Non-vanishing for quartic Hecke $L$-functions and ranks of elliptic curves

We show that a positive proportion of Hecke $L$-functions attached to the quartic residue symbols $(\frac{\cdot }{q}{)}_4$ for squarefree $q\in \mathbb{Z}[i]$ do not vanish at the central point. Our method also extends to the Hecke characters associated to quartic twists of the congruent number curve $E:y^2=x^3-x$. In particular, we prove that the elliptic curve $E^{(q)}:y^2=x^3-qx$ has Mordell-Weil rank $0$ over $\mathbb{Q}(i)$ for a positive proportion of squarefree $q\in \mathbb{Z}[i]$ ordered by norm.