Vector-valued Hirzebruch-Zagier series and class number sums

TL;DR

This paper corrects and extends the generating functions for Hurwitz class number sums to modular forms, explores their properties, and applies them to compute class number sums, connecting to classical Hirzebruch-Zagier series.

Contribution

It introduces a corrected modular form representation for Hurwitz class number sums and links these to classical Hirzebruch-Zagier series and class number computations.

Findings

Corrected generating functions as modular forms or quasimodular forms.

Derived properties of these forms in the Petersson scalar product.

Computed class number sums using Eisenstein series comparison.

Abstract

For any number $m \equiv 0,1 \, (4)$ we correct the generating function of Hurwitz class number sums $\sum_r H(4n - mr^2)$ to a modular form (or quasimodular form if $m$ is a square) of weight two for the Weil representation attached to a binary quadratic form of discriminant $m$ and determine its behavior in the Petersson scalar product. This modular form arises through holomorphic projection of the zero-value of a nonholomorphic Jacobi Eisenstein series of index $1/m$. When $m$ is prime, we recover the classical Hirzebruch-Zagier series whose coefficients are intersection numbers of curves on a Hilbert modular surface. Finally we calculate certain sums over class numbers by comparing coefficients with an Eisenstein series.