L-value at 0 [3]. His proof has restrictions similar to those in this work stemming
from the vanishing of the μ-invariant of a certain Iwasawa module. It would be
quite nice to prove compatibility of the conjecture with the functional equation for
this class of motives as well, as the combination of these results would give the
conjecture at any integer value of the L-function.
The only completely proven case of the equivariant Tamagawa number conjec-
ture is the proof of Burns, Flach, and Greither for abelian extensions of Q [9,10,16].
Huber and Kings proved independently a slightly weaker version of this cyclotomic
case [21] which has since been strengthened to a full proof by work of Witte [32].
Even partial results are quite more sparse. Burns and Flach give a proof for an infi-
nite family of quaternion extensions [8], and Navilarekallu gives a method of proof
for A4 extensions which he employs for a specific case [27]. There are also several
theorems that are not equivariant. Gealy recently proved a weakened version of
the Tamagawa number conjecture for modular forms of weight greater than 1 [18].
Kings also proved a weakened version for elliptic curves with CM by an imaginary
quadratic field of class number 1 [25]. In both of these cases, the conjecture must be
weakened because it is not known that the motivic cohomology groups are finitely
generated. By working with the constructible part of the group, however, a proof
can be given. Bars builds on work of Kings to give some non-equivariant results
for Hecke characters of imaginary quadratic fields [1]. The survey papers of Flach
[14,15] include a nice formulation of the local version of the equivariant Tamagawa
number conjecture for arbitrary motives over Q and discusses the proven cases. We
strive here to keep notation consistent with this overview.
This paper is an improvement of the main result in the author’s thesis, and
so many thanks are due her thesis advisor, Matthias Flach. She would also like to
thank Werner Bley, Matthew Gealy, and Guido Kings for very helpful conversations
and the referee for a careful reading of the manuscript.
1.1. Notation. Let K be an imaginary quadratic field with ring of integers
OK and let f be an integral ideal of OK . We will let K(f) denote the ray class field
of K of conductor f. By a CM pair of modulus f over a number field F , we mean a
pair (E, α) where E is an elliptic curve over F with complex multiplication by OK
and such that the inculsion of OK into F factors through End(E), and α E(C)
is a primitive f-division point. By [23, 15.3.1], there is a CM pair of modulus f
over K(f) which is isomorphic to (C/f, 1 mod f) over C. This pair is unique up to
isomorphism, and whenever OK
is injective the isomorphism is unique.
Denote (C/f, 1 mod f) the canonical CM pair.
We will make repeated use the graded determinant functor Det of Knudsen and
Mumford [26]. Let R be a commutative ring, and P a projective R-module. The
determinant of P is the invertible R-module
DetR P :=
rkR P
If C : · · · P
· · · is a perfect complex of projective R-
modules, the determinant of the complex is defined to be the graded invertible
DetR C :=
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