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

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.