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

xσ · 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