On the general transformation of the Wirtinger integral

Abstract

The Wirtinger integral is the uniformization to the upper half plane $H$ of the hypergeometric function defined on the complex projective line $\mathbb{P}^{1}$. In [5] we established the transformation formulas of the Wirtinger integral for the linear fractional transformations $\tau\to\tau+2$ and $\tau\to -1/\tau$ with the aide of the theory of theta functions. As a corollary we obtain the transformation formulas of the Wirtinger integral for the linear fractional transformations $\tau\to\tau+2$ and $\tau\to \tau/(-2\tau+1)$ which are identified with generators of the principal congruence subgroup $\varGamma(2)$ modulo center. These formulas correspond to the monodromy matrices of the hypergeometric function for generators of the fundamental group of $\mathbb{P}^{1}$ minus three points. The purpose of this paper is to generalize this result, that is, we establish the transformation formula of the Wirtinger integral for a general element $\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right)$ of $\varGamma(2)$, which corresponds to a general monodromy matrix of the hypergeometric function.