Heegner divisors in generalized Jacobians and traces of singular moduli

TL;DR

This paper extends the Gross-Kohnen-Zagier theorem by proving a modularity result for Heegner divisors in generalized Jacobians, linking harmonic Maass forms and traces of singular moduli.

Contribution

It generalizes known results by establishing a modularity framework for Heegner divisors and harmonic Maass forms, providing new proofs and geometric interpretations.

Findings

Generating series of Heegner divisor classes is a weakly holomorphic modular form of weight 3/2.

Harmonic Maass forms of weight 0 define functionals on generalized Jacobians.

New geometric interpretations for traces of singular moduli with non-positive index.

Abstract

We prove an abstract modularity result for classes of Heegner divisors in the generalized Jacobian of a modular curve associated to a cuspidal modulus. Extending the Gross-Kohnen-Zagier theorem, we prove that the generating series of these classes is a weakly holomorphic modular form of weight 3/2. Moreover, we show that any harmonic Maass forms of weight 0 defines a functional on the generalized Jacobian. Combining these results, we obtain a unifying framework and new proofs for the Gross-Kohnen-Zagier theorem and Zagier's modularity of traces of singular moduli, together with new geometric interpretations of the traces with non-positive index.