Science China Mathematics

, Volume 62, Issue 11, pp 2089–2102 | Cite as

Simple zero property of some holomorphic functions on the moduli space of tori

  • Zhijie Chen
  • Ting-Jung Kuo
  • Chang-Shou LinEmail author


We prove that some holomorphic functions on the moduli space of tori have only simple zeros. Instead of computing the derivative with respect to the moduli parameter τ, we introduce a conceptual proof by applying Painlevé VI equation. As an application of this simple zero property, we obtain the smoothness of the degeneracy curves of trivial critical points for some multiple Green function.


simple zero property Painlevé VI equation Green function 


34M55 34M35 



The first author was supported by National Natural Science Foundation of China (Grant No. 11701312). The authors thank Chin-Lung Wang very much for providing the file of Figure 1 to them.


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© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical Sciences and Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina
  2. 2.Department of MathematicsTaiwan Normal UniversityTaipeiChina
  3. 3.Taida Institute for Mathematical Sciences (TIMS), Center for Advanced Study in Theoretical Sciences (CASTS)Taiwan UniversityTaipeiChina

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