Tate–Shafarevich groups in anticyclotomic ℤp-extensions at supersingular primes

Abstract

AbstractLet E/ℚ be an elliptic curve and p a prime of supersingular reduction for E. Denote by $\mathrm {K}_\infty $ the anticyclotomic ℤp-extension of an imaginary quadratic field K which satisfies the Heegner hypothesis. Assuming that p splits in K/ℚ, we prove that ${\mbox {\textcyr {Sh}}} (\mathrm {K}_\infty , \mathrm {E})_{p^\infty }$ has trivial Λ-corank and, in the process, also show that $\mathrm {H^1_{Sel}}(\mathrm {K}_\infty , \mathrm {E}_{p^\infty })$ and $\mathrm {E}(\mathrm {K}_\infty )\otimes \mathbb {Q}_p/\mathbb {Z}_p$ both have Λ-corank two.