Sequence of families of lattice polarized \(K3\) surfaces, modular forms and degrees of complex reflection groups (Q6590682)

This paper studies a sequence of families of \(K3\) surfaces whose period mappings are closely related to complex reflection groups of exceptional type.\N\NConsider a sequence of even lattices \(A_3\subset A_2\subset A_1\subset A_0=A\) where \(A_0=U \oplus U \oplus A_2(-1) \oplus A_1(-1),\) \(A_1=U \oplus U \oplus A_2(-1),\) \(A_2=U \oplus U \oplus A_1(-1)\) and \(A_3=U \oplus\left(\begin{array}{cc} 2 & 1 \\ 1 & -2\end{array}\right)\). Set \(M_j=A_j^{\perp}\) in the \(K3\) lattice \(L_{K3}\) for \(j=0,1,2,3\). We see that \(A_j\) is of type \((5-j)\), and \(M_j\) of type \((1,14+j)\). This sequence of lattices induces a sequence of analytic spaces \(\mathfrak{A}_3\subset\mathfrak{A}_2\subset\mathfrak{A}_1\subset\mathfrak{A}_0\) of \(\text{dim}(\mathfrak{A}_j)=7-j\). This gives rise to a family of \(M_j\)-polarized \(K3\) surfaces \(\mathfrak{F}_j\). The period domain for \(\mathfrak{F}_j\) is a connected component \(\mathcal{D}_j\) of \(D_{M_j}\) of dimension \(5-j\).\N\NThe first main result of this paper is to study the period mapping of \(\mathfrak{F}_0\). The inverse of the period mapping yields a pair of meromorphic modular forms on \(\mathcal{D}_0\), and the generators of the ring of these modular forms are determined. Let \[z^2=y^3+(a_0x^5+a_4x^4+a_8x^3)y+(a_2x^7+a_6x^6+a_{10}x^5+a_{14}x^4)\] be a family of \(K3\) surfaces. Then the parameters \(a_2,a_4,a_8,a_{10}\) and \(a_{14}\) with positive weight are shown to induce a system of generators of the ring of modular forms. The period mappings for the subfamilies \(\mathfrak{F}_1, \mathfrak{F}_2\), and \(\mathfrak{F}_3\) are also described. These modular forms seem to have close relation with certain complex reflection groups.\N\NThere exists a double covering of every members of \(\mathfrak{F}_0\), which is again a \(K3\) surface. Let \(\mathfrak{G}_j\) \((j=0,1,2,3)\) be family of double covers. These \(K3\) surfaces turn out to be of hypergeometric type. The second main result is to determine the transcendental lattice \(B_j\) for \(\mathfrak{G}_j\) for \(j=0,1,2,3\). Indeed, \(B_j\) are given as follows: \(B_0=U(2) \oplus U(2) \oplus \left(\begin{array}{ccc} -2 & 0 & 1 \\ 0 & -2 & 1 \\ 1 & 1 & -4\end{array}\right),\) \(B_1=A_1(2),\, B_2=A_2(2)\) and \(B_3=A_3(2)\).