Relations between higher level Hurwitz class numbers

TL;DR

This paper explores the relationships between different generalizations of Hurwitz class numbers from distinct frameworks, leading to new bases for Eisenstein spaces and generalizations of classical formulas.

Contribution

It establishes connections between two frameworks of Hurwitz class numbers and derives new bases for Eisenstein spaces and generalized classical formulas.

Findings

New basis for Eisenstein space $E_{3/2}^{+}(4N, ext{id})$

Generalization of recent results of Beckwith and Mono

Extension of Gauss' classical formula

Abstract

We connect generalizations of the classical Hurwitz class numbers coming from two different frameworks: one introduced by Pei and Wang, arising from the generalized Cohen--Eisenstein series, and another by Li, Skoruppa, and Zhou, arising from Eichler orders of quaternion algebras. As applications, we obtain new basis for Eisenstein space $E_{3/2}^{+}(4N,\mathrm{id})$, a generalization of recent results of Beckwith and Mono, and a generalization of Gauss' formula.