Locally harmonic Maass forms and the kernel of the Shintani lift

TL;DR

This paper introduces a new class of modular objects related to harmonic Maass forms, providing explicit examples and a novel perspective on the rationality of Zagier's cusp form periods.

Contribution

It defines a new type of modular object, constructs explicit examples, and connects these to classical cusp forms and the Shintani lift, offering fresh insights into their properties.

Findings

Explicit examples of the new modular objects are constructed.

A new perspective on the rationality of Zagier's cusp form periods is provided.

The new functions share properties with harmonic weak Maass forms but also exhibit contrasting features.

Abstract

In this paper we define a new type of modular object and construct explicit examples of such functions. Our functions are closely related to cusp forms constructed by Zagier which played an important role in the construction by Kohnen and Zagier of a kernel function for the Shimura and Shintani lifts between half-integral and integral weight cusp forms. Although our functions share many properties in common with harmonic weak Maass forms, they also have some properties which strikingly contrast those exhibited by harmonic weak Maass forms. As a first application of the new theory developed in this paper, one obtains a new perspective on the fact that the even periods of Zagier's cusp forms are rational as an easy corollary.