Hecke relations for traces of singular moduli

Special values of the modular j function at imaginary quadratic points in the upper half-plane are known as singular moduli; these are algebraic integers that play many roles in number theory. Zagier proved that the traces (and more generally, the Hecke traces) of singular moduli are described by a multiply infinite family of weight 3/2 weakly holomorphic modular forms of level 4 (or, through what is sometimes called 'duality', by a multiply infinite family of weight 1/2 weakly holomorphic modular forms of level 4). Several authors have used this description to obtain relations and congruences for these traces modulo prime powers p ⁿ in various situations. We prove that the modular forms in question satisfy a simple relationship involving the Hecke operators T(p ²ⁿ) for n≥1. As a corollary we obtain uniform relations for the traces (some of which were known in particular cases).