K3 surfaces of finite height over finite fields

Details K3 surfaces of finite height over finite fields K3 surfaces of finite height over finite fields

2007-09-13

arxiv.org/abs/0709.1979v4

Mathematics

Algebraic Geometry

Cor.3.4 added, typos corrected, to appear in J. Math. Kyoto Univ

Scientific paper

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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