Heegner cycles in Griffiths groups of Kuga-Sato varieties
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866916531219202048 |
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| author | Lilienfeldt, David T. -B. G. |
| author_facet | Lilienfeldt, David T. -B. G. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2107_06731 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Heegner cycles in Griffiths groups of Kuga-Sato varieties Lilienfeldt, David T. -B. G. Number Theory Algebraic Geometry 11G15, 11F03, 14C25 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. |
| title | Heegner cycles in Griffiths groups of Kuga-Sato varieties |
| topic | Number Theory Algebraic Geometry 11G15, 11F03, 14C25 |
| url | https://arxiv.org/abs/2107.06731 |