On the Iwasawa Main Conjecture for generalized Heegner classes in a quaternionic setting

We prove one divisibility relation of the anticyclotomic Iwasawa Main Conjecture for a higher weight ordinary modular form $f$ and an imaginary quadratic field satisfying a “relaxed” Heegner hypothesis. Let $\mathrm{\Lambda }$ be the anticyclotomic Iwasawa algebra. Following the approach of Howard and Longo–Vigni, we construct the $\mathrm{\Lambda }$-adic Kolyvagin system of generalized Heegner classes coming from Heegner points on a suitable Shimura curve. As its application, we also prove one divisibility relation in the Iwasawa-Greenberg main conjecture for the $p$-adic $L$-function defined by Magrone.