Schwarz's map for Appell's second hypergeometric system with quarter integer parameters

Abstract

We study Schwarz's map for Appell's second system ${\mathcal F}_2$ of hypergeometric differential equations in two variables with parameters $a=c_1=c_2=\frac{1}{2}$, $b_1=b_2=\frac{1}{4}$. By using theta functions with characteristics, we give a defining equation of an analytic set in ${\mathbb C}^2\times {\mathbb H}$ of its image, and express its inverse.