Tate-Shafarevich groups under finite Galois field extensions

Question

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 $\text{Sha}(E/F)$.

Answer 1

The remark added to the question shows that the kernel of $Ш(E/F) \to Ш(E/L)^G$ is finite where $G$ is the finite Galois group of $L/F$. $\DeclareMathOperator{\coker}{coker}$

Here is an argument why the cokernel is finite. It may be too complicated. Let $p$ be a prime and let us show that the cokernel on the $p$-primary part is finite (and that it is trivial for all $p$ that do not divide $\lvert G\rvert$). Let $\alpha\colon S(E/F) \to S(E/L)^G$ where $S$ stands for the $p$-primary Selmer group in $H^1\bigl(F, E[p^{\infty}]\bigr)$. Consider the exact sequence $$ 0\to \Bigl( E(L) \otimes \mathbb{Q}_p/\mathbb{Z}_p \Bigr)^G \to S(E/L)^G \to Ш(E/L)[p^{\infty}]^G \to H^1\bigl( G, E(L) \otimes \mathbb{Q}_p/\mathbb{Z}_p\bigr) $$ Note that the last term is isomorphic to $H^2\bigl(G,E(L)\otimes \mathbb{Z}_p\bigr)$, which is finite. Comparing that sequence with the sequence with $S(E/F)$ in the middle, shows that the cokernel on Ш lies in an exact sequence between $\coker(\alpha)$ and $H^2\bigl(G,E(L)\otimes \mathbb{Z}_p\bigr)$.

Let $S$ be the finite set of places containing all places at $\infty$, all places of bad reduction and all places above $p$. Let $G_S(L)$ be the Galois group of the maximal extension of $L$ unramified outside $S$. Next the exact sequence $$ 0\to S(E/L)^G \to H^1\bigl(G_S(L), E[p^{\infty}]\bigr)^G \to \Bigl(\bigoplus_{w \in S} H^1\bigl(L_w, E\bigr)[p^{\infty}]\Bigr)^{G} $$ Again we compare it with the similar sequence defining $S(E/F)$. This shows that there is a map $\beta\colon\coker(\alpha)\to H^2\bigl(G,E(L)[p^{\infty}]\bigr)$. The target of $\beta$ is finite; its kernel can be shown to be a subquotient of kernel of the restriction map on the right hand side: $$\bigoplus_{v \in S_F} H^1\bigl( G_w, E(L_w)\bigr)[p^{\infty}]$$ where $S_F$ is the set of places below $S$ and for each $v$ we choose one $w$ above $v$. As this is a finite sum of finite groups, this is also finite.

For the second question. Since $Ш(E/L)\subset H^1\bigl(L,E)$, your limit is natrually a subgroup of the limit of these $H^1$, i.e of $H^1(\bar{L},E)=0$.