On the anticyclotomic Iwasawa Main Conjecture for modular forms

We prove a one-sided divisibility relation in the anticyclotomic Iwasawa main conjecture for a modular form f of weight k and level N, and an imaginary quadratic field K satisfying a "relaxed" Heegner hypothesis relative to N.

Let K_∞ be the anticyclotomic Zp-extension of K and Λ the Iwasawa of Gal(K_∞/K). Following the approach of Howard for bounding Selmer groups of elliptic curves over anticyclotomic towers, then generalized by Longo-Vigni to modular forms and imaginary quadratic fields satisfying the Heegner hypothesis, our main tool is the construction of a Kolyvagin system for the Λ-adic Galois representation attached to f using generalized Heegner classes coming from Heegner points on a suitable Shimura curve.