A $p$-arton model for modular cusp forms

To a modular form, we propose to associate (an infinite number of) complex-valued functions on $p$-adic numbers $\mathbb{Q}_p$ for each prime $p$. We elaborate on the correspondence and study its consequences in terms of the Mellin transform and the $L$-function related to the form. Further, we discuss the case of products of Dirichlet $L$-functions and their Mellin duals, which are convolution products of $\vartheta$-series. The latter are intriguingly similar to nonholomorphic Maass forms of weight zero as suggested by their Fourier coefficients.