Higher Weight Heegner Points

TL;DR

This paper proposes a higher weight generalization of the Gross-Kohnen-Zagier theorem, constructing a map from Heegner points to complex tori and conjecturing their alignment on a line related to Jacobi form coefficients, supported by numerical evidence.

Contribution

It introduces a new conjecture extending the Gross-Kohnen-Zagier theorem to higher weights and constructs a corresponding map from Heegner points to complex tori.

Findings

Verified the map as the Abel-Jacobi map for weight 4.

Numerical evidence supports the conjecture across multiple examples.

The map aligns with known cases at weight 2 and conjectural frameworks at higher weights.

Abstract

In this paper we formulate a conjecture which partially generalizes the Gross-Kohnen-Zagier theorem to higher weight modular forms. For f in S_k(N) satisfying certain conditions, we construct a map from the Heegner points of level N to a complex torus defined by f. We define higher weight analogues of Heegner divisors on this torus. 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 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 our map is the Abel-Jacobi for weight 4. We provide numerical evidence to support our conjecture for a variety of examples.