The singular moduli of \(\mathbb{Q} (\sqrt d)\), \(d<0\), are \(j(\tau)\), where the \(\tau\) are the roots of the \(h\) corresponding reduced forms. These moduli are roots of the class equation of degree \(h\) over \(\mathbb{Q}\), with coefficients of astronomical size, ill-suited for numerical work. Weber introduced his three “\(f\)-functions” (given by \((X-16)^3 =Xj\), with \(X= f^{24})\), leading to Weber’s class equation in \(f\), of more reasonable size. The authors give a definitive summary of this process. Weber’s class equation is found from the numerical values of \(f(\tau)\). There is a sign ambiguity in these values, apparently covered by a “principal square root” conjecture. Generally, \(j(\tau)\) and \(f(\tau)\) determine the same Hilbert class field over \(\mathbb{Q} (\sqrt d)\).
The authors also consider the extension of results of B. Gross and D. Zagier [J. Reine Angew. Math. 355, 191-220 (1985; Zbl 0545.10015)] on the factorization of the discriminant of the new class equation. This involves a complicated (and incomplete) network of special cases.