Shimura Lifts of Weakly Holomorphic Modular Forms

TL;DR

This paper extends the Shimura lift to weakly holomorphic modular forms of half-integral weight, using Borcherds' vector-valued theta lifts and regularization techniques, enabling new connections between modular forms of different weights and levels.

Contribution

It introduces a method to realize the Shimura lift for arbitrary level and character via Borcherds' theta lifts, including weakly holomorphic forms.

Findings

Extended Shimura lift to weakly holomorphic forms

Realized lift for arbitrary level and character

Connected half-integral and integral weight modular forms

Abstract

We show how to realize the Shimura lift of arbitrary level and character using the vector-valued theta lifts of Borcherds. Using the regularization of Borcherds' lift we extend the Shimura lift to take weakly holomorphic modular forms of half-integral weight to meromorphic modular forms of even integral weight having poles at CM points.