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

G°

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