Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 102, 13 pages      arXiv:2004.06035      https://doi.org/10.3842/SIGMA.2020.102

Triangle Groups: Automorphic Forms and Nonlinear Differential Equations

Sujay K. Ashok a, Dileep P. Jatkar b and Madhusudhan Raman c
a) Institute of Mathematical Sciences, Homi Bhabha National Institute (HBNI), IV Cross Road, C.I.T. Campus, Taramani, Chennai 600 113, India
b) Harish-Chandra Research Institute, Homi Bhabha National Institute (HBNI), Chhatnag Road, Jhunsi, Allahabad 211 019, India
c) Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Navy Nagar, Colaba, Mumbai 400 005, India

Received April 21, 2020, in final form October 05, 2020; Published online October 11, 2020; Acknowledgments corrected July 08, 2021

Abstract
We study the relations governing the ring of quasiautomorphic forms associated to triangle groups with a single cusp, thereby extending our earlier results on Hecke groups. The Eisenstein series associated to these triangle groups are shown to satisfy Ramanujan-like identities. These identities in turn allow us to associate a nonlinear differential equation to each triangle group. We show that they are solved by the quasiautomorphic weight-2 Eisenstein series associated to the triangle group and its orbit under the group action. We conclude by discussing the Painlevé property of these nonlinear differential equations.

Key words: triangle groups; Chazy equations; Painlevé analysis.

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