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INDIVISIBILITY OF HEEGNER CYCLES OVER SHIMURA CURVES AND SELMER GROUPS

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  15 February 2022

Haining Wang Open the ORCID record for Haining Wang [Opens in a new window]
Haining Wang*
Affiliation:
Shanghai Center for Mathematical Sciences, Fudan University, No. 2005 Songhu Road, Shanghai, 200438, China
*
wanghaining1121@outlook.com
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Abstract

In this article, we show that the Abel–Jacobi images of the Heegner cycles over the Shimura curves constructed by Nekovar, Besser and the theta elements contructed by Chida–Hsieh form a bipartite Euler system in the sense of Howard. As an application of this, we deduce a converse to Gross–Zagier–Kolyvagin type theorem for higher weight modular forms generalising works of Wei Zhang and Skinner for modular forms of weight 2. That is, we show that if the rank of certain residual Selmer group is 1, then the Abel–Jacobi image of the Heegner cycle is nonzero in this residual Selmer group.

Keywords

Heegner cycleEuler systemlevel raising

MSC classification

Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 22 , Issue 5 , September 2023 , pp. 2297 - 2336
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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