Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations

Abstract

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.