Modular vector fields and Calabi-Yau modular forms

This talk will mostly be concentrated on more general ideas, skipping proofs and technical details, to introduce my recent works on constructing a "modern" generalization of the "classical" theory of modular forms. More precisely, I will introduce a certain moduli space $T$ arising from a special family of Calabi-Yau (CY) varieties. There exists a unique vector field $R$ on $T$, called modular vector field, that satisfies a certain equation involving the Gauss-Manin connection. We observe that the modular vector field $R$, in some sense,  behaves similar to the Ramanujan vector field (Ramanujan relations between Eisenstein series). If we let $f$ to be any component of a solution of $R$, then surprisingly we see that the coefficients of the q-expansion of $f$ are integers and it carries a natural weight. By CY modular form we mean the elements of the space generated by the components of a solution of R. We observe that the space of the CY modular forms is endowed with a canonical Rankin-Cohen structure. 

Zoom Meeting ID: 919 6497 4060
For password see the email or contact Pieter Moree (moree@mpim...).