Higher-Weight Heegner Points

In this paper we formulate a conjecture which partially generalizes the Gross-Kohnen-Zagier theorem to higher weight modular forms. For f ∈ S2k(N) satisfying certain conditions, we construct a map from the Heegner points of level N to a complex torus, C/Lf , defined by f . We define higher weight analogues of Heegner divisors on C/Lf . We conjecture they all lie on a line, and their positions are given by the coefficients of a certain Jacobi form corresponding to f . In weight 2, our map is the modular parametrization map (restricted to Heegner points), and our conjectures are implied by GrossKohnen-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 our map is the AbelJacobi for weight 4. We provide numerical evidence to support our conjecture for a variety of examples.