CM-liftability of simple superspecial abelian surfaces over prime fields
Hsin-Yi Yang
TL;DR
The paper advances the CM-liftability program for simple abelian surfaces over prime fields by focusing on simple superspecial abelian surfaces and proving they admit CM liftings after base change to $\,\mathbb{F}_{p^2}$ via the residual reflex condition (RRC) and Lie-type analysis. It builds a framework that unifies CM types, good reductions, Dieudonné theory, and Serre–Tate deformation, and uses detailed Frobenius-valuation computations to classify Lie types in the superspecial isogeny classes. The results cover non-real and real Weil-$p$ numbers, with careful treatment of exceptional primes ($p=2,3$) and a dedicated explicit example at $p=7$, showing both CM liftings (CML) and strong CM liftings (sCML). By connecting RRC, Lie types, and lifting theory, the work contributes a step toward a complete CM-liftability picture for simple abelian surfaces over prime fields and complements existing ordinary and almost-ordinary cases. The methods have potential implications for understanding CM liftings in broader PEL-type contexts and for applications to endomorphism algebras and lifting criteria in arithmetic geometry.
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 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 $\mathbb{F}_{p}$.