Monodromy of Picard–Fuchs system for a family of K3 toric hypersurfaces and fixed points of the Hilbert modular group for Q(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}(\\sqrt{2})$$\\end{document}

Abstract

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\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-8$$\\end{document}. 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 P(1,2,3)\\{(1,0,0)}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}(1,2,3)\\hspace{1.111pt}{\\setminus }\\hspace{1.111pt}\\{(1,0,0)\\}$$\\end{document} which forms a moduli space of the family. We construct the isomorphism from P(1,2,3)\\{(1,0,0)}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}(1,2,3)\\hspace{1.111pt}{\\setminus }\\hspace{1.111pt}\\{(1,0,0)\\}$$\\end{document} to the symmetric Hilbert modular orbifold H2/⟨PSL2(OK),τ1⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^2/\\langle \ ext {PSL}_2(\\mathscr {O}_K),\ au _1\\rangle $$\\end{document} for K=Q(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K=\\mathbb {Q}(\\sqrt{2})$$\\end{document} 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 GL4(Z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {GL}_4(\\mathbb {Z})$$\\end{document} by deforming the domains for the period integrals expressing the solutions. We construct the isomorphism from the monodromy group to ⟨PSL2(OK),τ1⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \ ext {PSL}_2(\\mathscr {O}_K),\ au _1\\rangle $$\\end{document}. Using the monodromy, we obtain fixed points of the Hilbert modular group PSL2(OK)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {PSL}_2(\\mathscr {O}_K)$$\\end{document} and their respective isotropy groups, which are consistent with the results of the prior research taking an arithmetic approach.