Construction of cusp forms using Rankin-Cohen brackets

TL;DR

This paper introduces a new method for constructing cusp forms using Rankin-Cohen brackets, explicitly computing the associated linear map and relating Fourier coefficients to Rankin-Selberg convolutions.

Contribution

It defines a linear map on modular forms via Rankin-Cohen brackets, computes its adjoint explicitly, and links Fourier coefficients to Rankin-Selberg convolutions.

Findings

Explicit formula for the linear map T_{g,ν}

Fourier coefficients relate to Rankin-Selberg convolutions

Provides a new construction method for cusp forms

Abstract

For a fix modular form g and a non negative ineteger {\nu}, by using Rankin-Cohen bracket we first define a linear map $T_{g,{\nu}}$ on the space of modular forms. We explicitly compute the adjoint of this map and show that the n-th Fourier coefficients of the image of the cusp form f under this map is, upto a constant a special value of Rankin-Selberg convolution of f and g.