Let $E$ be an elliptic curve defined over $\mathbb{Q}$, let $$K:=\lim_{\stackrel{\rightarrow}{k\in\mathbb{Q}}[\mu_{{p}^{\infty}}]}\mathbb{Q}\left[\mu_{{p}^{\infty}},k^{{\frac{{1}}{{p}^{\infty}}}}\right]$$ and $G:=\operatorname{Gal}(\overline{K}/K)$. Suppose that ${E}_{p}(K)=0$.

Question: Is there always a $\tau\in G$ so that $$\operatorname{rank}_{\mathbf{Z}_{p}}((t-1)E_{{p}^{\infty}}(\overline{K}))=1$$

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, let $$K:=\varinjlim_{k\in\mathbb{Q}[\mu_{p^\infty}]} \mathbb{Q}\left[\mu_{p^\infty},k^{1/p^\infty}\right]$$ and $G:=\operatorname{Gal}(\overline{K}/K)$. Suppose that $E_p(K)=0$.

Question: Is there always a $\tau\in G$ so that $$\operatorname{rank}_{\mathbf{Z}_p}((t-1)E_{p^\infty}(\overline{K}))=1$$

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, let $$K:=\bigcup_{k\in\mathbb{Q}[\mu_{{p}^{\infty}}]}\mathbb{Q}\left[\mu_{{p}^{\infty}},k^{{\frac{{1}}{{p}^{\infty}}}}\right]$$ and $G:=\operatorname{Gal}(\overline{K}/K)$. Suppose that ${E}_{p}(K)=0$.

Question: Is there always a $\tau\in G$ so that $$\dim_{\mathbf{Z}_{p}}(E_{{p}^{\infty}}(\overline{K})/(\tau-1)E_{{p}^{\infty}}(\overline{K}))=1$$

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, let $$K:=\lim_{\stackrel{\rightarrow}{k\in\mathbb{Q}}[\mu_{{p}^{\infty}}]}\mathbb{Q}\left[\mu_{{p}^{\infty}},k^{{\frac{{1}}{{p}^{\infty}}}}\right]$$ and $G:=\operatorname{Gal}(\overline{K}/K)$. Suppose that ${E}_{p}(K)=0$.

Question: Is there always a $\tau\in G$ so that $$\operatorname{rank}_{\mathbf{Z}_{p}}((t-1)E_{{p}^{\infty}}(\overline{K}))=1$$

Let $E$ be an elliptic curve defined over $\mathbf{Q}$, let $$K:=\bigcup_{k\in\mathbf{Q}[\mu_{{p}^{\infty}}]}\mathbf{Q}[\mu_{{p}^{\infty}},k^{{\frac{{1}}{{p}^{\infty}}}}]$$ and $G:=Gal(\overline{K}/K)$. Suppose that ${E}_{p}(K)=0$.

Question: Is there always a $\tau\in G$ so that $$\dim_{\mathbf{Z}_{p}}(E_{{p}^{\infty}}(\overline{K})/(\tau-1)E_{{p}^{\infty}}(\overline{K}))=1$$

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, let $$K:=\bigcup_{k\in\mathbb{Q}[\mu_{{p}^{\infty}}]}\mathbb{Q}\left[\mu_{{p}^{\infty}},k^{{\frac{{1}}{{p}^{\infty}}}}\right]$$ and $G:=\operatorname{Gal}(\overline{K}/K)$. Suppose that ${E}_{p}(K)=0$.

Question: Is there always a $\tau\in G$ so that $$\dim_{\mathbf{Z}_{p}}(E_{{p}^{\infty}}(\overline{K})/(\tau-1)E_{{p}^{\infty}}(\overline{K}))=1$$