A modular framework for generalized Hurwitz class numbers III

TL;DR

The paper develops a higher-weight, higher-level generalization of Hurwitz class-number theory by constructing harmonic Maass preimages of Pei and Wang’s generalized Cohen–Eisenstein series under $\xi_{\frac{3}{2}-k}$ for $k>1$, and by realizing these preimages via regularized Shintani and Millson lifts at prime level. Central to the approach are regularized Siegel/Shintani/Millson theta lifts and their differential relations, which connect half-integral and integral weight objects through explicit Fourier expansions and quadratic traces. The results extend prior work at level 1 and odd primes to square-free levels, providing explicit formulas for Fourier coefficients in terms of generalized Hurwitz numbers $H_{k,\ell,N}(n)$ and Kloosterman data, and establishing precise links between theta lifts and Eisenstein series via $\xi$-images. These findings illuminate the arithmetic-geometric structure underlying generalized Hurwitz numbers and offer concrete tools for evaluating regularized lifts and their traces in higher-level settings, with potential connections to $L$-values and $p$-adic generalizations.

Abstract

In $2003$, Pei and Wang introduced higher level analogs of the classical Cohen--Eisenstein series. In recent joint work with Beckwith, we found a weight $\frac{1}{2}$ sesquiharmonic preimage of their weight $\frac{3}{2}$ Eisenstein series under $ξ_{\frac{1}{2}}$ utilizing a construction from seminal work by Duke, Imamoglu and Tóth. In further joint work with Beckwith, when restricting to prime level, we realized our preimage as a regularized Siegel theta lift and evaluated its (regularized) Fourier coefficients explicitly. This relied crucially on work by Bruinier, Funke and Imamoglu. In this paper, we extend both works to higher weights. That is, we provide a harmonic preimage of Pei and Wang's generalized Cohen--Eisenstein series under $ξ_{\frac{3}{2}-k}$, where $k > 1$. Furthermore, when restricting to prime level, we realize them as outputs of a regularized Shintani theta lift of a higher level holomorphic Eisenstein series, which builds on recent work by Alfes and Schwagenscheidt. Lastly, we evaluate the regularized Millson theta lift of a higher level Maass--Eisenstein series, which is known to be connected to the Shintani theta lift by a differential equation by earlier work of Alfes and Schwagenscheidt.

Theorems & Definitions (45)

• Theorem 1.1
• Remark
• Theorem 1.2
• Corollary 1.3
• Remark
• Theorem 1.4
• Remark
• Corollary 1.5
• Definition 2.1
• Lemma 2.3