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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