Locally harmonic Maass forms and periods of meromorphic modular forms

TL;DR

This paper introduces a new family of locally harmonic Maass forms linked to periods of modular forms, providing explicit decompositions and rational formulas for periods of meromorphic modular forms related to quadratic forms.

Contribution

It presents a novel class of locally harmonic Maass forms, with explicit splitting into harmonic and polynomial parts, and derives rational formulas for periods of meromorphic modular forms.

Findings

Explicit splitting of locally harmonic Maass forms into harmonic and polynomial parts

Finite rational formulas for periods of meromorphic modular forms

Connection between Maass forms and quadratic form periods

Abstract

We investigate a new family of locally harmonic Maass forms which correspond to periods of modular forms. They transform like negative weight modular forms and are harmonic apart from jump singularities along infinite geodesics. Our main result is an explicit splitting of the new locally harmonic Maass forms into a harmonic part and a locally polynomial part that captures the jump singularities. As an application, we obtain finite rational formulas for suitable linear combinations of periods of meromorphic modular forms associated to positive definite binary quadratic forms.