Numbers of points of surfaces in the projective $3$-space over finite fields

Abstract

In the previous paper, we established an elementary bound for numbers of points of surfaces in the projective $3$-space over ${\Bbb F}_q$. In this paper, we give the complete list of surfaces that attain the elementary bound. Precisely those surfaces are the hyperbolic surface, the nonsingular Hermitian surface, and the surface of minimum degree containing all ${\Bbb F}_q$-points of the $3$-space.