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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 product on De Rham cohomology.

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    $\begingroup$ I'm a Hodge theorist and not a modular forms guy, so you can take this with a grain of salt. A weight 2 modular form (with appropriate conditions at infinity) is a holomorphic differential form on a modular curve $X$, so it can be identified with an element of $H^0(X,\Omega^1)=F^1 H_{dR}^1(X,\C)$. For higher weight forms, you need to work de Rham cohomology with coefficients in variation of Hodge structure. I guess that's a partial answer. $\endgroup$
    – Donu Arapura
    Commented Jul 15, 2022 at 21:53
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    $\begingroup$ There's an account of this in section 6.1 of my paper with Kings and Zerbes "Rankin–Eisenstein classes for modular forms". $\endgroup$
    – David Loeffler
    Commented Jul 16, 2022 at 21:27

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