Shimura lift of Rankin-Cohen brackets of eigenforms and theta series

TL;DR

This paper generalizes the Shimura lift to Rankin-Cohen brackets involving eigenforms and theta series, showing that the lift of such brackets equals the bracket of the eigenform with itself.

Contribution

It extends the Shimura lift to a broader class of modular forms by incorporating Rankin-Cohen brackets, revealing new algebraic relationships.

Findings

Shimura lift of Rankin-Cohen brackets equals the bracket of the eigenform with itself

Generalization of the product map to Rankin-Cohen brackets

New algebraic identities involving theta series and eigenforms

Abstract

The Shimura lift of a Hekce eigenform multiplied by a theta series is the square of the form. We extend this result by generalizing the product map to the Rankin-Cohen bracket. We prove that the Shimura lift of Rankin-Cohen bracket of an eigenform and a theta series is given by Rankin-Cohen bracket of the eigenform and itself.