Eichler-Selberg relations for singular moduli

The Eichler-Selberg trace formula expresses the trace of Hecke operators on spaces of cusp forms as weighted sums of Hurwitz-Kronecker class numbers. We extend this formula to a natural class of relations for traces of singular moduli, where one views class numbers as traces of the constant function j₀(τ)=1. More generally, we consider the singular moduli for the Hecke system of modular functions \[ j_m(τ) := mT_m (j(τ)-744). \] For each ν≥ 0 and m≥ 1, we obtain an Eichler-Selberg relation. For ν=0 and m∈ \1, 2\, these relations are Kaneko's celebrated singular moduli formulas for the coefficients of j(τ). For each ν≥ 1 and m≥ 1, we obtain a new Eichler-Selberg trace formula for the Hecke action on the space of weight 2ν+2 cusp forms, where the traces of jₘ(τ) singular moduli replace Hurwitz-Kronecker class numbers. These formulas involve a new term that is assembled from values of symmetrized shifted convolution L-functions.