Notes on twisted period relations for the Wirtinger integral
Yoshiaki Goto, Genki Shibukawa
TL;DR
This work analyzes the Wirtinger integral, an integral representation of the Gauss hypergeometric function ${}_2F_1$, through the lens of twisted (co)homology on a punctured torus and its intersection theory. By exploiting an involution $\iota(u)=-u$, the authors decompose the twisted (co)homology into orthogonal eigenspaces, enabling a simplified, block-diagonal form of the twisted period relations (TPR) that relate theta-constant data to hypergeometric functions. They provide explicit intersection matrices, basis changes, and hypergeometric expressions for the period matrices, and derive a concrete $(2,2)$-entry identity that connects theta-derivative data to products of ${}_2F_1$ functions. A direct proof of this identity is given, using both cohomological computations and elliptic-function expansions, thereby clarifying the structural bridge between theta-functions and ${}_2F_1$ in the Wirtinger integral setting and simplifying the previously intricate TPR. The results illuminate how involutive symmetries and eigenbasis decompositions yield more tractable relations between modular theta data and classical hypergeometric functions.
Abstract
The Wirtinger integral is one of the integral representations of the Gauss hypergeometric function. Its integrand is given by a product of complex powers of theta functions. The twisted homology and cohomology groups associated with this integral yield twisted period relations. By studying the structure of the twisted homology and cohomology groups in detail, we obtain simple forms of the relations.