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 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.
Previous Page Next Page