LIFTED ROOT NUMBER CONJECTURE AND IWASAWA THEORY

3

in the Grothendieck group of all finitely generated ZG-modules having finite projective di-

mension, and conjectures that ft is the root number class of K/k.

1.4 The lifted-Q construction

Let Mi,M2 be finitely generated Z/G-modules and / : M

2

- M\ a Z/G-monomorphism

inducing an isomorphism Q/ g M

2

^ Qj g M\. Let K, € E x t |

G

(M2, Mi) have a realization

Mi — A — i? -» M2 in which A, B have finite projective dimension over Z\G and B has no

Zj-torsion. Following [GRW2, §2] we associate to this data an element

« ( / , K) in K0T(ZtG),

the Grothendieck group of finite Z/G-modules of finite projective dimension. In order to do

this first break any such realization of K, into two parts

M i ^ A ^ L , L^B ^ M

2

.

Then take Z/G-monomorphisms a,/3 : L - L which induce 0 in Ext

1

. The pull-back along

a and push-out along /3 give

Mi ^ L 0 M i ^ L L - B -» M

2

II

5J,

a I /U

/5|

||

Mi -+ A -» L L - L 0 M

2

-» M

2

The composite map

/ : £ A L 0 M

2

1

- ^ / L 0 Mi - ^ 4

is injective with finite cokernel and so [coker/] G KQT(Z[G). We define

Q(/, «) = [coker /] - d[Qt ®z, L, a(3]

with d : i^i(Q/G) — KQT(ZIG) from the localization sequence of X-theory

1TI(Z

Z

G) - ATi(QjG) - ^ KoTiZtG) - K0{ZiG) - K

0

(QiG).

The proofs of Lemma 1 and 2 in [GRW2] show that 0 ( / , « ) is independent of the choice of

a,a,/3,/3 and the realization of /c.

1.5 The Lifted Root Number Conjecture

If in the preceding section Z\ is replaced by Z and K is the Tate canonical class r, then the

construction gives

^ =

f

% , r ) G K

0

T ( Z G )

for any ZG-monomorphism y : A S ~ E, see [GRW2, §2]. Since K0T(ZG) = ®K0T(ZpG),

p

with p ranging over all prime numbers, we can speak of the /-component Q^/ of Vt^. Clearly

Oj/ = il((fi, 77) where ipi : Z\ g) A S ^ Z/ (g) £ is induced by (/? and 77 = Z/ (8 r.

The local Horn description is the isomorphism Det in

KifaG) - #i(Qz3) - KoTiZtG)

Det J

Hom^(fl

z

(G),F

z

x

)