*(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.