A configuration of six lines in \(\mathbb{P}^2\) determines a \(K3\) surface as the minimal model of a double covering of \(\mathbb{P}^2\) branched along these lines. Regarding an elliptic curve as a double covering of \(\mathbb{P}^1\) branched at four points, a classical theory relates the period integrals of an elliptic curve, Gauss hypergeometric function, theta functions, and limit formula of arithmetic-geometric means. Being an analogue of elliptic curves, peiod integrals of \(K3\) surfaces obtained as above are understood with certain hypergeometric functions and some mean iteration.
The paper under review focuses on \(K3\) surfaces correponding to six lines in \(\mathbb{P}^2\) as above and studies relations among their period integrals, hypergeometric functions, theta functions and limit of mean iterations. The main result is to give a relation between the period integrals of a \(K3\) surface and theta functions by writing down a Thomae-type formula that is induced from a \(2\tau\)-formula of the theta functions. As a corollary, defining hypergeometric functions \(F_S, F_T\), and expressing the period integrals by them, it is shown that the theta function is also written down with the functions \(F_S, F_T\). This result explains how the Thomae-type formula for Kummer locus relates to the limit formula for Borchardt’s mean iteration. Finally as the last application, a functional equation for the hypergeometric function \(F_S\) is given.