On the Fitting ideals of anticyclotomic Selmer groups of elliptic curves with good ordinary reduction

We give a short proof of the anticyclotomic analogue of the “strong” main conjecture of Kurihara on Fitting ideals of Selmer groups for elliptic curves with good ordinary reduction under mild hypotheses. More precisely, we completely determine the initial Fitting ideal of Selmer groups over finite subextensions of an imaginary quadratic field in its anticyclotomic ${\mathbb{Z}}_p$-extension in terms of Bertolini–Darmon’s theta elements.