LIFTED ROOT NUMBER CONJECTURE AND IWASAWA THEORY

15

lifted-0 constructions and obtain

pi : ZtS B — Zj 0 A

i :

z/C

- y

In order to get these maps related to each other, we observe that we can fill in the middle

vertical arrows

n

to obtain the commutative diagram with exact rows and columns

ZtS)L -• Z/ 0 B -» %i 0 A S

I I I

Z / 0 A * w (Z

z

G)

m

-» ZjS

i i i

Z/ 0 AG ~ Z/C -» Z/

As a consequence and putting things together, we arrive at the commuting diagram below

which has the 2-extensions of (D2.4) as its rows:

Z/ 0 E -• Zi®A - Z/ 0 5 -» Z/ 0 AS

I I I I

(D2.6) I I W ) - UC(VP) - (ZZG)TO - ZZ5

i 1 i i

G(M/K) ~ Y - Z/G -» Z|

The following observation now guarantees that we can simultaneously perform the lifted-^

constructions. Let

be a commutative diagram of Z[G-

lattices with exact rows.

\G\

Then the push-out along the endomorphism \G\ of the left end of each row, i.e.,

\G\

and Z' — Z', induces a natural cube together with a well-defined map X' 0 X" —• Z' 0 Z"

making it commute. Note that multiplication by \G\ is admissible as a map f3 (or a) in the

lifted-f2 construction, since \G\ annihilates G-cohomology.

Mutatis mutandis, the same applies to a diagram in which only the right ends, X" and

Zh',

are lattices, provided that now the pull-back along multiplication by \G\ on these lattices is

performed.

This being said, it is readily seen (as in the last paragraph of p.56 in [GRW2]) that the lifted-H

construction yields a commutative diagram

X'

i

Z'

~ X -» X"

1 1

~ Z -» Z"

$1

&c

i

: Z/ 0 B -»

I

: (Z,G)m -

i

: ZiG -

Z/ 0 L 0 Z/

I

Zi 0 A* 0

i

Z/ 0 AG (

)A S

iew

1©V?£

10$

J

s

i

- nc(vf:

s

Zi®AG®G(M/K) -» ]

xthe bottom one by inducing up the inclusion of augmentation sequences of Gp into that of G