Colloquium abstracts

Arvind Kumar
TIFR, Mumbai
October 25, 2018

Certain identities among eigenforms:  Identities among modular forms (in particular, Hecke eigenforms) have attracted the attention of many mathematicians since they imply nice identities among their Fourier coefficients. We investigate when the product and more generally Rankin-Cohen brackets of two Hecke eigenforms is an eigenform. Duke and Ghate independently addressed this topic (the product case) for eigenforms of the full modular group, proving there are only 16 such identities. In this talk, we will give a brief survey of the existing results in this direction after introducing the spaces of quasimodular and nearly holomorphic modular forms. These two spaces are isomorphic generalizations of the space of modular forms. In the main result, we classify all the cases when the Rankin-Cohen bracket of two quasimodular eigenforms results in an eigenform. In the process, we obtain some new polynomial identities among quasimodular eigenforms. We extensively use the Rankin?s method and the non-vanishing properties of modular