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Abstract

The singular moduli of \bbfQ (\sqrt d), dlt;0, are j(τ), where the τ are the roots of the h corresponding reduced forms. These moduli are roots of the class equation of degree h over \bbfQ, 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(τ). There is a sign ambiguity in these values, apparently covered by a ``principal square root'' conjecture. Generally, j(τ) and f(τ) determine the same Hilbert class field over \bbfQ (\sqrt d).\par The authors also consider the extension of results of \it B. Gross and \it 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.