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Generalised Heegner cycles and the complex Abel–Jacobi map

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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.

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Acknowledgements

David Lilienfeldt would like to thank Michele Fornea, James Rickards and Jan Vonk for helpful discussions. The authors thank the anonymous referee for their valuable feedback which helped to improve the quality of the exposition.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Strasse 9, 45127, Essen, Germany

    Massimo Bertolini

  2. Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, QC, H3A 0G4, Canada

    Henri Darmon & David Lilienfeldt

  3. Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, MI, 48109, USA

    Kartik Prasanna

Authors
  1. Massimo Bertolini
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  2. Henri Darmon
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  3. David Lilienfeldt
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  4. Kartik Prasanna
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Correspondence to Henri Darmon.

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During the preparation of this article, KP was supported partially by NSF grants DMS-1015173 and DMS-0854900 and DL was supported partially by the Institut des Sciences Mathématiques, Montréal, Canada.

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Bertolini, M., Darmon, H., Lilienfeldt, D. et al. Generalised Heegner cycles and the complex Abel–Jacobi map. Math. Z. 298, 385–418 (2021). https://doi.org/10.1007/s00209-020-02603-8

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  • DOI: https://doi.org/10.1007/s00209-020-02603-8

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