Enriques' characterization of Abelian surfaces in positive characteristic

Extending Enriques’ characterization to algebraically closed fields of characteristic $p\ge 7$, we show that every smooth projective surface $X$ with $h^1(X,{\mathcal{O}}_X)=2$ and $p_1(X)=p_2(X)=1$ is birational to an Abelian surface. This characterization fails if $p\le 5$, and we give a sharp alternative.