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Let $q$ be any prime $\equiv 7 \mod 16$, $K = \mathbb{Q}(\sqrt{-q})$, and let$H$ be the Hilbert class field of $K$. Let $A/H$ be the Gross elliptic curvedefined over $H$ with complex multiplication by the ring of integers of $K$. Weprove the existence of a large explicit infinite family of quadratic twists of$A$ whose complex $L$-series does not vanish at $s=1$. This non-vanishingtheorem is completely new when $q > 7$. Its proof depends crucially on theresults established in our earlier paper for the Iwasawa theory at the prime$p=2$ of the abelian variety $B/K$, which is the restriction of scalars from$H$ to $K$ of the elliptic curve $A$.

Non-vanishing theorems for central $L$-values of some elliptic curves with complex multiplication II