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[1



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[]\OT1/cmr/m/n/10.95 Denote com-plex con-ju-ga-tion by $\OML/cmm/m/it/10.95 ^^\
$\OT1/cmr/m/n/10.95 . We will de-com-pose $\OML/cmm/m/it/10.95 E\OT1/cmr/m/n/10
.95 (\OML/cmm/m/it/10.95 K\OT1/cmr/m/n/10.95 )$ into its eigenspaces
 []

[2]

LaTeX Warning: Citation `mazur-swinnerton-dyer:arithmetic' on page 3 undefined 
on input line 76.


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ine 83.


LaTeX Warning: Citation `kolyvagin:euler_systems' on page 3 undefined on input 
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LaTeX Warning: Citation `MR92e:11050' on page 3 undefined on input line 89.


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\OT1/cmr/m/n/10.95 quadratic ex-ten-sion $\OML/cmm/m/it/10.95 K$ \OT1/cmr/m/n/1
0.95 sat-si-fy-ing the Heeg-ner hy-poth-e-sis and such that $\OML/cmm/m/it/10.9
5 L[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 E=K; \OT1/cmr/m/n/10.95 1) \OMS/cm
sy/m/n/10.95 6\OT1/cmr/m/n/10.95 =
 []

[3]
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\OML/cmm/m/it/10.95 H[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 K[]=K; E\OT1/cmr
/m/n/10.95 (\OML/cmm/m/it/10.95 K[]\OT1/cmr/m/n/10.95 )[\OML/cmm/m/it/10.95 p\O
T1/cmr/m/n/10.95 ]) \OMS/cmsy/m/n/10.95 ! \OML/cmm/m/it/10.95 H[]\OT1/cmr/m/n/1
0.95 (\OML/cmm/m/it/10.95 K; E\OT1/cmr/m/n/10.95 [\OML/cmm/m/it/10.95 p\OT1/cmr
/m/n/10.95 ]) \OMS/cmsy/m/n/10.95 ! \OML/cmm/m/it/10.95 H[]\OT1/cmr/m/n/10.95 (
\OML/cmm/m/it/10.95 K[]; E\OT1/cmr/m/n/10.95 [\OML/cmm/m/it/10.95 p\OT1/cmr/m/n
/10.95 ])[] \OMS/cmsy/m/n/10.95 ! \OML/cmm/m/it/10.95 H[]\OT1/cmr/m/n/10.95 (\O
ML/cmm/m/it/10.95 K[]=K; E\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 K[]\OT1/cmr/m
/n/10.95 )[\OML/cmm/m/it/10.95 p\OT1/cmr/m/n/10.95 ])\OML/cmm/m/it/10.95 :
 []


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\OML/cmm/m/it/10.95 H[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 L=K; E\OT1/cmr/m
/n/10.95 (\OML/cmm/m/it/10.95 L\OT1/cmr/m/n/10.95 )[\OML/cmm/m/it/10.95 p\OT1/c
mr/m/n/10.95 ]) \OMS/cmsy/m/n/10.95 ! \OML/cmm/m/it/10.95 H[]\OT1/cmr/m/n/10.95
 (\OML/cmm/m/it/10.95 K; E\OT1/cmr/m/n/10.95 [\OML/cmm/m/it/10.95 p\OT1/cmr/m/n
/10.95 ]) \OMS/cmsy/m/n/10.95 ! \OML/cmm/m/it/10.95 H[]\OT1/cmr/m/n/10.95 (\OML
/cmm/m/it/10.95 L; E\OT1/cmr/m/n/10.95 [\OML/cmm/m/it/10.95 p\OT1/cmr/m/n/10.95
 ])[] \OMS/cmsy/m/n/10.95 ! \OML/cmm/m/it/10.95 H[]\OT1/cmr/m/n/10.95 (\OML/cmm
/m/it/10.95 L=K; E\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 L\OT1/cmr/m/n/10.95 )
[\OML/cmm/m/it/10.95 p\OT1/cmr/m/n/10.95 ])\OML/cmm/m/it/10.95 :
 []


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[4]
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[]\OT1/cmr/m/n/10.95 In par-tic-u-lar, the sec-ond the-o-rem will im-ply $[](\O
ML/cmm/m/it/10.95 E=K\OT1/cmr/m/n/10.95 )$ has no $\OML/cmm/m/it/10.95 `$\OT1/c
mr/m/n/10.95 -torsion if $\OML/cmm/m/it/10.95 ` [] \U/msb/m/n/10.95 -  []\OT1/c
mr/m/n/10.95 [\OML/cmm/m/it/10.95 E\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 K\OT
1/cmr/m/n/10.95 ) :  []\OT1/cmr/bx/n/10.95 Z\OML/cmm/m/it/10.95 y[]\OT1/cmr/m/n
/10.95 ]$. 
 []


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[5]

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58.


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\OT1/cmr/bx/n/10.95 Proposition 4.3. \OT1/cmr/m/it/10.95 Sup-pose $\OML/cmm/m/i
t/10.95 E$ \OT1/cmr/m/it/10.95 has no $\OT1/cmr/bx/n/10.95 Q$\OT1/cmr/m/it/10.9
5 -rational $\OML/cmm/m/it/10.95 p$\OT1/cmr/m/it/10.95 -isogeny. Then $\OML/cmm
/m/it/10.95 H[]\OT1/cmr/m/n/10.95 (\OT1/cmr/bx/n/10.95 Q\OT1/cmr/m/n/10.95 (\OM
L/cmm/m/it/10.95 E\OT1/cmr/m/n/10.95 [\OML/cmm/m/it/10.95 p\OT1/cmr/m/n/10.95 ]
)\OML/cmm/m/it/10.95 =\OT1/cmr/bx/n/10.95 Q\OML/cmm/m/it/10.95 ; E\OT1/cmr/m/n/
10.95 [\OML/cmm/m/it/10.95 p\OT1/cmr/m/n/10.95 ]) =
 []


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[9]

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