Harmonic Maass forms associated to real quadratic fields

TL;DR

This paper constructs explicit harmonic Maass forms linked to real quadratic fields and their theta series, revealing detailed arithmetic information in their Fourier coefficients, thus confirming a key conjecture.

Contribution

It provides an explicit construction of harmonic Maass forms associated with real quadratic fields and clarifies the arithmetic nature of their Fourier coefficients.

Findings

Explicit harmonic Maass forms are constructed for real quadratic fields.

Fourier coefficients encode significant arithmetic information.

Main conjecture about the structure of these coefficients is established.

Abstract

In this paper, we explicitly construct harmonic Maass forms that map to the weight one theta series associated by Hecke to odd ray class group characters of real quadratic fields. From this construction, we give precise arithmetic information contained in the Fourier coefficients of the holomorphic part of the harmonic Maass form, establishing the main part of a conjecture of the second author.