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Frobenius polynomial for Calabi-Yau equations. (English) Zbl 1166.14026

The paper considers a linear differential operator \(P\) of order \(n\) defined over \(\mathbb{Q}\), called a \(CY(n)\)-operator. Such an operator has the following properties:
(1) \(P\) has maximal unipotent monodromy at \(0\) (MUM)
(2) \(P\) is self-dual, and
(3) \(P\) has a convergent power series solution \(f_0(x)\in\mathbb{Z}[[x]]\) with \(f_0(0)=1\).
The question addressed in this paper is:
Given a \(CY(4)\)-operator \(P\) of a family of Calabi-Yau threefolds \(f : X\to \mathbb{P}^1\) defined over \(\mathbb{Z}\), is there a way to calculate the Frobenius polynomials \(P_s(T)\)?
The paper describes a method to solve this problem. \(CY(4)\)-operators arise from families of Calabi-Yau threefolds with \(h^{1,2}=1\) (so that \(B_3=4\)). In the paper, it is assumed that the \(CY(4)\)-operator is the Picard-Fuchs operator on a rank \(4\) submodule in \(H^3_{dR}\) of some family of smooth Calabi–Yau threefolds. Here the Frobenius polynomial is referred to the characteristic polynomial of the Frobenius morphism, which is of the form \[ P_s(T)=1+aT+bpT^2+ap^3T^t+p^6T^4 \] It has four different roots \(r_1, pr_2, p^2/r_2, p^3/r_1\) where \(r_1\) and \(r_2\) are \(p\)-adic units. Therefore, giving a formula for \(P_s(T)\) is equivalent to determining \(r_1\) and \(r_2\). The paper under review gives \(p\)-adic analytic formulas for the unit roots \(r_1\) and \(r_2\) assuming some conjectures on Dwork’s congruences. Many examples are produced computing the Frobenius polynomials. At singular points, the Frobenius polynomial splits into a product of two linear factors and a quadratic part of the form \(1-a_pT+p^3T^2\). In the examples, the coefficients \(a_p\) are identified with the Fourier coefficients of modular forms of weight \(4\). This suggests that there is a rigid Calabi-Yau motive corresponding to the quadratic factor.

MSC:

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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