Contemporary Mathematics
Volume 606, 2013
The Local Equivariant Tamagawa Number Conjecture for
Almost Abelian Extensions
Jennifer Johnson-Leung
Abstract. We prove the local equivariant Tamagawa number conjecture for
the motive of an abelian extension of an imaginary quadratic field with the
action of the Galois group ring for all split primes p = 2, 3 at all integer values
s 0.
1. Introduction
Since Dirichlet’s remarkable proof of the analytic class number formula in the
first half of the nineteenth century, conjectures on the relationship between the val-
ues of L-functions and invariants of arithmetic objects have motivated a great deal
of research. The equivariant Tamagawa number conjecture (ETNC) is a unifying
statement concerning the special values of motivic L-functions which encompasses
both the Birch and Swinnerton-Dyer conjecture and the generalized Stark conjec-
tures. It is a deep and sweeping assertion which has yielded to proof in very few
cases. The Tamagawa number conjecture builds on the conjectures of Beilinson
[2], predicting that the L-values of smooth projective varieties over Q are given by
period integrals and regulator maps, up to a rational factor, q. Bloch and Kato
[4], further predicted that the rational number q is given in terms of Tamagawa
numbers and the order of a certain Tate-Shafarevic group. This conjecture was
reformulated by Fontaine and Perrin-Riou [17] in a language that was naturally
extended to motives with extra symmetries by Burns and Flach [7, 8]. There are
two equivalent formulations of the conjecture. The first is a global formulation that
concerns of the vanishing of a certain element in relative K-theory. The second is
a local formulation that concerns the equality of two lattices.
In this paper, we study the conjecture for the motive of an abelian extension
of an imaginary quadratic field. We call these almost abelian extensions because
there are many similarities to the case of absolutely abelian extensions stemming
from the fact that an imaginary quadratic field has only one Archimedean place.
Notice that as this place is complex, the local conjecture at the prime 2 will be less
complicated than in the case of absolutely abelian extensions [16]. However, we
do not consider the prime 2 in this paper. Our main result is a proof of the local
ETNC at all split primes 6 at negative integer values of the L-function. Bley
also considers the case of abelian extensions of imaginary quadratic fields for the
c 2013 American Mathematical Society
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