Speaker: Gabriele Bogo
Date: 10:30 24/06/2022
Title: Ramanujan systems associated to hyperbolic triangles 

Abstract:
In the talk we construct systems of three nonlinear ODEs from hypergeometric differential
equations associated to hyperbolic triangles. The main tool in the construction is the 
existence of modular embeddings for triangle groups. The solutions P,Q,R are holomorphic 
on the upper half-plane and algebraically independent. In particular, Q and R are (twisted) 
modular forms of rational weight on a suitable triangle group. A classical example is given
by Ramanujan's differential relations for Eisenstein series. 
As a corollary, we find new identities for hypergeometric functions. Finally, we discuss 
when these systems are of Rankin-Cohen type, some examples, and possible generalizations.