4
JURGEN PUTTER AND ALFRED WEISS
and its implied representation of elements of KQT(ZIG). We say that / G H o m ^ (i?/(G), F*)
represents a G KQT{ZIG) if there exists an x G K\(QiG) so that Det(x) = / and d(x) = a
(see [GRW2, Appendix A]).
Defining Alp G H o m ^ (i?/(G),
Fx)
by Ap(x) =
s
( ^ (
s _ 1
x ) ) allows us to formulate the
L I F T E D R O O T N U M B E R C O N J E C T U R E AT / . x ^ ^ 4 (x) represents 0,$.
The conjecture does not depend on the choice of s : Q
c
^-
Q^c.
The canonical map KQT(ZG) KQ(ZG) takes fi^, to Chinburg's 0 and the Lifted Root
Number Conjecture for all / implies (indeed strengthens) Chinburg's Root Number Conjecture.
Define cu^ to be the class in KQT(ZIG) so that Oj / + UJ^ is represented by x ^ A\p'{x)-
Expressing the Lifted Root Number Conjecture in terms of UJ^ gives more invariant formulas,
because wO is independent of (p [GRW2, Theorem 2'].
1.6 The translation functor
We will need to change group extensions with abelian kernel, as typically arising from Galois
theory, into module sequences. This is done by means of the translation functor described in
[RW1, p. 154/155] of which an /-adic version will be needed.
Let A^X^G be a short exact sequence of finite groups, with A an abelian /-group. With
it comes the short exact sequence A Y A^G of Z/G-modules in which A/G is the
augmentation ideal in Z/G and Y A\X / AiA A[X. The map Y A/G is the obvious
one and the map A —» Y is induced by a i— a 1.
The translation functor also works for exact sequences of profinite groups A X G with
A an abelian pro-/ group. This is described in the Appendix 4A of chapter 4.
The special case when A is a Z^G-module, with G finite, will often be used. Then there is
the following analogue to [RW1, Lemma 3, p.155]: if x G
H2(G,A)
is the extension class
of A - X G and y G Ext^
G
(A/G , A) that of its translate A - F A/G, then the
connecting isomorphism
Hl{G,
Hom(A;G, A)) - i #
2
( G , A ) , induced by A/G Z/G Zj,
has (5(y) = x (because Hom(Z/,.A) = A).
1.7 Leopoldt's conjecture
We denote by ifp the completion of K at the finite prime p, and by t/p and
Upl
its group of
units, respectively principal units. Let E^ be the group of all global units in K. The inclusion
Kx
Ylpii
^pX
takes Eoo into f]pu ^ P
an^ s o
induces a natural map Zi^E^ —• n
p
u ^p\ since
/ f [f/p : C7p]. Leopoldt conjectures that this map is injective (see e.g. [Wa, p.75]). By a
theorem of Brumer [Brl] this is the case when K/Q is abelian (see [Wa, p.77]).
A We now turn to the feet of the tripod.They arise as lifted-Q classes emerging from
the ingredients in the construction of a realization of Tate's canonical class r. Namely, Tate
sequences are obtained from the matching of the local fundamental classes in S with the global
fundamental class of K/k. We perform functorial processes of /-completion and 5-truncation
to these sequences in order to get three 2-extensions to which the lifted-0 construction applies.
These processes do not preserve exactness in general. We avoid the difficulties caused by this
by assuming that Leopoldt's conjecture holds for K. It will also be convenient to have K
totally real.
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