Modularity of Galois traces of Weber's resolvents

Abstract

The values of the classical j-invariant at CM points are called singular moduli. Zagier [15] proved that the traces of singular moduli are Fourier coefficients of a weakly holomorphic modular form of weight 3/2 and Bruinier-Funke [1] generalized his result to the sums of the values at Heegner points of modular functions on modular curves of arbitrary genus. Extending the work of [9], we construct the real-valued class invariants by using the modular functions defined by Weber's resolvents and identify their Galois traces with Fourier coefficients of a weight 3/2 weakly holomorphic modular form by using Shimura's reciprocity law.