Heegner cycles and higher weight specializations of big Heegner points

Let f be a p-ordinary Hida family of tame level N, and let K be an imaginary quadratic field satisfying the Heegner hypothesis relative to N. By taking a compatible sequence of twisted Kummer images of CM points over the tower of modular curves of level Γ₀(N) ∩ Γ₁(p^s), Howard has constructed a canonical class Z in the cohomology of a self-dual twist of the big Galois representation associated to f. If a p-ordinary eigenform f on Γ₀(N) of weight k > 2 is the specialization of f at ν, one thus obtains from Z_ν a higher weight generalization of the Kummer images of Heegner points. In this paper we relate the classes Z_ν to the étale Abel-Jacobi images of Heegner cycles when p splits in K.