For any prime $p>0$, we prove that simple superspecial abelian surfaces over $\mathbb{ F}_{ p}$ admit CM liftings after base change at most to $\mathbb{ F}_{ p^2}$, by using the residual reflex condition (RRC) and Lie types. The CM-liftability of ordinary simple abelian surfaces is proved by Serre-Tate, and the CM-liftability of almost ordinary simple abelian surfaces is proved by Oswal-Shankar and Bergstr\"om-Karemaker-Marseglia, respectively. As there can only be ordinary, almost ordinary, or supersingular simple abelian surfaces over $\mathbb{ F}_{ p}$, our work is another step to complete the CM-liftability of simple abelian surfaces over $\mathbb{ F}_{ p}$.
Categories: math.NT