Differential modular forms attached to newforms mod p

TL;DR

This paper extends the concept of duality between classical newforms and modular forms to higher weights, attaching specific $ ext{delta}_ ext{p}$-modular forms to newforms mod p of weights between 3 and p, broadening the scope of the theory.

Contribution

It introduces a higher weight version of the $ ext{sharp}$-duality, attaching $ ext{delta}_ ext{pi}$-modular forms to newforms mod p of weights between 3 and p.

Findings

Constructs $ ext{delta}_ ext{pi}$-modular forms for newforms mod p of specified weights.

Establishes a duality relationship extending previous weight 2 results.

Provides a framework for analyzing modular forms of higher weights in the mod p setting.

Abstract

In a previous paper we attached to classical complex newforms $f$ of weight $2$ certain $\delta_p$-modular forms $f^{\sharp}$ of order $2$ and weight $0$; the forms $f^{\sharp}$ can be viewed as "dual" to $f$ and played a key role in some of the applications of the theory. The aim of this paper is to provide a higher weight version of this "$\sharp$-duality," by attaching to classical newforms mod $p$, $\overline{f}$, of weight $\kappa$ between $3$ and $p$, $\delta_{\pi}$-modular forms $f^{\sharp}$ of order $2$ and weight $-\kappa'$, with $\kappa'$ between $1$ and $p-2$.