A Gross--Kohnen--Zagier Type Theorem for Higher-Codimensional Heegner Cycles

TL;DR

This paper establishes a new relation between higher-codimensional Heegner cycles and modular forms, extending classical results and introducing a Borcherds-style theta lift to connect different types of modular forms.

Contribution

It proves a Gross--Kohnen--Zagier type theorem for higher-codimensional Heegner cycles and introduces a novel theta lift that links weakly holomorphic and meromorphic modular forms.

Findings

Heegner cycles are coefficients of modular forms of specific weights.

A Borcherds-style theta lift is constructed for generating relations.

A new Shimura-type correspondence is established between different modular form spaces.

Abstract

We prove that Heegner cycles of codimension m+1 inside Kuga-Sato type varieties of dimension 2m+1 are coefficients of modular forms of weight 3/2+m in the appropriate quotient group. The main technical tool for generating the necessary relations is a Borcherds style theta lift with polynomials. We also show how this lift defines a new singular Shimura-type correspondence from weakly holomorphic modular forms of weight 1/2-m to meromorphic modular forms of weight 2m+2.