Pure and Applied Mathematics Quarterly

Volume 17 (2021)

Number 5

The geometry of generalized Lamé equation, III: One-to-one of the Riemann–Hilbert correspondence

Pages: 1619 – 1668

DOI: https://dx.doi.org/10.4310/PAMQ.2021.v17.n5.a2

Authors

Zhijie Chen (Department of Mathematical Sciences, Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)

Ting-Jung Kuo (Department of Mathematics, Taiwan Normal University, Taipei, Taiwan)

Chang-Shou Lin (Department of Mathematics, Taiwan University, Taipei, Taiwan)

Abstract

In this paper, the third in a series, we continue to study the generalized Lamé equation $\mathrm{H}(n_0, n_1, n_2, n_3; B)$ with the Darboux–Treibich–Verdier potential\[{y^{\prime\prime }}(z)=\bigg[{\sum \limits_{k=0}^{3}}{n_{k}}({n_{k}}+1)\wp (z+\frac{{\omega _{k}}}{2}|\tau )+B\bigg]y(z),\hspace{1em}{n_{k}}\in {\mathbb{Z}_{\ge 0}}\]and a related linear ODE with additional singularities $\pm P$ 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\[\{\text{H}({n_{0}},{n_{1}},{n_{2}},{n_{3}};B)|B\in \mathbb{C}\}\mathrm{\cup }\{\text{H}({n_{0}}+2,{n_{1}},{n_{2}},{n_{3}};B)|B\in \mathbb{C}\}\]to the set of group representations $\rho : \pi_1 (E_\tau) \to SL(2,\mathbb{C})$ is one-to-one. We emphasize that this result is not trivial at all. There is an example that for $\tau = \frac{1}{2} + i \frac{\sqrt{3}}{2}$, there are $B_1, B_2$ such that the monodromy representations of $\mathrm{H} (1, 0, 0, 0; B_1)$ and $\mathrm{H} (4, 0, 0, 0; B_2)$ are the same, namely the Riemann–Hilbert correspondence from the set\[\{\text{H}({n_{0}},{n_{1}},{n_{2}},{n_{3}};B)|B\in \mathbb{C}\}\mathrm{\cup }\{\text{H}({n_{0}}+3,{n_{1}},{n_{2}},{n_{3}};B)|B\in \mathbb{C}\}\]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 CP1 with finite singularities.

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Received 25 November 2020

Accepted 2 February 2021

Published 26 January 2022