The arithmetic mirror symmetry and Calabi-Yau manifolds
In this paper the authors try to find a relation between their version of mirror symmetry for K3 surfaces [\textit{V. A. Gritsenko} and \textit{V. V. Nikulin}, Math. Res. Let. 3, No. 2, 211-229 (1996; Zbl 0871.14031)] and mirror symmetry for Calabi-Yau manifolds fibered in K3 surfaces. The difference between the authors' version of mirror symmetry for K3's and the version of \textit{I. V. Dolgachev} [J. Math. Sci., New York 81, No. 3, 2599-2630 (1996; Zbl 0890.14024)] is that they take into account the existence of an automorphic form on the \(B\)-model side which is conjectured to fulfill a Borcherds type identity. In this way they reflect information about effective curves on K3 surfaces by properties of functions on the moduli space of the mirror family. In the paper under review a few examples are mentioned where such automorphic forms are known to exist. The authors formulate the problem to find examples and to classify Calabi-Yau 3-folds fibered in K3 surfaces whose Picard or transcendental lattice has a certain property (2-reflective). This short paper does not contain any proof. At the end the authors present a very vague formulation of a conjecture relating the automorphic forms considered above to the Yukawa coupling and quantum cohomology of a Calabi-Yau manifold fibered in K3 surfaces.
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