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

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.