Estimating the growth in Mordell–Weil ranks and Shafarevich–Tate groups over Lie extensions

Abstract

Let $$E_{/\mathbb {Q}}\!$$ be an elliptic curve, $$p>3$$ a good ordinary prime for E, and $$K_\infty $$ a p-adic Lie extension of a number field k. Under some standard hypotheses, we study the asymptotic growth in both the Mordell–Weil rank and Shafarevich–Tate group for E over a tower of extensions $$K_n/k$$ inside $$K_\infty $$ ; we obtain lower bounds on the former, and upper bounds on the latter’s size.