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Modular and automorphic forms & beyond

Posted in
Hossein Movasati
Tue, 2017-01-17 14:00 - 15:00
MPIM Lecture Hall

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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