On anticyclotomic Selmer groups of elliptic curves

Abstract

Let $p\ge 5$ be a prime number and let $K$ be an imaginary quadratic field where $p$ is unramified. Under mild technical assumptions, in this paper we prove the non-existence of non-trivial finite $\hat{I}›$-submodules of Pontryagin duals of signed Selmer groups of a $p$-supersingular rational elliptic curve over the anticyclotomic ${\mathbb{Z}}_p$-extension of $K$, where $\hat{I}›$ is the corresponding Iwasawa algebra. In particular, we work under the assumption that our plus/minus Selmer groups have $\hat{I}›$-corank $1$, so they are not $\hat{I}›$-cotorsion. Our main theorem extends to the supersingular case analogous non-existence results by Bertolini in the ordinary setting; furthermore, since we cover the case where $p$ is inert in $K$, we refine previous results of Hatley-Lei-Vigni, which deal with $p$-supersingular elliptic curves under the assumption that $p$ splits in $K$.

Slight revision following the referee's report; 13 pages. Final version, to appear in Mathematical Research Letters