Skip to main content
SpringerLink
Log in

On reducible monodromy representations of some generalized Lamé equation

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

In this note, we compute the explicit formula of the monodromy data for a generalized Lamé equation when its monodromy is reducible but not completely reducible. We also solve the corresponding Riemann–Hilbert problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Babich, M.V., Bordag, L.A.: The Elliptic form of the Sixth Painlevé equation. Preprint NT Z25/1997, Leipzig (1997)

  2. Chen, Z., Kuo, T.J., Lin, C.S.: Hamiltonian system for the elliptic form of Painlevé VI equation. J. Math. Pures Appl. 106, 546–581 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, Z., Kuo, T.J., Lin, C.S.: Simple zero property of some holomorphic functions on the moduli space of tori. Preprint 2016. arXiv: 1703.05521v1 [math. CV]

  4. Chen, Z., Kuo, T.J., Lin, C.S.: Painlevé VI Equation. Modular Forms and Application to Integral Lamé equation, Preprint (2016)

  5. Hitchin, N.J.: Twistor spaces, Einstein metrics and isomonodromic deformations. J. Differ. Geom. 42(1), 30–112 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Iwasaki, K., Kimura, H., Shimomura, S., Yoshida, M.: From Gauss to Painlevé: A Modern Theory of Special Functions, vol. E16. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  7. Lisovyy, O., Tykhyy, Y.: Algebraic solutions of the sixth Painlevé equation. J. Geom. Phys. 85, 124–163 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Manin, Y.: Sixth Painlevé quation, universal elliptic curve, and mirror of \(\mathbb{P} ^{2}\). Am. Math. Soc. Transl. 186, 131–151 (1998)

    MATH  Google Scholar 

  9. Okamoto, K.: Studies on the Painlevé equations. I. Sixth Painlevé equation \(P_{VI}\). Ann. Math. Pura Appl. 146, 337–381 (1986)

    Article  Google Scholar 

  10. Takemura, K.: The Hermite–Krichever Ansatz for Fuchsian equations with applications to the sixth Painlevé equation and to finite gap potentials. Math. Z. 263, 149–194 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors wish to thank the anonymous referee very much for his/her careful reading and comments.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Yau Mathematical Sciences Center, Beijing, 100084, China

    Zhijie Chen

  2. Department of Mathematics, National Taiwan Normal University, Taipei, 11677, Taiwan

    Ting-Jung Kuo

  3. Taida Institute for Mathematical Sciences (TIMS), Center for Advanced Study in Theoretical Sciences (CASTS), National Taiwan University, Taipei, 10617, Taiwan

    Chang-Shou Lin

  4. School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK

    Kouichi Takemura

  5. Department of Mathematics, Faculty of Science and Engineering, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo, 112-8551, Japan

    Kouichi Takemura

Authors
  1. Zhijie Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ting-Jung Kuo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chang-Shou Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Kouichi Takemura
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhijie Chen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, Z., Kuo, TJ., Lin, CS. et al. On reducible monodromy representations of some generalized Lamé equation. Math. Z. 288, 679–688 (2018). https://doi.org/10.1007/s00209-017-1906-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-017-1906-z

Keywords

Search

Navigation