Iwasawa theory of plus/minus Selmer groups with non-co-free plus/minus local conditions

We often use the plus/minus Selmer groups to study Iwasawa theory for elliptic curves with good supersingular reduction at p. But, if the plus/minus local conditions are not co-free, it can be difficult to use them effectively. In this paper, we introduce some technical improvements so that we can use the plus/minus Selmer groups even when the plus/minus local conditions are not co-free. In particular, we show that the plus/minus Selmer groups have no proper submodule of finite index and also they satisfy algebraic functional equations even when the said local conditions are not co-free, thus improve our previous results [B. Kim, The algebraic functional equation of an elliptic curve at supersingular primes, Math. Res. Lett. 15(1) (2008) 83-94; The plus/minus Selmer groups for supersingular primes, J. Aust. Math. Soc. 95(2) (2013) 189-200].