Heegner cycles in Griffiths groups of Kuga-Sato varieties

TL;DR

This paper proves that the subgroup generated by Heegner cycles in the Griffiths group of Kuga-Sato varieties has infinite rank, extending classical results and using explicit Abel-Jacobi maps linked to cusp forms.

Contribution

It generalizes Schoen's classical result to higher-dimensional Kuga-Sato varieties and provides explicit formulas for Abel-Jacobi images of Heegner cycles.

Findings

Heegner cycles generate an infinitely ranked subgroup in Griffiths groups.

Explicit formulas relate Abel-Jacobi images to line integrals of cusp forms.

Infinite rank of Griffiths group for products with CM elliptic curves is established.

Abstract

The aim of this article is to prove, using complex Abel-Jacobi maps, that the subgroup generated by Heegner cycles associated with a fixed imaginary quadratic field in the Griffiths group of a Kuga-Sato variety over a modular curve has infinite rank. This generalises a classical result of Chad Schoen for the Kuga-Sato threefold, and complements work of Amnon Besser on complex multiplication cycles over Shimura curves. The proof relies on a formula for the images of Heegner cycles under the complex Abel-Jacobi map given in terms of explicit line integrals of even weight cusp forms on the complex upper half-plane. The latter is deduced from previous joint work of the author with Massimo Bertolini, Henri Darmon, and Kartik Prasanna by exploiting connections with generalised Heegner cycles. As a corollary, it is proved that the Griffiths group of the product of a Kuga-Sato variety with powers of an elliptic curve with complex multiplication has infinite rank. This recovers results of Ashay Burungale by a different and more direct approach.