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arXiv · 1711.10519

Vector-valued Hirzebruch-Zagier series and class number sums

Brandon Williams
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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.

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Brandon Williams. 2017-11-28. Vector-valued Hirzebruch-Zagier series and class number sums. https://doi.org/10.1007/s40687-018-0142-4

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