Generalised Heegner cycles and the complex Abel–Jacobi map

Abstract

Generalised Heegner cycles were introduced in Bertolini et al. (Duke Math J 162(6), 1033–1148, 2013) as a variant of Heegner cycles on Kuga–Sato varieties. The first main result of this article is a formula for the image of these cycles under the complex Abel–Jacobi map in terms of explicit line integrals of modular forms on the complex upper half-plane. The second main theorem uses this formula to show that the Chow group and the Griffiths group of the product of a Kuga–Sato variety with an elliptic curve with complex multiplication are not finitely generated. More precisely, it is shown that the subgroup generated by the image of generalised Heegner cycles has infinite rank in the group of null-homologous cycles modulo both rational and algebraic equivalence.