Some remarks on Hurwitz–Kronecker class numbers and traces of singular moduli

Abstract

Applying the Kudla–Millson theta lift developed by Bruinier and Funke (J Reine Angew Math 594:1–33, 2006) and theory of mock Jacobi forms (Dabholkar et al. in http://arxiv.org/abs/1208.4074v2, 2014), we express certain combinations of Hurwitz–Kronecker class numbers as Fourier coefficients of holomorphic modular forms of weight \(3/2\). Moreover, we extend the results of Choi et al. (Int Math Res Not 2007:17, 2007) and Choi and Kim (Acta Arith 145:155–169, 2010) to express the class numbers as traces of singular moduli of the modular function \(j_{p^2,n}^*\) with \(p\in \{3,5,7\}\) and \(n\ge 1\).