LIFTED ROOT NUMBER CONJECTURE AND IWASAWA THEORY

13

and look at the two cokernel sequences of the (injective) southwest arrows. Because of (D2.1/)

and Lemma 2.3 they yield the diagram

G(K^l/K)

~ C(vo) -» Aj3

(D2.2) / / S=

G(M/K) - Y -» A/G

defining Y. This information is summarized in the commutative diagram with exact rows and

columns

Z/ ® E - Zi® A -» Z/ (8) L

I I I

(D2.3) r W £ ( # p X ) ^ I l p

6

s ^ p ) ^ Z * ® A ,

G{M/K) - y -» A,G

Collecting everything so far, we have almost established

L E M MA 2.4. Assume that K satifies LeopoldVs conjecture and that either I is odd or K/k is

split at all infinite primes. Then

1. Y has finite projective dimension over Z/(7;

2. the translation functor transforms G(M/K) -* G(M/k) -» G into

G(M/K) -» Y -» Z, g AG.

PROOF. Claim 1. is an immediate consequence of the middle column of (D2.3) and of 4. in

Lemma 2.1.

Claim 2. is due to the fact that (D2.2) is a push-out diagram for Y. Moreover, in

H2(G,CK) - H2(G,G{K^l/K)) - H2(G,G(M/K))

II II II

ExtJ

G

(AG,CA:) -+ E x t ^ ( A 2 , G ( t f a b ' 7 * 0 ) -* ExtJ

G

(AG,G(M/A') )

II II

ExtlaiAtGiGiK^/K)) - E x t ^

G

( A ^ , G ( M / ^ ) ) ,

with the bottom equalities because

G(Kah'l/K)

and G(M/K) are Z/-modules, the fundamen-

tal class in H2(G, CK) is sent to the extension class of G(M/K) — G/M/k) -» G in the top

line, by Safarevic-Weil.

2.4 The totally real case

Remembering Tate's canonical class Z{ g) r and observing that A* fits into

A , ~ 0 indg

p

Z«G

p

- 0 indg

p

Z, ,

pes* pes*

we enlarge each row in (D2.3) to a 2-extension

ZiS)E - Zi®A - Z/(g)B -» A/ 5

(D2.4) I W W ) ^ I l p e S ^ ) - (Z,G)™ - ZZS

G(M/A-) ~ Y -+ ZiG ^ Zt ,

with m = 15* |, having middle terms of finite projective dimension and the third term without

Z/-torsion. This will be exploited below assuming that Leopold^s conjecture holds for K and

that K is totally real.