The Arithmetic Mirror Symmetry and¶Calabi–Yau Manifolds

We extend our variant of mirror symmetry for K3 surfaces [GN3] and clarify its relation with mirror symmetry for Calabi–Yau manifolds. We introduce two classes (for the models A and B) of Calabi–Yau manifolds fibrated by K3 surfaces with some special Picard lattices. These two classes are related with automorphic forms on IV type domains which we studied in our papers [GN1]–[GN6]. Conjecturally these automorphic forms take part in the quantum intersection pairing for model A, Yukawa coupling for model B and mirror symmetry between these two classes of Calabi–Yau manifolds. Recently there were several papers by physicists where it was shown on some examples. We propose a problem of classification of introduced Calabi–Yau manifolds. Our papers [GN1]–[GN6] and [N3]–[N14] give hope that this is possible. They describe possible Picard or transcendental lattices of general K3 fibers of the Calabi–Yau manifolds.