Asymptotic distribution of traces of singular moduli

Abstract: Singular moduli are special values of the modular j-invariant (or, more generally, any modular function) at CM points in the complex upper half-plane. Traces of singular moduli appear as coefficients of modular forms of half-integral weight. We determine the asymptotic behavior of twisted traces of singular moduli with a power-saving error term in both the discriminant and the order of the pole at infinity. Using this asymptotic formula, we obtain an exact formula for these traces involving the class number and a finite sum involving the exponential function evaluated at CM points. This is joint work with Bill Duke.