  We prove non-vanishing theorems for the central values of $L$-series of
certain quadratic twists of the Gross elliptic curve with complex
multiplication by the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$, where
$q$ is any prime congruent to $7$ modulo $8$. This completes the non-vanishing
theorems proven by Coates and the second author in which the primes $q$ were
taken to be congruent to $7$ modulo $16$. From this, we obtain the finiteness
of the Mordell-Weil group and the Tate-Shafarevich group for these curves. For
a special prime $\mathfrak{P}$ lying above the prime $2$, we also prove a rank
zero converse theorem and the $\mathfrak{P}$-part of the Birch-Swinnerton-Dyer
conjecture for the higher-dimensional abelian varieties obtained by restriction
of scalars.

Number Theory