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 ƒ ∊ S _2k (N) satisfying certain conditions, we construct a map from the Heegner points of level N to a complex torus /L _ƒ defined by ƒ. We define higher-weight analogues of Heegner divisors on /L _ƒ.

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 ƒ. 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.

Keywords:

• Modular forms
• elliptic curves
• Gross–Kohnen–Zagier formula
• Shimura–Kohnen correspondence