Traces of Singular Moduli

Introduction. “Singular moduli” is the classical name for the values assumed by the modular invariant j(τ) (or by other modular functions) when the argument is a quadratic irrationality. These values are algebraic numbers and have been studied intensively since the time of Kronecker and Weber. In [5], formulas for their norms, and for the norms of their differences, were obtained. Here we obtain instead a result for their traces, and a number of generalizations. The results are closely related to a theorem of Borcherds [1] of which we will give a new proof and a generalization.