On the anticyclotomic Iwasawa main conjecture for Hilbert modular forms of parallel weights

In this article, we study the Iwasawa theory for Hilbert modular forms over anticyclotomic extension of a CM field. We prove an one-sided divisibility result toward main conjecture in setting. The proof relies on first and second reciprocity laws relating theta elements to Heegner point Euler systems Shimura curves. As by-product also towards rank 0 case certain Bloch–Kato parity conjecture.