  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.
