Non-vanishing of central $L$-values of the Gross family of elliptic curve

We prove non-vanishing theorems for the central values of $L$-series of 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 prime $\mathfrak{P}$ lying above the prime $2$, we also prove a converse theorem in the rank $0$ case and the $\mathfrak{P}$-part of the Birch-Swinnerton-Dyer conjecture for the higher-dimensional abelian varieties obtained by restriction of scalars.