On the theta operator for modular forms modulo prime powers

TL;DR

This paper investigates the action of the theta operator on modular forms modulo prime powers, establishing the precise weight shift and its optimality, which impacts the understanding of Galois representations associated with these forms.

Contribution

It provides the first explicit description of the theta operator mod p^m, including the exact weight shift and its optimality, clarifying its effect on modular forms and Galois representations.

Findings

Theta mod p^m maps weight k forms to weight k+2+2p^{m-1}(p-1)

The identified weight shift is optimal in certain cases for m ≥ 2

Application to Galois representations shows explicit bounds on cyclotomic twists

Abstract

We consider the classical theta operator $\theta$ on modular forms modulo $p^m$ and level $N$ prime to $p$ where $p$ is a prime greater than 3. Our main result is that $\theta$ mod $p^m$ will map forms of weight $k$ to forms of weight $k+2+2p^{m-1}(p-1)$ and that this weight is optimal in certain cases when $m$ is at least 2. Thus, the natural expectation that $\theta$ mod $p^m$ should map to weight $k+2+p^{m-1}(p-1)$ is shown to be false. The primary motivation for this study is that application of the $\theta$ operator on eigenforms mod $p^m$ corresponds to twisting the attached Galois representations with the cyclotomic character. Our construction of the $\theta$-operator mod $p^m$ gives an explicit weight bound on the twist of a modular mod $p^m$ Galois representation by the cyclotomic character.