Hsin-Yi Yang: CM-liftability of simple superspecial abelian surfaces over F_p

Abstract: In 1992, Oort asked a question: Does there exist a CM lifting of an abelian variety over a field with characteristic \(p>0\)? 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.

Our work shows the CM-liftability of simple superspecial abelian surfaces over \(\mathbb F_p\) by using the so-called residual reflex condition and Lie types. 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 \(\mathbb F_p\).

In this talk, we will explain the main idea of the proof by an example: a simple superspecial abelian surface over \(\mathbb F_7\) in the isogeny class, which is determined by the Weil-7 number \(\sqrt{7}\zeta_{3}\).