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**thesis
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] [1] [2]) (ack.tex) (foreword.tex [3] [4] [5]) [6] (thesis.toc
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 (notations.tex [8]) [9]
Chapter 1.
(class.res.tex [10

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[12] [13]
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[]\cmr/bx/n/17 A re-fined class num-ber for-mula for deriva-

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Chapter 2.
(class.prf.tex [18

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[]\cmr/bx/n/17 Divisibility prop-er-ties of the cir-cu-lar units 

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[] []\cmr/m/it/12 The great-est com-mon di-vi-sor of $\cmm/m/it/12 n\cmr/m/n/12
 (\cmm/m/it/12 S\cmr/m/n/12 )$

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[]\cmr/m/n/12 Since $\cmm/m/it/12 ^^T\cmr/m/n/12 (\cmm/m/it/12 l\cmr/m/n/12 ) =
 \cmm/m/it/12 ^^L\cmr/m/n/12 (\cmm/m/it/12 l\cmr/m/n/12 )\cmm/m/it/12 x[]$\cmr/
m/n/12 , where $\cmm/m/it/12 x$ \cmr/m/n/12 be-longs to $\cmm/m/it/12 K[]$\cmr/
m/n/12 , and since $[][](\cmm/m/it/12 ^^L\cmr/m/n/12 (\cmm/m/it/12 l\cmr/m/n/12
 )) =

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Chapter 3.
(bsd.res.tex [31

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[]\cmr/bx/n/17 A con-jec-ture of Mazur-Tate type for deriva-

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[] []$\cmm/m/it/12 z[]\cmr/m/n/12 (\cmm/m/it/12 A\cmr/m/n/12 (\cmm/m/it/12 E; S
l\cmr/m/n/12 )[]) = []$ 

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[] []\cmr/m/it/12 Suppose that $\cmm/m/it/12 S$ \cmr/m/it/12 is a prod-uct of p
rimes which

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Chapter 4.
(bsd.prf.tex [45

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\cmr/m/n/12 write $\cmm/m/it/12 s[] \cmr/m/n/12 = [][](\cmm/m/it/12 s\cmr/m/n/1
2 ) \cmsy/m/n/12 2 \cmm/m/it/12 H[]\cmr/m/n/12 (\cmm/m/it/12 K[]; E[]\cmr/m/n/1
2 )$, and let $[]$ be the lift of $\cmm/m/it/12 s[]$ \cmr/m/n/12 to $\cmm/m/it/
12 E\cmr/m/n/12 (\cmm/m/it/12 K[]\cmr/m/n/12 )\cmm/m/it/12 =qE\cmr/m/n/12 (\cmm
/m/it/12 K[]\cmr/m/n/12 )$.

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\cmm/m/it/12 a[]$\cmr/m/n/12 , and then $\cmm/m/it/12 p$ \cmr/m/n/12 di-vides t
he or-der of each eigenspace so that $\cmm/m/it/12 r[]\cmr/m/n/12 (\cmm/m/it/12
 E\cmr/m/n/12 (\cmm/m/it/12 k[]\cmr/m/n/12 )\cmm/m/it/12 =pE\cmr/m/n/12 (\cmm/m
/it/12 k[]\cmr/m/n/12 )[]) =

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[] []\cmr/m/it/12 If $[]\cmr/m/n/12 ([][]) \cmm/m/it/12 < []\cmr/m/n/12 (\cmm/m
/it/12 r[]\cmr/m/n/12 ([][])\cmm/m/it/12 ; r[]\cmr/m/n/12 ([][])) \cmsy/m/n/12 
^^@ \cmr/m/n/12 1$\cmr/m/it/12 , then $[][]\cmm/m/it/12 y\cmr/m/n/12 (\cmm/m/it
/12 S\cmr/m/n/12 ) =

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[]\cmr/bx/n/17 The first non-vanishing deriva-tives of Heeg-

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\cmr/m/n/12 4.7.7[], the cok-er-nel $\cmm/m/it/12 J[]$ \cmr/m/n/12 is triv-ial,
 so that $[][]$ sur-jects onto $[][]\cmm/m/it/12 E\cmr/m/n/12 (\cmm/m/it/12 K[]
\cmr/m/n/12 )\cmm/m/it/12 =pE\cmr/m/n/12 (\cmm/m/it/12 K[]\cmr/m/n/12 )$.

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