Monodromy of Picard–Fuchs system for a family of K3 toric hypersurfaces and fixed points of the Hilbert modular group for $$\mathbb {Q}(\sqrt{2})$$
Year: 2024
Type: article
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 $$-8$$ - 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 $$\mathbb {P}(1,2,3)\hspace{1.111pt}{\setminus }\hspace{1.111pt}\{(1,0,0)\}$$ P ( 1 , 2 , 3 ) \ { ( 1 , 0 , 0 ) } which forms a moduli space of the family. We construct the isomorphism from $$\mathbb {P}(1,2,3)\hspace{1.111pt}{\setminus }\hspace{1.111pt}\{(1,0,0)\}$$ P ( 1 , 2 , 3 ) \ { ( 1 , 0 , 0 ) } to the symmetric Hilbert modular orbifold $$\mathbb {H}^2/\langle \text {PSL}_2(\mathscr {O}_K),\tau _1\rangle $$ H 2 / ⟨ PSL 2 ( O K ) , τ 1 ⟩ for $$K=\mathbb {Q}(\sqrt{2})$$ K = Q ( 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 $$\text {GL}_4(\mathbb {Z})$$ GL 4 ( Z ) by deforming the domains for the period integrals expressing the solutions. We construct the isomorphism from the monodromy group to $$\langle \text {PSL}_2(\mathscr {O}_K),\tau _1\rangle $$ ⟨ PSL 2 ( O K ) , τ 1 ⟩ . Using the monodromy, we obtain fixed points of the Hilbert modular group $$\text {PSL}_2(\mathscr {O}_K)$$ PSL 2 ( O K ) and their respective isotropy groups, which are consistent with the results of the prior research taking an arithmetic approach.
Source: European Journal of Mathematics
Author: Toshimasa Ishige
Institution: Chiba University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
Award: Coupling理論の代数解析と対角埋め込み法
OpenAlex ID: W4404853606