  The paper uses Iwasawa theory at the prime $p=2$ to prove non-vanishing
theorems for the value at $s=1$ of the complex $L$-series of certain quadratic
twists of the Gross family of elliptic curves with complex multiplication by
the field $K = \BQ(\sqrt{-q})$, where $q$ is any prime $\equiv 7 \mod 8$. Our
results establish some broad generalizations of the non-vanishing theorem first
proven by D. Rohrlich using complex analytic methods. Such non-vanishing
theorems are important because it is known that they imply the finiteness of
the Mordell-Weil group and the Tate-Shafarevich group of the corresponding
elliptic curves over the Hilbert class field of $K$. It is essential for the
proofs to study the Iwasawa theory of the higher dimensional abelian variety
with complex multiplication which is obtained by taking the restriction of
scalars to $K$ of the particular elliptic curve with complex multiplication
introduced by Gross.
