LIFTED ROOT NUMBER CONJECTURE AND IWASAWA THEORY
9
and det( £ ^ ( l o g ^ a , ) * "
1
| i n d g
( X / Q
V
x
) .
xeG(K/q)
The elements bp in the first expression generate integral normal bases of the local extensions
Kp/kp. Exploiting results of Taylor we get
T H E O R E M E. n
NSpkp/ql(spbP/Xp) maV
be replaced by S T ( S
_ 1
X ) ndet(—Fr
p
| Vx
p
) ~ \ with
pk\i
the product over all primes p^ in S or dividing d^/q- Here, r is the global Galois Gaufi
sum.
Finally, the element ai in the second expression above is a unit in Q^ whose image in (
under the reciprocity map equals the image of 7^ under G(koc/k) G(Qoo/Q). It is only
now that we must specialize to the cyclotomic case.
In chapter 8 we take for £ a Ramachandra unit which allows that expression to be written in
terms of L/(l,x) via Leopoldt's formula and A^(x) in terms of 1/(0, x). Combining various
representing homomorphisms we then get at the deviation u/'' of Q^ from Alp .
THEOREM F. Let K be a totally real cyclotomic field and let I be odd and tame in K/k. Then

Jl) is represented by x ^ x(T) / Y[ {Npk)d[mV* * .
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