Rankin-Cohen brackets of eigenforms and modular forms

TL;DR

This paper investigates when the Rankin-Cohen brackets of eigenforms and modular forms are eigenforms, using Maeda's Conjecture to establish conditions and provide evidence for the conjecture.

Contribution

It characterizes the circumstances under which Rankin-Cohen brackets of eigenforms remain eigenforms, linking these conditions to space dimensions and operator injectivity, and offers evidence for Maeda's Conjecture.

Findings

Rankin-Cohen brackets of eigenforms are eigenforms only under specific dimension-related conditions.

The injectivity of the Rankin-Cohen operator influences whether the brackets produce eigenforms.

The work provides new insights and evidence supporting Maeda's Conjecture.

Abstract

We use Maeda's Conjecture to prove that the Rankin-Cohen bracket of an eigenform and any modular form is only an eigenform when forced to be because of the dimensions of the underlying spaces. We further determine when the Rankin-Cohen bracket of an eigenform and modular form is not forced to produce an eigenform and when it is determined by the injectivity of the operator itself. This can also be interpreted as using the Rankin-Cohen bracket operator of eigenforms to create evidence for Maeda's Conjecture.