Contemporary Mathematics

Volume 606, 2013

http://dx.doi.org/10.1090/conm/606/12137

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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