An Asymptotic formula for Tate-Shafarevich groups of CM elliptic curves at supersingular primes

Abstract

Let $K$ be an imaginary quadratic field and $E/\mathbb{Q}$ an elliptic curves with complex multiplication by ${\mathcal{O}}_K$. Let $K_{\infty }/K$ be the anticyclotomic ${\mathbb{Z}}_p$-extension of $K$ and $K_n$ the intermediate layers. Under additional assumptions on Kobayashi's signed Selmer groups we prove an asymptotic formula for the Tate-Shafarevich group over $K_n$.