Eichler-Selberg relations for singular moduli

TL;DR

This paper extends the Eichler-Selberg trace formula to relate traces of singular moduli for modular functions to class numbers and new L-function terms, providing a unified framework for understanding Hecke operator traces.

Contribution

It introduces new Eichler-Selberg relations for traces of singular moduli of modular functions, generalizing classical formulas and connecting them to shifted convolution L-functions.

Findings

Derived new trace formulas involving singular moduli and Hecke operators.

Connected singular moduli traces to symmetrized shifted convolution L-functions.

Generalized classical formulas to higher weights and levels.

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 \[ j_m(\tau) := mT_m \left(j(\tau)-744\right). \] 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. These formulas involve a new term that is assembled from values of symmetrized shifted convolution $L$-functions.