Zeta functions of alternate mirror Calabi-Yau families
An invertible polynomial is a polynomial of the form \[ F_A=\sum_{i=0}^n\prod_{j=0}^n x_j^{a_{ij}}\in{\mathbb{Z}}[x_0,\dots,x_n] \] where the matrix of exponents \(A=(a_{ij})\) is a square matrix of size \(n+1\) with nonnegative integer entries satisfying (1) \(\det(A)\neq 0\), (2) \(F_A\) is quasi-homogeneous, and (3) \(F_A:{\mathbb{C}}^{n+1}\to{\mathbb{C}}\) is nonsingular at the origin. When \(F_A\) is invertible and homogeneous of degree \(d=n+1\), \(F_A=0\) defines a Calabi-Yau variety \(X_A\) in \({\mathbb{P}}^n\). For an invertible polynomial \(F_A\), there is the (dual) polynomial \[ F_{A^T}:=\sum_{i=0}^n\prod_{j=0}^n x_j^{a_{ji}}. \] Then \(F_{A^T}\) is again an invertible polynomial, quasihomogeneous with weights \(q_0,\dots, q_n\) (\(\gcd(q_0,\dots, q_n)=1\)), so that \(F_{A^T}=0\) defines a hypersurface \(X_{A^T}\) in \({\mathbf {WP}}(q_0,\dots, q_n)\) with the dual weights \((q_0,\dots, q_n)\). Let \(d^T=\sum_{i=0}^nq_i\). Define a one-parameter deformation of \(F_A\) by \[ F_{A,\psi}:=\sum_{i=0}^n\prod_{j=0}^n x_j^{q_{ij}}-d^T\psi\prod_{i=0}^n x_j \in{\mathbb{Z}}[\psi][x_0,\dots, x_n]. \] Then \(X_{A,\psi}: F_{A,\psi}=0\) is a family of hypersurfaces in \({\mathbb{P}}^n\) in the parameter \(\psi\), which is called an invertible pencil. \textit{S. Gährs} [in: Arithmetic and geometry of \(K3\) surfaces and Calabi-Yau threefolds. Proceedings of the workshop, Toronto, Canada, August 16--25, 2011. New York, NY: Springer. 285--310 (2013; Zbl 1302.14034)] showed the Picard-Fuchs differential equation for \(X_{A,\psi}\) is determined by the \((n+1)\)-tuple of dual weights \((q_0,\dots, q_n)\), and that its order \(D(q_0,\dots, q_n)\) depends only on the dual weights. This paper shows that invertible pencils whose Berglund-Hübsch-Krawits (BHK) mirrors have common properties share arithmetic properties as well. In particular, invertible pencils whose BHK mirrors are hypersurfaces in quotients of the same weighted projective space have the same Picard-Fuchs equation associated to theire holomorphic form. This implies that the zeta-functions of mirror pairs have the same factor corresponding to the Picard-Fuchs equation, which is of hypergeometric type. A more precise formulation of this fact is described in the following theorem: Theorem. Let \(A_{A,\psi}\) and \(X_{B,\psi}\) be invertible pencils of Calabi-Yau \((n-1)\)-folds in \({\mathbb{P}}^n\). Suppose that \(A\) and \(B\) have the same dual weights \((q_0,\dots, q_n)\). Then for each \(\psi\in{\mathbb{F}}_q\) such that \(\gcd(q,(n+1))=1\) and the fiber \(X_{A,\psi}\) and \(X_{B,\psi}\) are nondegenerate and smooth, the polynomials \(P_{X_{A,\psi}}(T)\) and \(P_{X_{B,\psi}}(T)\) have a common factor \(R_{\psi}(T)\in {\mathbb{Q}}[T]\) with \(\deg(R_{\psi}(T))\geq D(q_0,\dots, q_n).\) Moreover, the common factor \(R_{\psi}(T)\) is attached to the holomorphic form on \(X_{A,\psi}\) and \(X_{B,\psi}\). As an application, several pencils of \(K3\) surfaces and their relation to the Dwork pencil are studied obtaining new cases of arithmetic mirror symmetry.
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