$S$-Tate-Shafarevich groups of elliptic curves

Question

Let $S$ be a finite set of places of a number field $k$ and let $E$ be an elliptic curve over $k$. Define the ''$S$-Tate-Shafarevich group" of $E$ to be

$$Ш(E,S) = \ker\left(H^1(k,E) \to \prod_{v \not \in S}H^1(k_v,E_v)\right).$$

Note that the normal Tate-Shafarevich group is $Ш(E) = Ш(E,\emptyset)$. Recall that the Tate-Shafarevich conjecture states that $Ш(E)$ is finite. My question concerns the corresponding problem for $Ш(E,S)$.

Is $Ш(E,S)$ conjecturally finite?

If the answer to this is yes, then an obvious next question is whether this is equivalent to finiteness of $Ш(E)$, i.e.

Is it known that $Ш(E) \subset Ш(E,S)$ has finite index?

Answer 1

Global duality (as for instance on page 29 of Rubin's "Euler systems") gives you a description of the cokernel of your inclusion. Let $p$ be a prime. Then the $p$-primary part of the quotient of $Ш(E,S)$ by $Ш(E)$ is dual to the cokernel of the map $$ \mathfrak S_p(E/k) = \varprojlim_n \,\mathrm{Sel}_{p^n}(E/k) \ \to \ \bigoplus_{v\in S} E(k_v)^{*}.$$ Here the source is the compact $p$-adic Selmer group of $E/k$, which is a finitely generated $\mathbb{Z}_p$-module of rank $r$, believed to be the rank of $E(k)$. The target is the sum of the $p$-adic completions of the local points $E(k_v)$. If $v$ is not above $p$, then this group is finite and otherwise it is a finitely generated $\mathbb{Z}_p$-module of rank $[k_v:\mathbb{Q}_p]$.

One can make quite precise conjectures as to what the corank of the $p$-primary part of $Ш(E,S)$ should be in terms of the rank of $E(k)$ and $k/\mathbb{Q}$ by the above duality. Roughly speaking some sort of independence of elliptic logarithms tells you that the above map should have image as large as possible. For instance over $k=\mathbb{Q}$, it is clear that the corank of the $p$-primary part of $Ш(E,S)$ will be $0$ is $p$ if is not in $S$ or if $r\geq 1$, and $1$ otherwise. For larger $k$ it is a bit harder to formulate as it will depend on the field of definition of $E$, but generally speaking it tends to be large as soon as a $p$-adic place of large degree is in $S$.