A $p$-arton Model for Modular Cusp Forms

TL;DR

This paper introduces a novel $p$-adic framework for modular cusp forms, linking their properties to Mellin transforms and $L$-functions, and exploring connections to Maass forms and Dirichlet $L$-functions.

Contribution

It develops a $p$-arton model associating modular forms with $p$-adic functions, revealing new insights into their Mellin transforms and $L$-function relationships.

Findings

Establishes a correspondence between modular forms and $p$-adic functions.

Analyzes the Mellin transforms and $L$-functions related to these forms.

Draws parallels between convolution products of $ heta$-series and Maass forms.

Abstract

We propose to associate to a modular form (an infinite number of) complex valued functions on the $p$-adic numbers $\mathbb{Q}_p$ for each prime $p$. We elaborate on the correspondence and study its consequence in terms of the Mellin transforms and the $L$-functions related to the forms. 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 non-holomorphic Maass forms of weight zero as suggested by their Fourier coefficients.