CM-liftability of simple superspecial abelian surfaces over prime fields

Abstract

For any prime $p>0$, we prove that simple superspecial abelian surfaces over ${\mathbb{F}}_p$ admit CM liftings, 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 Bergström-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 ${\mathbb{F}}_p$.

33 pages, with minor modifications, concise version, and more complete results