The growth of Tate-Shafarevich groups of $p$-supersingular elliptic curves over anticyclotomic $\mathbb{Z}_p$-extensions at inert primes

TL;DR

This paper investigates the growth of Tate-Shafarevich groups of certain supersingular elliptic curves over anticyclotomic extensions, proving boundedness of ranks and providing asymptotic formulas for Sha growth under specific Iwasawa-theoretic assumptions.

Contribution

It establishes boundedness of Mordell-Weil ranks and derives asymptotic formulas for Tate-Shafarevich groups in the context of supersingular elliptic curves over anticyclotomic extensions, assuming cotorsion properties.

Findings

Mordell-Weil ranks are bounded over anticyclotomic extensions.

Asymptotic growth formulas for Tate-Shafarevich groups.

Results depend on cotorsion assumptions for signed Selmer groups.

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $K$ be an imaginary quadratic field. Consider an odd prime $p$ at which $E$ has good supersingular reduction with $a_p(E)=0$ and which is inert in $K$. Under the assumption that the signed Selmer groups are cotorsion modules over the corresponding Iwasawa algebra, we prove that the Mordell-Weil ranks of $E$ are bounded over any subextensions of the anticyclotomic $\mathbb{Z}_p$-extension of $K$. Additionally, we provide an asymptotic formula for the growth of the $p$-parts of the Tate-Shafarevich groups of $E$ over these extensions.