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Chapter 1.
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[]\OT1/cmr/bx/n/14.4 Galois Rep-re-sen-ta-tions (Taniyama, Shimura, Mumford-Tat
e) 
 []

[1


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$[] \OMS/cmsy/m/n/12 !$
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[2] [3] [4

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Chapter 2.
[5]
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$[] \OMS/cmsy/m/n/12 !$
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[6] [7] [8]
Chapter 3.
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$[] \OMS/cmsy/m/n/12 !$
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[10] [11] [12] [13] [14

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Chapter 4.
[15] [16] [17] [18

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Chapter 5.
[19] [20] [21] [22

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Chapter 6.
[23] [24] [25] [26] [27] [28

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Chapter 7.
[29] [30] [31] [32

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Chapter 8.
[33] [34] [35] [36]
Chapter 9.

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[]\OT1/cmr/m/n/12 Suppose $^^@ = ^^@[](\OML/cmm/m/it/12 N\OT1/cmr/m/n/12 )$, th
en the de-gree of the cov-er-ing is the in-dex $([] \OML/cmm/m/it/12 =\OMS/cmsy
/m/n/12 f^^F\OT1/cmr/m/n/12 1\OMS/cmsy/m/n/12 g \OT1/cmr/m/n/12 :
 []

[37

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[38] [39] [40]
Chapter 10.
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[]\OT1/cmr/m/it/12 Let $\OML/cmm/m/it/12 E[]$ \OT1/cmr/m/it/12 and $\OML/cmm/m/
it/12 E[]$ \OT1/cmr/m/it/12 be el-lip-tic curves de-fined over $\OML/cmm/m/it/1
2 K$ \OT1/cmr/m/it/12 with equal $\OML/cmm/m/it/12 j$\OT1/cmr/m/it/12 -invarian
ts,
 []

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[44] [45] [46]
Chapter 11.
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Chapter 12.
[59] [60] [61]
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[]\OT1/cmr/m/it/12 Let the no-ta-tion be as above. Then $[]\OT1/cmr/m/n/12 (\OM
L/cmm/m/it/12 ^^Z[]\OT1/cmr/m/n/12 ([][])) = \OML/cmm/m/it/12 a[]$ \OT1/cmr/m/i
t/12 and $[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 ^^Z[]\OT1/cmr/m/n/12 ([][])) =
 []

[62] [63] [64] [65] [66

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Chapter 13.
[67]
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[]\OT1/cmr/m/n/12 The Hecke op-er-a-tors $\OML/cmm/m/it/12 T[]$ \OT1/cmr/m/n/12
 act on the space $\OML/cmm/m/it/12 S[]\OT1/cmr/m/n/12 (^^@[](\OML/cmm/m/it/12 
N\OT1/cmr/m/n/12 ))$. Let $\OT1/cmr/bx/n/12 T \OT1/cmr/m/n/12 = \OT1/cmr/bx/n/1
2 Z\OT1/cmr/m/n/12 [[]  \OML/cmm/m/it/12 ; T[]; [] \OT1/cmr/m/n/12 ] \OMS/cmsy/
m/n/12 ^^Z []\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 S[]\OT1/cmr/m/n/12 (^^@[](\OML/c
mm/m/it/12 N\OT1/cmr/m/n/12 )))$
 []


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\OT1/cmr/m/n/12 and re-call that $\OT1/cmr/bx/n/12 T$ \OT1/cmr/m/n/12 is a com-
mu-ta-tive ring which is a $\OT1/cmr/bx/n/12 Z$\OT1/cmr/m/n/12 -module of rank 
equal to $[][] \OML/cmm/m/it/12 S[]\OT1/cmr/m/n/12 (^^@[](\OML/cmm/m/it/12 N\OT
1/cmr/m/n/12 ))$.
 []

[68] [69] [70] [71]
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[]\OT1/cmr/m/n/12 The \OT1/cmr/bx/n/12 Pon-tr-ja-gin dual \OT1/cmr/m/n/12 of a 
mod-ule $\OML/cmm/m/it/12 M$ \OT1/cmr/m/n/12 is the mod-ule $\OML/cmm/m/it/12 M
[] \OT1/cmr/m/n/12 := [](\OML/cmm/m/it/12 M; \OT1/cmr/bx/n/12 Q\OML/cmm/m/it/12
 =\OT1/cmr/bx/n/12 Z\OT1/cmr/m/n/12 )$
 []

[72] [73]
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\OT1/cmr/m/n/12 But $[][] \OMS/cmsy/m/n/12 
[]\OT1/cmr/bx/n/12 C \OT1/cmr/m/n/12 = \OML/cmm/m/it/12 L \OMS/cmsy/m/n/12 
[] \OT1/cmr/bx/n/12 C$ \OT1/cmr/m/n/12 is free of rank $2$ over $\OT1/cmr/bx/n/
12 T \OMS/cmsy/m/n/12 
 \OT1/cmr/bx/n/12 C$\OT1/cmr/m/n/12 . There-fore the prod-uct $[][] [][] \OMS/c
msy/m/n/12 
[]\OT1/cmr/bx/n/12 C$
 []

[74] [75] [76] [77] [78] [79] [80

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Chapter 14.

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\OT1/cmr/m/n/12 For each $\OML/cmm/m/it/12 w\OMS/cmsy/m/n/12 j\OML/cmm/m/it/12 
p$ \OT1/cmr/m/n/12 let the in-er-tia group $\OML/cmm/m/it/12 I[]$ \OT1/cmr/m/n/
12 be the ker-nel of the map from $\OML/cmm/m/it/12 D[]$ \OT1/cmr/m/n/12 into $
[](\OMS/cmsy/m/n/12 O[]\OML/cmm/m/it/12 =w; \OT1/cmr/bx/n/12 Z\OML/cmm/m/it/12 
=p\OT1/cmr/bx/n/12 Z\OT1/cmr/m/n/12 )$.
 []

[81] [82] [83]
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[]\OT1/cmr/bx/n/17.28 Adelic Rep-re-sen-ta-tions As-so-ci-ated to Mod-u-lar For
ms 
 []

[84] [85] [86] [87] [88] [89] [90

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Chapter 15.
[91] [92] [93] [94] [95] [96] [97] [98]
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\OT1/cmr/m/sl/12 15.6.  THE WEIGHT IN SERRE'S CONJECTURES ON MODULAR REPRESENTA
TIONS \OT1/cmr/m/n/12 99
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[104]
Chapter 16.
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Chapter 17.
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Chapter 18.
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\OML/cmm/m/it/12 "$\OT1/cmr/m/n/12 , ($\OML/cmm/m/it/12 "$ \OT1/cmr/m/n/12 Diri
ch-let char-ac-ter ram-i-fied at primes in $\OML/cmm/m/it/12 N[]$\OT1/cmr/m/n/1
2 ). Get a form of level $[]([][] \OML/cmm/m/it/12 p[]; N\OT1/cmr/m/n/12 (\OML/
cmm/m/it/12 f\OT1/cmr/m/n/12 ))\OMS/cmsy/m/n/12 j\OML/cmm/m/it/12 N[]$\OT1/cmr/
m/n/12 .
 []

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