Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications
In this paper, we study third-order modular ordinary differential equations (MODE for short) of the following form ′′′ + ₂()′ + ₃() = 0, ∈ ℍ = { ∈ ℂ | Im > 0}, where ₂() and ₃() − 1/2′₂() are meromorphic modular forms on SL(2, ℤ) of weight 4 and 6, respectively. We show that any quasimodular for...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2022 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2022
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211532 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications. Zhijie Chen, Chang-Shou Lin and Yifan Yang. SIGMA 18 (2022), 013, 50 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862699068435726336 |
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| author | Chen, Zhijie Lin, Chang-Shou Yang, Yifan |
| author_facet | Chen, Zhijie Lin, Chang-Shou Yang, Yifan |
| citation_txt | Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications. Zhijie Chen, Chang-Shou Lin and Yifan Yang. SIGMA 18 (2022), 013, 50 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this paper, we study third-order modular ordinary differential equations (MODE for short) of the following form ′′′ + ₂()′ + ₃() = 0, ∈ ℍ = { ∈ ℂ | Im > 0}, where ₂() and ₃() − 1/2′₂() are meromorphic modular forms on SL(2, ℤ) of weight 4 and 6, respectively. We show that any quasimodular form of depth 2 on SL(2, ℤ) leads to such a MODE. Conversely, we introduce the so-called Bol representation ^: SL(2, ℤ) → SL(3, ℂ) for this MODE and give the necessary and sufficient condition for the irreducibility (resp. reducibility) of the representation. We show that the irreducibility yields the quasimodularity of some solution of this MODE, while the reducibility yields the modularity of all solutions and leads to solutions of certain SU(3) Toda systems. Note that the SU( + 1) Toda systems are the classical Plücker infinitesimal formulas for holomorphic maps from a Riemann surface to ℂℙᴺ.
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| first_indexed | 2026-03-18T10:46:51Z |
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| id | nasplib_isofts_kiev_ua-123456789-211532 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-18T10:46:51Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
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| spelling | Chen, Zhijie Lin, Chang-Shou Yang, Yifan 2026-01-05T12:27:19Z 2022 Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications. Zhijie Chen, Chang-Shou Lin and Yifan Yang. SIGMA 18 (2022), 013, 50 pages 1815-0659 2020 Mathematics Subject Classification: 11F11; 34M03 arXiv:2106.12438 https://nasplib.isofts.kiev.ua/handle/123456789/211532 https://doi.org/10.3842/SIGMA.2022.013 In this paper, we study third-order modular ordinary differential equations (MODE for short) of the following form ′′′ + ₂()′ + ₃() = 0, ∈ ℍ = { ∈ ℂ | Im > 0}, where ₂() and ₃() − 1/2′₂() are meromorphic modular forms on SL(2, ℤ) of weight 4 and 6, respectively. We show that any quasimodular form of depth 2 on SL(2, ℤ) leads to such a MODE. Conversely, we introduce the so-called Bol representation ^: SL(2, ℤ) → SL(3, ℂ) for this MODE and give the necessary and sufficient condition for the irreducibility (resp. reducibility) of the representation. We show that the irreducibility yields the quasimodularity of some solution of this MODE, while the reducibility yields the modularity of all solutions and leads to solutions of certain SU(3) Toda systems. Note that the SU( + 1) Toda systems are the classical Plücker infinitesimal formulas for holomorphic maps from a Riemann surface to ℂℙᴺ. We would like to thank the referees for their many valuable comments and for pointing out some references. The research of Z. Chen was supported by NSFC (No. 12071240). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications Article published earlier |
| spellingShingle | Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications Chen, Zhijie Lin, Chang-Shou Yang, Yifan |
| title | Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications |
| title_full | Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications |
| title_fullStr | Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications |
| title_full_unstemmed | Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications |
| title_short | Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications |
| title_sort | modular ordinary differential equations on sl(2, ℤ) of third order and applications |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211532 |
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