Classification of Zeta Functions of Bielliptic Surfaces over Finite Fields

Let $S$ be a bielliptic surface over a finite field, and let an elliptic curve $B$ be the Albanese variety of $S$; then the zeta function of the surface $S$ is equal to the zeta function of the direct product $\mathbb P^1\times B$. Therefore, the classification problem for the zeta functions of bielliptic surfaces is reduced to the existence problem for surfaces of a given type with a given Albanese curve. In the present paper, we complete this classification initiated in [1].