Fields of definition of K3 surfaces with complex multiplication

Abstract

Let ∕ℂ be a K3 surface with complex multiplication by the ring of integers of a CM number field . Under some natural conditions on the discriminant of the quadratic form ( ), we produce a model can of over an explicit abelian extension ∕ with the property that ( can ∕ ) = ( ∕ℂ). We prove that can ∕ is canonical in the following sense: if ∕ is another model of such that ( ∕ ) = ( ∕ℂ), then ⊂ and can ≅ . If is fixed, our theorem applies to all but finitely many surfaces with complex multiplication by . In case is not one of those, we still provide necessary and sufficient conditions for a model enjoying the same properties of can to exist. As an application to our work, we give necessary and sufficient conditions for a singular K3 surfaces with CM by the ring of integers of an imaginary quadratic field to have a model with all Picard group defined over , and provide an alternative proof of a finiteness result obtained by Shafarevich and later generalised by Orr and Skorobogatov. CONTENTS 18 6. Applications and open questions 21 References 24