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Have Kuga-Satake correspondences been investigated in characteristic $p$?

(I'm being intentionally vague about what this would mean.)

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closed as too broad by Piotr Achinger, Pace Nielsen, Dima Pasechnik, András Bátkai, Venkataramana Jul 14 at 2:43

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Yes. Have you tried google? ;) It starts with Deligne's proof of the Weil conjectures for K3 surfaces. Later, Nygaard and Ogus proved the Tate conjecture for K3 surfaces of finite height, also using Kuga-Satake. The recent proofs of the Tate conjecture also use Kuga-Satake, sometimes disguised in the form of an integral model of some Shimura variety (Madapusi Pera). Please take a look at some of those papers and come back if you have more specific questions. $\endgroup$ – Piotr Achinger Jul 10 at 6:17
  • $\begingroup$ I thought that these works involve lifting to characteristic zero and applying K-S there. Is that a misconception? @PiotrAchinger $\endgroup$ – rj7k8 Jul 10 at 6:23
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    $\begingroup$ I think yes, except for Madapusi Pera. But in those works the Kuga-Satake variety is reduced back modulo p, becoming a Kuga-Satake variety in char p. Your question is quite vague and in particular it does not indicate that these existing results are not what you want. Please make your question more specific and indicate what you already know. $\endgroup$ – Piotr Achinger Jul 10 at 6:43
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    $\begingroup$ @PiotrAchinger Ok thanks, your remarks have already been helpful. I'll think this over, look at those papers more carefully, and hopefully improve this question later. $\endgroup$ – rj7k8 Jul 10 at 6:49