On reducible monodromy representations of some generalized Lamé equation

Chen, Zhijie; Kuo, Ting Jung; Lin, Chang–Shou; Takemura, Kouichi

doi:10.1007/s00209-017-1906-z

Cited by 3 publications

(3 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This f C (τ) appears in the expression of solutions of certain Painlevé VI equation (cf. [3,5]). The following result proves the existence and uniqueness of τ(C) as zeros of f C (τ).…”

Section: Existence and Uniqueness Of τ(C)mentioning

confidence: 99%

Critical points of the classical Eisenstein series of weight two

Chen

Lin

2019

J. Differential Geom.

Self Cite

View full text Add to dashboard Cite

In this paper, we completely determine the critical points of the normalized Eisenstein series E 2 (τ) of weight 2. Although E 2 (τ) is not a modular form, our result shows that E 2 (τ) has at most one critical point in every fundamental domain of Γ 0 (2). We also give a criteria for a fundamental domain containing a critical point of E 2 (τ). Furthermore, under the Möbius transformation of Γ 0 (2) action, all critical points can be mapped into the basic fundamental domain F 0 and their images are contained densely on three smooth curves. A geometric interpretation of these smooth curves is also given. It turns out that these smooth curves coincide with the degeneracy curves of trivial critical points of a multiple Green function related to flat tori.

show abstract

“…This f C (τ) appears in the expression of solutions of certain Painlevé VI equation (cf. [3,5]). The following result proves the existence and uniqueness of τ(C) as zeros of f C (τ).…”

Section: Existence and Uniqueness Of τ(C)mentioning

confidence: 99%

Critical points of the classical Eisenstein series of weight two

Chen

Lin

2019

J. Differential Geom.

Self Cite

View full text Add to dashboard Cite

show abstract

Section: From the Viewpoint Of The Monodromy Datamentioning

confidence: 99%

“…Without knowing the root structure of Q(A; p, τ), the proof of the assertion (2) is to apply the connection between GLE (4.4) and Painlevé VI equation. Since the proof of Theorem 4.A is long and has nothing related to Theorem 1.2, we refer the proof to[5].How to apply Theorem 4.A to obtain Theorem 1.2? For k ∈ {1, 2, 3}, we denote C k to be the unique solution of equation(4.6).…”

mentioning

confidence: 99%

Real-root property of the spectral polynomial of the Treibich–Verdier potential and related problems

Chen¹,

Kuo

Lin

et al. 2018

Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

We study the spectral polynomial of the Treibich-Verdier potential. Such spectral polynomial, which is a generalization of the classical Lamé polynomial, plays fundamental roles in both the finite-gap theory and the ODE theory of Heun's equation. In this paper, we prove that all the roots of such spectral polynomial are real and distinct under some assumptions. The proof uses the classical concept of Sturm sequence and isomonodromic theories. We also prove an analogous result for a polynomial associated with a generalized Lamé equation. Differently, our new approach is based on the viewpoint of the monodromy data.

show abstract