Mordell-Weil ranks and Tate-Shafarevich groups of elliptic curves with mixed-reduction type over cyclotomic extensions

Let E be an elliptic curve defined over a number field K where p splits completely. Suppose that E has good reduction at all primes above p. Generalizing previous works of Kobayashi and Sprung, we define multiply signed Selmer groups over the cyclotomic Z(p)-extension of a finite extension F of K where p is unramified. Under the hypothesis that the Pontryagin duals of these Selmer groups are torsion over the corresponding lwasawa algebra, we show that the Mordell-Weil ranks of E over a subextension of the cyclotomic Z(p)-extension are bounded. Furthermore, we derive an aysmptotic formula of the growth of the p-parts of the Tate-Shafarevich groups of E over these extensions.