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 |$p$|⁠.

中文翻译：

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K3表面之间的同构

我们对| $ {\ mathbb {C}} $ |上的K3表面之间的同构性进行了Mukai和Shafarevich的同构定义的概括。到任意理想场，并描述如何在| $ \ bar {{\ mathbb {F}}} _ p $ |上构造同构K3曲面 通过在| $ p $ |时指定线性代数数据 大。主要步骤是，在Shimura品种的完整模型的背景下，表明Kuga-Satake阿贝尔品种之间的同质性会诱导K3表面之间的同质性。作为副产品，我们表明在| $ p $ |⁠的轻微假设下，每个有限高度的K3曲面都允许CM提升。