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\T1/aer/m/n/10.95 duce the gen-eral case from the iden-tity $\OML/cmm/m/it/10.9
5 H[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 G; A \OMS/cmsy/m/n/10.95 ^^H \OML/
cmm/m/it/10.95 B\OT1/cmr/m/n/10.95 ) = \OML/cmm/m/it/10.95 H[]\OT1/cmr/m/n/10.9
5 (\OML/cmm/m/it/10.95 G; A\OT1/cmr/m/n/10.95 ) \OMS/cmsy/m/n/10.95 ^^H \OML/cm
m/m/it/10.95 H[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 G; B\OT1/cmr/m/n/10.95 
)$\T1/aer/m/n/10.95 .
 []


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[]\T1/aer/m/n/10.95 We now ex-plic-itly con-sider $\OML/cmm/m/it/10.95 H[]\OT1/
cmr/m/n/10.95 (\OML/cmm/m/it/10.95 G; A\OT1/cmr/m/n/10.95 )$ \T1/aer/m/n/10.95 
as a quo-tient of co-cy-cles by cobound-
 []

[8]
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[]\T1/aer/m/n/10.95 Recall that $[][]\OT1/cmr/m/n/10.95 (\OMS/cmsy/m/n/10.95 ^^
@\OML/cmm/m/it/10.95 ; A\OT1/cmr/m/n/10.95 )$ \T1/aer/m/n/10.95 is a con-travar
i-ant func-tor and let $\OML/cmm/m/it/10.95 K[] \OT1/cmr/m/n/10.95 = [][](\OML/
cmm/m/it/10.95 P[]; A\OT1/cmr/m/n/10.95 )$\T1/aer/m/n/10.95 .
 []


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[]\T1/aer/m/n/10.95 Suppose that $\OML/cmm/m/it/10.95 f \OMS/cmsy/m/n/10.95 2 [
][]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 P[]; A\OT1/cmr/m/n/10.95 )$ \T1/aer/
m/n/10.95 and de-fine the set-theoretic map []
 []


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[] 
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[]\T1/aer/m/n/10.95 The map $\OML/cmm/m/it/10.95 d [] [] P[] \OMS/cmsy/m/n/10.9
5 ! \OML/cmm/m/it/10.95 P[]$ \T1/aer/m/n/10.95 in-duces a map $[][]\OT1/cmr/m/n
/10.95 (\OML/cmm/m/it/10.95 P[]; A\OT1/cmr/m/n/10.95 ) \OMS/cmsy/m/n/10.95 ! []
[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 P[]; A\OT1/cmr/m/n/10.95 )$
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[9]
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\T1/aer/m/n/10.95 A mor-phism of pairs $\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95
 G; A\OT1/cmr/m/n/10.95 ) \OMS/cmsy/m/n/10.95 ! \OT1/cmr/m/n/10.95 (\OML/cmm/m/
it/10.95 G[]; A[]\OT1/cmr/m/n/10.95 )$ \T1/aer/m/n/10.95 to-gether with ho-mo-m
or-phisms $\OML/cmm/m/it/10.95 f [] [] G[] \OMS/cmsy/m/n/10.95 !
 []

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[]\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 G[]; \U/msb/m/n/10.95 Z\OML/cmm/m/it/
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[35]
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\T1/aer/m/n/10.95 Another stan-dard fact about el-lip-tic curves---the (weak) M
ordell-Weil theorem---
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\T1/aer/m/n/10.95 Recall that if $\OML/cmm/m/it/10.95 G$ \T1/aer/m/n/10.95 is a
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[]\T1/aer/m/n/10.95 Since, as was men-tioned above, $[]\OT1/cmr/m/n/10.95 (\OML
/cmm/m/it/10.95 G; A\OT1/cmr/m/n/10.95 ) \OMS/cmsy/m/n/10.95 ^^Y []\OT1/cmr/m/n
/10.95 (\OML/cmm/m/it/10.95 G; A\OT1/cmr/m/n/10.95 )$ \T1/aer/m/n/10.95 and $[]
\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 G; A\OT1/cmr/m/n/10.95 ) \OMS/cmsy/m/n/
10.95 ^^Y
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[]\T1/aer/bx/n/10.95 Theorem 20.1 \T1/aer/m/n/10.95 (Lang--Tate, 1958)\T1/aer/b
x/n/10.95 . []\T1/aer/m/it/10.95 There is a nat-u-ral iso-mor-phism $[]\OT1/cmr
/m/n/10.95 (\OML/cmm/m/it/10.95 A=k\OT1/cmr/m/n/10.95 ) \OMS/cmsy/m/n/10.95 !
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5 [\OML/cmm/m/it/10.95 n\OT1/cmr/m/n/10.95 ])) = [][]([\OML/cmm/m/it/10.95 n\OT
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[]\T1/aer/bx/n/10.95 Definition 25.3 \T1/aer/m/n/10.95 (�tale Co-ho-mol-ogy)\T1
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cmr/m/n/10.95 (\OML/cmm/m/it/10.95 X[]; F\OT1/cmr/m/n/10.95 )$
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t/10.95 M[]\OT1/cmr/m/n/10.95 )\OML/cmm/m/it/10.95 ; I\OT1/cmr/m/n/10.95 ) = []
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