On ODEs Satisfied by Modular Forms
Explore the fascinating world of modular forms and their associated ordinary differential equations (ODEs) in this comprehensive lecture. Delve into the algebraic dependence of modular forms and their derivatives, examining how these forms satisfy third-order nonlinear ODEs invariant under SL(2, R) and fourth-order nonlinear ODEs invariant under GL(2, R). Discover how to express these ODEs using differential invariants and their connection to plane algebraic curves. Investigate examples of nonlinear ODEs satisfied by classical modular forms, including Eisenstein series, modular forms on congruence subgroups, theta constants, and newforms. Extend your understanding to Jacobi forms and their invariant third-order partial differential equation systems. Gain insights into the SCREAM project, focusing on Cartan and parabolic geometries, and their applications in various fields such as mechanical systems, integrable systems, and cosmology.