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 [] []\OT1/cmr/bx/n/12 Mordell-Weil The-o-rem, Shafarevich-Tate Group and Selme
r Groups 
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\OT1/cmr/m/n/12 nec-tion be-tween the Birch and Swinnerton-Dyer con-jec-ture, a
nd the Tate-Shafarevich
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[]\OT1/cmr/m/n/12 The main goal of this the-sis is to in-tro-duce and study a s
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Chapter 1.
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[]\OT1/cmr/bx/n/17.28 Computing the Weak Mordell-Weil Group and
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[]\OT1/cmr/m/n/12 The main idea for the com-pu-ta-tion of the weak Mordell-Weil
 group $\OML/cmm/m/it/12 E\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 K\OT1/cmr/m/n/12 )\
OML/cmm/m/it/12 =mE\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 K\OT1/cmr/m/n/12 )$
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[]
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[]\OT1/cmr/m/n/12 Our ul-ti-mate goal is com-put-ing the im-age of $\OML/cmm/m/
it/12 E[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 K\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 =
^^^\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 E\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 K\OT1/c
mr/m/n/12 ))$ in $\OML/cmm/m/it/12 H[]\OT1/cmr/m/n/12 ([]([]\OML/cmm/m/it/12 =K
\OT1/cmr/m/n/12 )\OML/cmm/m/it/12 ; E\OT1/cmr/m/n/12 [\OML/cmm/m/it/12 ^^^\OT1/
cmr/m/n/12 ])$,
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\OT1/cmr/m/n/12 We also con-sider the Shafarevich-Tate group of $\OML/cmm/m/it/
12 E=K$ \OT1/cmr/m/n/12 to be the sub-group of $\OML/cmm/m/it/12 WC\OT1/cmr/m/n
/12 (\OML/cmm/m/it/12 E=K\OT1/cmr/m/n/12 )$
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[]\OT1/cmr/m/n/12 The the-o-rem im-plies that for all $\OML/cmm/m/it/12 P[]; P[
]; [] ; P[]; Q[]; Q[]; [] ; Q[]$\OT1/cmr/m/n/12 , there ex-ist $\OML/cmm/m/it/1
2 R[]; [] ; R[]$\OT1/cmr/m/n/12 ,
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\OT1/cmr/bx/n/12 Corollary 4.1.2. \OT1/cmr/m/it/12 For any em-bed-ding $\OML/cm
m/m/it/12 i \OT1/cmr/m/n/12 : \OML/cmm/m/it/12 A \OMS/cmsy/m/n/12 ! \OML/cmm/m/
it/12 J$\OT1/cmr/m/it/12 , the vis-i-ble sub-group Vis$[]\OT1/cmr/m/n/12 (\OML/
cmm/m/it/12 H[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 K; A\OT1/cmr/m/n/12 ))$
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[]\OT1/cmr/bx/n/17.28 Producing Vis-i-ble El-e-ments of the Shafarevich-
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[]\OT1/cmr/m/n/12 If $\OML/cmm/m/it/12 ^^W$ \OT1/cmr/m/n/12 is real archi-me-di
an, we have $\OML/cmm/m/it/12 H[]\OT1/cmr/m/n/12 ([]([][])\OML/cmm/m/it/12 =K[]
; A\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 K[]\OT1/cmr/m/n/12 )) = \OML/cmm/m/it/12 H
[]\OT1/cmr/m/n/12 ([](\U/msb/m/n/12 C\OML/cmm/m/it/12 =\U/msb/m/n/12 R\OT1/cmr/
m/n/12 )\OML/cmm/m/it/12 ; A\OT1/cmr/m/n/12 (\U/msb/m/n/12 R\OT1/cmr/m/n/12 ))$
.
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[]\OT1/cmr/bx/n/17.28 Algorithms for Com-put-ing with Mod-u-lar Abelian
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[]\OT1/cmr/m/n/12 Conversely, sup-pose that $\OML/cmm/m/it/12 ^^K[]$ \OT1/cmr/m
/n/12 and $\OML/cmm/m/it/12 ^^K[]$ \OT1/cmr/m/n/12 sat-isfy the con-di-tion. Fi
rst, choose $\OML/cmm/m/it/12 s[]; r[]; s[]; r[] \OMS/cmsy/m/n/12 2
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\OT1/cmr/m/it/12 where by $\OT1/cmr/m/n/12 ^^H(\OML/cmm/m/it/12 H[]\OT1/cmr/m/n
/12 (\OML/cmm/m/it/12 X[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 N\OT1/cmr/m/n/12 ); 
\U/msb/m/n/12 Z\OT1/cmr/m/n/12 ))[]$ \OT1/cmr/m/it/12 we mean the pos-i-tive ei
genspace of $[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 S[]\OT1/cmr/m/n/12 (^^@[](\OML
/cmm/m/it/12 N\OT1/cmr/m/n/12 ))[\OML/cmm/m/it/12 I[]\OT1/cmr/m/n/12 ]\OML/cmm/
m/it/12 ; \U/msb/m/n/12 C\OT1/cmr/m/n/12 )$
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\OT1/cmr/m/n/12 Thus, $\OMS/cmsy/m/n/12 j[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 A[
]=\U/msb/m/n/12 Q\OT1/cmr/m/n/12 )\OMS/cmsy/m/n/12 j \OT1/cmr/m/n/12 = 2[] \OMS
/cmsy/m/n/12 ^^A \OT1/cmr/m/n/12 5[]$. Us-ing the in-jec-tion from Step 6, we g
et $5[]\OMS/cmsy/m/n/12 jj[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 A[]=\U/msb/m/n/12
 Q\OT1/cmr/m/n/12 )\OMS/cmsy/m/n/12 j$\OT1/cmr/m/n/12 ,
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[][]\OT1/cmr/m/n/12 , \OT1/cmr/m/it/12 Ja-co-bian va-ri-eties\OT1/cmr/m/n/12 , 
Arith-metic ge-om-e-try (Storrs, Conn., 1984),
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