The Wirtinger-type integral for a genus two curve

TL;DR

This paper extends the classical Wirtinger integral, originally for elliptic curves, to genus two hyperelliptic curves, analyzing the related twisted homology and cohomology groups.

Contribution

It introduces an analogue of the Wirtinger integral for genus two curves and studies its twisted homology and cohomology using hyperelliptic involution.

Findings

Analysis of twisted homology groups on genus two curves

Investigation of intersection forms on hyperelliptic curves

Extension of integral representations to higher genus

Abstract

The Wirtinger integral is one of the integral representations of the Gauss hypergeometric function. Its integrand can be regarded as a multivalued function on an elliptic curve. In this paper, we study an analogue of the Wirtinger integral on a hyperelliptic curve of genus two, introduced by Mizutani and Watanabe. We investigate the associated twisted homology and cohomology groups using the hyperelliptic involution and intersection forms.