TY - JOUR

T1 - The geometry of generalized Lamé equation, III

T2 - one-to-one of the Riemann–Hilbert correspondence

AU - Chen, Zhijie

AU - Kuo, Ting Jung

AU - Lin, Chang Shou

N1 - Funding Information:
We thank Prof. Treibich for his interest and valuable comments to this work. The research of Z. Chen was supported by NSFC (No. 12071240, 11871123). The research of T-J. Kuo was supported by MOST (MOST 107-2628-M-003-002-MY4).
Publisher Copyright:
© 2021, International Press, Inc.. All rights reserved.

PY - 2021

Y1 - 2021

N2 - In this paper, the third in a series, we continue to study the generalized Lamé equation H(n0, n1, n2, n3; B) with the Darboux–Treibich–Verdier potential [3∑ ] y′′ (z) = nk (nk + 1)℘(z +ωk 2 |τ ) + B y(z), k=0 nk ∈ 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(n0, n1, n2, n3; B)|B ∈ C} ∪ {H(n0 + 2, n1, n2, n3; B)|B ∈ C} to the set of group representations ρ: π1 (Eτ ) → SL(2, C) is one-to-one. We emphasize that this result is not trivial at all. There is an example that for τ =12+i √ 3 2, there areB1, B2 such that the monodromy representations of H(1, 0, 0, 0; B1 ) and H(4, 0, 0, 0; B2 ) are the same, namely the Riemann–Hilbert correspondence from the set {H(n0, n1, n2, n3; B)|B ∈ C} ∪ {H(n0 + 3, n1, n2, n3; 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 CP1 with finite singularities.

AB - In this paper, the third in a series, we continue to study the generalized Lamé equation H(n0, n1, n2, n3; B) with the Darboux–Treibich–Verdier potential [3∑ ] y′′ (z) = nk (nk + 1)℘(z +ωk 2 |τ ) + B y(z), k=0 nk ∈ 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(n0, n1, n2, n3; B)|B ∈ C} ∪ {H(n0 + 2, n1, n2, n3; B)|B ∈ C} to the set of group representations ρ: π1 (Eτ ) → SL(2, C) is one-to-one. We emphasize that this result is not trivial at all. There is an example that for τ =12+i √ 3 2, there areB1, B2 such that the monodromy representations of H(1, 0, 0, 0; B1 ) and H(4, 0, 0, 0; B2 ) are the same, namely the Riemann–Hilbert correspondence from the set {H(n0, n1, n2, n3; B)|B ∈ C} ∪ {H(n0 + 3, n1, n2, n3; 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 CP1 with finite singularities.

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U2 - 10.4310/PAMQ.2021.v17.n5.a2

DO - 10.4310/PAMQ.2021.v17.n5.a2

M3 - Article

AN - SCOPUS:85125636959

SN - 1558-8599

VL - 17

SP - 1619

EP - 1668

JO - Pure and Applied Mathematics Quarterly

JF - Pure and Applied Mathematics Quarterly

IS - 5

ER -