Injectivity of p-adic Abel–Jacobi maps on Heegner-cycle lines

Prove that the p-adic Abel–Jacobi map attached to a proper smooth algebraic variety over a number field is injective on the line spanned by the relevant Heegner cycle, including the specific map AJ_{p,K,k} on the line generated by the Heegner cycle z_{f,K}.

Background

The proof requires the injectivity of AJ_{p,K,k} on the Q_f-vector subspace generated by the Heegner cycle z_{f,K}. This condition ensures that the nontriviality of the Heegner cycle is detected in the relevant Selmer group and is then propagated through the Hida family.

The paper identifies this assumption as a special case of a broader conjecture concerning injectivity of p-adic Abel–Jacobi maps for proper smooth varieties over number fields. The broader conjectural statement is explicitly described as unresolved in the paper’s discussion of the arithmetic-geometric hypotheses.