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The Wirtinger-type integral for a genus two curve

Yoshiaki Goto

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.

The Wirtinger-type integral for a genus two curve

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.
Paper Structure (12 sections, 10 theorems, 58 equations, 1 figure)

This paper contains 12 sections, 10 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperelliptic curve of genus two
  4. Expression by theta functions
  5. Twisted homology and cohomology groups
  6. Bases of $H_1(Y;\mathcal{L}_0)$ and $H^1(Y;\nabla _0)$
  7. Hyperelliptic involution
  8. Twisted cohomology group
  9. Twisted homology group
  10. Twisted cycles
  11. Eigenvectors
  12. Linear relations

Key Result

Theorem 4.1

The intersection matrix $C$ of $\varphi _1 ,\dots \varphi _8$ is where

Figures (1)

  • Figure 1: Octagon associated with $\bar{X}$

Theorems & Definitions (20)

  • Theorem 4.1
  • proof
  • Corollary 4.2
  • Remark 4.3
  • Proposition 5.1: Mizutani-Watanabe-J, Mizutani-Watanabe-E
  • proof
  • Theorem 5.2
  • proof
  • Corollary 5.3
  • proof
  • ...and 10 more