Let $E/K$ be an elliptic curve over a number field $K$. Let $L=K(\sqrt{D})/K$ be a quadratic extension. Let $E_D/K$ be a twist of $E/K$ by $D$.
Let $\text{Sha}(E/K)$ be a Tate-Shafarevich group of $E/K$. Let $\text{Sha}(E/K)[2]=\{a\in \text{Sha}(E/K)\mid 2a=0\}$.
There is a natural trace map $\text{Sha}(E/L)[2]\to \text{Sha}(E/K)[2]$ which sends $P$ to $P+P^{\sigma}$ where $\sigma$ is generator of $\text{Gal}(L/K)$.
My question is,
What can we say about the size of $\text{trace}(\text{Sha}(E/L)[2])$ ? Is it bounded by the value which does not depend on $L$ (only dependent on $K$) or does it become bigger arbitrary?
About $\text{trace}(\text{Sha}(E/L))$, $2\text{Sha}(E/K)\subset \text{trace}(\text{Sha}(E/L)$ holds clearly, but about 2-part, $2\text{Sha}(E/K)[2]=0$. So we can say nothing about 2-part $\text{trace}(\text{Sha}(E/L)[2] )$.
Thank you for you ideas.