K3 surfaces with algebraic period ratios have complex multiplication.

Let Ω be a non-zero holomorphic 2-form on a K3 surface S. Suppose that S is projective algebraic and is defined over . Let be the -vector space generated by the numbers given by all the periods ∫γ Ω, γ ∈ H2(S, ℤ). We show that, if , 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. [ABSTRACT FROM AUTHOR]