Let $K$ be a complete local field of characteristic $0$ with ring of integers $R=\mathcal{O}_{K}$ and residue field $k=R/\mathfrak{m}_R$ of characteristic $p>0$. Let $E/K$ be an elliptic curve of split multiplicative reduction type $I_m, m >1$, ie the special fibre $\mathcal{C}_k$ of its minimal proper regular model $\mathcal{C}/R$ has Kodaira type $I_n$, ie isomorphic to "cycle" of $n$ components $C_k^r \cong \Bbb P^1, r=1,..., m$ intersecting transversally with two "neighbour components" pw in a unique point.
As $E$ has split multiplicative reduction type on can use Tate model $\mathbb G_m / \langle q \rangle$ for appropr $q \in K^*$ of positive valuation $v(q) =m$ as auxilary tool which is isomorphic to $E$ as analytic space.
I would like to understand better the structure of torsion points of $E$ in terms of combinatorics of components of special fibre using techniques provided by the Tate model. More concretely I would like to understand following passage of the answer from this answer by Will Sawin:
[...] It is convenient here to use the Tate curve model where a curve with split multiplicative reduction is $\mathbb G_m / \langle q \rangle$ for $q$ an element of $K$ with positive $p$-adic valuation and its $n$-torsion points are the $n$th roots of unity times powers of the $n$th roots of $q$. The components are given by the different possible valuations modulo the valuation of $q$. A torsion point lies in any component at all if and only if it has an integer valuation, and the elements of the subgroup lie on distinct components if and only if they have distinct integer valuations.
Firstly about the statement that "components (of special fibre) are given by the different possible valuations modulo the valuation of $q$": Does this have to do with the fact that there is an iso $\mathcal{E}_k(k)/\mathcal{E}_k^0(k) \cong \Bbb Z/m$ "counting" irred components of the special fibre (see eg Silverman's AEC C.15 or for elaboration the ATEC)?
(Here $\mathcal{E}/R \subset \mathcal{C}/R$ is the Neron model of $E$ obtained roughly by discarding non-smooth points from $\mathcal{C}$ and $\mathcal{E}^0$ the connected component of neutral element $e \in \mathcal{C}(R) =E(K)$ (standard fact). Then $\mathcal{E}_k^0 \cong \Bbb{G}_{m,k}$ obtained roughly by discarding the two points of $C_k^0$ where the latter intersect transversally with two "neighbour" components. )
Then, so far I understand the idea correctly, in terms of Tate model, the points in $E(K) \cong K^*/ q^{\Bbb Z}$ which specialize (=intersection of schematic closure of $p$ inside $\mathcal{C}$ with special fibre) inside the same component $C_k^j$ of spec fibre, are mapped to same element in $\Bbb Z/m$, so the above mapinduces a map $ K^* \to K^*/ q^{\Bbb Z} \to \Bbb Z/m$ given by valuation at $v$.
Especially so far I understand, this quoted statement is at this stage only about the $K$-valued points of $E$, and so tells nothing about "other" points in $E(\overline{K})$, right?
And so one could phrase it after making this identification with Tate model as that points in $E(K)$ have precisely same valuation wrt $v$ modulo $v(q)=m$ iff their specialization is contained in same component.
Firstly, is this correct interpretation/ expansion of this quoted statement?
But the next part confuses me even more.
Main Problem: What does it mean that a "torsion point lies in any component at all if and only if it has an integer valuation"? Say $g \in E(L)$ is some torsion point of $E$ (where $L/K$ in appropr finite extension of $K$) and we asking "in which component it lies".
For a torsion point the "lying in certain component" (of special fibre) presumably means we ask with which components $C_k^j$ of the special fibre the schematic closure of $g$ inside $\mathcal{C}$ (= a $1$-dim subscheme) intersect nontrivially. Does it make sense?
But how it can be detected using techniques on the Tate curve and why the quoted characterization Will invoking there holds?
Once $E$ has a Tate model, the $L$-points of $E$ can be identified with $L^*/q^{\Bbb Z}$ and this we can interpret $g$ naturally living there, but how does it help to decide in which components in sense above it?
