We solve the monodromy problem about the Picard–Fuchs system for the two-parameter family of the anticanonical hypersurfaces in the toric three-fold arising from a specific reflexive polytope. The members of the family are resolved to be K3 surfaces polarized by a lattice of signature (1, 17) with discriminant \(-8\) . The Picard–Fuchs system consists of two hypergeometric differential equations of order 2 with the four-dimensional solution space. The independent variables are affine coordinates of the punctured weighted projective space \(\mathbb {P}(1,2,3)\hspace{1.111pt}{\setminus }\hspace{1.111pt}\{(1,0,0)\}\) which forms a moduli space of the family. We construct the isomorphism from \(\mathbb {P}(1,2,3)\hspace{1.111pt}{\setminus }\hspace{1.111pt}\{(1,0,0)\}\) to the symmetric Hilbert modular orbifold \(\mathbb {H}^2/\langle \text {PSL}_2(\mathscr {O}_K),\tau _1\rangle \) for \(K=\mathbb {Q}(\sqrt{2})\) and its inverse as a triplet of symmetric Hilbert modular forms. Here (1, 0, 0) corresponds to the cusp. As the fundamental set of solutions, we take a set of series convergent near the cusp. We obtain the generators of the monodromy group in \(\text {GL}_4(\mathbb {Z})\) by deforming the domains for the period integrals expressing the solutions. We construct the isomorphism from the monodromy group to \(\langle \text {PSL}_2(\mathscr {O}_K),\tau _1\rangle \) . Using the monodromy, we obtain fixed points of the Hilbert modular group \(\text {PSL}_2(\mathscr {O}_K)\) and their respective isotropy groups, which are consistent with the results of the prior research taking an arithmetic approach.
