The Tamagawa number conjecture for CM elliptic curves

Abstract

In this paper we prove the Tamagawa number conjecture of Bloch and Kato for CM elliptic curves using a new explicit description of the specialization of the elliptic polylogarithm. The Tamagawa number conjecture describes the special values of the L-function of a CM elliptic curve in terms of the regulator maps of the K-theory of the variety into Deligne and etale cohomology. The regulator map to Deligne cohomology was computed by Deninger with the help of the Eisenstein symbol. For the Tamagawa number conjecture one needs an understanding of the $p$-adic regulator on the subspace of K-theory defined by the Eisenstein symbol. This is accomplished by giving a new explicit computation of the specialization of the elliptic polylogarithm sheaf. It turns out that this sheaf is an inverse limit of $p^r$-torsion points of a certain one-motive. The cohomology classes of the elliptic polylogarithm sheaf can then be described by classes of sections of certain line bundles. These sections are elliptic units and going carefully through the construction one finds an analog of the elliptic Soul\'e elements. Finally Rubin's ``main conjecture'' of Iwasawa theory is used to compare these elements with etale cohomology.Comment: 60 pages, Latex2

On the tamagawa number conjecture for hecke characters

“…We observe that the elliptic units C ∞,f , which are the ones that appear in [25] and [22], satisfies the theorem of Iwasawa main conjecture of [24] for any ∆-representation under the hypothesis of the theorem in [24] (personal communication of Rubin).…”

“…This paper need to deal with negative twists. This problem does not appear in [22]. For negative twists, we modify Deninger's elements [8] in order to apply the p-adic techniques of [22].…”

“…Let us observe that with this notation we can follow straightforward all the results and proofs of §3 and §4. In §5 we need to compute is part of the definition of K M = e θ • δ * with l := |l| > 0 given at [22]. We want only to study these projector maps on the Galois cohomology.…”

A note on polylogarithms on curves and abelian schemes

“…Most notably, they play a decisive role in the study of the Tamagawa number conjecture for abelian number fields ([Be2], [Del], [HuW] [Hu-Ki]), CM elliptic curves ( [Den], [Ki2]) and modular forms ([Be1], [Ka]). …”

On Jannsen’s conjecture for Hecke characters of imaginary quadratic fields

“…The same arguments of the proof of [11,Prop.5.2.5] can be used with the diagram [3, Lemma 4.11] to obtain the result.…”