ENTC FOR ALMOST ABELIAN EXTENSIONS 3
and depends only on the quasi-isomorphism class of C. Indeed, if the cohomology
groups
Hi(C)
are all perfect, one has
DetR C =
i∈Z
DetR
(−1)i
Hi(C).
2. The main theorem
Let F be an abelian extension of K with Galois group G. We consider the
Chow motive M = h0
(
Spec(F )
)
(j) where j is a negative integer.
2.1. The local statement of the ETNC. We will formulate the equivariant
Tamagawa number conjecture (ETNC) for this motive. M carries an action of the
semisimple Q-algebra A = Q[G]. We study M via its realizations and the action of
A on these spaces, focusing on the Betti realization
MB =
H0
(
Spec(F )(C), Q(j)
)
,
which carries an action of complex conjugation, the de Rham realization
MdR = HdR(Spec(F
0
)/Q)(j),
with its Hodge filtration, and the -adic realization
M = Het
0
(
Spec(F ) ×Q Q, Q (j)
)
,
which is a continuous representation of Gal(Q/Q). The A-equivariant L-function
of M is defined via an Euler product
L(AM, s) =
p
DetA(1 Frobp
−1 ·p−s
| M
Ip
)−1.
The leading term of the Taylor expansion at s = 0 decomposes over the characters
of G
L∗(AM)
=
(
L (η, j)
)
η∈G
(A ⊗Q
R)×.
We can now introduce one of the key objects in this formulation of the Tamagawa
number conjecture: the fundamental line is the A-module
Ξ(AM) = DetA(K1−2j(OF
)∗
⊗Z Q) ⊗A
DetA1(MB +),
where + denotes the invariants under complex conjugation and K1−2j(OF
)∗
is the
dual of the algebraic K-group K1−2j(OF ) = K1−2j(F ). This “line” is the tool
which enables the comparison of L-value with algebraic invariants of number field.
Borel’s regulator [5], is an isomorphism
K1−2j(OF ) ⊗Z R
ρ
−∞
σ∈T
C/R ·
(2πi)1−j
· σ
+
where T = Hom(F, C). Since j 0, K1−2j(OF ) K1∑ −2j(F ). For an element,

σ∈T
· σ, the Galois group acts via g · (

σ∈T
x · σ) =
σ∈T
x ·
g−1σ.
With this
action, ρ∞ is A-equivariant just as in the case of the Dirichlet regulator. Now,the
R-dual of (
σ∈T
C/R·(2πi)1−j ·σ)+
is identified with MB
+⊗QR
by taking invariants
in the Gal(C/R)-equivariant perfect pairing
σ∈T
R ·
(2πi)j
×
σ∈T
C/R ·
(2πi)1−j

σ∈T
C/2πi · R
Σ
R
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