e needed . (Cf . Exercis e 7. )
A globa l stud y i s the n neede d t o sho w tha t H^
A
(F,En) i s finite. Th e ke y
ingredient i n this finiteness result (cf . Exercis e 8) is the Hermite-Minkowski theore m
asserting tha t ther e ar e onl y finitely man y extension s o f a given numbe r field wit h
bounded degre e an d ramification .
To recapitulate, th e proof o f the Mordell-Weil theore m sketche d abov e (and , i n
particular, th e proof o f the weak Mordell-Weil theorem) ha s led to the introductio n
of tw o fundamenta l invariants , th e n-Selme r grou p o f E/F an d th e Shafarevich -
Tate grou p o f E/F, fitting int o th e fundamenta l exac t sequenc e (1.13) . Th e wea k
Mordell-Weil theore m follow s fro m th e finiteness o f Sel n(E/F) whos e proof i n tur n
relies o n th e Hermite-Minkowsk i theorem , whic h i s itsel f on e o f th e ke y genera l
finiteness result s o f algebrai c numbe r theory .
Since Se\
n
(E/F) i s effectively calculable, the followin g questio n emerge s natu -
rally fro m th e proo f o f theorem 1.2.
QUESTION
1.8. How good an approximation to E(F)/nE(F) is Sel n(E/F),
i.e., how large can III(E/F)
n
be?
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