A $p$-arton model for modular cusp forms

Abstract

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.

Appendix: Wavelets on $$\pmb{\mathbb{Q}}_p^\times$$

We modify the Kozyrev wavelets to define

$$ \boldsymbol{\Psi}^{(p)}_{n,m,j}(x)=|x|_p^{1/2}\psi^{(p)}_{n,m,j}(x),$$

(A.1)

which are naturally defined on $\mathbb{Q}_p^\times$ because they are orthonormal with respect to the scale invariant multiplicative Haar measure $d^\times x$:

$$ \int_{\mathbb{Q}_p^\times}\frac{dx}{|x|_p}\boldsymbol{\Psi}^{(p)}_{n,0,1}(x)\boldsymbol{\Psi}^{(p)}_{n',0,1}(x)= \int_{\mathbb{Q}_p} dx\,\psi^{(p)}_{n,0,1}(x)\psi^{(p)}_{n',0,1}(x)=\delta_{nn'}.$$

(A.2)

We note that the wavelets above, being different from Kozyrev wavelets (13) by a coordinate dependent factor, are not equal to the constant $p^{-n/2}$ for $|x|_p<p^n$, although they are still locally constant functions on $\mathbb{Q}_p$. The raising and lowering operators in (18) and (20) act as before:

$$\mathbf{a}^{(p)}_\pm\boldsymbol{\Psi}^{(p)}_{n,0,1}(x)=\boldsymbol{\Psi}^{(p)}_{n\pm 1,0,1}(x),\qquad \mathbf{a}^{(p)}_{+}\boldsymbol{\Psi}^{(p)}_{1,0,1}(x)=0,$$

but we choose to use a different notation to emphasize the fact that they act on a different space of functions. Mellin transform (25) of these modified wavelets

$$\mathcal M_{(p,\omega)}[\boldsymbol{\Psi}^{(p)}_{n,0,1}](s)= \mathbf c_p(\ell,s)p^{ns}= -\biggl(\frac{1}{p(1-p^{-s-1/2})}-\frac{1}{p^{s+1/2}-1}\,\delta_{\ell,0} -\delta_{\ell,p-1}\biggr)p^{ns}$$

likewise differs somewhat from (27). We note that

$$\sum_{\ell}|\mathbf c_p(\ell,s)|^2=1+\frac{1-|p^s|^2}{|p^{s+1/2}-1|^2},$$

and hence, if the argument $s$ is purely imaginary, the sum is 1.