Abstract:

This talk will mostly be concentrated on more general ideas, skipping proofs and technical details, of my 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 the Dwork 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 an sl_2(C)-module and canonical algebraic Rankin-Cohen structure.