Transcendental lattices of certain singular \(K3\) surfaces (Q6619054)

Denote by \(M_n\) the lattice of rank \(11\) that is the direct sum of the hyperbolic lattice, the negative-definite even unimodular lattice of rank \(8\), and the lattice of rank one with norm \(2n\). The aim of the paper under review is to study a relation between families of \(K3\) surfaces polarized by \(M_6\) (Apéry-Fermi pencil), by \(M_{12}\), and by \(M_3\) (Verrill's family). They are motivated by the determination of relations between the Mahler measure and \(L\)-series of the transcendental lattice of singular members of these families.\N\NAs the first main result, they give the transcendental lattice of the generic members of the family of \(M_{12}\)-polarized \(K3\) surfaces by investigating their structure as a genus-\(1\) fibration, and then, their Jacobian surface, being the generic member of the Verrill's family, are also given its transcendental lattice.\N\NIn the second part, the transcendental lattices (partly conjectural) of special members of each family are given. For Apéry-Fermi pencil and Verrill's family, they can find generators for the lattice by using an expression of the members in terms of the Dedekind eta function. For the family polarized by \(M_{12}\), they use the result in the first part, and by investigating singular fibres, they give the lattices.\N\NAs a final remark, they conclude that there is also a relation with respect to the Shioda-Inose srtucture among members in the families.\N\NFor the entire collection see [Zbl 1540.11003].