  We study the properties of Eisenstein-Kronecker numbers, which are related to
special values of Hecke $L$-function of imaginary quadratic fields. We prove
that the generating function of these numbers is a reduced (normalized or
canonical in some literature) theta function associated to the Poincare bundle
of an elliptic curve. We introduce general methods to study the algebraic and
$p$-adic properties of reduced theta functions for CM abelian varieties. As a
corollary, when the prime $p$ is ordinary, we give a new construction of the
two-variable $p$-adic measure interpolating special values of Hecke
$L$-functions of imaginary quadratic fields, originally constructed by
Manin-Vishik and Katz. Our method via theta functions also gives insight for
the case when $p$ is supersingular. The method of this paper will be used in
subsequent papers to study the precise $p$-divisibility of critical values of
Hecke $L$-functions associated to Hecke characters of quadratic imaginary
fields for supersingular $p$, as well as explicit calculation in two-variables
of the $p$-adic elliptic polylogarithm for CM elliptic curves.
