Eichler-Selberg relations for 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 $j_0(\ddot{I}„)=1$. More generally, we consider the singular moduli for the Hecke system of modular functions \[ j_m(τ) := mT_m \left(j(τ)-744\right). \] For each $\hat{I}½\ge 0$ and $m\ge 1$, we obtain an Eichler-Selberg relation. For $\hat{I}½=0$ and $m\in \{1,2\},$ these relations are Kaneko's celebrated singular moduli formulas for the coefficients of $j(\ddot{I}„).$ For each $\hat{I}½\ge 1$ and $m\ge 1,$ we obtain a new Eichler-Selberg trace formula for the Hecke action on the space of weight $2\hat{I}½+2$ cusp forms, where the traces of $j_m(\ddot{I}„)$ singular moduli replace Hurwitz-Kronecker class numbers. These formulas involve a new term that is assembled from values of symmetrized shifted convolution $L$-functions.

21 pages