### Abstract

In literature, it is known that any solution of Painlevé VI equation governs the isomonodromic deformation of a second order linear Fuchsian ODE on CP^{1}. In this paper, we extend this isomonodromy theory on CP^{1} to the moduli space of elliptic curves by studying the isomonodromic deformation of the generalized Lamé equation. Among other things, we prove that the isomonodromic equation is a new Hamiltonian system, which is equivalent to the elliptic form of Painlevé VI equation for generic parameters. For Painlevé VI equation with some special parameters, the isomonodromy theory of the generalized Lamé equation greatly simplifies the computation of the monodromy group in CP^{1}. This is one of the advantages of the elliptic form.

Original language | English |
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Pages (from-to) | 546-581 |

Number of pages | 36 |

Journal | Journal des Mathematiques Pures et Appliquees |

Volume | 106 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2016 Sep 1 |

Externally published | Yes |

### Keywords

- Hamiltonian system
- Isomonodromy theory
- Painlevé VI equation
- The elliptic form

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Journal des Mathematiques Pures et Appliquees*,

*106*(3), 546-581. https://doi.org/10.1016/j.matpur.2016.03.003