Heegner cycles and higher weight specializations of big Heegner points

Abstract

Let $${\mathbf{{f}}}$$ 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 $$\Gamma _0(N)\cap \Gamma _1(p^s)$$ , Howard has constructed a canonical class $$\mathfrak{Z }$$ in the cohomology of a self-dual twist of the big Galois representation associated to $${\mathbf{{f}}}$$ . If a $$p$$ -ordinary eigenform $$f$$ on $$\Gamma _0(N)$$ of weight $$k>2$$ is the specialization of $${\mathbf{{f}}}$$ at $$\nu $$ , one thus obtains from $$\mathfrak{Z }_{\nu }$$ a higher weight generalization of the Kummer images of Heegner points. In this paper we relate the classes $$\mathfrak{Z }_{\nu }$$ to the etale Abel-Jacobi images of Heegner cycles when $$p$$ splits in $$K$$ .