CM-liftability of simple superspecial abelian surfaces over prime fields

TL;DR

This paper proves that simple superspecial abelian surfaces over finite fields can be lifted to complex multiplication (CM) structures after a quadratic extension, advancing the understanding of their liftability properties.

Contribution

It establishes the CM-liftability of simple superspecial abelian surfaces over prime fields using residual reflex conditions and Lie types, filling a gap in the liftability classification.

Findings

Superspecial abelian surfaces admit CM liftings over _{p^2}.

The work complements existing results on ordinary and almost ordinary surfaces.

It advances the classification of CM-liftability for all simple abelian surfaces.

Abstract

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 $\mathbb{F}_{p}$.