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].

我们经常使用正/负 Selmer 群来研究具有良好超奇异约简的椭圆曲线的 Iwasawa 理论。$p$。但是，如果当地的正/负条件不是共自由的，则可能很难有效地利用它们。在本文中，我们介绍了一些技术改进，以便即使在正/负局部条件不是无余的情况下，我们也可以使用正/负 Selmer 群。特别是，我们证明正/负 Selmer 群没有有限指数的真子模，并且即使当所述局部条件不是共自由时它们也满足代数函数方程，从而改进了我们之前的结果 [B. Kim，超奇异素数处椭圆曲线的代数函数方程，数学。资源。莱特。 15（1）（2008）83-94；超奇异素数的正/负 Selmer 群，J. Aust。数学。苏克。 95（2）（2013）189-200]。