K3 surfaces of finite height over finite fields

2007/09/13 by Jeng-Daw Yu, Yu, J. -D., Noriko Yui +1
Computer Science · Mathematics · #14G15 #14J28 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.0709.1979

openalex publication_date 2007/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Arithmetic of K3 surfaces defined over finite fields is investigated. In particular, we show that any K3 surface of finite height over a finite field k of characteristic p > 3 has a quasi-canonical lifting to characteristic 0, and that for any such lifting, the endormorphism algebra of the transcendental cycles, as a Hodge module, is a CM field. The Tate conjecture for the product of certain two K3 surfaces is also proved. We illustrate by examples how to determine explicitly the formal Brauer group associated to a K3 surface over k. Examples discussed here are all of hypergeometric type.

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• Special lifts of ordinary K3 surfaces and applications
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• Complex multiplication and Brauer groups of K3 surfaces