Thomae type formula for K3 surfaces given by double covers of the projective plane branching along six lines

TL;DR

This paper derives a Thomae type formula for certain K3 surfaces represented as double covers of the projective plane branched along six lines, connecting theta constants, period integrals, and hypergeometric functions.

Contribution

It establishes a new Thomae type formula relating theta constants and period integrals for K3 surfaces with explicit hypergeometric function expressions.

Findings

Derived a Thomae type formula for K3 surfaces as double covers of the plane

Expressed period integrals using a four-variable hypergeometric function

Connected mean iterations of sequences to hypergeometric functions

Abstract

In this paper, we give Thomae type formula for \KK surfaces $\cS$ given by double covers of the projective plane branching along six lines. This formula gives relations between theta constants on the bounded symmetric domain of type $I_{22}$ and period integrals of $X$. Moreover, we express the period integrals by using the hypergeometric function $F_S$ of four variables. As an application of our main theorem, we define $\R^4$-valued sequences by mean iterations of four terms, and express their common limits by the hypergeometric function $F_S$.