Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

On the non-triviality of the $p$-adic Abel–Jacobi image of generalised Heegner cycles modulo $p$, I: Modular curves


Author: Ashay A. Burungale
Journal: J. Algebraic Geom. 29 (2020), 329-371
DOI: https://doi.org/10.1090/jag/748
Published electronically: January 8, 2020
MathSciNet review: 4069652
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Abstract: Generalised Heegner cycles are associated with a pair of an elliptic newform and a Hecke character over an imaginary quadratic extension $K/\mathbf {Q}$. Let $p$ be an odd prime split in $K/\mathbf {Q}$ and let $\ell \neq p$ be an odd unramified prime. We prove the non-triviality of the $p$-adic Abel–Jacobi image of generalised Heegner cycles modulo $p$ over the $\mathbf {Z}_\ell$-anticylotomic extension of $K$. The result is evidence for the refined Bloch–Beilinson and the Bloch–Kato conjecture. In the case of weight two and $\ell$ an ordinary prime, it provides a non-trivial refinement of the results of Cornut and Vatsal on Mazur’s conjecture regarding the non-triviality of Heegner points over the $\mathbf {Z}_\ell$-anticylotomic extension of $K$. In the case of weight two and $\ell$ a supersingular prime, it settles Mazur’s conjecture earlier known just for $\ell$ ordinary.


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Ashay A. Burungale
Affiliation: Institute for Advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540; and California Institute of Technolology, 1200 East California Boulevard, Pasadena California 91125
MR Author ID: 890463
Email: ashayburungale@gmail.com

Received by editor(s): July 9, 2017
Received by editor(s) in revised form: July 29, 2018, and June 25, 2019
Published electronically: January 8, 2020
Article copyright: © Copyright 2020 University Press, Inc.