  This note shows how to use the framework of Euler characteristic formulae to
study Selmer groups of abelian varieties in certain dihedral or anticyclotomic
extensions of CM fields via Iwasawa main conjectures, and in particular how to
verify the p-part of the refined Birch and Swinnerton-Dyer conjecture in this
setting. When the Selmer group is cotorsion with respect to the associated
Iwasawa algebra, we obtain the p-part of formula predicted by the refined Birch
and Swinnerton-Dyer conjecture. When the Selmer group is not cotorsion with
respect to the associated Iwasawa algebra, we give a conjectural description of
the Euler characteristic of the cotorsion submodule, and explain how to deduce
inequalities from the associated main conjecture divisibilities of Perrin-Riou
and Howard.
