The Wirtinger-type integral for a genus two curve

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.