LIFTED ROOT NUMBER CONJECTURE AND IWASAWA THEORY
lifted-0 constructions and obtain
pi : ZtS B — Zj 0 A
In order to get these maps related to each other, we observe that we can fill in the middle
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
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-^
be a commutative diagram of Z[G-
lattices with exact rows.
Then the push-out along the endomorphism \G\ of the left end of each row, i.e.,
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
are lattices, provided that now the pull-back along multiplication by \G\ on these lattices is
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 -» X"
~ Z -» Z"
: Z/ 0 B -»
: (Z,G)m -
: ZiG -
Z/ 0 L 0 Z/
Zi 0 A* 0
Z/ 0 AG (
Zi®AG®G(M/K) -» ]
xthe bottom one by inducing up the inclusion of augmentation sequences of Gp into that of G