A modular framework for functions of Knopp and indefinite binary quadratic forms

TL;DR

This paper develops a modular framework for functions related to indefinite binary quadratic forms, introducing non-holomorphic bimodular forms and harmonic Maass forms with new properties and local splittings.

Contribution

It extends Knopp's functions to non-holomorphic bimodular forms and introduces locally harmonic Maass forms with local splittings, advancing the theory of quadratic forms.

Findings

Introduction of non-holomorphic bimodular forms related to indefinite quadratic forms

Development of locally harmonic Maass forms with removable singularities

Establishment of local splittings via Eichler integrals

Abstract

We study functions introduced by Knopp and complete them to non-holomorphic bimodular forms of positive integral weight related to indefinite binary quadratic forms. We investigate further properties of our completions, which in turn motivates certain local cusp forms. We then define modular analogues of negative weight of our local cusp forms, which are locally harmonic Maass forms with continuously removable singularities. We show that they admit local splittings in terms of Eichler integrals.