  We extend our variant of mirror symmetry for K3 surfaces \cite{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
\cite{GN1}-\cite{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 \cite{GN1}-\cite{GN6} and \cite{N3}-\cite{N14} give a hope that this is
possible. They describe possible Picard or transcendental lattices of general
K3 fibers of the Calabi-Yau manifolds.
