Suppose $L/F$ is a finite Galois extension of number fields. Let $E$ be an elliptic curve over $F$ and $E_L$ its base change to $L$.

Do we have that $\text{Sha}(E/F)$ is finite if and only if $\text{Sha}(E_L/L)^{\text{Gal}(L/F)}$ is finite?

Analogously, write $\text{Sha}(E_{\overline{F}}/\overline{F})$ for the direct limit of the system $\{\text{Sha}(E_L/L)\}$ with $L$ running across the finite Galois extensions of $F$, with transition maps induced by functoriality.

Same question with $\text{Sha}(E_{\overline{F}}/\overline{F})^{\text{Gal}(\overline{F}/F)}$ replacing $\text{Sha}(E_L/L)^{\text{Gal}(L/F)}$.

I was reading about the relation between the BSD conjecture for $E$ and that for $E_L$ and the question came up.

**Remark**

I expect the inflation-restriction exact sequence in group cohomology to be key here. For example, in the first question the kernel of the restriction map $\text{Sha}(E/F)\to \text{Sha}(E_{L}/L)^{\text{Gal}(L/F)}$ is contained in $H^1(\text{Gal}(L/F), E_L(L))$. This implies that if $\text{Sha}(E_{L}/L)^{\text{Gal}(L/F)}$ is finite, so is $\text{Sha}(E/F)$.