1 results
ON THE NON-TRIVIALITY OF THE
$p$ -ADIC ABEL–JACOBI IMAGE OF GENERALISED HEEGNER CYCLES MODULO 
$p$ , II: SHIMURA CURVES
- Part of
-
- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 16 / Issue 1 / February 2017
- Published online by Cambridge University Press:
- 07 May 2015, pp. 189-222
- Print publication:
- February 2017
-
- Article
- Export citation
-
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 
$K$ .
