Integrality Properties of Class Polynomials for Non-Holomorphic Modular Functions

Abstract

In his paper Traces of Singular Moduli [14], Zagier studied values of certain modular functions at imaginary quadratic points known as singular moduli. He proved that "tracesa" of these algebraic integers are Fourier coefficients of certain half-integral weight modular forms. In this paper, he obtained similar results for certain non-holomorphic modular functions. However, he observed that these "singular moduli" are not necessarily algebraic integers. Based on numerical examples, the "class polynomials" whose roots are these singular moduli seem to have predictable denominators. Here we explain this phenomenon and provide a sharp bound on these denominators.