Hecke relations for traces of singular moduli

Abstract

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 pn in various situations. We prove that the modular forms in question satisfy a simple relationship involving the Hecke operators T(p2n) for n⩾1. As a corollary we obtain uniform relations for the traces (some of which were known in particular cases).