K3 surfaces with algebraic period ratios have complex multiplication

Paula Tretkoff

doi:10.1142/s1793042115400217

K3 surfaces with algebraic period ratios have complex multiplication

Paula Tretkoff

Department of Mathematics, Texas A&M University, College Station, TX 77843, USA

説明

<jats:p>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 ∫<jats:sub>γ</jats:sub>Ω, γ ∈ H<jats:sub>2</jats:sub>(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 H<jats:sup>2</jats:sup>(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.</jats:p>