Locally harmonic Maa{\ss} forms of positive even weight

TL;DR

This paper constructs locally harmonic Maaß forms of positive even weight by twisting Zagier's functions with sign functions and genus characters, correcting modularity obstructions, and providing new representations via twisted trace formulas.

Contribution

It introduces a method to produce locally harmonic Maaß forms of positive even weight by twisting Zagier's functions, addressing modularity obstructions and offering alternative trace-based representations.

Findings

Successfully corrects modularity obstructions using sign twists

Constructs locally harmonic Maaß forms of positive even weight

Provides new trace integral representations of the functions

Abstract

We twist Zagier's function $f_{k,D}$ by a sign function and a genus character. Assuming weight $0 < k \equiv 2 \pmod{4}$, and letting $D$ be a positive non-square discriminant, we prove that the obstruction to modularity caused by the sign function can be corrected obtaining a locally harmonic Maa\ss form or a local cusp form of the same weight. In addition, we provide an alternative representation of our new function in terms of a twisted trace of modular cycle integrals of a Poincar\'e series due to Petersson.