Shimura's Vector-Valued Modular Forms, Weight Changing Operators, and Laplacians

TL;DR

This paper explores weight modification operators and Laplacian eigenfunctions in the context of Shimura's vector-valued modular forms, enhancing understanding of their structure and spectral properties.

Contribution

It systematically analyzes weight changing operators and characterizes all eigenfunctions of associated Laplacians for Shimura's modular forms.

Findings

Classification of weight raising and lowering operators

Complete description of Laplacian eigenfunctions

Insights into the spectral theory of vector-valued modular forms

Abstract

We investigate the various types of weight raising and weight lowering operators on quasi-modular forms, or equivalently on Shimura's vector-valued modular forms involving symmetric power representations. We also present all the eigenfunctions of the two possible Laplacian operators.