K3 surfaces with algebraic period ratios have complex multiplication
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Let ω be a non-zero holomorphic 2-form on a K3 surface S. Suppose that S is projective algebraic and is defined over Q. Let P be the Q-vector space generated by the numbers given by all the periods γ ω, γ H2(S,). We show that, if Q P = 1, then S has complex multiplication, meaning that the Mumford-Tate group of the rational Hodge structure on H;bsupesup(S,) 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. 2015 World Scientific Publishing Company.
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