Eichler integrals for Maass cusp forms of half-integral weight

TL;DR

This paper extends the theory of Eichler integrals to Maass cusp forms of half-integral weight, exploring their associated nearly periodic and period functions and their relationships.

Contribution

It introduces Eichler integrals for half-integral weight Maass cusp forms and relates nearly periodic functions to period functions, extending prior work on weight 0 forms.

Findings

Defined Eichler integrals for half-integral weight forms

Established relationships between nearly periodic and period functions

Extended Lewis and Zagier's framework to new form classes

Abstract

In this paper, we define and discuss Eichler integrals for Maass cusp forms of half-integral weight on the full modular group. We discuss nearly periodic functions associated to the Eichler integrals, introduce period functions for such Maass cusp forms, and show that the nearly periodic functions and the period functions are closely related. Those functions are extensions of the periodic functions and period functions for Maass cusp forms of weight 0 on the full modular group introduced by Lewis and Zagier.