4 J. JOHNSON-LEUNG

induced by multiplication. Hence, the dual of the Borel regulator induces an A-

equivariant isomorphism

ϑ∞ : A ⊗Q R → Ξ(AM) ⊗Q R.

Note that the

L∗(AM)

lies in the domain of this isomorphism, and Gross conjec-

tured that its image lies in the rational space Ξ(AM) ⊗Q 1 [28]. This “Stark-type”

conjecture says that up to a rational factor, the L-value is given by the Borel regu-

lator and was proved by Deninger [12] in his work on the Beilinson conjectures for

Hecke characters of imaginary quadratic fields.

Fix a prime number , let S be a set of primes containing , ∞, and the primes

which ramify in F , and let A = A ⊗Q Q . We now concern ourselves with the

-part of the rational factor by considering the isomorphism induced by the Chern

class map and the cycle class map

ϑ : Ξ(AM) ⊗Q A → DetA RΓc Z

1

S

, M .

where the right-hand side denotes the cohomology with compact supports as defined

by the mapping cone

RΓc Z

1

S

, M → RΓ Z

1

S

, M →

p∈S

RΓ(Qp,M ).

The conjecture then compares a natural lattice in the right hand side of this isomor-

phism to the lattice generated by the image of

L∗(AM).

To construct the lattice,

we choose the order Z[G] in A and the Gal(Q/Q)-stable projective Z [G]-lattice

T =

H0

(

Spec(F ⊗Q Q), Z (j)

)

.

Conjecture (Local ETNC). There is an equality of lattices

ϑ

ϑ∞(L∗(AM)−1)

· Z [G] = DetZ

[G]

RΓc

Z

1

S

, T .

inside of DetA RΓc(Z[1/S],M ).

The ETNC for number fields is equivalent to the statement that the local

conjecture holds at every prime number . This determines L∗(AM) up to a unit

in Z[G]. Notice that the ETNC depends on the choice of order but is independent

of the choice of S and T [13]. This indepence of lattice is exploited to prove the

main results of [22] which will be an important ingredient in the proof of our main

theorem.

Main Theorem. Let F be an abelian extension of an imaginary quadratic field

K with Galois group G. the local equivariant Tamagawa number conjecture

is valid for the motive

h0

(Then

Spec(F )

)

(j) for j 0 at every rational prime p 6 which

splits in K.

Remark. The restriction to split primes can be lifted whenever the μ-invariant

of a certain Iwasawa module can be shown to vanish, as discussed in Section 4.