Eichler integrals and harmonic weak Maass forms

TL;DR

This paper extends the connection between Eichler integrals and harmonic weak Maass forms from the full modular group to more general H-groups, using supplementary functions theory, and explores relations among period functions.

Contribution

It generalizes the known relationship to H-groups and employs supplementary functions theory to relate Eichler integrals with harmonic weak Maass forms.

Findings

Eichler integrals with polynomial period functions match holomorphic parts of harmonic weak Maass forms.

Non-holomorphic parts of these forms are period integrals of cusp forms.

Relations among period functions for harmonic weak Maass forms are established.

Abstract

Recently, K. Bringmann, P. Guerzhoy, Z. Kent and K. Ono studied the connection between Eichler integrals and the holomorphic parts of harmonic weak Maass forms on the full modular group. In this article, we extend their result to more general groups, namely, $H$-groups by employing the theory of supplementary functions introduced and developed by M. I. Knopp and S. Y. Husseini. In particular, we show that the set of Eichler integrals, which have polynomial period functions, is the same as the set of holomorphic parts of harmonic weak Maass forms of which the non-holomorphic parts are certain period integrals of cusp forms. From this we deduce relations among period functions for harmonic weak Maass forms.