Is it worth to elaborate a (new) mathematical theory which is a huge generalization of the theory of (holomorphic)

modular/automorphic forms, without knowing if at some point you will have fruitful applications similar to those

of modular forms? If your answer is yes, this talk might be useful for you. This new theory starts with a moduli

space of projective varieties enhanced with elements in their algebraic de Rham cohomology and with some

compatibility with the Hodge filtration and the cup product. These moduli spaces are conjectured to be affine

varieties, and their ring of functions are candidates for the generalization of automorphic forms. I will explain

this picture in three examples.

1. The case of elliptic curves and the derivation of the algebra of quasi-modular forms

(due to Kaneko and Zagier).

2. The case of principally polarized abelian surfaces and the derivation of Igusa's generators

for the algebra of genus $2$ Siegel modular forms.

3. The case of Calabi-Yau varieties and the derivation of generating function for Gromov-Witten invariants.

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