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

SIGMA 16 (2020), 001, 26 pages      arXiv:1810.07919      https://doi.org/10.3842/SIGMA.2020.001

Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy 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 May 06, 2019, in final form December 29, 2019; Published online January 01, 2020

Abstract
We study various relations governing quasi-automorphic forms associated to discrete subgroups of ${\rm SL}(2,\mathbb{R}) $ called Hecke groups. We show that the Eisenstein series associated to a Hecke group ${\rm H}(m)$ satisfy a set of $m$ coupled linear differential equations, which are natural analogues of the well-known Ramanujan identities for quasi-modular forms of ${\rm SL}(2,\mathbb{Z})$. Each Hecke group is then associated to 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 $E_2^{(m)}$ associated to ${\rm H}(m) $ and its orbit under the Hecke group. We conclude by demonstrating that these nonlinear equations possess the Painlevé property.

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

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