On the Tamagawa number conjecture for modular forms twisted by anticyclotomic Hecke characters

Let $f\in S_{2r}({\mathrm{\Gamma }}_0(N))$ be a normalized newform of weight $2r$ which is good at $p$. Let $K$ be an imaginary quadratic field of class number one in which every prime divisor of $pN$ splits. Let $\chi$ be an anticyclotomic Hecke character of $K$ which is crystalline at the primes above $p$ and such that $L(f,\chi ,r)\mathrm{\ne }0$. We prove that the Tamagawa number conjecture for the critical value $L(f,\chi ,r)$ follows from the Iwasawa main conjecture for the Bertolini-Darmon-Prasanna $p$-adic $L$-function.