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Generalised Heegner cycles are associated to a pair of an elliptic Heckeeigenform and a Hecke character over an imaginary quadratic extension $K/\Q$.Let $p$ be an odd prime split in $K/\Q$ and $l\neq p$ an odd unramified prime.We prove the non-triviality of the $p$-adic Abel-Jacobi image of generalisedHeegner cycles modulo $p$ over the $\Z_l$-anticylotomic extension of $K$. Theresult is an evidence for the refined Bloch-Beilinson and the Bloch-Katoconjecture. In the case of two, it provides a refinement of the results ofCornut and Vatsal on the non-triviality of Heegner points over the$\Z_l$-anticylotomic extension of $K$.

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