Eichler-Selberg Relations for Traces of Singular Moduli

Abstract

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 \( j_0(\tau) = 1 \). More generally, we consider the singular moduli for the Hecke system of modular functions. For each \( \nu \geq 0 \) and \( m \geq 1 \), we obtain an Eichler–Selberg relation. For \( \nu = 0 \) and \( m \in \{1, 2\} \), these relations are Kaneko’s celebrated singular moduli formulas for the coefficients of \( j(\tau) \). For each \( \nu \geq 1 \) and \( m \geq 1 \), we obtain a new Eichler–Selberg trace formula for the Hecke action on the space of weight \( 2\nu + 2 \) cusp forms, where the traces of \( j_m(\tau) \) singular moduli replace Hurwitz–Kronecker class numbers.