Appell's Hypergeometric Function F₂ and Periods of Certain Elliptic K3 Surfaces

SEIJI NISHIYAMA relative to the modular group Γ(2) which ṙs the principal congruence subgroup of level 2.(3) The map λ induces a biholomorphic equivalence between (H/Γ(2))* and P₁(C) , where (H/Γ(2))* denotes the compactification of the space H/Γ(2) which is obtained by attaching three cusp points \0,1, ∞\ .We shall show, using some properties of the period map for a family of certain elliptic K3 surfaces, the properties similar to the above (1), (2) and (3) for F₂(1/2,1/2,1/2,1,1, x, y) .Now, we sketch our method.The function F₂(1/2,1/2,1/2,1,1, x, y) is represented by the following double integral:So we consider the following surface:and the 2-form:where the parameters $(x, y)$ move in the domain Λ :Λ=\(x, y)eC²:xy(1-x)(1-y)(1-x-y)≠ 0\ , (see 1, (1.5), (1.5') and Figure 1.1).We compactify the surface (0.1) in a certain fibre space and denote it by $S(x, y)$ .The surface $S(x, y)$ has 11 normal two-dimensional singu- larities: one of them is of type A₈ and the others are of type A₁ .Let S̃(x, y) be the minimal nonsingular model of $S(x, y)$ , let μ:S̃(x, y)→ S(x, y) be the resolution map and put ψ=μ*φ .The surface S̃(λ)(x=(x, y)eΛ) is an elliptic K3 surface with 5 singular fibres of type I₀*, I₀*, I₂, I₂, I₂*; and the 2-form ψ is a non-vanishing holomorphic 2-form on S̃(x, y) (see 2, Propositions 2.1, 2.2).Since H₂(S̃(λ), Z) is a free Z-module of rank 22, we have a basis \Γ₁(λ), ⋯, Γ₂₂(λ)\ of H₂(S̃(x), Z) .And we can always take eighteen of them as algebraic cycles, so let us say that they are Γf(λ), ⋯, Γ₂₂(λ) .Therefore if we put η(λ)=∫Γ(λ)ψ(i=1, ⋯, 22) , then we have η(λ)≡ 0(i=5, ⋯, 22) .Hence we define the period map Φ₁ for