## Abstract

In this paper, the third in a series, we continue to study the generalized Lamé equation H(n_{0}, n_{1}, n_{2}, n_{3}; B) with the Darboux–Treibich–Verdier potential [_{3∑} ] y^{′′} (z) = n_{k} (n_{k} + 1)℘(z +^{ω}k 2 |τ ) + B y(z), k=0 n_{k} ∈ Z_{≥0} and a related linear ODE with additional singularities ±p from the monodromy aspect. We establish the uniqueness of these ODEs with respect to the global monodromy data. Surprisingly, our result shows that the Riemann–Hilbert correspondence from the set {H(n_{0}, n_{1}, n_{2}, n_{3}; B)|B ∈ C} ∪ {H(n_{0} + 2, n_{1}, n_{2}, n_{3}; B)|B ∈ C} to the set of group representations ρ: π_{1} (E_{τ} ) → SL(2, C) is one-to-one. We emphasize that this result is not trivial at all. There is an example that for τ =^{1}2^{+i √} 3 2^{, there areB}1, B_{2} such that the monodromy representations of H(1, 0, 0, 0; B_{1} ) and H(4, 0, 0, 0; B_{2} ) are the same, namely the Riemann–Hilbert correspondence from the set {H(n_{0}, n_{1}, n_{2}, n_{3}; B)|B ∈ C} ∪ {H(n_{0} + 3, n_{1}, n_{2}, n_{3}; B)|B ∈ C} to the set of group representations is not necessarily one-to-one. This example shows that our result is completely different from the classical one concerning linear ODEs defined on CP^{1} with finite singularities.

Original language | English |
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Pages (from-to) | 1619-1668 |

Number of pages | 50 |

Journal | Pure and Applied Mathematics Quarterly |

Volume | 17 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2021 |

## ASJC Scopus subject areas

- General Mathematics