The growth of Tate–Shafarevich groups of pp‐supersingular elliptic curves over anticyclotomic Zp {Z}ₚ‐extensions at inert primes

Abstract Let be an elliptic curve defined over , and let be an imaginary quadratic field. Consider an odd prime at which has good supersingular reduction with and which is inert in . Under the assumption that the signed Selmer groups are cotorsion modules over the corresponding Iwasawa algebra, we prove that the Mordell–Weil ranks of are bounded over any subextensions of the anticyclotomic ‐extension of . Additionally, we provide an asymptotic formula for the growth of the ‐parts of the Tate–Shafarevich groups of over these extensions.