Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations
We study various relations governing quasi-automorphic forms associated with discrete subgroups of SL(2, ℝ), namely the Hecke groups. We show that the Eisenstein series associated to a Hecke group H(m) satisfy a set of m coupled linear differential equations, which are natural analogues of the well-...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2020 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2020
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/210609 |
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| Cite this: | Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations. Sujay K. Ashok, Dileep P. Jatkar and Madhusudhan Raman. SIGMA 16 (2020), 001, 26 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862694401753481216 |
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| author | Ashok, Sujay K. Jatkar, Dileep P. Raman, Madhusudhan |
| author_facet | Ashok, Sujay K. Jatkar, Dileep P. Raman, Madhusudhan |
| citation_txt | Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations. Sujay K. Ashok, Dileep P. Jatkar and Madhusudhan Raman. SIGMA 16 (2020), 001, 26 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We study various relations governing quasi-automorphic forms associated with discrete subgroups of SL(2, ℝ), namely the Hecke groups. We show that the Eisenstein series associated to a Hecke group H(m) satisfy a set of m coupled linear differential equations, which are natural analogues of the well-known Ramanujan identities for quasi-modular forms of SL(2, ℤ). Each Hecke group is then associated with a (hyper-)elliptic curve, whose coefficients are determined by an anomaly equation. For the m = 3 and 4 cases, the Ramanujan identities admit a natural geometric interpretation as a Gauss-Manin connection on the parameter space of the elliptic curve. The Ramanujan identities also allow us to associate a nonlinear differential equation of order m to each Hecke group. These equations are higher-order analogues of the Chazy equation, and we show that they are solved by the quasi-automorphic Eisenstein series E⁽ᵐ⁾₂ associated to H(m) and its orbit under the Hecke group. We conclude by demonstrating that these nonlinear equations possess the Painlevé property.
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| id | nasplib_isofts_kiev_ua-123456789-210609 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:20Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
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| spelling | Ashok, Sujay K. Jatkar, Dileep P. Raman, Madhusudhan 2025-12-12T10:41:57Z 2020 Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations. Sujay K. Ashok, Dileep P. Jatkar and Madhusudhan Raman. SIGMA 16 (2020), 001, 26 pages 1815-0659 2010 Mathematics Subject Classification: 34M55; 11F12; 33E30 arXiv:1810.07919 https://nasplib.isofts.kiev.ua/handle/123456789/210609 https://doi.org/10.3842/SIGMA.2020.001 We study various relations governing quasi-automorphic forms associated with discrete subgroups of SL(2, ℝ), namely the Hecke groups. We show that the Eisenstein series associated to a Hecke group H(m) satisfy a set of m coupled linear differential equations, which are natural analogues of the well-known Ramanujan identities for quasi-modular forms of SL(2, ℤ). Each Hecke group is then associated with a (hyper-)elliptic curve, whose coefficients are determined by an anomaly equation. For the m = 3 and 4 cases, the Ramanujan identities admit a natural geometric interpretation as a Gauss-Manin connection on the parameter space of the elliptic curve. The Ramanujan identities also allow us to associate a nonlinear differential equation of order m to each Hecke group. These equations are higher-order analogues of the Chazy equation, and we show that they are solved by the quasi-automorphic Eisenstein series E⁽ᵐ⁾₂ associated to H(m) and its orbit under the Hecke group. We conclude by demonstrating that these nonlinear equations possess the Painlevé property. We would like to thank Suresh Govindarajan for the discussions and for the collaboration during an early stage of the project. We also thank Renjan John for helpful comments on an earlier version of the manuscript, and the anonymous referees for valuable comments and feedback. MR acknowledges support from the Infosys Endowment for Research into the Quantum Structure of Spacetime. This research was supported in part by the International Centre for Theoretical Sciences (ICTS) during a visit to participate in the program Quantum Fields, Geometry and Representation Theory (Code: ICTS/qftgrt/2018/07). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations Article published earlier |
| spellingShingle | Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations Ashok, Sujay K. Jatkar, Dileep P. Raman, Madhusudhan |
| title | Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations |
| title_full | Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations |
| title_fullStr | Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations |
| title_full_unstemmed | Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations |
| title_short | Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations |
| title_sort | aspects of hecke symmetry: anomalies, curves, and chazy equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210609 |
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