Summary: We formulate a conjecture that partially generalizes the Gross-Kohnen-Zagier theorem to higher-weight modular forms. For \(f \in S_{2k}(N)\) satisfying certain conditions, we construct a map from the Heegner points of level \(N\) to a complex torus \(\mathbb{C}/L_f\) defined by \(f\). We define higher-weight analogues of Heegner divisors on \(\mathbb{C}/L_f\).
We conjecture that they all lie on a line and that their positions are given by the coefficients of a certain Jacobi form corresponding to \(f\). In weight 2, our map is the modular parameterization map (restricted to Heegner points), and our conjectures are implied by Gross-Kohnen-Zagier. For any weight, we expect that our map is the Abel-Jacobi map on a certain modular variety, and so our conjectures are consistent with the conjectures of Beilinson-Bloch. We have verified that our map is the Abel-Jacobi map for weight 4. We provide numerical evidence to support our conjecture for a variety of examples.