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

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) = n_k (n_k + 1)℘(z +^ωk 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(n₀, n₁, n₂, n₃; B)|B ∈ C} ∪ {H(n₀ + 2, n₁, n₂, n₃; B)|B ∈ C} to the set of group representations ρ: π₁ (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 =¹2^{+i √} 3 2^{, there areB}1, 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.