Traces of CM values of modular functions

TL;DR

This paper generalizes Zagier's result on traces of singular moduli to CM values of modular functions on higher genus curves, linking them to modular forms and arithmetic intersection numbers.

Contribution

It extends the trace formula for singular moduli to arbitrary genus modular curves using theta correspondence and relates generating series of intersection numbers to Eisenstein series.

Findings

Generalization of trace formulas to higher genus modular curves

Realization of intersection number series as derivatives of Eisenstein series

Connection between CM value traces and modular form coefficients

Abstract

Zagier proved that the traces of singular moduli, i.e., the sums of the values of the classical j-invariant over quadratic irrationalities, are the Fourier coefficients of a modular form of weight 3/2 with poles at the cusps. Using the theta correspondence, we generalize this result to traces of CM values of (weakly holomorphic) modular functions on modular curves of arbitrary genus. We also study the theta lift for the weight 0 Eisenstein series for SL(2,Z) and realize a certain generating series of arithmetic intersection numbers as the derivative of Zagier's Eisenstein series of weight 3/2. This recovers a result of Kudla, Rapoport and Yang.