Class invariants by Siegel resolvents and the modularity of their Galois traces (Q2070382)

In the paper under review, the authors define \textit{Siegel resolvents} as the quadratic polynomials of Siegel functions of level \(3\). There are two main results. First of all, they show that they are modular functions of level \(3\). They construct real-valued class invariants over imaginary quadratic fields by using the singular values of Siegel resolvents at imaginary quadratic irrationals. Secondly, they also prove that the generating series of their Galois traces become a weakly holomorphic modular form with weight \(3/2\). The motivation of the paper is coming from Kaneko's work. In that work, the modular trace of the normalized Hauptmodul has been extended to the Galois trace of a class invariant. Hence it is natural to search for class invariants for which the Galois traces have modular properties. Furthermore, the paper under review can be considered as a continuation work of Don Zagier as traces of singular moduli can be extended to the modular functions of higher level. Proofs of the two main results follow from a series of lemmas.