Let $E$ be a elliptic curve without complex multiplication over a number field $F$. Let $F_n=F[E_{p^n}]$ and $F_{\infty}=F[E_{p^\infty}]$. So by a well know theorem of Serre, the Galois group $Gal(F_\infty/F)$ is an open subgroup of $GL_2(\mathbb{Z}_p)$.
Now suppose $E$ has good ordinary reduction over any place $v$ of $F$ over $p$. Let $\omega$ be a prime of $F_{\infty}$ over $v$. Then I know from the work of Coates and Howson here, that the Decomospostion subgroup $D(\omega|v)$ embeds into upper triangular matrices of $GL(2,\mathbb{Z}_p)$ and that the cohomological $p$-dimension of $D(\omega|v)$ is $3$. This is also the dimension as $p$-adic Lie group by Lazard since it contains no elements of order $p$)
Is anything known about the Inertia subgroup $I(\omega|v)$. What is its cohomological $p$-dimension (or dimension as a $p$-adic Lie group)??
I guess that $I(\omega|v)$ should have dimension $2$ because $D(\omega|v)/I(\omega|v) \cong Gal(k_{\omega}/k_v) $ where $k_{\omega}$ (and $k_v$) are residue fields of the completion of $F_{\infty}$ (and $F$) at prime $\omega$ (and $v$). The Galois group of the residue fields should be generated topologically by some Frobenius and hence will be of dimension $1$. Therefore the inertia group $I(\omega|v)$ should have dimension $2$ as a $p$-adic Lie group.
Is my argument correct? Thanks for help.
