论文标题
$λ$ -Submodules的反旋转率和减去椭圆曲线的Selmer组
$Λ$-submodules of finite index of anticyclotomic plus and minus Selmer groups of elliptic curves
论文作者
论文摘要
让$ p $是一个奇怪的素数，$ k $是一个虚构的二次场，其中$ p $ splits。在适当的假设下，贝托利尼（Bertolini）表明，在反通风$ \ mathbb z_p $ - $ k $ extension $ p $ - 非凡的椭圆曲线的selmer组不承认有限的$λ$ -SBMODULE有限指数，其中$λ$是合适的Iwasawa algebra。我们将此结果推广到$ p $ simpersingular椭圆曲线的加号和减去Selmer组（从Kobayashi的意义上）。特别是，在我们的环境中，plus/sinus selmer组具有$λ$ -Corank One，因此它们不是$λ$ cotorsion。作为我们的主要定理的应用，我们证明了$ p $ - 诸如 - 连通椭圆形曲线的伊瓦沙瓦（Iwasawa）不变的格林伯格·瓦特萨尔（Greenberg-Vatsal）的脉络，从而扩展到$ p $ p $ p $ p $ p $ - 非凡的椭圆形曲线的替代病例结果。
Let $p$ be an odd prime and $K$ an imaginary quadratic field where $p$ splits. Under appropriate hypotheses, Bertolini showed that the Selmer group of a $p$-ordinary elliptic curve over the anticyclotomic $\mathbb Z_p$-extension of $K$ does not admit any proper $Λ$-submodule of finite index, where $Λ$ is a suitable Iwasawa algebra. We generalize this result to the plus and minus Selmer groups (in the sense of Kobayashi) of $p$-supersingular elliptic curves. In particular, in our setting the plus/minus Selmer groups have $Λ$-corank one, so they are not $Λ$-cotorsion. As an application of our main theorem, we prove results in the vein of Greenberg-Vatsal on Iwasawa invariants of $p$-congruent elliptic curves, extending to the supersingular case results for $p$-ordinary elliptic curves due to Hatley-Lei.
