On the p-adic variation of Heegner points

Let $f$ be a newform of weight $k\geq 2$ of trivial nebentypus, let $V_f$ be the self-dual Tate twist of the $p$-adic Galois representation associated to $f$, and let $\chi$ be an anticyclotomic Hecke character of an imaginary quadratic field. The conjectures of Bloch and Kato predict the equality between the order of vanishing of the $L$-function of $V_f\otimes\chi$ and the dimension of its Selmer group. In this paper, using Heegner points and their variation in $p$-adic families, we prove many "rank 0" cases of this conjecture when the infinity type of $\chi$ is $z^{-n}\bar{z}^{n}$ with $n\geq k/2$, as well as one of the divisibilities in the Iwasawa--Greenberg main conjecture for $V_f\otimes\chi$.