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arXiv · 2107.06731

Heegner cycles in Griffiths groups of Kuga-Sato varieties

David T. -B. G. Lilienfeldt
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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.

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David T. -B. G. Lilienfeldt. 2021-07-14. Heegner cycles in Griffiths groups of Kuga-Sato varieties. https://doi.org/10.4310/mrl.241024234657

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