On anticyclotomic Selmer groups of elliptic curves
Matteo Longo, Jishnu Ray, Stefano Vigni
TL;DR
This paper advances the Iwasawa-theoretic understanding of signed Selmer groups for p-supersingular elliptic curves over the anticyclotomic $\mathbb{Z}_p$-extension, specifically in the inert case where $p$ is unramified in the imaginary quadratic field $K$. Building on Bertolini and Hatley–Lei–Vigni, and leveraging the local-unit results of Burungale–Kobayashi–Ota, the authors establish that under a Λ-corank-1 hypothesis and suitable Sha and Heegner-point hypotheses, the Pontryagin dual $\mathfrak X_p^\pm(E/K_\infty)$ admits no non-trivial finite $\Lambda$-submodules. The core mechanism is a fine analysis of universal norms $US_p^\pm(E/K)$ inside the p-adic Tate module of the signed Selmer group, showing $US_p^\pm(E/K)\cong \mathbb{Z}_p$ and that the quotient $S_p^\pm(E/K)/US_p^\pm(E/K)$ is torsion-free. The results complete the picture by treating inert primes, refine previous split-case conclusions, and sharpen the link between Heegner-point arithmetic and Λ-module structure in the anticyclotomic setting.
Abstract
Let $p\geq5$ 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 $Λ$-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 $Λ$ is the corresponding Iwasawa algebra. In particular, we work under the assumption that our plus/minus Selmer groups have $Λ$-corank $1$, so they are not $Λ$-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 $K$.