$$\Lambda $$-submodules of finite index of anticyclotomic plus and minus Selmer groups of elliptic curves

Abstract

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 \(\Lambda \)-submodule of finite index, where \(\Lambda \) 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 \(\Lambda \)-corank one, so they are not \(\Lambda \)-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.