Abstract
Let Ω be a non-zero holomorphic 2-form on a K3 surface S. Suppose that S is projective algebraic and is defined over [Formula: see text]. Let [Formula: see text] be the [Formula: see text]-vector space generated by the numbers given by all the periods ∫γΩ, γ ∈ H2(S, ℤ). We show that, if [Formula: see text], then S has complex multiplication, meaning that the Mumford–Tate group of the rational Hodge structure on H2(S, ℚ) is abelian. This result was announced in [P. Tretkoff, Transcendence and CM on Borcea–Voisin towers of Calabi–Yau manifolds, J. Number Theory 152 (2015) 118–155], without a detailed proof. The converse is already well known.
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