An Analogue of Weil's Converse Theorem for Harmonic Maass Forms of Polynomial Growth

TL;DR

This paper constructs harmonic Maass forms of polynomial growth linked to Eisenstein series, analyzes associated Dirichlet series, and proves an analogue of Weil's converse theorem for these forms.

Contribution

It introduces a new class of harmonic Maass forms with polynomial growth and establishes an analogue of Weil's converse theorem for them.

Findings

Constructed harmonic Maass forms of polynomial growth for any level and cusp.

Analyzed the analytic properties of Dirichlet series attached to these forms.

Proved an analogue of Weil's converse theorem for harmonic Maass forms.

Abstract

We construct a family of harmonic Maass forms of polynomial growth of any level corresponding to any cusp whose shadows are Eisenstein series of integral weight. We further consider Dirichlet series attached to a harmonic Maass form of polynomial growth, study its analytic properties, and prove an analogue of Weil's converse theorem.