A modular framework for generalized Hurwitz class numbers II

TL;DR

This paper constructs a sesquiharmonic Maass form related to Hurwitz class numbers, realizes it via a regularized theta lift, and computes its Fourier coefficients, revealing explicit identities with quadratic traces.

Contribution

It provides a new realization of a sesquiharmonic Maass form as a regularized theta lift and computes its Fourier coefficients explicitly.

Findings

Explicit formulas for Fourier coefficients of the Maass form.

Identities linking Fourier coefficients to quadratic traces.

Evaluation of the Millson theta lift and spectral deformations.

Abstract

In a recent preprint, we constructed a sesquiharmonic Maass form $\mathcal{G}$ of weight $\frac{1}{2}$ and level $4N$ with $N$ odd and squarefree. Extending seminal work by Duke, Imamo\={g}lu, and T\'{o}th, $\mathcal{G}$ maps to Zagier's non-holomorphic Eisenstein series and a linear combination of Pei and Wang's generalized Cohen--Eisenstein series under the Bruinier--Funke operator $\xi_{\frac{1}{2}}$. In this paper, we realize $\mathcal{G}$ as the output of a regularized Siegel theta lift of $1$ whenever $N=p$ is an odd prime building on more general work by Bruinier, Funke and Imamo\={g}lu. In addition, we supply the computation of the square-indexed Fourier coefficients of $\mathcal{G}$. This yields explicit identities between the Fourier coefficients of $\mathcal{G}$ and all quadratic traces of $1$. Furthermore, we evaluate the Millson theta lift of $1$ and consider spectral deformations of $1$.