Sharp $L^2\to L^4$ extension inequality for quadratic surfaces in finite fields

In this work, we establish the sharp $L^2\to L^4$ extension inequality for a large class of quadratic surfaces in ${\mathbb{F}}^3$, where $\mathbb{F}$ is a finite field of odd characteristic. These surfaces include the cones without the origin and the spheres with nonzero radius $j$ such that $-j$ is not a square in $\mathbb{F}$. We also show that the maximizers of these inequalities must have constant modulus.