Computing Differential Equation Of Deformation Theory About Calabi-Yau Varieties

The purpose of this paper is to give an efficient method for solving the differentialequations of Calabi-Yau varietiesF(Γ) = x₁ⁿ +···+ x_nⁿ +Γnx₁···x_nwith the help of A. G. B. Lauder's idea in [17].Firstly, we introduce some definitions and facts, including p-adic Dwork theoremand Dwork's Lefschetz fixed points thoerem.Secondly, we introduce the deformation theory. The basic idea is to make thesehypersurfaces move along an affine line. One can select a special fiber, whose zetafunction is easy to compute, and compute the zeta functions of those fibers near it byDwork's deformation theory. The deformation theory can be expressed by a differentialequation which is not easy to be written clearly. In this paper, we present an effectivealgorithm for computing the differential equation of Calabi-Yau varieties.At last, we explain the algorithm by showing an example with n=3 and the varietybeing given byF(Γ) = x₁³+ x₂³ + x₃³ + 3Γx₁x₂x₃.we compute B(Γ) by some C programme with n=4 and the variety being given byF(Γ) = x₁⁴ + x₂⁴ + x₃⁴ + x₄⁴ + 4Γx₁x₂x₃x₄.