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Picard-Fuchs uniformization and modularity of the mirror map. (English) Zbl 1047.11044

This paper is a continuation of the author’s article [in The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), CRM Proc. Lect. Notes 30, 27–35 (2001; Zbl 1047.11043)], where the modularity of the mirror maps of elliptic curve families and of one-parameter families of rank-19 lattice polarized K3 surfaces was studied.

Summary: Arithmetic properties of mirror symmetry (type IIA-IIB string duality) are studied. We give criteria for the mirror map q-series of certain families of Calabi-Yau manifolds to be automorphic functions. For families of elliptic curves and lattice polarized K3 surfaces with surjective period mappings, global Torelli theorems allow one to present these criteria in terms of the ramification behavior of natural algebraic invariants - the functional and generalized functional invariants respectively. In particular, when applied to one parameter families of rank 19 lattice polarized K3 surfaces, our criterion demystifies the Mirror-Moonshine phenomenon of Lian and Yau and highlights its non-monstrous nature. The lack of global Torelli theorems and presence of instanton corrections makes Calabi-Yau threefold families more complicated. Via the constraints of special geometry, the Picard-Fuchs equations for one parameter families of Calabi-Yau threefolds imply a differential equation criterion for automorphicity of the mirror map in terms of the Yukawa coupling. In the absence of instanton corrections, the projective periods map to a twisted cubic space curve. A hierarchy of “algebraic” instanton corrections correlated with the differential Galois group of the Picard-Fuchs equation is proposed.

11F23Relations of number theory with algebraic geometry and topology
14D05Structure of families
14J32Calabi-Yau manifolds
32G20Period matrices, variation of Hodge structure; degenerations