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An automorphic form is a well-behaved function from a topological group $G$ to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup $\Gamma \subset G$ of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.

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Modular forms and Petersson inner product via De Rham cohomology, Hodge filtration and cup p...

I'm looking for an explanation on how and why you can define modular forms through De Rham cohomology via the Hodge filtration and especially how the Petersson inner product is related to the cup prod …