Hecke Group Anomalies: Detecting and Classifying Unexpected Modularity in Higher Genus Curves

Building on Ashok, Jatkar, and Raman (2018), who extend Ramanujan-style identities and anomaly equations to Hecke groups and hyperelliptic curves, this idea pushes further: What other “anomalies” or unexpected modular patterns arise when we systematically analyze families of curves (say, genus > 2) linked to non-congruence subgroups? The project would develop computational and theoretical tools to search for, classify, and explain any surprising instances where modular or quasi-modular relations appear—especially those not predicted by the current theory. This diverges from the explicit constructions in the cited work by focusing on systematic detection and classification of anomalies, potentially revealing new landscapes of modularity. The impact would be a deeper understanding of the interplay between group symmetry, anomalies, and modular forms in previously unexplored settings.

References:

1. Aspects of Hecke Symmetry:Anomalies, Curves, and Chazy Equations. S. Ashok, Dileep P. Jatkar, M. Raman (2018). Symmetry, Integrability and Geometry: Methods and Applications.