Higher-Weight Heegner Points

Abstract

In this paper 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.