Isogenies between K3 Surfaces over $\bar{\mathbb{F}}_p$

TL;DR

This paper extends the concept of isogenies between K3 surfaces to arbitrary perfect fields, providing methods to construct isogenous K3 surfaces over algebraic closures of finite fields and exploring their relation to Kuga-Satake abelian varieties.

Contribution

It generalizes the definition of isogenies for K3 surfaces over any perfect field and links these to isogenies of Kuga-Satake abelian varieties, with applications to CM liftings.

Findings

Isogenies between Kuga-Satake abelian varieties induce isogenies between K3 surfaces.

Every finite height K3 surface admits a CM lifting under mild conditions on p.

Construction of isogenous K3 surfaces over algebraic closures of finite fields.

Abstract

We generalize Mukai and Shafarevich's definitions of isogenies between K3 surfaces over $\mathbb{C}$ to an arbitrary perfect field and describe how to construct isogenous K3 surfaces over $\bar{\mathbb{F}}_p$ by prescribing linear algebraic data when $p$ is large. The main step is to show that isogenies between Kuga-Satake abelian varieties induce isogenies between K3 surfaces, in the context of integral models of Shimura varieties. As a byproduct, we show that every K3 surface of finite height admits a CM lifting under a mild assumption on $p$.