Appell's hypergeometric function \(F_ 2\) and periods of certain elliptic K3 surfaces
In 1880 P. Appell introduced four two-dimensional generalizations of the Gauß hypergeometric function F(\(\alpha\),\(\beta\),\(\gamma\),z), which he denoted by \(F_ 1, F_ 2, F_ 3\) and \(F_ 4\). There are several classical and recent generalizations of the elliptic modular function to the K3 modular function and other automorphic functions using Appell's hypergeometric function \(F_ 1\), e.g., by \textit{E. Picard} (1883, 1885), \textit{T. Terada} (1983), \textit{H. Shiga} (1979, 1981), and \textit{P. Deligne} and \textit{G. Mostow} [Publ. Math., Inst. Hautes Étud. Sci. 63, 5-89 (1986; Zbl 0615.22008)], but there are almost no attempts of using Appell's other hypergeometric functions \(F_ 2, F_ 3\) and \(F_ 4\) for such a purpose. The author constructs and investigates an automorphic function in two variables derived from Appell's hypergeometric function \(F_ 2(,,,1,1,x,y)\). It is shown that this function gives rise to a two- dimensional family of elliptic K3 surfaces (with five specified singular fibres and a naturally defined non-vanishing holomorphic 2-form on each fibre). The corresponding period map of that family is completely described. More precisely, the image of the period map is shown to be a subspace of the Cartesian product of the upper Siegel half plane \({\mathfrak H}\) with itself, the inverse map of the period map is shown to be an automorphic function on \({\mathfrak H}\times {\mathfrak H}\) with respect to some modular group - which turns out to be the monodromy group of the family - and, concludingly, a suitable compactification of the period domain is shown to be holomorphically equivalent, via the inverse period map, to the complex projective plane.
scientific article; zbMATH DE number 4023473