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ON THE NON-TRIVIALITY OF THE $p$ -ADIC ABEL–JACOBI IMAGE OF GENERALISED HEEGNER CYCLES MODULO $p$ , II: SHIMURA CURVES

Published online by Cambridge University Press:  07 May 2015

Ashay A. Burungale*
Affiliation:
Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA (ashayburungale@gmail.com)

Abstract

Generalised Heegner cycles are associated to a pair of an elliptic newform and a Hecke character over an imaginary quadratic extension $K/\mathbf{Q}$ . The cycles live in a middle-dimensional Chow group of a Kuga–Sato variety arising from an indefinite Shimura curve over the rationals and a self-product of a CM abelian surface. Let $p$ be an odd prime split in $K/\mathbf{Q}$ . We prove the non-triviality of the $p$ -adic Abel–Jacobi image of generalised Heegner cycles modulo $p$ over the $\mathbf{Z}_{p}$ -anticyclotomic extension of  $K$ . The result implies the non-triviality of the generalised Heegner cycles in the top graded piece of the coniveau filtration on the Chow group, and proves a higher weight analogue of Mazur’s conjecture. In the case of weight 2, the result provides a refinement of the results of Cornut–Vatsal and Aflalo–Nekovář on the non-triviality of Heegner points over the $\mathbf{Z}_{p}$ -anticyclotomic extension of  $K$ .

Type
Research Article
Copyright
© Cambridge University Press 2015. This is a work of the U.S. Government and is not subject to copyright protection in the United States. 

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ON THE NON-TRIVIALITY OF THE $p$ -ADIC ABEL–JACOBI IMAGE OF GENERALISED HEEGNER CYCLES MODULO $p$ , II: SHIMURA CURVES
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