About quasi-modular forms, differential operators and Rankin-Cohen algebras

In this talk we treat two problems which are parts of an ongoing work. The first one is
a generalization of the canonical Rankin-Cohen structure to the graded algebras which
contain elements of negative degree, namely a generalization of Proposition 1 of
"D. Zagier, Modular forms and differential operators. Proceedings Mathematical Sciences,
104(1):57{75, 1994". The second problem is to give some sufficient conditions under which
a certain combination of a weight 2 quasi-modular form with itself or with an arbitrary modular
form and their derivatives will be again a modular form. Indeed, we solve this problem in
a more algebraic context, namely in Rankin-Cohen algebras.

The idea of these problems are coming from previous works of the speaker, where he is endowing
the space of Calabi-Yau modular forms, which is the space generated by solution components of a
certain vector field on a special moduli space of Calabi-Yau varieties, with a Rankin-Cohen algebra
structure. In fact, the space of Calabi-Yau modular forms that are defined in this way contains
elements of negative weights.