The growth of Tate-Shafarevich groups of p-supersingular elliptic curves over anticyclotomic Zₚ-extensions at inert primes

Let E be an elliptic curve defined over Q, and let K be an imaginary quadratic field. Consider an odd prime p at which E has good supersingular reduction with aₚ (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 Zₚ-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.