A Gross–Kohnen–Zagier type theorem for higher-codimensional Heegner cycles

Shaul Zemel

doi:10.1007/s40993-015-0025-3

A Gross–Kohnen–Zagier type theorem for higher-codimensional Heegner cycles

Shaul Zemel^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

We prove that the Heegner cycles of codimension m+1 inside Kuga-Sato type varieties of dimension 2m+1 are coefficients of modular forms of weight 3/2+m in the appropriate quotient group. The main technical tool for generating the necessary relations is a Borcherds style theta lift with polynomials. We also show how this lift defines a new singular Shimura-type correspondence from weakly holomorphic modular forms of weight 1/2−m to meromorphic modular forms of weight 2m+2.

Original language	American English
Article number	23
Journal	Research in Number Theory
Volume	1
Issue number	1
DOIs	https://doi.org/10.1007/s40993-015-0025-3
State	Published - 1 Dec 2015

Bibliographical note

Funding Information:
Authors’ information: The initial stage of this research has been carried out as part of my Ph.D. thesis work at the Hebrew University of Jerusalem, Israel. The final stage of this work was carried out at the Technical University of Darmstadt, Germany and supported by the Minerva Fellowship (Max-Planck-Gesellschaft).

Publisher Copyright:
© 2015, The Author(s).

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abstract = "We prove that the Heegner cycles of codimension m+1 inside Kuga-Sato type varieties of dimension 2m+1 are coefficients of modular forms of weight 3/2+m in the appropriate quotient group. The main technical tool for generating the necessary relations is a Borcherds style theta lift with polynomials. We also show how this lift defines a new singular Shimura-type correspondence from weakly holomorphic modular forms of weight 1/2−m to meromorphic modular forms of weight 2m+2.",

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N2 - We prove that the Heegner cycles of codimension m+1 inside Kuga-Sato type varieties of dimension 2m+1 are coefficients of modular forms of weight 3/2+m in the appropriate quotient group. The main technical tool for generating the necessary relations is a Borcherds style theta lift with polynomials. We also show how this lift defines a new singular Shimura-type correspondence from weakly holomorphic modular forms of weight 1/2−m to meromorphic modular forms of weight 2m+2.

AB - We prove that the Heegner cycles of codimension m+1 inside Kuga-Sato type varieties of dimension 2m+1 are coefficients of modular forms of weight 3/2+m in the appropriate quotient group. The main technical tool for generating the necessary relations is a Borcherds style theta lift with polynomials. We also show how this lift defines a new singular Shimura-type correspondence from weakly holomorphic modular forms of weight 1/2−m to meromorphic modular forms of weight 2m+2.

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A Gross–Kohnen–Zagier type theorem for higher-codimensional Heegner cycles

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