Horizontal variation of Tate--Shafarevich groups

Ashay A. Burungale, Haruzo Hida, Ye Tian

Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be an odd prime and $\iota :\overline{\mathbb{Q}}\hookrightarrow {\mathbb{C}}_p$ an embedding. Let $K$ be an imaginary quadratic f…