Refined conjectures on Fitting ideals of BDP Selmer groups

Under mild assumptions, we prove the strong main conjecture for BDP Selmer groups of elliptic curves with good reduction in the sense of Kurihara’s refined Iwasawa theory. In particular, the structure of the initial Fitting ideal of BDP Selmer groups over finite subextensions in the anticyclotomic ${\mathbb{Z}}_p$-extension is determined by the values of the corresponding $p$-adic modular form at CM points.