Let $E: y^2 + y = x^3 - x^2 - 10x - 20$ be an elliptic curve over $\mathbb{Q}$, it is known that its $E(\mathbb{Q}) = \mathbb{Z}/5\mathbb{Z}$ is finite and its Shafarevich-Tate group $\mathrm{Sha}(E/\mathbb{Q})$ is trivial. Let $\mathbb{Q}_\infty/\mathbb{Q}$ be the (cyclotomic) $\mathbb{Z}_p$-extension of $\mathbb{Q}$ with the $n$-th layer $\mathbb{Q}_n/\mathbb{Q}$. Then I am hoping to show:
Goal: $\dim_{\mathbb{F}_p} \mathrm{Sha}(E/\mathbb{Q}_n)[p] > p^n - c$ for all $n$, where $c$ is some constant.
The purpose of this post is asking how to prove this.
My attempts: We consider the short exact sequence $$ 0 \rightarrow E(\mathbb{Q}_n) \otimes \mathbb{Z}/p \rightarrow \mathrm{Sel}(E/\mathbb{Q}_n)[p] \rightarrow \mathrm{Sha}(E/\mathbb{Q})[p] \rightarrow 0. $$
Then
$$ \dim_{\mathbb{F}_p} \mathrm{Sha}(E/\mathbb{Q}_n)[p] = \dim_{\mathbb{F}_p} \mathrm{Sel}(E/\mathbb{Q}_n)[p] - \mathrm{rank}E(\mathbb{Q}_n). $$
We assume $E$ has good ordinary reduction at $p$. Then we know from a corollary of Mazur's control theorem that $\mathrm{rank} E(\mathbb{Q}_n)$ is bounded from above by some constant $c$ as $n$ varies. So we have
$$ \dim_{\mathbb{F}_p} \mathrm{Sha}(E/\mathbb{Q}_n)[p] \geq \dim_{\mathbb{F}_p} \mathrm{Sel}(E/\mathbb{Q}_n)[p] - c. $$
Therefore a key is to compute $\dim_{\mathbb{F}_p} \mathrm{Sel}(E/\mathbb{Q}_n)[p] \approx p^n$. This is where I got stuck, since there are two giant steps for me to go through:
The control theorem of Mazur said that there is a pseudo-isomorphism of $\Lambda$-modules $$ \mathrm{Sel}(E/\mathbb{Q}_n)[p^{\infty}] \rightarrow \mathrm{Sel}(E/\mathbb{Q}_\infty)[p^{\infty}]^{\mathrm{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q}_n)}. $$ Actually I cannot see how to equip these objects with $\mathbb{F}_p$-linear structure since they are all "$p$-primary parts", not "$p$-torsion parts". I can see that the $\mu$-invariant is $1$ and $\lambda$-invariant is $0$ at $p=5$, but how do these lead to the computation of the $\mathbb{F}_p$-dimension?
There is still a gap between $\mathrm{Sel}(E/\mathbb{Q}_n)[p^{\infty}]$ and $\mathrm{Sel}(E/\mathbb{Q}_n)[p]$ that I am not able to go through.
So I got stuck here.
Remark: This is Exercise 4.5 in Ralph Greenberg's article "Introduction to Iwasawa Theory for Elliptic Curves".
Remark: Another problem is that to use Mazur's control theorem, we assumed that $E$ is good ordinary at $p$. But how to get the result for all the other primes, since in the exercise, there is actually no assumption on the reduction type.