On the non-triviality of the $p$-adic Abel-Jacobi image of generalised Heegner cycles modulo $p$, II: Shimura curves

Abstract

Generalised Heegner cycles are associated to a pair of an elliptic newform and a Hecke character over an imaginary quadratic extension $K/\Q$. 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 $K/\Q$. We prove the non-triviality of the $p$-adic Abel-Jacobi image of generalised Heegner cycles modulo $p$ over the $\Z_p$-anticylotomic 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 two, the result provides a refinement of the results of Cornut-Vatsal and Aflalo-Nekov\'{a}\v{r} on the non-triviality of Heegner points over the $\Z_p$-anticylotomic extension of $K$.